MathScape: Seeing and Thinking Mathematically, Course 3, Mathematics in Motion, Student Guide (Mathscape: Seeing and Thinking Mathematically)

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GLENCOE MATHEMATICS

™

Seeing and Thinking Mathematically

COURSE 3

interactive student edition w w w . m a t h s c a p e 3 . c o m

MathScape©05 C3 TP 1/15/04 2:20 PM Page 2 mac85 Mac 85:1st shift: 1268_tm:6037a:6037a:

Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240 ISBN: 0-07-860468-0 1 2 3 4 5 6 111/058 08 07 06 05 04

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TABLE OF CONTENTS Looking Behind the Numbers

. . . . . .2

Correlations, Rankings, and Permutations ➤ PHASE ONE Statistical Measures Lesson 1

Sports Opinions and Facts . . . . . . . . . . . . . . .6

Lesson 2

Who’s the Best? . . . . . . . . . . . . . . . . . . . . . . .8

Lesson 3

Mystery Data . . . . . . . . . . . . . . . . . . . . . . . .10

Lesson 4

Top Teams . . . . . . . . . . . . . . . . . . . . . . . . . . .12

➤ PHASE TWO Data, Scatter Plots, and Correlations Lesson 5

Comparing Sizes . . . . . . . . . . . . . . . . . . . . . .16

Lesson 6

Is There a Relationship? . . . . . . . . . . . . . . . . .18

Lesson 7

What Type of Relationship Is It? . . . . . . . . . .20

Lesson 8

The Mysterious Footprint . . . . . . . . . . . . . . .22

➤ PHASE THREE Probability, Combinations, and Permutations Lesson 9

On Tour . . . . . . . . . . . . . . . . . . . . . . . . . . . .26

Lesson 10 Lunch Specials . . . . . . . . . . . . . . . . . . . . . . . .28 Lesson 11 The Battle of the Bands . . . . . . . . . . . . . . . .30 Lesson 12 The Demo CD . . . . . . . . . . . . . . . . . . . . . . .32 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

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Mathematics of Motion . . . . . . . . . . . .46 Distance, Speed, and Time ➤ PHASE ONE Measuring, Estimating, and Representing Motion Lesson 1

Moving, Measuring, and Representing . . . . . .50

Lesson 2

Going the Distance . . . . . . . . . . . . . . . . . . . .52

Lesson 3

Reporting Live from the Parade . . . . . . . . . .54

Lesson 4

The Parade Continues . . . . . . . . . . . . . . . . .56

➤ PHASE TWO Distance-Time Graphs Lesson 5

Walk This Way . . . . . . . . . . . . . . . . . . . . . . .60

Lesson 6

Stories, Maps, and Graphs . . . . . . . . . . . . . . .62

Lesson 7

A Graphing Matter . . . . . . . . . . . . . . . . . . . .64

Lesson 8

Juan and Marina Go Walking . . . . . . . . . . . . .66

➤ PHASE THREE Using Graphs to Solve Problems Lesson 9

The Race Announcer . . . . . . . . . . . . . . . . . .70

Lesson 10 How Fast? How Far? How Long? . . . . . . . . .72 Lesson 11 The Race Is On! . . . . . . . . . . . . . . . . . . . . . .74 Lesson 12 Final Project . . . . . . . . . . . . . . . . . . . . . . . . .76 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

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TABLE OF CONTENTS • MATHSCAPE

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Shapes and Space

. . . . . . . . . . . . . . . . .90

Thinking Three-Dimensionally ➤ PHASE ONE The Ins and Outs of Cubes Lesson 1

Nets That Catch Cubes . . . . . . . . . . . . . . . .94

Lesson 2

Any Way You Slice It . . . . . . . . . . . . . . . . . . .96

Lesson 3

Take One for Good Measure! . . . . . . . . . . . .98

Lesson 4

Move the Cube . . . . . . . . . . . . . . . . . . . . . .100

➤ PHASE TWO Prisms and Cylinders Lesson 5

All Boxed Up . . . . . . . . . . . . . . . . . . . . . . .104

Lesson 6

Wrapping Up Prisms . . . . . . . . . . . . . . . . . .106

Lesson 7

Outside and Inside . . . . . . . . . . . . . . . . . . .108

Lesson 8

Putting It Together . . . . . . . . . . . . . . . . . . .110

➤ PHASE THREE Solids with Points Lesson 9

Pyramid Tips . . . . . . . . . . . . . . . . . . . . . . . .114

Lesson 10 Cutting the Cone . . . . . . . . . . . . . . . . . . . .116 Lesson 11 Does the Unit Fit? . . . . . . . . . . . . . . . . . . .118 Lesson 12 New Heights in Architecture . . . . . . . . . . .120 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122

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What Comes Next?

. . . . . . . . . . . . . .134

Modeling and Predicting ➤ PHASE ONE Exploring Growth Sequences Lesson 1

Predicting World Population . . . . . . . . . . . .138

Lesson 2

Target Practice . . . . . . . . . . . . . . . . . . . . . .140

Lesson 3

Growing Spirals . . . . . . . . . . . . . . . . . . . . .142

Lesson 4

Make a Model . . . . . . . . . . . . . . . . . . . . . . .144

➤ PHASE TWO Representing Growth Sequences Lesson 5

Growing in One Step . . . . . . . . . . . . . . . . .148

Lesson 6

Growing in Two Steps . . . . . . . . . . . . . . . . .150

Lesson 7

What Do You See? . . . . . . . . . . . . . . . . . . .152

Lesson 8

Which Model Fits the Data Better? . . . . . .154

➤ PHASE THREE Modeling with Growth Sequences Lesson 9

Making Plans . . . . . . . . . . . . . . . . . . . . . . . .158

Lesson 10 Looking Ahead . . . . . . . . . . . . . . . . . . . . . .160 Lesson 11 Rates and Relationships . . . . . . . . . . . . . . .162 Lesson 12 Focus on the Future . . . . . . . . . . . . . . . . . .164 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166

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Exploring the Unknown

. . . . . . . . . .178

Writing and Solving Equations ➤ PHASE ONE Working with Expressions Lesson 1

Gearing Up . . . . . . . . . . . . . . . . . . . . . . . . .182

Lesson 2

What’s in the Bag? . . . . . . . . . . . . . . . . . . .184

Lesson 3

Cornered! . . . . . . . . . . . . . . . . . . . . . . . . . .186

Lesson 4

Lab Problem No. 1 . . . . . . . . . . . . . . . . . . .188

➤ PHASE TWO Minuses and Parentheses Lesson 5

Extending the Lab Gear Model . . . . . . . . . .192

Lesson 6

Making Long Expressions Shorter . . . . . . . .194

Lesson 7

Grouping and Ungrouping . . . . . . . . . . . . . .196

Lesson 8

Lab Problem No. 2 . . . . . . . . . . . . . . . . . . .198

➤ PHASE THREE Solving Equations Lesson 9

Polynomial Arithmetic . . . . . . . . . . . . . . . . .202

Lesson 10 Simplify and Solve . . . . . . . . . . . . . . . . . . . .204 Lesson 11 Staying Balanced . . . . . . . . . . . . . . . . . . . . .206 Lesson 12 Lab Problem No. 3 . . . . . . . . . . . . . . . . . . .208 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .210

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Roads and Ramps

. . . . . . . . . . . . . . . .222

Slopes, Angles, and Ratios ➤ PHASE ONE Slope as an Angle Lesson 1

Slopes and Slope-o-meters . . . . . . . . . . . . .226

Lesson 2

Working for Scale . . . . . . . . . . . . . . . . . . . .228

Lesson 3

Sighting and Angle of Elevation . . . . . . . . . .230

➤ PHASE TWO Right-Triangle Relationships Lesson 4

Right Triangles . . . . . . . . . . . . . . . . . . . . . . .234

Lesson 5

Exploring the Pythagorean Theorem . . . . . .236

Lesson 6

Using the Pythagorean Theorem . . . . . . . . .238

➤ PHASE THREE Slope as a Ratio Lesson 7

Stairs and Ratios . . . . . . . . . . . . . . . . . . . . .242

Lesson 8

The 12-Inch Tread . . . . . . . . . . . . . . . . . . . .244

Lesson 9

7% Grade Ahead! . . . . . . . . . . . . . . . . . . . .246

➤ PHASE FOUR Road Design Lesson 10 The Tangent Ratio . . . . . . . . . . . . . . . . . . . .250 Lesson 11 A Mathematical Hill . . . . . . . . . . . . . . . . . .252 Lesson 12 The Road Project . . . . . . . . . . . . . . . . . . . .254 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256

Angle of elevation

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Family Portraits

. . . . . . . . . . . . . . . . . .268

Comparing Function Families ➤ PHASE ONE Direct and Inverse Variation Lesson 1

Inputs and Outputs . . . . . . . . . . . . . . . . . . .272

Lesson 2

Keeping a Steady Pace . . . . . . . . . . . . . . . .274

Lesson 3

How Long Is a Meter? . . . . . . . . . . . . . . . .276

➤ PHASE TWO Linear Functions Lesson 4

A New Slant on Linear Functions . . . . . . . .280

Lesson 5

What’s in an Equation? . . . . . . . . . . . . . . . .282

Lesson 6

The Shortest Distance Between Two Points . . . . . . . . . . . . . . . . . . . . . . . . .284

➤ PHASE THREE Quadratic Functions Lesson 7

The Area of a Projected Image . . . . . . . . . .288

Lesson 8

Going Around a Curve . . . . . . . . . . . . . . . .290

Lesson 9

Fenced In . . . . . . . . . . . . . . . . . . . . . . . . . .292

➤ PHASE FOUR Exponential Functions Lesson 10 Folds and Regions . . . . . . . . . . . . . . . . . . . .296 Lesson 11 Rules of the Road for Exponents . . . . . . . .298 Lesson 12 The Very Large and the Very Small . . . . . . .300 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .302

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QUICK REFERENCE CONTENTS MACINTOSH QuickStart Installation: Power Macintosh Increasing Memory Allocated to Acrobat Reader WINDOWS QuickStart Installation: Windows 95/98/ME/NT/2000/XP Accelerating System Performance ACROBAT MENU ICONS AND FUNCTIONS GUIDE

COPYRIGHT

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001-003_MS G8 U1_866820-4 12/12/03 12:50 PM Page 2 mac85 Mac 85:1st shift: 1268_tm:6037a:G8 Mathscapes Student Guide U1:

How can math help you describe and analyze data?

LOOKING

BEHIND THE

N UMBERS


PHASEONE

PHASETWO

PHASETHREE

Statistical Measures

Data, Scatter Plots, and Correlations You will begin by making some measurements of your own body. Then you will have a chance to use this data throughout the phase. You will see how to use scatter plots, lines of best fit, and correlations to describe relationships in data. At the end of the phase, you will use everything you have learned to help solve a mystery.

Probability, Combinations, and Permutations Imagine that you and your classmates have formed a musical group. You will have many choices to make about concert locations and song sequences. The investigations in this phase will help you use combinations, permutations, and probability in answering these types of questions.

In this phase you will be collecting and analyzing sports data. You will find means, medians, modes, and ranges. You will also see how stem-andleaf plots can help you display a lot of information very compactly. All of these tools will help you to make fair comparisons.

PHASE ONE



In this phase, you will be collecting and analyzing sports data. By using means, medians, modes, and ranges, you will be able to compare the performances of individuals and teams. Stem-and-leaf plots will give you a powerful way of displaying data. Batting averages are just one example of how data and statistics are used in sports. What other examples of sports statistics can you think of?

Statistical Measures WHAT’S THE MATH? Investigations in this section focus on: STATISTICS: DATA COLLECTION and REPRESENTATION ■

Collecting and organizing data

■

Making stem-and-leaf plots

STATISTICS: DATA ANALYSIS ■

Calculating means, medians, modes, and ranges

■

Finding data sets that fit a given mean, median, mode, or range

■

Making conclusions that are based on data

NUMBER ■

Using percentages to describe data

mathscape3.com/self_check_quiz

LOOKING BEHIND THE NUMBERS

5


1

EXPLORING MEANS, PERCENTS, AND GRAPHS

Sports Opinions and Facts Surveys are useful tools for collecting data about people’s opinions. You will take a survey about sports and use averages and percentages to analyze the class data.Then you will interpret tables and graphs to find out the facts about in-line skating injuries.

Collect and Analyze the Class Data How can you use means and percents to analyze data?

The class will take a survey to find out students’ opinions about in-line skating. Then analyze the data by finding the following: 1

How many students chose each rating?

2

What percent of the students in the class gave each rating?

3

Add up all the ratings to find the total rating.

4

What is the class average, or mean rating? To find the average, add up all the rating values that students selected and then divide by the number of students who gave these ratings.

Sports Opinions Survey

Rate in-line skating on these scales. Write your ratings on small pieces of paper. Rating Scale A: How much fun do you think in-line skating (rollerblading) is? 1 2 No fun at all

3 OK

4

5 Great fun

Rating Scale B: How likely do you think it is for people to get injured when they go in-line skating? 1 2 Very unlikely to get injuries

6

3

4

5 Very likely to get injuries

LOOKING BEHIND THE NUMBERS • LESSON 1


Interpret Graphs and Tables How can you interpret data from multiple graphs to get an in-depth picture of a topic?

The handout In-Line Skating Data provides information on in-line skating injuries. You might be surprised by what you find out. For each question, write at least one sentence and give specific data to support your answer. 1

Did more injuries happen outdoors or indoors?

2

Were more injuries on arms or legs?

3

How much experience did the injured in-line skaters have skating?

4

5

6

Count the Lessons

Did more injuries happen to people who didn’t take lessons or to those who took a lot of lessons?

How many skating lessons did the injured in-line skaters take?

What percent of the people injured were wearing all the recommended safety gear? What percent were wearing no safety gear? How do the number of injuries compare for in-line skating and gymnastics? Why do you think this is the case?

6 or more lessons 13%

50% 1 to 5 lessons

0 lessons

37%

Write Questions About Data ■

Write two new questions about in-line skating that you could answer by using data from the graphs and table.

■

Write two questions on in-line skating that you would like to find out about but do not have the data to answer.

hot words

mean percent

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2

EXPLORING MEANS, MEDIANS, MODES, AND RANGES

Who’s the Best? Means, medians, modes, and ranges are statistical measures that are useful for figuring out what’s typical and what’s unusual. How can you use these measures to determine which basketball player is the best?

Determine Means, Medians, Modes, and Ranges How can you use means, medians, modes, and ranges to analyze athletic performance?

The first table of Basketball Data shows how many points three basketball players scored in seven different games. 1

Look at the example for Player A in the second table. Then make a copy of the table and figure out the missing statistical measures for Players B and C.

2

After you complete the second table, use these clues to help you figure out which player is which. The three players’ names are Sheryl, Neisa, and Tina. Sheryl’s median score is the same number as Neisa’s mean score.

Basketball Data

Game

8

Player A’s Player B’s Player C’s Points Points Points

1

12

18

24

2

13

21

14

3

12

15

14

4

14

13

22

5

11

16

25

6

20

18

16

7

16

18

11


Player A Player B Player C High Score

20

?

?

Median

13

?

?

Mean

14

?

?

Mode

12

?

?

Range

9

?

?


Compare and Rank the Players If you compare the players’ high scores, then Player C is the best because 25 is the highest score. Will Player C continue to be the “best” when you use the other statistical measures? Figure out the other rankings and put them in a table like the one below. Different Ways to Rank the Basketball Players

Do different statistical measures lead to different rankings of who’s the best?

Ranked by Ranked by Ranked by High Median Mean Ranked by Ranked by Score Score Score Mode Range Highest

C, 25

2nd

B, 21

3rd

A, 20

After you complete the table, decide which player or players you think are best overall. Who do you think is second best? third best? Be prepared to explain your conclusions.

Write Conclusions The basketball coach is planning to give an award to the best player, but she can’t decide which of the three players is the best. She has asked your class for help. Write a letter to the coach to convince her of one of the following arguments: ■

that one of the players is the best

■

that two of the players are the best

■

that all the players are equally good

■

that it doesn’t make sense to give out this kind of award

Make sure to use the data to support your conclusions.

hot words

median mode

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3

WORKING BACKWARDS WITH STATISTICS

Mystery Data If you know that a basketball player’s mean score is 10 points, what might each of his or her individual scores have been? Working backwards will help you build a better understanding of means, medians, modes, and ranges.

Solve Clues About Data How can you create data sets that have specific statistical measures?

You and a partner will play a Mystery Data game like the one below. For each game, use a calculator to help you find a data set that matches all the clues. Write your solution with the values in order from least to greatest. Be prepared to prove to the class that your solutions work.

Sample Clues for a Mystery Data Game A. Tanya played 5 basketball games and kept track of how many points she scored in each one. B. Her mean score was 8 points. C. She never scored 0 points. What might each of her scores be? Find a data set to match all the clues. ? ,

? ,

? , ? , ?

Sample Solutions 6, 7, 7, 7, 13

5, 7, 8, 9, 11

What other solutions can you find? Rule Data sets like 8, 8, 8, 8, 8 are not allowed in the game because they are too easy to find.

10



Experiment with Means, Medians, and Modes Seven students kept track of how many hours they played sports during August. Their mean was 21 hours. How many hours might each of the students have played sports? Your challenge is to make data sets to match the descriptions below. Choose four of the descriptions and make a different data set to match each one. All the data sets should have 7 values and a mean of 21. In each of your data sets, circle the median and put a square around the mode (if any).

How can you create data sets that have different relationships among means, medians, and modes?

A. Mean is larger than median. B. Median is larger than mean. C. Mean is larger than mode. D. Mode is larger than mean. E. Median is larger than mode. F. Mode is larger than median. G. Mean, median, and mode are equal.

Write Your Own Clues Follow the clue guidelines below to create Mystery Data games like the ones you played. Write two different solutions to your game on a separate sheet of paper. Then, exchange your games with your classmates. Remember, there are likely to be multiple correct solutions. Clues should provide the following information: ■

type of data; for example, scores, hours, or prices

■

number of values in the data set (Choose a number between 5 and 10.)

■

the mean of the values

■

at least one of the following: median, mode, range, lowest value, or highest value

hot words

mean range

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4

CREATING AND COMPARING STEMAND-LEAF PLOTS

Top Teams Stem-and-leaf plots are useful tools for representing and analyzing data. You can find out a lot by examining the shape of the data. You will use stem-and-leaf plots to compare sports teams.

Interpret Stem-and-Leaf Plots What types of information can you get from stem-andleaf plots?

Two teams of students timed how long they could balance on one foot with their eyes closed and arms extended overhead. Follow these steps to compare the teams. 1

On graph paper, make a stem-and-leaf plot for Team B. Be sure to put the numbers in order from least to greatest.

2

Find the median for each team.

3

How are the shapes of the two stem-and-leaf plots different? What does that tell you about the differences between the teams?

4

Which team do you think is better at balancing? Use data to support your conclusions.

Balancing Data Team A: This stem-and-leaf plot shows how many seconds each student balanced on one foot.

Team B: Here are the numbers of seconds each student balanced on one foot. 33 18 41 26 30 35 38 49 27 46 47 29 36 44 45

12


Stems 0 1 2 3 4 5 6

Leaves 9 24566689 03589

9

Note: 2 4 means 24 seconds


Compare Stem-and-Leaf Plots Your challenge is to figure out which of two mystery baseball teams is better at hitting home runs. 1

Use the data below to make a stem-and-leaf plot for each team.

2

Find the following for each team. a. least value

b. greatest value

d. median

e. mean

How can you use stem-and-leaf plots to make comparisons?

c. range

3

What are the differences between the two teams? Write at least six comparison statements.

4

Which team do you think is better overall at hitting home runs? Why? Home-Run Data The lists show the numbers of home runs hit by players on each team in a recent season. For example, a 6 means that one player hit 6 home runs during the season. Team X 6

30 23 28 23 34 12

4

5

10 10

2

11

5

3

9

5

14

6

5

6

5

Team Y 0

42 22

5

19

2

2

3

Write Data-Based Statements Take on the role of a professional baseball player for Team X or Team Y. An interviewer asks you these questions: 1

You have the honor of hitting the median number of home runs. How do you compare with the other players?

2

How does the best home-run hitter on your team compare with the other players?

3

Is your team better at hitting home runs than the other team? Give me the data to prove it.

hot words

stem-and-leaf plot

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P H A S E TWO



In this phase, you will investigate questions like,“Can you use foot measurements to make predictions about students’ heights?” You will collect data and make scatter plots to see if there is a relationship between variables. You will also learn about different types of correlations.

Data, Scatter Plots, & Correlations WHAT’S THE MATH? Investigations in this section focus on: STATISTICS: DATA COLLECTION and REPRESENTATION ■

Collecting and organizing data

■

Making back-to-back stem-and-leaf plots

■

Making scatter plots to represent paired data

STATISTICS: DATA ANALYSIS ■

Analyzing the relationship between pairs of data

■

Identifying positive and negative correlations

■

Sketching and interpreting lines of best fit

■

Distinguishing between correlation, and cause and effect

■

Making conclusions that are based on data

MEASUREMENT ■

Estimating and measuring



15


5

COLLECTING AND REPRESENTING MEASUREMENT DATA

Comparing Sizes Back-to-back stem-and-leaf plots are useful for making comparisons. You will collect measurement data about your body and plot it with data from your classmates. Then you will make comparisons and write your conclusions.

Collect Measurement Data How can you make accurate measurements so that your data is useful for making comparisons?

When you collect data to compare sizes, it’s important to make careful, accurate measurements. Sloppy measurements could throw off the class data and lead to false conclusions. 1

Starting with Line A, estimate its length to the nearest half centimeter and record your estimate. Then measure the line to the nearest half centimeter and record your measurement. Repeat this process with the other lines. Did your estimates improve?

A B C

D E F

2

16

Your teacher will provide a Measurement Survey listing the body parts you should measure. Remember to measure to the nearest half centimeter.



Make Back-to-Back Stem-and-Leaf Plots Now that your class has collected lots of data, it’s time to make comparisons. Which are longer, students’ feet or their forearms? 1

On graph paper, make a back-to-back stem-and-leaf plot for the class’s foot and forearm data. Before you begin, make sure to put each data set in order from least to greatest values.

2

How do the shapes of the data compare for feet and forearms?

3

Find the mean, median, and range for lengths of feet and forearms.

4

In your class, which are longer, feet or forearms? Are they the same size? Use data to support your conclusions.

How can you use back-to-back stemand-leaf plots to make comparisons?

A Sample Back-to-Back Stem-and-Leaf Plot In this type of plot, the stems are in the center of the plot.

Data from 23 Students in Ms. Rodriguez’s Class Foot Lengths (cm) 1 8 • 5.5 5 4 4 4 2 2 0 2 9 9 8 8 8 7 7 6 6 6 6 6 • 4 1.5 3

0 2 represents 20 cm 2 0 represents 20 cm

Forearm Lengths (cm)

Leaves

0 0 2 2 4 4 5 5 5 5 5.5 6 6 6 6.5 7 8 8 0 • 6 6 7 4 5 Stems Leaves

Write Comparison Statements When you make comparisons, it’s important to use specific examples from the data. ■

In your class data, how do the lengths of students’ feet and forearms compare? Write at least 5 data-based comparisons.

■

How does the back-to-back stem-and-leaf plot for your class compare with the sample one? Would your conclusions be the same or different for the two classes? Why?

hot words

stem-and-leaf plot median

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6 INTRODUCING SCATTER PLOTS

Is There a Relationship? Scatter plots are useful for showing the relationship between two variables, such as heights and weights. They are also called scattergrams and scatter diagrams. You will analyze a scatter plot and then create your own using data on mother animals and their babies.

Interpret a Scatter Plot How can you identify points on a scatter plot?

Do students with longer feet tend to have longer forearms? You will interpret a scatter plot to see if there is a relationship between the two measurements. 1

Five of the points on the scatter plot have been labeled, A, B, C, D, E. Which of the following students does each point represent?

Scatter Plot of Foot and Forearm Data for 23 Students in Ms. Rodriguez’s Class E 46 44 42 40 38 D 36 34 32 Length of Forearm 30 C 28 (cm) 26 24 B 22 A 20 18 16 16 18 20 22 24 26 28 30 32 34 Length of Foot (cm)

18

Simon: “I have the longest forearm and foot in the class.” Tanya:

“My foot is the same length as my forearm.”

Heather: “My forearm is about 5 centimeters longer than my foot.” Miguel: “My foot is the same length as those of four other students, but my forearm is longer than theirs.” 2

The extra labeled point is for David’s data. Write a description for him.

3

Based on the scatter plot, what do you think the relationship is between the lengths of students’ feet and forearms? Why?



Make a Scatter Plot How can you make a scatter plot to show the relationship between pairs of data?

Is there a relationship between the lengths of mother animals and their babies? Your teacher will give you a table of animal data. Use this data to investigate the following. 1

Choose two variables from the table. Make a scatter plot to show the relationship between the two variables.

2

Analyze the scatter plot. Do the points suggest a line? What is the relationship between the two variables? Why? Guidelines for Making Scatter Plots

To make a scatter plot, you need pairs of data.The data can be number counts, such as the number of free throws made, or measurement data, such as heights. Scatter plots cannot be used for category data, such as colors or foods. ■

The axes do not need to begin at zero.The two axes can have different scales. Choose scales that makes sense for your data.

■

The intervals between the numbers on a scale need to be the same size. For example, 1, 2, 3, 4, or 2, 4, 6, or 10, 20, 30.

■

If there is more than one pair of data for one point on a grid, you can use a number instead of a point. For example, if three pairs of data have the values 5 ft and 90 lbs, then use the number 3 instead of a point.

Write About Points on a Scatter Plot Label four of the points on your animal scatter plot with the letters, A, B, C, D. Write a description of each point similar to the descriptions on the opposite page.

hot words

scatter plot ordered pair

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7 EXPLORING POSITIVE AND NEGATIVE CORRELATIONS

What Type of Relationship Is It? Scatter plots can help you see relationships in data. First, you will look at pairs of variables to see how their values are related, then you will make some scatter plots and look for correlations.You will see how a line of best fit can help you figure out if a correlation is weak or strong.

Discuss Relationships Between Variables What types of relationships can two variables have?

Pairs of variables can be related in different ways: ■ positive correlation: one variable increases when the other increases ■ negative correlation: one variable increases when the other decreases ■ no correlation Look over the pairs of variables in the table below and answer these questions.

20

1

Which of the three relationships do you think each pair is most likely to have? Why?

2

Come up with your own examples for each type of relationship.

Pair

Variable 1

Variable 2

A

amount of trash a family recycles

amount of trash a family sends to the dump

B

number of rainy days per year

number of dry days per year

C

length of an object in inches

length of an object in centimeters

D

age

length of hair

E

money spent on bags of pretzels

number of bags purchased

F

weight of package

cost of postage

G

number of days a student is absent from school

number of days a student attends school

H

number of teenagers in house

amount of time phone is in use in house

I

number of candy bars eaten in a typical week

number of cavities



Identify Different Correlations Is there a correlation between the popularity of a food and the amount of fat, sugars, or sodium in a serving? Make a scatter plot to find out. 1

Use your class data to make a scatter plot.

2

What type of correlation does the scatter plot show?

3

Draw a line of best fit for the food scatter plot. If it doesn’t make sense to draw this line for your scatter plot, explain why.

4

How close are the points to the line of best fit? What does this tell you about the relationship between the two variables?

How can you determine what type of correlation a scatter plot shows?

Different Types of Correlations These scatter plots show different types of correlations. Strong Positive Correlation Variable 2

Perfect Negative Correlation Variable 2

Variable 2

Perfect Positive Correlation

Strong Negative Correlation

Weak Positive Correlation

No Correlation

Variable 1

Variable 2

Variable 1

Variable 2

Variable 1

Variable 2

Variable 1

Variable 1

Variable 1

Drawing a line of best fit can help you figure out the relationship between two variables in a scatter plot. Use a ruler and pencil to draw a line that follows the trend of the data. In some scatter plots, the line will connect many of the points. In others, you will need to draw a line so that about half the points are above the line and half are below. If most of the points are on the line or close to the line, that shows that there is a strong correlation between the variables. See page 22 for an example.

hot words

correlation line of best fit

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8

APPLYING SCATTER PLOTS AND CORRELATIONS

The Mysterious Footprint This is your chance to be a detective. You will use your knowledge of scatter plots and correlations to investigate a mystery.

Make Reasonable Estimates

1

Analyze the scatter plot on your handout. What type of relationship does it show between heights and foot lengths?

2

What would you estimate the thief ’s height to be for each of the following foot lengths? a. 27 cm

c. 32 cm

d. 23 cm

8

9

10 11 12 13 14 15 16 17

35 34 33 32 31 30 29 Foot 28 Length (cm) 27 26 25 24 23 22 21 142 146 150 154 158 162 166 170 174 178 182 186 Height (cm) Estimated height is about 180 cm 1

2

3

4

5

If there is a correlation between the two variables, you can use the line of best fit to make estimates. For example, if a footprint is 33 cm long, you could estimate that the thief’s height is about 180 cm.The stronger the correlation, the more accurate the estimates are likely to be.

b. 31 cm

7

How to Make an Estimate

On the day of the Baker Middle School bake sale, a large pan of brownies was stolen from the cooking room. Fortunately, the floor was coated with flour, and the thief left a floury footprint 27 cm long. Your challenge is to use this clue and your knowledge of correlations to make a reasonable estimate about the thief’s height.

6

How can you use a line of best fit to make predictions?

22



Investigate Class Data There were two more clues left at the scene of the Baker Middle School crime: a wristwatch and a chocolate-covered, long-sleeved shirt (see below). What estimates can you make about the thief ’s measurements based on these clues and your class data?

Watch: 16 cm

Footprint: 27 cm

How can you make and test hypotheses about which pairs of measurement data have strong correlations?

Shirt Sleeve from shoulder to wrist: 51 cm Neck circumference: 32 cm

Planning and Organizing 1

Choose a variable from one of the clues (foot, wrist, neck, or arm measurements). Then choose a different variable (height, forearm, calf, or ankle) from your class measurement survey.

2

Make a hypothesis. What kind of relationship do you think that these variables have? Why?

3

Make a table to organize the pairs of data.

Representing and Analyzing the Data 4

Make a scatter plot of the data. How did you set up the axes?

5

Draw a line of best fit. If it doesn’t make sense to draw one, explain why not.

6

How would you describe the points? What trends do you see?

Making Conclusions and Estimates 7

What type of correlation does your scatter plot show: positive, negative, or no correlation? Why? Give examples of at least four points that have this relationship.

8

Do your conclusions support your hypothesis? Why?

9

Write an explanation of how to use a line of best fit to make an estimate about the thief ’s measurements. If your scatter plot shows a correlation, make an estimate. For example, if you plotted data on wrists and necks, estimate the thief ’s neck measurement. Draw lines on your scatter plot to show how you made the estimate.

hot words

scatter plot correlation

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23

PHASE THREE


usic a m to be f o se er emb an choo . You m a s of ger s, ou c dent role tour. Y r mana bination e u t e S h o t To: on com o on n th eer, take out to g d engin ability, arise o l l i w b n ob at You hat is a er, sou s of pr ems th g l c t i n b p si pro mat grou sician, athe o solve m u a m se the ions t t u will ermuta p and . tour


In this phase, you will be playing games of chance and figuring out the probabilities of events in the games. You will learn some useful techniques, such as making tree diagrams and using the Permutation Theorem. These tools will help you to find all the different combinations of a set of objects.

Probability, Combinations, and Permutations WHAT’S THE MATH? Investigations in this section focus on: PROBABILITY and STATISTICS ■

Collecting and analyzing data

■

Determining theoretical probabilities

■

Determining experimental probabilities

NUMBER ■

Figuring out different combinations of objects

■

Figuring out different permutations of objects

■

Using systematic listings

■

Making tree diagrams

■

Using the Permutation Theorem



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EXPLORING PROBABILITY AND COMBINATIONS

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On Tour Have you ever wanted to go on tour with a band? You will play a game of chance to collect data on different events that can happen while on tour. Then you will analyze your data.

Play a Game of Chance How can you analyze data to find out about the probabilities of events?

In the Concert Tour Game, you will use a number cube, spinner, and coin to randomly determine what happens when your group goes on tour. Keep track of your data in the table on the handout. Analyze the game rules and the data to answer these questions: 1

How many times did you get each of the following: a. ideal location?

b. ideal audience size?

c. ideal audience reaction? 2

Based on your data, what is the experimental probability of each of the above events?

3

What is the theoretical probability of each of the above events?

4

How many times did all three of your ideal events come up on one turn? Probabilities

Theoretical probabilities are found by analyzing a situation, such as looking at game pieces or game rules.You can use the following ratio: number of favorable outcomes total number of possible outcomes Experimental probabilities are based on data collected by conducting experiments, playing games, and researching statistics in books, newspapers, and magazines.You can use the following ratio: number of times a particular outcome occurred total number of tries or turns

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List All the Possible Combinations In order to figure out the theoretical probability of getting all three of your ideal events on one turn, you need to know how many possible combinations of events there are. 1

Start by simplifying the problem. List all the possible combinations for just two events: locations and audience reactions. For example: 1H (Boston, Happy), 1T (Boston, Thumbs Down). The order of the events doesn’t matter, since 1H is the same combination as H1.

2

How many different combinations did you find? How can you make sure you found them all?

3

List all the possible combinations of the three events. How many combinations did you get?

4

What is the theoretical probability of all three of your ideal events coming up on one turn?

How can you use systematic listing to figure out all the different combinations?

Improve the Game The game designers would like your help in changing the Concert Tour Game. They want to use the three spinners below instead of a number cube, a coin, and a spinner. 1

How would you label each of the spinner parts? Brainstorm ideas. Then copy the spinners and fill in your labels.

2

What are your three ideal events in this new game?

3

What is the theoretical probability of getting all three of your ideal events on one turn? Explain how you figured it out. Concert locations

Transportation: How will you get there?

What will you wear?

hot words

experimental probability theoretical probability

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Lunch Specials Is a restaurant’s special as good a deal as it claims? You can use tree diagrams to help answer this question.

INTRODUCING TREE DIAGRAMS

Make a Tree Diagram How can you use tree diagrams to figure out all the possible combinations of things?

When your music group is on tour, you are on a tight budget. That’s why you’re always on the lookout for restaurants with special deals. Figure out whether the special shown below is as good as it is claimed to be. 1

Make a tree diagram like the one on your handout to figure out all the possible lunch combinations.

2

Circle the path that shows the combination that you like best. How much money would you save by ordering the special instead of buying each item separately?

3

How many different combinations are there? Count the bottom branches to find out.

4

Is the advertisement’s claim true? Why or why not?

The Unique Diner’s Lunch Special Choose one main dish, one side dish, and one beverage for just $2.50. There are over 21 unique combinations, and each one saves you $1 or more!

28

Main Dish Choice

Side Dish Choices

Beverage Choices

Hamburger $2.25

French Fries $0.99

Milk $0.65

Burrito $1.99

Salad $1.25

Juice $0.75

Pizza $1.80

Corn $0.55


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Compare the Numbers of Combinations The table shown here depicts how many different lunch choices are offered at six different restaurants. Customers can choose one main course, one beverage, and one side dish for a special price.

1

Restaurant

Number of Main Dishes

Number of Side Dishes

Number of Beverages

How many combinations?

A

3

2

2

?

B

3

3

2

?

C

4

3

2

?

D

4

3

3

?

E

4

3

4

?

F

5

3

4

?

How does adding more choices affect the total number of different combinations?

Find the number of different combinations for each restaurant. Tip: Make a tree diagram for Restaurant A. Either decide what the dishes will be (tacos, spaghetti, hot dogs, and so on) or use letters and numbers to stand for each dish. For example, Restaurant A has three main dishes (M1, M2, M3), two side dishes (S1, S2), and two beverages (B1, B2).

2

Keep track of the number of combinations for each restaurant by making a table as shown. What patterns do you see?

Create Dinner Specials Create and investigate dinner menus by following these steps: 1

Make a dinner menu that has 4 main dishes, 2 side dishes, 2 beverages, and 2 desserts. Exchange menus with a classmate.

2

For your classmate’s menu, make a tree diagram to figure out all the possible combinations consisting of 1 main dish, 1 beverage, 1 side dish, and 1 dessert.

3

Use the Fundamental Counting Principle to check that you found all the combinations. This principle states that if the first category has m options and the second category has n options, and if you choose one option from each category, then the number of combinations is m • n.

hot words

combination tree diagram

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The Battle of the Bands Your music group is going to compete in a “Battle of the Bands.” Do you want to perform first, second, third, or last? The order will be chosen randomly. You will figure out how many different orderings there are and calculate the probability of getting the one you want.

Experiment with Permutations How many different orderings can you get if you randomly choose the order of four things?

Do an experiment to find out how many times your music group will be randomly chosen to perform first, second, third, or fourth. Follow the directions on the handout What’s the Order? After you do the experiment, answer these questions. 1

How many different orderings or permutations did you get?

2

Did you get the same ordering more than once?

3

Do you think you got all the possible orderings in your experiment? Why or why not?

4

Would you prefer to perform first, second, third, or fourth? How many times did you get your preference?

Permutations A permutation is an arrangement of names, letters, or objects in a particular order. For example, imagine that your group is going to perform in three different cities: Atlanta, Boston, and Chicago. You get to choose the order in which you visit the cities. Here are three different permutations. 1. Atlanta

Boston

Chicago

2. Boston

Chicago

Atlanta

3. Chicago

Atlanta

Boston

What other permutations can you find?

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Find All the Permutations How can you use a tree diagram to find all permutations of four objects?

Find all the possible orderings or permutations for the four music groups by following these steps. 1

Copy the tree diagram below and complete it for the four bands. Each letter stands for a different band. Decide which letter represents your band.

2

How many different permutations are there?

3

Find the theoretical probability of randomly choosing each of the following orderings.

Unfinished Tree Diagram of Possible Orders for Bands First

A

Second

C

B

B

C

D

D

a. Your band is first. b. Your band is last.

Third

C

D

c. Your band is second or third. d. Band B is first, and Band C is last.

Fourth

e. D, C, B, A

Permutations

D

C

ABCD ABDC

Trace down the branches to find the different permutations. Count the branches at the bottom to find the number of permutations.

Create Word Puzzles with Permutations You can use what you know about permutations to write and solve word puzzles. 1

Here are permutations of the letters in two words: a. trposs

b. rigsne

What are the words? 2

Write your own word puzzle. Choose a word that has 5 to 10 letters. Try out at least three different permutations of the letters. Choose one of them for your puzzle.

3

Exchange puzzles with a classmate.

hot words

permutation tree diagram

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The Demo CD Suppose a large number of bands are going to compete in a contest. How many different orderings of the bands are there? You will see how the Permutation Theorem can help answer this question.

Use the Permutation Theorem How can you use the Permutation Theorem to figure out the number of possible orderings for a large number of objects?

Five more bands want to enter the Battle of the Bands contest. It would be difficult to make a tree diagram to show all of the orderings for nine bands. Another way to solve the problem is to use the Permutation Theorem. Use the Permutation Theorem and a calculator to figure out the number of different orderings possible for each of the following numbers of bands: a. 2

b. 3

c. 4

d. 5

e. 6

f. 7

g. 8

h. 9

The Permutation Theorem

The Permutation Theorem says that there are n! (n factorial) possible permutations of n objects. (Each object can be used only once.) For example, for 4 bands, there are 4! (or 4 factorial) permutations. To calculate 4!, multiply 4 3 2 1.

32

Number of bands that can go first: Number that can go second:

4 3

(After one is picked to go first, you are choosing from 3 bands to go second.) Number that can go third: Number that can go fourth: 4 3 2 1 24 possible permutations or orderings


2 1

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Investigate Different Situations 1

How can you apply what you know about combinations, permutations, and probability?

The band is getting offers to perform all over the country. On each tour they arrange, they will play at 6–10 different locations. There may be 2 or 3 different audience reactions. To allow for each set of possibilities, a revised Concert Tour Game is played with 3 spinners, each divided into equal parts. Make a table like the one here and complete the missing information. Explain how you figured it out. Number of Locations

Number of Audience Sizes

Number of Audience Reactions

Number of Possible Combinations

Original

6

3

2

?

Version B

8

2

2

?

Game Version

Version C

9

4

3

?

Version D

?

?

?

90

2

For each version of the game, what is the probability that you would get your ideal location, audience size, and audience reaction on one turn? Rank the games in order from least to greatest probability. Explain how you figured this out.

3

Your band gets a contract to make a demo CD. Which of your three favorite songs should go first, second, or third? Write down the song names, then make a tree diagram to show all the possible permutations, or orderings, of these songs.

4

Your group has 11 songs to put on the demo CD. Use a calculator to figure out how many possible orderings there are for 11 songs. Explain how you figured it out.

Write About the Mathematics Write a letter to a student who will be doing this unit next year. Describe the investigations and the mathematics you learned.

hot words

factorial probability

W Homework page 45


33


1

Sports Opinions and Facts

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HW

Applying Skills

In Mr. Jackson’s class, 25 students rated how much they enjoyed playing soccer. 1 (Not at all) 2 3 4 5 (Very much)

1. How many students chose each rating? 2. What percentage of students in the class

gave a rating of 1? of 3? of 5?

lll l lll lll l lll l l ll lll l

3. Add up all the ratings to find the total. 4. What is the class average, or mean

rating?

Extending Concepts Number of Participants in Exercise Programs 5

Ages of People Who Played Soccer in 1994

Exercising with equipment Running/jogging

7–11 years 44% 12–17 yrs 28%

4 Number of 3 Participants (millions) 2

8% 35

1 0

1988

1990

1992

1994

8% 11%

25–34 18–24 yrs yrs

Give specific data to support your answer to each question below.

6. Of the people who played soccer in 1994,

35 30 25 Percentage of Football 20 15 Players 10 5 0

7– 12– 18– 25– 35+ 11 17 24 34

7. Which group is generally older, the

people who played soccer in 1994 or the people who played football in 1994?

5. Of the people who played football in

1994, what percentage was over 25 years of age?

Ages of People Who Played Football in 1994

8. In 1990 which was more popular,

jogging or exercising with equipment? 9. During the period 1988–1994, did

were more of them between 12 and 17 years of age or over 18?

jogging gain in popularity? Did exercising with equipment gain in popularity?

Writing

10. Write two questions of your own that can be answered using the graphs above and two

that cannot be answered using these graphs.

34

LOOKING BEHIND THE NUMBERS • HOMEWORK 1


2

Who’s the Best?

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HW

Applying Skills

Extending Concepts

Find the mean and median for each data set.

13. Which player (A, B, or C) was the most

consistent in the number of points scored? How can you tell?

1. 12, 19, 16, 30, 11

14. Bill claimed that Player A was the best

2. 2, 8, 11, 7, 17, 5, 5, 1

player. Maria claimed that Player C was the best player. Use the data to write an argument in support of each person’s claim.

3. 1.5, 6.2, 9.9, 2.8

Find the range for each data set. 4. 2, 5, 7, 14, 9, 12

15. Which player (A, B, or C) would you

5. 1, 26, 87, 2, 90, 55 6. 5.0, 4.8, 5.0, 5.1, 4.9 7. In nine basketball games, Marta scored

11, 18, 20, 15, 18, 8, 6, 13, and 26 points. Find her mean, median, mode, high score, and range. 8. In six gymnastics competitions, Jacy

scored 9.1, 9.8, 9.5, 8.8, 9.1, and 9.9 points. Find his mean, median, mode, high score, and range. The table shows statistical measures for three basketball players. The statistics are based on the data for nine games. High Mean Median Mode Score Range

choose to put in a game if you are playing an excellent team? Only an outstanding performance will give you a chance of winning. Give reasons for your choice.

Making Connections

16. These are the approximate speeds (in

miles per second) at which the planets orbit the sun. Find the mean, median, mode, and range of the speeds. Mercury

30

Venus

22

Player A

16

16

15

18

4

Earth

19

Player B

9

10

6

14

9

Mars

15

Player C

18

18

9

28

22

Jupiter

8

Saturn

6

Uranus

4

10. median

Neptune

3

12. high score

Pluto

3

Rank the players from best to third best by each of the following: 9. mean 11. mode


35


3 ork w e om

HW

Mystery Data

Applying Skills

For each data set, find the mean, median, mode, and range.

Extending Concepts

7. a. When averages are used to make

comparisons, everyone wants to be above average, and no one wants to be below average. Make a data set that has seven values, a mean of 50, and only one value that is below the mean.

1. 3, 5, 9, 14, 3, 2, 10, 10, 3, 15, 8 2. 2.5, 3.1, 4.0, 7.2, 1.7, 2.5

Create a data set that matches each set of clues.

b. Do you think it is possible to create a

data set that has seven values, a median of 50, and only one value below the median? Explain your thinking.

3. Tyler played 5 basketball games.

The mean was 10 points. The median was 12 points.

8. Create a data set that has five values, a

What might each of his scores be?

median of 30 and:

4. Desrie played 6 basketball games.

a. a mean that is greater than its median.

The mean was 18 points.

b. a median that is greater than its mean.

The range was 8 points.

Explain how you created the data sets.

What might each of her scores be? 5. Six students kept track of how much

time they spent reading in a week. The mean time was 6 hours.

Writing

9. Answer the letter to Dr. Math.

The mode was 7 hours. What might each of their times be? 6. Francisco bought 5 books.

The mean price was $23.80. The most expensive book cost $30. What might each of the books cost?

Dear Dr. Math: I’m confused. We’ve been studying means, medians, and modes. I’m always getting them confused because they all begin with the letter m. What’s the difference between these things? Mark M. Morrison

36



4

Top Teams

ork w e om

HW

Applying Skills

Extending Concepts

The stem-and-leaf plot below shows the number of home runs for each player on a baseball team during one season. Stems 0 1 2 3 3

Leaves 2 3 5 5 7 8 8 8 1 4 6 0 2 5 1 1 = 31 home runs

1. What is the greatest number of home

The stem-and-leaf plots show the average number of points per game for basketball players of the New York Knicks and the Orlando Magic during one playing season. New Stems 0 1 2 2

York Leaves 2 3 4 4 5 6 0 1 1 3 4 5 3 3 = 23 points

Orlando Stems Leaves 0 2 3 4 4 5 5 6 9 1 3 5 8 2 2 7 1 3 = 13 points

8. Find each team’s mean, median, and range.

runs? the least? 2. What is the median number of home

runs?

9. Write a paragraph comparing the two

teams. Use your answers to item 8 and compare the shapes of the two plots.

3. What is the mean number of home

runs? Making Connections

4. What is the range?

10. The table shows the average life span of

5. How many players got 8 home runs?

11 mammal species. Make a stem-andleaf plot to represent this data. Use your plot to find the median life span.

fewer than 10? more than 19? The following are the number of home runs scored by the players of a second baseball team during the same season.

Animal

6, 17, 12, 11, 6, 21, 5, 12, 9, 14, 24, 4, 25, 14, 18 6. Make a stem-and-leaf plot for this

second team. 7. Use your plot to find the median, highest

and lowest values, and range.

Life Span (years)

Elk

15

Deer

8

Tiger

16

Possum

1

Fox

7

Camel

12

Mouse

3

Elephant

35

Wolf

5

Monkey

15

Grizzly Bear

25


37


5

Comparing Sizes

ork w e om

HW

Applying Skills

Making Connections

The back-to-back stem-and-leaf plot shows the heights of 15 college football players and of 15 college basketball players.

The table shows the 1995 life expectancy for females in ten European countries. Country

Heights of Football Players (in.)

Heights of Basketball Players (in.) 6

5

4

3

3

2 8

2 7

1 7

1 6

9 0 6

• 8 7

• 8

•

3 6 0 6

0 7 = 70 inches

5 6 0 7

6 1

7 4

8

9

7 3 = 73 inches

1. What is the height of the tallest football

player? 2. How many basketball players are over

Life Expectancy

Sweden

81

Portugal

79

Germany

80

Hungary

76

United Kingdom

80

France

82

Poland

77

Italy

80

Belgium

81

Denmark

79

82 inches tall? 3. Find the mean, median, and range of the

7. Find the mean, median, and range of the

data set.

basketball players’ heights. 4. Find the mean, median, and range of the

8. What additional data could you look up

football players’ heights. 5. In general, are the football players or the

basketball players taller?

Extending Concepts

6. How do the heights of the football

players compare with the heights of the basketball players? Use the back-to-back stem-and-leaf plot to write five databased comparison statements. Be sure to include statements comparing the shapes of the two data sets.

38


in order to make a back-to-back stemand-leaf plot with the data in the above table? (The data in the table would be one side on the plot; your new data would be on the other side.) What might you be able to find out from your backto-back stem-and-leaf plot?

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HW

6:33 AM


Is There a Relationship? a. As one variable increases, the other

Applying Skills

increases.

Use the scatter plot for items 1 and 2. 26 24 22 20 18 Hours Spent 16 Watching 14 Sports on 12 TV in a 10 Typical Week 8 6 4 2 0

b. As one variable increases, the other

decreases.

C

c. There is no relationship between the

variables. 4. What conclusions would you draw

from the graph about the relationship between the number of hours a student plays sports and the number of hours he or she watches sports on TV?

E A D B 0 2 4 6 8 10 12 14 16 18 20 Hours Spent Playing Sports or Exercising in a Typical Week

Making Connections

1. Which of the students below does each

of the points A, B, C, D, E represent? Matthew: “Each week I spend more than 12 hours playing sports. I also spend more than 12 hours watching sports on TV.”

The table shows the latitude and a typical minimum temperature in January for each of eight U.S. cities. City

Latitude Minimum Temperature (°N) in January (°F)

Anchorage

61

8

Miami

26

59

Jong: “I spend the same amount of time playing sports as watching them on TV.”

Helena

47

10

Buffalo

43

17

Lyn: “I spend twice as much time watching sports on TV as playing them.”

Reno

39

21

Memphis

35

31

LaToya: “I spend time playing sports, but I don’t watch them on TV.”

Houston

30

40

Chicago

42

13

2. One of the points doesn’t have a match. 5. Make a scatter plot showing the

Write a description for that point.

relationship between the two variables. Extending Concepts

6. What trend does this set of data show? 7. What can you say about the relationship

3. According to the scatter plot, which

statement best describes the relationship between time spent playing sports and time spent watching sports on TV?

between the two variables?


39

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What Type of Relationship Is It?

ork w e om

HW

Applying Skills

Extending Concepts

For each of the following, determine whether or not you can make a scatter plot. If you can, draw and label the axes. Describe what type of relationship there would be. If you cannot make a scatter plot, explain why not.

8. Which plot from items 5–7 shows the

strongest correlation? How can you tell?

Writing


1. lengths and weights of cats 2. students’ favorite sports and favorite

Dear Dr Math,

music groups 3. heights of 8th graders 4. students’ ages and the amount of

allowance they receive Tell whether each scatter plot shows positive, negative, or no correlation. 5.

6.

I’m having trouble figuring out how to read a scatter plot. There are so many little dots all over the place. How can you tell if there is a correlation? Please give me some examples and draw some pictures to help me figure this out.

Variable 2

Variable 2

Dottie Variable 1

Variable 1

7. Variable 2

Variable 1

40


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6:33 AM


The Mysterious Footprint

ork w e om

HW

Applying Skills

Writing

This scatter plot shows ages and weights of some monkeys at a zoo. 150


Dear Dr. Math,

140 130

Weight 120 (lbs) 110 100 90 80

8

10

12

14

16

18

20

Age (yrs)

1. Use the scatter plot to estimate the

weight of monkeys of the following ages: a. 13

b. 16

c. 19

A friend of mine told me that I could use scatter plots to make predictions. Then he said I couldn’t make predictions from just any scatter plot. He also said certain scatter plots are better for making predictions than others. I don’t know what he was talking about. Can you help me sort this out? Scattered in Scranton

2. Use the scatter plot to estimate the age of

a monkey whose weight is the following: a. 90 lb

b. 110 lb

c. 130 lb

Extending Concepts

3. Give an example of a pair of variables

that are likely to have no correlation and a pair that are likely to have negative correlation. 4. Give an example of a pair of variables

that are likely to have a strong positive correlation.


41


9

On Tour

ork w e om

HW

3. What is the theoretical probability of

Applying Skills

getting each of the above events?

On each turn of a game, players spin the two spinners shown and flip a coin. They get a point for a 1, a point for blue, a point for tails, and a bonus point for getting all three in one turn. 1

2

4

3

Blue Pink

4. How many times did Pei get 1, blue,

and tails on one turn? What is the experimental probability of getting all three events on one turn?

Extending Concepts

Red

5. List all the possible combinations for the

two spinners. For example: 1B (1, blue) is one combination. How many combinations did you find?

Pei’s results for 12 turns are shown here. Spinner 1

Spinner 2

Coin

2

blue

H

3

blue

H

4

red

T

2

pink

H

1

red

T

3

blue

H

3

red

T

4

pink

H

1

blue

T

4

pink

T

1

blue

H

4

red

H

6. List all possible combinations for the

two spinners and the coin. 1BH (1, blue, heads) is one combination. How many did you find? 7. Find the theoretical probability of

getting 1, blue, and tails in one turn.

Making Connections

8. The Native American game of Totolospi

1. How many times did Pei get: a. a 1?

b. blue?

c. tails?

2. Based on Pei’s data, what is the

experimental probability of getting each of the above events?

42


was played by the Moki Indians of New Mexico. Players would throw three flat throwing sticks, which were plain on one side and colored on the other, and score points if all sticks fell the same way. List all the possible combinations when the sticks are thrown. For example, CCP is one combination (stick 1: colored, stick 2: colored, stick 3: plain). What is the theoretical probability that all three sticks land the same way?


ork w e om

HW

Lunch Specials

Applying Skills

Extending Concepts

The tree diagram shows the possible combinations for a lunch special. Soup Choices

Sandwich Choices

Miso

Tuna

9. Do you think the advertising claim

below is true? Why or why not? Unique Diner’s $1.99 Breakfast Special!

Onion

Pastrami

Tuna

Chicken

Pastrami Chicken

Choose one item from each category: 4 types of pancake, 3 kinds of egg, 3 kinds of cereal, and 3 kinds of juice. You could eat breakfast here for 100 days and have a different combination each day!

1. Write down each combination using

initials.

Writing

2. How many different combinations are

there? Make a tree diagram to show the different lunch combinations. Find the total number of combinations if you can choose from the following. In each case, you can choose one item from each category. 3. 3 soups (tomato, leek, bean),

3 sandwiches (egg, tuna, cheese) 4. 2 sandwiches (cheese, turkey), 3 desserts

(pie, ice cream, cake), 2 beverages (soda, juice) Without making a tree diagram, find the number of lunch combinations if you can choose from the following. In each case, you can choose one item from each category.


Dear Dr. Math: I would like to offer 50 unique combinations for my lunch special at the Unique Diner. How many main dishes, side dishes, and/or beverages do I need? Please explain how you figured this out, so I can do it on my own. The owner of the Unique Diner

5. 3 sandwiches, 4 beverages 6. 2 sandwiches, 3 soups, 5 beverages 7. 2 soups, 6 sandwiches, 4 desserts 8. 4 soups, 7 sandwiches, 8 desserts


43


ork w e om

HW

The Battle of the Bands

Applying Skills

Extending Concepts

1. Copy and complete the tree diagram to

find all possible permutations of the letters X, Y, Z. How many permutations are there? First Letter

X

Y

Z

5. Five bands (A, B, C, D, E) have offered to

play at a party. Only two bands will be selected to play. The organizer will decide which band will play first and which band will play second by picking names from a bag. a. Make a tree diagram to show all

Second Letter

Y

possible orderings of 2 bands.

Z

b. How many possibilities are there?

Third Letter

Z

Permutation

XYZ

Emma, Fiona, and Gina are three skaters in a competition. The order in which they skate will be picked randomly.

c. Find the theoretical probability that

band C will play first, followed by band B.

Making Connections

6. a. Find all the possible permutations of

the letters a, i, l, r. How many permutations are there?

2. Make a list of all possible orderings

(permutations) for the skaters.

b. How many of the permutations in

3. How many permutations are there? 4. Find the theoretical probability of each

of the following: a. Gina will skate first. b. Fiona will skate either first or last. c. Emma will skate last. d. The order will be Emma, Fiona, Gina.

44


item 6a represent English words? Which ones? (Note: The aril of a seed is its exterior covering. The lira is the currency in Italy, and the rial is the currency in Iran.)


k wor e m o

HW

The Demo CD 13. What is the probability that in a single

Applying Skills

turn a player will score the following:

Use a calculator to find each of the following: 1. 3!

2. 5!

4. 10!

5. 12!

a. 4 points?

3. 8!

Use the Permutation Theorem and a calculator to find the number of possible orderings for the following number of skaters in a contest: 6. 4

7. 6

9. 10

10. 13

b. 7 points?

c. 10 points?

The winner of the game show continues to round 2. In round 2, each of the letters R, O, M, and E is put in a bag. The player selects the letters randomly one at a time. If the letters are picked in the order R-O-M-E, the player wins a trip to Rome. 14. How many possible orderings are there

for the letters R, O, M, E?

8. 9

15. Find the theoretical probability that a

participant will win round 2.

11. Seven bands (A, B, C, D, E, F, G) are

playing in a contest. The order in which they play will be selected randomly. How many different orderings are there?

Writing

16. Answer the letter to Dr. Math. Extending Concepts

Dear Dr. Math,

Five TV game show contestants play a game of chance. Each player spins the spinner shown and rolls a number cube (with the numbers 1–6 on it). 12. Use a tree diagram to

show all the possible combinations for the spinner and the number cube. How many combinations are there?

A

B

How can you tell if a problem is about finding combinations or permutations? What’s the difference between these two things anyway? Please give me some examples. Combi Nation

C

On each turn of the game show, points are scored as follows: spin an A 4 points spin a B and roll an even number 7 points roll 6 and spin a C 10 points LOOKING BEHIND THE NUMBERS • HOMEWORK 12

45

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How can mathematics be used to analyze motion?

MATHEMATICS of

MOTION

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PHASEONE

PHASETWO

PHASETHREE

Measuring, Estimating, and Representing Motion In this phase, you will practice measuring motion and estimating how fast you move, how long it takes, and how far you go. Making good measurements is an important part of this beginning study of motion. You will also find new ways to represent motion in words, pictures, and diagrams.

Distance-Time Graphs First, you will walk at different speeds to collect motion data. Then you will use your data to make graphs that show how far you moved versus how long it took—distance-time graphs. You will also compare graphs with maps and stories that describe motion to find the best way to represent motion in any situation.

Using Graphs to Solve Problems For more practice graphing, you will convert a sports announcer’s race commentary into a graph. You will learn about average speed and find out it might be different than speed at any particular time. You will also find out about a powerful mathematical expression for calculating distance, speed, or time of travel. A final project brings together all of your new skills in dealing with motion.

PHASE ONE

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ing oach ore c s r e e n th unn he m ic R we o tion. T ss. The p , e m c mo an Oly gre form e about our pro ture r u e F p or to ur ky To: rate e yo learn m an trac e accu so need v o r k l c a to ow ta u mp To i ant you ore yo o is to un. You you kn ways d f m r i e u d to fw t al r yo ne spee ay no so be staf now, th u need a f l m k i w o d you thing y s of ho determ se you hould a me, an i t s u t , t n n a s firs ureme you ca e. Bec on, you tance, ession s s i s i m t i a mo ing me out how and t of d our imates r train and e c y d n n e i a u r t f e s dist s yo easu crib te e your le to m accura o discus to des b t w be a to make l want now ho l i k e to o. ew abl d. W portant otion to e e p s ur m ’s im so it sent yo e repr


Get ready to hop, skip, and jump! In this phase you will learn how the motions that we make can be analyzed and described with mathematics. Studying motion is not only fun, it can be an important part of your job if you are an engineer, astronaut, or athlete, for example.

Measuring, Estimating, and Representing Motion WHAT’S THE MATH? Investigations in this section focus on: MEASUREMENT and GEOMETRY ■

Estimating and measuring distance, time, and speed

■

Using units of measurement

NUMBER and COMPUTATION ■

Plotting positions of parade characters on a Snapshot Sequence Sheet

■

Using a Snapshot Sequence Sheet to determine direction, change in speed, and distance traveled over an interval

ALGEBRA FUNCTIONS ■

Determining distance, time, or speed when the other two measures are known


MATHEMATICS OF MOTION

49


1

Moving, Measuring, and Representing

INVESTIGATING RATES OF MOTION

Fast or slow, up or down, turning, twisting, or straight— there are many kinds of motion. You will move your body in different ways, estimate and measure speed and distance, and use different methods to represent motion.

Move at Precise Speeds How closely can you estimate the speed of motion?

In the following activities, work with your partner. Take turns being the mover and the measurer. 1

The mover tries to move at exactly the speed given in the List of Motions without looking at a clock or watch. The measurer measures, times, and checks the mover.

2

Make up new motions at precise speeds. Choose what the mover should do and how fast the mover should go. Record what motions you tried and what happened.

3

Write a description of something that might move at each of the following speeds: a. 5 feet per second b. 3 meters per second c. 50 meters per hour d. 10 miles per hour e. 50 miles per hour f. 1,200 miles per hour

List of Motions Tap a pencil at exactly 1 tap per second for 10 seconds. Turn the pages of a book at exactly 2 pages per second for 10 seconds. Move at a speed of exactly 1 foot per second for 30 seconds. Walk from your seat to the classroom door at a speed of 2 feet per second.

50

MATHEMATICS OF MOTION • LESSON 1

Tap a pencil every 5 seconds for 1 minute. Turn the pages of a book at exactly 1 page per second for 20 seconds. Move at a speed of exactly 1 meter per second for 10 seconds. Walk the length of the classroom in exactly 15 seconds.What was your speed?


Represent Your Trip to School If you need to explain how something moves, there are different kinds of representations to choose from. For example, consider how you can represent the motion of your daily trip to school.

How can words and visual images be used to represent motion?

1

Describe how you get to school in writing. You can use a story, a set of directions, or any other way to describe your trip in words. Make sure someone could tell from your description whether you walk or ride to school, how long it takes you to get there, and how far and how fast you travel.

2

Show your trip to school visually. You can use a map, a diagram, a graph, pictures, or any other kind of visual representation.

3

Use the list Things to Know About Your Trip to answer the following two questions.

■

Each turn that you make along the way

a. What could someone learn about your trip by reading your description of it?

■

The names of the streets

■

The mode of transportation (walk, bicycle, bus, car, train)

■

The distance of the entire trip

■

The distance of each part of the trip

■

The time of the entire trip

■

The time of each part of the trip

■

Whether you go with anyone else

■

The speed you travel for each part of the trip

Things to Know About Your Trip

b. What could they learn by looking at your visual representation?

hot words

estimate

W Homework page 78


51


2

ESTIMATING, MEASURING, AND CALCULATING DISTANCE

Going the Distance To find out how far it is from one point to another, do you estimate the distance? measure it? calculate it? In this lesson, you will have the chance to use all three ways of finding distance.

Find Out How Far When you move from here to there, how do you know how far you have gone?

1

Copy the table Knowing the Distance in your notebook or on another sheet of paper. Without actually walking or measuring, estimate how far you will go in 10 seconds when you move each way listed in the table. Record your estimates in your table.

2

Working with your partners, measure how far you actually go in 10 seconds for each way of moving. How do your estimates compare with your measurement results?

3

Using the data from the second column, calculate how far you would go in 1 second, 1 minute, and 1 hour; and how long it would take you to go 1 mile or 1 kilometer.

nce : the Dista Calculate Knowing lculate: Time to Go 1 a C : te alcula ce in km ulate: C in Distanour Mile or 1 re: Calc nce in Distance H su 1 a e M ta : 1 Minute Estimate Distance in DisSecond in Distance ds 10 Seconds 1 10 Secon Walking at r your regula pace Walking quickly Hopping Walking backwards A way of moving thatup you make

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Relate How Long? How Far? and How Fast? Copy the table Going Places in your notebook or on a sheet of paper. Include all the information already given. Then fill in the rest of your copy of the table. Each row should show places you could start from and go to, how you could get there (by foot, by bike, by bus, etc.), and the distance, time, and speed of the trip. You may have to estimate some of the information. Some of it you will find by making calculations.

From your desk

To classroom door our school

ces Going Pla How Far?) How? (distance .) etc (foot, bus, foot

g? How Lon (time)

Is there a relationship between time, distance, and speed?

? How Fast (speed)

about 1 mile 30 minutes

50 miles per hour

Explain Your Solutions After you have completed the table, choose one distance, one time, and one speed that you were not able to measure directly. Describe in writing why you couldn’t measure it directly and how you figured out the numbers you put in the table.

hot words

equation distance

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3

REPRESENTING MOTION WITH PICTURE COORDINATES

Reporting Live from the Parade Have you ever seen the film used in movie projectors? Each frame, or “snapshot,” shows the scene at a different time.Taken together, the snapshots show action.You can use written snapshots to create a visual representation of the action at a parade.

Use Snapshots to Represent Motion How can snapshots be used to show the motions of parade characters?

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When you read the handout Reporting Live from the Parade, you learned that Monica Chang is a radio reporter covering a parade. JT Diaz, the photographer, takes a snapshot every 5 seconds, capturing the changing positions of parade characters. Latasha Williams is JT’s assistant. She takes detailed notes describing the scenes in JT’s photos. 1

Use the information in the handout Latasha’s Field Notes I, snapshots 1–8, to complete a Snapshot Sequence Sheet. For each snapshot, fill in the snapshot number and the time.

2

Record the position of the rollerblader, the drum major, and the dragon float when JT took snapshots 1–8. Choose a letter, symbol, or color to represent each parade character.



Practice Problem Solving with the Snapshots Read the following bulletin just in from the parade:

Live from the Parade “This is Monica Chang again, from KMTH radio, bringing you more live coverage from the Main Street parade. JT has been taking pictures during our commercial break, and he tells me he’s getting ready to take his 17th picture. The dragon float is still in view. We can also see a juggler on a unicycle and Lisa, a photographer friend of JT’s from the newspaper. Lisa is jogging with her camera. That is unusual. I wonder why she’s in such a hurry?”

■

Make another visual representation of the action at the parade. This time, use the handout Latasha’s Field Notes I for snapshots 17–24 and another Snapshot Sequence Sheet.

■

Answer the question, “Why is Lisa in such a hurry?”

Write Equations for Motion Use your Snapshot Sequence Sheets to write equations describing motion at the parade. 1

Write a rule you can use to figure out how many snapshots JT would take in any given number of minutes. Use P to stand for the number of snapshots, and M to stand for the number of minutes.

2

Assuming the drum major moves at a constant rate, write a rule you can use to figure out how far in meters the drum major would march in a given number of minutes. Use D to stand for the distance in meters he marches and M to stand for the number of minutes.

3

Assuming they move at constant rates, write rules to find the distance in meters the rollerblader and the dragon float move in any given number of minutes.

hot words

coordinates equation

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4

FINDING DATA FROM A SNAPSHOT SEQUENCE

The Parade Continues Things have changed at the parade. There are new characters on the scene, and Monica Chang, the “Eye in the Sky,” is having a hard time making her news report. Can you help Monica figure out what’s going on?

Find Missing Data How can you determine unknown positions on a snapshot sequence?

Read Monica’s report below, Reporting Live from the Clouds. Then read the handout Latasha’s Field Notes II. Notice that clouds prevented Latasha from taking notes some of the time. On a Snapshot Sequence Sheet, fill in the picture number and the time for each photograph. Then record the positions of the tumbling team, the clown, and the pig in each snapshot. What would JT’s pictures have looked like if the clouds had not blocked the view in the 4th, 5th, 6th, and 8th pictures?

Reporting Live from the Clouds “This is Monica Chang, again, bringing you live coverage of the parade from our Eye on the Scene hot air balloon. We’re still looking down on Main Street, between Avenue A and Avenue B. JT has just loaded his camera with a new roll of film, and we’re awaiting the next group of parade characters to come into our view.” “Now, JT, it’s gotten pretty cloudy all of a sudden,” says Monica, “Do you think you’ll be able to take any good pictures?”

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“I hope so, Monica. The clouds are going to make it difficult for me to take pictures, and for Latasha to see what’s going on well enough to take notes, but they seem to be moving in and out, so I should be able to get a clear view for at least some of the pictures.” “Well, JT, get your camera ready. I can just barely see a clown on a skateboard approaching the Gas Station, and just behind her I think I see the Twisted Toes Tumbling Team turning cartwheels. They must be getting dizzy! And . . . what’s that? There’s a pig heading from the Stop Sign at Avenue B toward the Gas Station! That’s strange. Maybe it escaped from the Creature Corral Petting Farm float.”


Imagine Using a Faster Camera Assuming the characters are moving at constant rates, what happened to the pig and the clown between 1:04:15 P.M. and 1:04:20 P.M.? Imagine that you have a faster camera than JT, and you can take a picture every 1 second between 1:04:15 and 1:04:20. 1

Create a snapshot sequence with a picture every 1 second.

2

Write an explanation of what happened to the pig and the clown between 1:04:15 P.M. and 1:04:20 P.M.

What happens between the snapshots in a snapshot sequence?

Write Your Own Snapshot Story Write a story explaining the snapshot sequence that your teacher gives you. Use your imagination. Your story can be about anything you want! 1

Before you write your story, do the following: a. Decide what the story is about. Who or what are Characters A, B, and C? What person, place, or thing is at each marker along the road? b. Decide how much time there is between each snapshot and fill in the time of day that each snapshot was taken. c. Decide how much distance there is between each of the 5 landmarks on the road or path.

2

Your story should also include information about the directions and speeds of the characters, and why they are moving that way.

hot words

distance

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P H A S E TWO



Distance-time graphs are powerful: they give you a lot of information in a very compact form. With a distance-time graph, you can quickly see how long it took someone to go someplace, how fast they went, and how far they traveled. You can see, at a glance, which contestant won a race, who lost, and who didn’t finish. You can see whether everyone started at the same time, or if somebody had a head start. The distance-time graph is one mathematical tool that belongs in everyone’s mathematical tool kit.

Distance-Time Graphs WHAT’S THE MATH? Investigations in this section focus on: MEASUREMENT and GEOMETRY ■

Setting appropriate scales on graphs

■

Interpreting slopes on graphs


Creating and interpreting distance-time graphs

■

Relating different types of distance-time graphs to each other and to stories

■

Reading information from maps


Finding distance, time, and speed data in stories



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5

INTRODUCING DISTANCE-TIME GRAPHS

Walk This Way As you have discovered, motion can be represented visually in a variety of ways. There are drawings and maps, and diagrams such as the snapshot sequences you created in Lessons 3 and 4. A graph is another way to represent motion visually.

Create Distance-Time Graphs How can you make graphs representing walks at different speeds?

Apply what you learned about making distance-time graphs by graphing your data from the three walks at different speeds. 1

Collect data to graph by working with partners to measure the speeds of a slow walk, a regular walk, and a fast walk. a. Decide who will walk and who will measure each walk. You must also measure the distance walked and the time it takes in order to calculate the speed. b. Copy the table, Walks of Various Speeds, to store your data.

2

Make three distance-time graphs—one for each of the slow, regular, and fast walks. Label the axes to show the distance and time measures.

ds ious Spee r a V f o s Walk Time Distance Slow Regular Fast

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Speed


Connect Stories and Graphs Sometimes, data is gathered from an experiment before it is graphed. Other times, graphs can be prepared from information given in a story. 1

Choose any three of the characters in the parade in Lessons 3 and 4 and make a distance-time graph showing how they moved. Remember to label the axes to show the distance and time measures.

2

Read the story “Tanya and Pedro’s Trips to School.” Create a graph to show what happens in the story. Be sure your graph shows about how far each person went and how much time they spent.

How can you make a distance-time graph to represent the activities of characters in a story?

Tanya and Pedro’s Trips to School Tanya’s house is 1 mile away from her school. She started out to school at 8:00 A.M. On her way to school,Tanya walked for 5 minutes, going about 14 of a mile. She then spent 5 minutes waiting for her friend Pedro. Pedro wasn’t ready, and Tanya was worried that she would be late

for school. She walked for 15 more minutes to get to school. Pedro started from his house and ran to school in 10 minutes. He got there at the same time as Tanya arrived.

Relate Graphs to Motion Write your answers to these questions about the distance-time graphs you made. ■

How did you decide what scale to use on the axis of each of your graphs?

■

If you used a different scale, how would your graph change? How would it stay the same?

■

Are all the lines on your graphs straight? Why or why not?

■

When two lines are on the same graph, what does it tell you when one is steeper than the other? Why?

hot words

coordinate graph point

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6

COMPARING AND CONTRASTING REPRESENTATIONS

Stories, Maps, and Graphs As you know, there are several useful ways to represent information about motion. Stories, maps, and graphs are some of these ways. But do stories, maps, and graphs provide the same information? Are they all equally useful?

Interpret a Story, Map, and Graph Read the story, map, and graph about Jessica’s travels. Then answer the questions on the sheet your teacher will give you that asks what Jessica was doing. Information About Jessica’s Travels She grabbed the milk and went to the express line.There were several people in front of her, so she had to wait.

At 10:00 A.M., her mother asked her to go to the corner grocery store to get milk for her brother. She jumped on her bike and rushed out, biking two blocks along Avenue A and then going on 3rd Street to the grocery store.

When she left the grocery store, she discovered she had a flat tire. She walked her bike home along Avenue C and 1st Street. When she got home, she had to wait for her brother to drink his milk.Then her mother drove her to 10 school, but she was late, and her friends weren’t there. Her mother 5 drove her to the park to look for her friends. She finally caught up to her friends at the park at 11:00 A.M. 10 A.M. 10:30 A.M. 11 A.M.

Park

4th Street

Avenue B

3rd Street

1st Street

Avenue A

Avenue C Grocery Avenue D

2nd Street

Avenue E Avenue F School

A Map of Jessica’s Travels

62


Blocks from home

Jessica lives at the corner of 1st Street and Avenue A. She was planning to meet her friends at 10:30 A.M. at the school.

Time A Graph of Jessica’s Travels


Graph Other Things That Change Over Time Graphs can be used to show how an object’s position—or almost anything else—changes over time. Draw a graph that shows how Jessica felt during different times in the story. You can graph how happy she felt, how annoyed, or any feeling you choose.

What kinds of changes over time can be shown on a graph?

Create Stories, Maps, and Graphs Choose a trip that you are familiar with, or make up a fantasy trip. For your trip, do each of the following. ■

Write a description or story.

■

Draw a map.

■

Make a distance-time graph.

■

Make one other graph showing how you or a character in your story felt during the trip.

What kinds of travels can you describe with a story, map, and graph?

hot words

graph picture graph

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7

MAKING THREE TYPES OF DISTANCETIME GRAPHS

A Graphing Matter Distance-time graphs come in several different sorts. You’ve had some experience with graphs that show how far something is from its starting point. Now you will learn how to make two other useful distance-time graphs.

Make Three Types of Distance-Time Graphs What are some useful kinds of distance-time graphs?

In this activity, you and your partners will make three different kinds of graphs—Graph A, Graph B, and Graph C. Either your teacher or someone in your group will be the mover. 1

The mover does the following: Select a place where you can walk in a straight line while the others watch. When walking, walk at a steady pace. Others will need to count the number of steps you take, so don’t go too quickly. When everyone is ready: a. Walk from the starting point to the ending point. b. Wait for about 5 seconds. c. Walk halfway back to the starting point. d. Wait about 5 seconds. e. Walk back to the ending point. f. Wait about 5 seconds. g. Walk back to the starting point.

2

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Make the graphs. Each member of the group should make a different graph—either A, B, or C. Each graph will show something different about the number of steps the mover has taken. Graph A shows how many steps the mover is from the starting point throughout the walking time. Graph B shows how many steps the mover is from the ending point throughout the walking time. Graph C shows the total number of steps the mover has taken throughout the walking time.



Graph New Routes Practice making the three kinds of distance-time graphs with new walking paths. 1

Decide who will be the mover and what the mover’s path will be. If your teacher doesn’t provide a path, make one up. Here is an example:

Can you make three kinds of distance-time graphs for new walking routes?

a. Start at the door to the classroom. b. Walk slowly to the teacher’s desk. c. Wait 10 seconds. d. Walk quickly to the bookcase. 2

Decide which graph you will make. Make either an A graph (distance from the starting point), a B graph (distance to the end point), or a C graph (total distance).

3

Watch the walker walk the path and gather the data you need to make your graph. Make your A, B, or C graph.

4

Repeat the process twice more but rotate which graph you make, so that each member of your group makes each kind of graph.

Convert Among Types of Graphs Using the story, map, and graph of Jessica’s travels from Lesson 6, page 62, create each of the following graphs: ■

a graph showing the total distance Jessica traveled

■

a graph showing how Jessica’s distance from the school changed over time

■

a graph showing how Jessica’s distance from the park changed over time

hot words

total distance graph distance-from graph

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8

INTERPRETING GRAPHS OF MOTION

Juan and Marina Go Walking “A picture is worth a thousand words.” Have you ever heard that saying? As you will see in this lesson, graphs are pictures that can contain a lot of information in a compact form.

Write a Story to Match Four Graphs What story can you write based on four different graphs?

Write an imaginative story that matches the information on the graphs below. Use information from all four of the graphs. Juan’s and Marina’s Walks Juan’s Graphs Distance (miles) from Home Against Time How Hungry Against Time 6 Starving 5 4

Hungry

Miles 3 2 1 0 2 P.M.

O.K.

4 P.M.

6 P.M.

8 P.M.

Full 2 P.M.

4 P.M.

6 P.M.

8 P.M.

Marina’s Graphs Distance (miles) Walked Against Time 12

How Hungry Against Time Starving

10 Hungry

8 Miles 6

O.K.

4 2 0 2 P.M.

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4 P.M.

6 P.M.

8 P.M.

Full 2 P.M.

4 P.M.

6 P.M.

8 P.M.


Find the Impossible Graphs Examine the distance-time graphs shown. Then answer the following questions. 1

Which of the graphs below could show total distance traveled? How did you decide?

2

Some of the other graphs could show distance from a starting point, but some of them cannot possibly be correct! Which graphs cannot represent the journey of a single person or vehicle? Why?

3

For each of the graphs that is possible, describe in one sentence what the graph might show. For example, could any of the graphs represent throwing a ball? riding a bike?

Which of the graphs shown are impossible?

Write Science Fiction from Impossible Graphs Extra challenge: Make up a science fiction story that makes one of the impossible graphs possible.

Distance-Time Graphs c Distance

b Distance

Distance

a

d

e

f Distance

Time

Distance

Time

Distance

Time

Time

g

h

i

Time

hot words

Distance

Distance

Time

Distance

Time

Time

Time

dependent events

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PHASE THREE


cers oun n n nts. king ts A -ma unceme r o h p p gra no HS new orts an stance ces, KMT l a e p nn ased w di g ra ur s Cha urch with yo hat sho runnin graphs To: p s for he ou hs t n ha atio o help y es grap us time vents. T to your t s rs n se eat The uter t r cr peed ve r sport ormatio aphs of p e t m u o s e f p r c h n g d i t m o co an he ! se ting The s time ces, and e preci ading t interes up to it u e r a vers ming r give mo actice r e much ills are r d k swim you to wers. P ll provi ading s i e e w allo ision vi they w graph-r r telev aces, fo if your r the mation— r info


Who will make it first to the finish line? Graphs and equations allow you to see patterns that help you predict the results of a race. Using mathematics allows you to do much more than predict the outcome of a race. With the techniques you will learn in this phase, you can calculate how long it will take to ride your bike someplace you have never been before, or how fast you should walk to get to your friend’s house in 10 minutes.

Using Graphs to Solve Problems WHAT’S THE MATH? Investigations in this section focus on: MEASUREMENT and GEOMETRY ■

Determining or estimating average speeds


Interpreting and creating distance-time graphs

■

Interpreting and creating speed-time graphs

■

Relating distance-time and speed-time graphs

■

Comparing constant and variable speeds on distance-time graphs


Applying graphs to solving problems

■

Applying equations relating distance, time, and speed to solving problems

■

Applying average speed to solving problems



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9

CONVERTING WORDS TO GRAPHS AND BACK AGAIN

The Race Announcer In a race, do all the runners move at a constant speed? As sports announcers know, racers normally speed up and slow down as a race progresses. Distance-time graphs also show the changing speeds in a race.

Play the Role of a Sports Announcer How does speed vary during a race?

You will receive a graph of a running, swimming, or three-legged race. Your job is to write a commentary describing the race. Then your partner will use your commentary to recreate the original distance-time graph of the race. 1

Imagine that you are the TV announcer for the race. Describe what is happening as fully and as accurately as you can. Be sure your commentary matches the graph. Include enough detail in your commentary so that your partner will be able to make a graph from it.

2

Work with a partner who wrote a commentary on a different race. Use your partner’s commentary to make a distance-time graph. Try to make your graph just like the original one. When you are finished, write your answers to the following questions. a. How closely does your graph match the original graph? Summarize the features that your graph shares with the original graph. Then, summarize the features that are different in your graph and the original graph. b. Compare the original graph of the race to the commentary you used to make your graph. Describe any errors the commentary contains. How could they be corrected?

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Find Out How Fast a Robot Should Run The following graph shows how Antonio bicycled in a five-mile race. Use the graph to answer the questions below. 5 4

At what speed should you program a robot so it ties Antonio in a race?

3 Miles from start 2 1 0 5

10

15

Time (min)

1

For which part of the race did Antonio bicycle the fastest? the slowest?

2

A robot can be programmed to roll at exactly the same speed for the entire race. How fast should the robot go if you want it to tie Antonio exactly in the race?

3

How did you figure out the speed at which to program the robot so it tied Antonio?

4

Draw a graph of the robot’s race. How does it compare to the graph of Antonio’s race?

5

How should the robot be programmed so that it will tie the winner of the race for which you wrote a commentary? Add the robot’s race to your graph.

hot words

average average speed total distance graph

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10 EXPLORING SPEED-TIME GRAPHS

How Fast? How Far? How Long? Study the graphs in this lesson carefully. They contain information that you have not seen in a graph before.Your job in the activities ahead is to learn how to create and interpret these new kinds of graphs.

Problem Solve with New Kinds of Graphs What does each graph show?

Read the story about the Lin’s family trip. Then examine the four graphs of their trip and answer the following questions. 1

What does each graph show?

2

How should the vertical axis be labeled?

3

What scale should go on the vertical axis?

The Lins Go on an Outing On Sunday, the Lin family set out on a car ride at 10 A.M. For the first hour, they drove at a speed of 40 miles per hour. In the second hour, traffic was heavy, so they only drove at 20 miles per hour. From 12 P.M. to 1 P.M., they stopped for lunch and did not drive at all. After lunch, it started to rain, so they decided to go home.They drove at 30 miles per hour to get home.

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Graph A

10 A.M.

Graph B

11

12

1

2

3

A.M.

P.M.

P.M.

P.M.

P.M.

10

11

12

1

2

3

A.M.

A.M.

P.M.

P.M.

P.M.

P.M.

Graph C

Graph D

10

11

12

1

2

3

10

11

12

1

2

3

A.M.

A.M.

P.M.

P.M.

P.M.

P.M.

A.M.

A.M.

P.M.

P.M.

P.M.

P.M.



Create Distance-Time and Speed-Time Graphs 1

Choose three different motions you have done or seen. Here are some motions you could use, or you can think up your own. a. someone going down a water slide

What do the graphs of some familiar motions look like?

b. someone on a bicycle climbing to the top of a hill and then coasting down the other side c. the movement of a child in a swing d. an airplane taking off e. someone who jumps out of an airplane and opens her parachute when halfway down f. an athlete in the high-jump event at the Olympics g. a baseball being pitched and then hit for a home run h. a cat trying to catch a mouse 2

For each motion, create one type of distance-time graph and a speed-time graph.

3

For each motion, also find or estimate the average speed and the fastest speed reached during the motion. When was the person, animal, or object moving the fastest?

Decide Which Graph to Use Consider the three types of graphs about motions. For each question, say which graph or graphs you would use to find the information you need, and how you would find it. Questions 3, 4, and 5 are for graphs that show more than one person moving. Information from Graphs of Motion 1. How far did someone travel? 2. How much time did someone spend traveling?

hot words

3. Who is traveling the fastest?

slope speed-time graph

4. Is one person moving twice as fast as another? 5. When are two people moving in opposite directions?

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6. What is someone’s average speed over a whole trip?

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11 SOLVING DISTANCE, SPEED, AND TIME PROBLEMS

The Race Is On! Most things don’t move with constant speed. Take your bike, for example.You start off slowly, pedal to a coasting speed, slow down soon and pedal again. Constant speed is rare. So in problem solving, you often use average speed.

Problem Solve with Average Speed What equations relate distance, speed, and time?

Use the table to answer the following questions. 1

For each distance, what is the average speed of the world champion runner?

2

How does the average speed change as the distance of the race increases?

3

What average speed would you predict as the world record for the following women’s races: a. 1,000 meter

b. 5,000 meter

World’s Running Records for Women

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Distance

Time

Runner and Year

100 meters

10.5 seconds

Florence Griffith-Joyner, 1988

200 meters

21.3 seconds

Florence Griffith-Joyner, 1988

400 meters

47.6 seconds

Marita Koch, 1985

800 meters

1 minute, 53 seconds

J. Kratochvilova, 1983

1,500 meters

3 minutes, 51 seconds

Qu Yunxia, 1993

3,000 meters

8 minutes, 6 seconds

Wang Junxia, 1993

10,000 meters

29 minutes, 32 seconds Wang Junxia, 1993

42,206 meters (marathon)

2 hours, 15 minutes


Paula Radcliffe, 2003

c. 25,000 meter


Invent the Story of a Graph The graph shows three people, animals, or objects (A, B, and C) that move over time. It shows their distances from a certain point. 1

Write a story that goes with the graph. The story can be about anything you choose—people, animals, vehicles, or asteroids. Your story should include details about the speeds and directions of the three people, animals, or objects. Also say where the distances are measured from (zero on your graph).

2

Decide on scales for the time and distance axes that match your story. Mark your scales on the graph.

3

Draw a graph that shows the total distance traveled over the given time by A, B, and C. How far did each of A, B, and C travel?

4

Draw a graph that shows the speeds at which each of A, B, and C traveled during your story.

5

Find the average speed for each of A, B, and C.

What story will the graph tell when finished?

A Graph to Finish

C

Distance from B A

0

Time in

hot words

average speed distance

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12 COMBINING MOTION DATA, GRAPHS, AND EQUATIONS

Final Project Here is your chance to use everything you have learned about motion to analyze, compare, and describe motions of your own choosing.

Collect the Motion Data What kind of motion should you study?

Gather data about a motion of your own choosing. 1

Pick a situation from Sample Motions that allows you to compare the motion of two or more people, animals, or objects.

2

Collect as much data on the motion as you can. For each of the people or objects you will be comparing, be sure to find as many times, speeds, and distances as possible.

Sample Motions ■

■

■

■

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An express train and a local train that pass through your town The winner of the Indy 500 and the driver who came in second The results of two participants in your school’s 100-meter race Olympic track and field events, such as the 100meter sprint, the hurdles, the long jump, or the high jump

■

Walking and rollerblading

■

Flying from Atlanta to Los Angeles on an airplane versus taking a train

■

A jaguar and a cheetah

■

A jaguar and a tortoise

■

The speed of a hockey puck versus the speed of a baseball

■

Balls rolling down ramps with different slopes

■

Bicycling in different gears



Analyze the Motion For each of the people or objects that you are comparing, analyze the data you collected and determine each of the following. ■

the total distance traveled and the total time

■

the average speed over the entire trip

■

intermediate speeds, whenever possible

■

equations that can be used to calculate time, speed, and/or distance for each person/object

■

graphs displaying total distance traveled, distance from a starting point, and speed versus time

■

graphs that compare the motion of the two people/objects

Describe and Display the Motion Prepare a chart, booklet, or other presentation that describes and displays your findings as fully as possible. Be sure to include the following: ■

your initial data about the motion

■

all of the equations you developed

■

graphs of all motions

■

maps, diagrams, or illustrations describing each motion

■

written descriptions or stories for each motion

Your final project should show how much you know about motion.

hot words

total distance (formula d s t) statistics

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Applying Skills

1. a jet plane flying 2. a cheetah running at full speed 3. an athlete sprinting 4. a snail crawling

What units could you use for each motion?

A tsunami (tidal wave) may be caused by an earthquake beneath the ocean. Its speed depends on the depth of the ocean. As it approaches shallow water along a coast, it slows and may increase in height to as much as 115 ft. Each picture below gives information about a tsunami. Height (ft)

For each motion, tell which of these speeds are possible: 200 meters per second, 27 feet per second, 20 miles per hour, 90 feet per second, 50 meters per hour, 500 miles per hour, 300 feet per second, 5 feet per hour.

Making Connections

100 75 50 25

Kodiak San Francisco Honolulu 2 hrs

Tokyo

100 200 300 Speed (mph)

5. a train traveling at full speed 6. the turning of a phonograph record 7. the rotation of the hour hand of a clock

10 hrs 18 h rs

HW

Moving, Measuring, and Representing

5h rs

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Santiago sea level speed/mph 519

299

depth/ft 18,000 6,000

212 –

94

30

600

60

8. a spider walking on a wall ocean floor

Extending Concepts

9. Use the second hand of your watch to

measure your pulse rate for 1 minute. Give the rate together with the appropriate units. 10. Find a space where you can walk for

10 seconds. Walk at your normal pace for 10 seconds and count how many steps you take. Estimate your stride length in feet. Estimate the distance that you would cover in 1 minute. Estimate your walking speed in feet per minute.

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MATHEMATICS OF MOTION • HOMEWORK 1

11. What information is given in each

representation? For each one, give an example of a question that could be answered using that representation. 12. Which representation, if any, would you

use to answer the question, “How fast was the tsunami traveling when it was 500 miles from Hawaii?” Explain. 13. Write three questions about tsunamis

that cannot be answered using any of the representations above.


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HW

Going the Distance b. In item 6a, when you calculated how

Applying Skills

Estimate how far you would go in 20 seconds by: 1. walking quickly 2. hopping

7. Daniel runs the length of his backyard in

3. walking backward

10 seconds.

4. Copy and complete the table below.

Distance in 10 seconds

Walking

Jogging

Cycling

50 feet

85 feet

280 feet

half Daniel’s speed, how long will it take him? How long will it take him at twice Daniel’s speed? the yard in 30 seconds, how does her speed compare to Daniel’s?

Distance in 1 minute

c. If Doug runs twice the length of the

yard in 5 seconds, how does his speed compare to Daniel’s speed?

Distance in 1 hour Time to go 1 mile

Making Connections

5. Copy and complete the table below.

Include the appropriate units. Time

100 miles

Speed 40 miles per hour

8 seconds 160 meters

a. If Scott runs the length of the yard at

b. If Kemi runs three times the length of

Distance in 1 second

Distance

far Bill would run in one hour, what assumption did you make? Do you think he would really run that far in an hour? Why or why not? How far do you think he would run?

6 feet per second

4 minutes

Extending Concepts

6. a. Bill runs as fast as he can for 12

seconds and covers a distance of 280 feet. At that pace, how far would he run in one minute? in one hour? What is his speed in miles per hour?

8. a. The speed of light in a vacuum is

about 186,000 miles per second. How far would light travel in a vacuum in one hour? What is its speed in miles per hour? b. The speed of a glacier depends on its

steepness and its temperature. A steep, warm glacier may flow as much as 100 feet per day. At this rate, how far would it flow in a year? in one hour? What would its speed be in miles per hour?


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3 ork w e om

HW

Reporting Live from the Parade

Applying Skills

The field notes below give information about the position of characters in a parade. Use this and your Snapshot Sequence Sheets to answer the questions. The distance between each pair of landmarks on your Snapshot Sequence Sheet is 15 meters. A = Acrobat AD = African Drummers CT = Capoeira Troupe 1st Snapshot, 10:30:00 A.M. A: At Gas Station AD: 10 meters past Gas Station CT: At Taco Zone 2nd Snapshot, 10:30:10 A.M. A: 5 meters past Dog Lovers’ Club AD: 5 meters past Dog Lovers’ Club CT: At Taco Zone 3rd Snapshot, 10:30:20 A.M. A: 10 meters past Taco Zone AD: At Taco Zone CT: At Taco Zone 4th Snapshot, 10:30:30 A.M. A: At U.F.O. Chronicle AD: 10 meters past Taco Zone CT: At Taco Zone

1. Use the field notes to complete the first

four pictures of your Snapshot Sequence Sheet. In each picture, record the positions of the characters. 2. How far is it from the Taco Zone to the

T-Shirt Hut? from the TreeWatch booth to the stop sign at Avenue B? 3. How much time is there between

consecutive snapshots?

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4. How much time is represented by

Snapshots 1–4 all together? 5. How far does each character travel

between snapshots? 6. How fast, in meters per second, is each

character moving? Who is traveling the fastest? Extending Concepts

7. Write a rule you can use to calculate how

far in meters the acrobat would move in any given number of minutes. Use D to stand for the distance and M for the number of minutes. Use your rule to figure out how far the acrobat would move in 9 minutes. 8. Write a rule you can use to calculate how

many snapshots would be taken in any given number of minutes. Use P to stand for the number of snapshots and M for the number of minutes. How many snapshots would be taken in 9 minutes? 9. Write your own snapshot sequence for

Snapshots 5–8 to show how the parade characters might continue. Use your sequence to complete Pictures 5–8 of your Snapshot Sequence Sheet. Writing

10. Write a short narrative describing what

the three parade characters might have been doing in Snapshots 1–8. Explain how you can tell.


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HW

The Parade Continues

Applying Skills

The field notes below give information about the position of characters in a parade. Use this and your Snapshot Sequence Sheet to answer the questions. The distance between each pair of landmarks on your Snapshot Sequence Sheet is 15 meters. SW = Stilt Walker, S = Saxophonist, D = Danny 1st Snapshot, 2:20:00 P.M. SW: At T-Shirt Hut S: At TreeWatch D: 5 meters past Gas Station 2nd Snapshot, 2:20:05 P.M. SW: At U.F.O. Chronicle S: 5 meters past TreeWatch D: At Dog Lovers’ Club

Extending Concepts

5. What happened between 2:20:10 and

2:20:15? Create a snapshot sequence with a picture every 1 second between 2:20:10 and 2:20:15. At what point did Danny and the stilt walker pass? 6. Explain how you can use a snapshot

sequence to figure out: a. the speed at which an object is

traveling b. when one object is traveling faster

than another c. what probably happened between two

snapshots Writing

3rd and 4th Snapshots Too cloudy 5th Snapshot, 2:20:20 P.M. SW: At Dog Lovers’ Club S: 5 meters past U.F.O. Chronicle D: At TreeWatch 1. Use the field notes to complete Pictures

1, 2, and 5 of your Snapshot Sequence Sheet. In each picture, record the positions of the characters.

7. Write a story explaining the following

snapshot sequence. Explain who the characters are and what is at each marker along the road. Give the distance between markers and time between snapshots. Give the speeds and directions of the characters. Explain why they are moving that way. A B

2. Figure out what the 3rd and 4th pictures

would have looked like and complete Pictures 3 and 4 of your Snapshot Sequence Sheet. 3. Find the speed and direction of each

character.

BC

A B C

B AC

C A

Picture

1

2

3

4

Time

4. How far would each character travel in

5 minutes? MATHEMATICS OF MOTION • HOMEWORK 4

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5 ork w e om

HW

Walk This Way

Applying Skills

Mary walked 200 feet along a straight line in 40 seconds. Follow the guidelines to make a distance-time graph.

Liam’s Graph 100 Distance from 50 home (ft) 0 10 20 Time (sec)

Emma’s Graph 400 Distance 300 from 200 home 100 (ft) 0 10 20 Time (sec)

1. Draw axes for a graph. Label the

horizontal axis “Time (seconds)” and the vertical axis “Distance from starting point (feet).” 2. Set the scale for each axis. Mark equal

divisions along each axis. 3. How long did it take her to walk the

entire distance? Mark that point. 4. Assuming that Mary’s speed was

constant, how long did it take her to walk half the distance? Mark that point. 5. Draw the line showing the entire walk.

Extending Concepts

8. How far did Liam and Emma walk in

20 seconds? Who walked faster? Whose graph looks steeper? Why is it not true in this case that a steeper graph indicates a faster motion? 9. Which graph, Liam’s or Emma’s, has a

more suitable scale? Why? Tyler set out from the trailhead at 9:00 A.M. He walked for 1 hour, going 3 miles, and then rested for 20 minutes. He continued a little more slowly and reached Mirror Lake at 12:20 P.M. Suzi left the trailhead at 10:00 A.M. She walked steadily without breaks and also reached the lake at 12:20 P.M. 10. Create a graph to show what happens in

The graphs for items 6–9 all represent motion along a straight line. Toni Distance from home

Lynn Rob

Time

6. In Toni, Lynn, and Rob’s graph, who

walked the fastest? the slowest? How can you tell? 7. Why does Lynn’s graph slope downward?

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the above story. Be sure to show the distances and times for each person. Estimate how far each person walked. Assume that each person walks along a straight line. Making Connections

11. The Pronghorn antelope is the swiftest

North American mammal. It can run 200 meters in about 7.5 seconds. The fastest human can run 200 meters in about 20 seconds. Using the same axes, make distance-time graphs for a Pronghorn antelope and a human.


6

Stories, Maps, and Graphs

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HW

Applying Skills

Use the story, map, and graph below to answer the questions. Tell whether each answer is exact or approximate and whether you used the story and map below or graph in column 2 to find it.

1. At what time did David return home? 2. How many blocks did David walk along

Hill Street before turning? 3. Why did David want to buy a CD? 4. When was David traveling the fastest?

It was Friday evening. David remembered that his sister’s birthday was the next day. He decided to buy her a CD. He left his house on Hill Street at 5:30 P.M. to walk to the store. A few moments later, he ran into a friend and stopped to chat for a while. When he left his friend, he remembered that the CD store would close at 6:00 P.M., and he ran the rest of the way. He reached the store in time, bought a CD, and walked home.

the slowest? 5. Why did David hurry as he left his

friend? 6. How long did David spend talking to his

friend? 7. In what direction was David traveling

when he first left his house? 8. During what time period(s) was David

moving closer to his house? David’s home

8th St.

7th St.

6th St.

5th St.

4th St.

Hill St. North CD store

Extending Concepts

9. Write a story to describe a fantasy trip.

Make a map and a distance-time graph to go with your story. Then make a second graph showing how a character in your story felt during the trip.

5 4 Distance 3 from home 2 1 0 5:30

5:45

6:00 Time

6:15

6:30

Writing

10. Make up a question about your story

that can be answered most easily using your map. Make up another question that can be answered most easily using your distance-time graph.


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Extending Concepts

John walked 100 feet in a straight line from the starting point to the ending point in 15 seconds, then 40 feet back toward the starting point in 5 seconds. He waited 5 seconds, then walked all the way back to the start in 10 seconds. His “Distance from Start” graph is shown here. Distance from Start 100 80 Distance (ft) 60 40 20 0

10 20 30 Time (sec)

Follow the guidelines to make John’s “Distance to End Point” graph.

5. The three graphs at

right show total distance, distance to ending point, and distance from starting point. Which is which? How can you tell? 6. Which two graphs

could represent the same motion? For each pair of graphs that could not represent the same motion, explain why not.

Distance (ft)

Applying Skills

80 40 20

160

5 10 15 20 Time (sec) B

120 80 40 0 80

5 10 15 20 Time (sec) C

60 40 20 0

1. Draw axes for a graph. Label each axis

A

60

0

Distance (ft)

HW

A Graphing Matter

Distance (ft)

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5 10 15 20 Time (sec)

appropriately. Set the scale. 2. Find John’s distance to the ending point

after 0, 15, 20, 25, and 35 seconds. Mark those points. Draw the line showing the entire walk. Follow the guidelines to make John’s “Total Distance” graph. 3. Draw axes for a graph. Label each axis

appropriately. What is the total distance that John walked? What will be the largest number on the vertical axis? Set the scale on each axis. 4. Find the total distance John had walked

after 0, 15, 20, 25, and 35 seconds. Mark those points. Draw the line showing the entire walk.

Making Connections

The gray whale migrates seasonally between its breeding ground in Baja California and its feeding ground in the Arctic. The graph shows distance to the Arctic for the gray whale for a one-year period. 5,000

Distance to Arctic

Distance (mi) 0 Jan

Mar May July

Sep Nov

7. A part of the graph is sloped upward.

What are the whales doing during this time period? Where are they during February? How can you tell? 8. Draw the corresponding “Distance from

Baja” and “Total Distance” graphs. Explain how you figured them out. 84



8

Juan and Marina Go Walking

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HW

6. How far did Jamie walk from 1 to 5 P.M.?

Applying Skills

7. Why is there a flat portion on Jamie’s a.

b.

graph between 11 A.M. and 1 P.M.? Distance

Distance

8. Why does Leah’s graph slope up and Time

Time

c.

9. Can you tell where Jamie was at 5 P.M.?

d.

Why? Where was Leah at 2 P.M.?

Distance

Distance

then down?

Time

Time

1. Which of the graphs above could show

total distance traveled? Why? 2. Which graphs could not show the

journey of a single person or vehicle? 3. Which graphs could represent distance

from a point? 4. Sketch a distance-time graph that could

show total distance traveled.

Explain why each story is inconsistent with the graphs in Extending Concepts. 10. “Leah and Jamie left home at 9 A.M.,

walked together until 11 A.M., rested for two hours, and walked home together.” 11. “Leah and Jamie left home at 9 A.M.

Jamie walked faster than Leah until 11 A.M. Both rested between 11 A.M. and 1 P.M. Then they walked home.” 12. “Leah and Jamie left home at 9 A.M. and

Extending Concepts Leah’s Graph Distance (mi) from Home Against Time

walked 4 miles north. After lunch they walked another mile north. Then Jamie walked to his friend’s house, and Leah walked home.”

4 3

Writing

2 1 0 9 A.M. 11 A.M. 1 P.M.

13. Write your own imaginative story that 3 P.M.

5 P.M.

Jamie’s Graph Distance (mi) Walked Against Time

matches the information on the two graphs in item 5 and the graph below. Use information from all three graphs.

8

Jamie’s Graph How Tired Against Time

6 4 2 0 9 A.M. 11 A.M. 1 P.M.

Exhausted Tired 3 P.M.

5 P.M.

5. What is graphed on the horizontal axis of

O.K. Not tired 9 A.M. 11 A.M. 1 P.M.

3 P.M.

5 P.M.

each graph shown? on the vertical axis? Are the scales of the two graphs the same? MATHEMATICS OF MOTION • HOMEWORK 8

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9

The Race Announcer

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HW

Applying Skills

Extending Concepts

Refer to the graph of the running race for items 1–4. Running Race 400 Distance from starting line (m) 200 0

7. a. In the running race, what is the total

distance that Tom runs? What is his total time? his average speed? b. What is Tom’s speed at 10 seconds?

Lee Bob Tom 20 40 60 Time (sec)

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25 seconds? 40 seconds? Is there some point in the race when he was running at the average speed that you calculated in item 7a? Explain. c. If you program a robot to roll at a

1. At what times were there changes in

speed? 2. At what time did Lee pass Bob? 3. What were the relative positions of the

racers after 10 seconds? 4. How long did each person take to

complete the race? Who won? Refer to the graph of the swimming race for items 5 and 6. 5. Who is in the lead at 20 seconds? at

40 seconds? at 58 seconds? What is happening at 30 seconds? 6. How long does each person take to

complete the first 30-meter lap? Who completes the first lap more quickly? Swimming Race 30 Distance from one end of 15 pool (m) 0

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constant speed in the race, at what speed should it roll to tie Tom? Draw a graph of the robot’s race. d. Find Kate’s average speed in the

swimming race. How did you find it? The report below doesn’t give enough information to tell the final outcome. Draw two graphs that match the report but show different outcomes. Rewrite the report so that it fully and accurately describes one of your graphs. 8. “As they start on this 200-meter race,

Sam is in the lead, Eric is behind him, then Raphael. Now, just 10 seconds into the race, Eric has tripped, and he’s down. Raphael has passed him. This is amazing, with 50 meters to go Raphael passes Sam to win the race in 25 seconds.”

Kate Jen 20 40 60 80 Time (sec)


Writing

9. Use the graph in item 5 to write a

commentary for the swimming race. Describe what is happening as fully and as accurately as you can.

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HW

8:44 AM


How Fast? How Far? How Long? 8. If a speed-time graph stays flat, what

Applying Skills

For each question, tell which graph or graphs shown here could be used to find the information. Then answer the question. 1. How far was each

Distance-Time Graph

person from home at 3:00 P.M.? 2. Who was traveling

faster in each time period?

50 40 30 Miles 20 10 0

Amy

1 P.M. 3 P.M. 5 P.M. Time

3. What was each

Total Distance Graph

person’s average speed for the whole trip? 4. What was each

person’s exact speed at 4:30 P.M.?

100 80 60 Miles 40 20 0

Brad

Amy

1 P.M. 3 P.M. 5 P.M. Time Speed-Time Graph

5. During which

time period(s) was Brad moving twice as fast as Amy? 6. How far did each

Brad

50 40 Miles per 30 hour 20 10 0

Brad Amy 1 P.M. 3 P.M. 5 P.M. Time

person travel?

does that tell you about the motion? What will the corresponding distancetime graph look like? If a speed-time graph slopes upward, what does that tell you about the motion? Why are there no sloping lines on your speed-time graph in item 7? Do you think this is realistic? Why or why not?

Making Connections

9. An object dropped off a building

accelerates because of gravity. Each second its speed increases by 32 feet per second—after 1 second its speed is 32 feet per second, after 2 seconds it is 64 feet per second, and so on. Which graph below could be a speed-time graph for an object dropped off a building? Which could be a distancetime graph? How you can tell? a.

b.

?

?

Time

Time

Extending Concepts

7. Find Thomas’s speed during each time

period. How did you figure it out? Make a speed-time graph to represent Thomas’s trip. Thomas Total distance traveled (mi)

40 30 20 10 0 1 P.M.

c.

d.

?

?

Time

Time

3 P.M. 5 P.M. Time


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HW

The Race Is On!

Applying Skills

Extending Concepts

Below are the world’s running records for men in different distances. Distance

Time

100 m

9.8 sec

400 m

43.2 sec

800 m

1 min, 41 sec

1,500 m

3 min, 26 sec

1. For each distance, find the average speed

of the world champion runner. 2. How does the average speed change as

the length of the race increases? 3. What would you predict for the world

record for the men’s 600-meter race? 1,200-meter race? 4,000-meter race? 4. Joel drove for 30 minutes at 40 miles per

hour, for 1 hour at 60 miles per hour, and for 20 minutes at 54 miles per hour. a. How far did he travel at each speed? b. What was his total distance? his total

time? his average speed? c. Create a total distance graph for Joel’s

trip.

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5. In a 20-mile bicycle race, Lois gave Sarah

a head start of 15 minutes. Lois rode at an average speed of 22 miles per hour. Sarah rode at an average speed of 18 miles per hour. Who won the race? How did you find the answer? Make a total distance graph to show the race.

Writing

6. Write a story to go with the speed-time

graph shown here. Give speeds, distances, and directions for each person. Mark scales on the graph to match your story. A Speed

B

Time in

Draw a graph showing how the distance of each person from a certain point changes during the story. Tell where the distances are measured from. Find the average speed for each person.


k wor e m o

HW

Final Project

Applying Skills

Felix hiked up a mountain. During the first 3 hours, the trail was steep and he hiked only 4 miles. To hike the remaining 2 miles to the top took him 1 hour. It took him 2 hours to hike down again. 1. Find the total distance he walked and the

7. Graphs showing total distance traveled

and speed versus time. 8. Who was going faster during each period,

at what times one person was passed by the other, and who won the race. 9. Equations that can be used to calculate

time and/or distance for each person.

total time. 2. Find his average speed for the whole trip. 3. Find his speed for each part of the trip. 4. For each part of the trip, find an

equation that can be used to calculate distance traveled. 5. Make a speed-time graph, a graph

showing the total distance traveled, and a graph showing distance from the starting point.

Extending Concepts

Jim and Rajan ran a 400-meter race. Jim ran at 6 meters per second for the first 200 meters, 9 meters per second for the next 100, and at 10 meters per second for the last 100 meters. Rajan ran the first 200 meters in 22 seconds, the next 100 in 18 seconds, and the last 100 meters in 22 seconds. Use this data to write a report. Include the following:

Making Connections

Artificial satellites, which are placed in orbit around the earth, may be used for communications, research, meteorology, or navigation. 10. Suppose a satellite travels in a circular

orbit about the earth at constant speed. Without showing any scale on the axes, sketch graphs showing total distance traveled, distance from the earth, and speed versus time while the satellite is in orbit. 11. Can you assume that an object is not

moving if its distance-from graph is flat? if its total distance graph is flat? Explain your thinking.

6. For each person: the time and speed for

each section of the race and the total time and average speed for the whole race.


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What mathematical ideas can we expect to consider when we explore three-dimensional shapes?

SHAPES AND

SPACE


PHASEONE

PHASETWO

PHASETHREE

The Ins and Outs of Cubes The first solid you investigate is the cube. You will see what twodimensional patterns, or nets, you can use to construct a cube. You then explore cross sections of a cube and sketch cross sections of other solids. Next you estimate the surface areas and volumes of familiar objects and discover what arrangement of centimeter cubes gives the greatest surface area for a given volume.

Prisms and Cylinders You begin this phase by exploring rectangular prisms, constructing nets, and finding surface areas and volumes. Then you use strategies to find the surface areas and volumes of nonrectangular prisms and investigate a formula for these solids. Next you investigate the surface areas and volumes of cylinders. Finally, you explore figures made up of prisms and cylinders.

Solids with Points First, you explore the surface areas and volumes of pyramids. You will see how the volume of a pyramid compares to the volume of a prism. Then you explore and develop a formula for the volume of a cone. Next you discuss appropriate units of measurement and estimate the volume of a large object. Finally, you make scale models of skyscrapers and decide which is the best building design.

PHASE ONE



In this phase, you begin to think three-dimensionally. You explore nets that can be folded into a cube or other solid. You also investigate cross sections of a cube. By exploring surface area and volume, you begin to realize the connection between the outside and the inside of a solid. Finding surface areas and volumes is an important skill that industrial designers, architects, and engineers use in their work. What other careers can you think of that rely on this skill?

The Ins and Outs of Cubes WHAT’S THE MATH? Investigations in this section focus on: GEOMETRY and MEASUREMENT ■

Identifying and drawing nets for a cube

■

Identifying and drawing cross sections of solids

■

Estimating and finding surface area and volume

■

Relating volume and surface area

STATISTICS and PROBABILITY ■

Recording data in a table

ALGEBRA and FUNCTIONS ■

Graphing results on a grid

■

Writing an equation to describe the relationship between the number of cubes and the maximum surface area


SHAPES AND SPACE

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1

REPRESENTING SOLIDS WITH NETS

Nets That Catch Cubes When you wrap a gift box or some other kind of package, —a flat surface that folds up you are using the idea of a net— into a three-dimensional shape. In this lesson you will explore how nets are used to represent different geometric figures and pay special attention to nets that make cubes.

Use Nets to Make Three-Dimensional Figures How do different nets fold up to make three-dimensional shapes?

You may have flattened out a box for recycling. The flattened figure is a net for the original box. In this investigation you will do the reverse. You will make three-dimensional figures from two-dimensional nets.

SHOES SHOES 1

Cut out the nets on A Set of Nets No. 1 and A Set of Nets No. 2. Before you begin folding, think about the different types of nets you see. Then write a brief description of each type.

2

Fold and tape the nets to make three-dimensional figures. Describe in writing the different shapes you made.

Can you look at a net and predict the three-dimensional figure it will make?

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Find Nets for a Cube You have folded up some ready-made nets. Now it’s time to draw nets of your own. 1

One net that folds up into a cube is shown below. Draw this pattern on a sheet of grid paper and cut it out. Then see how you can use the net to make a cube. You will see that each grid square represents a face of the cube.

How many different nets can there be for a cube?

Vertex Face Edge

2

There are other nets for a cube. See how many you can find. Be sure to keep a record of the different nets you find.

hot words

edge face net

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2

EXPLORING CROSS SECTIONS

Any Way You Slice It X rays, MRI’s, and CAT scans help medical doctors make accurate diagnoses. These images show cross sections of the parts of the human body that physicians want to investigate. In the upcoming activities, you will have a chance to investigate cross sections of several different solids.

Investigate Cross Sections of Cubes How many different two-dimensional figures can you make by slicing a cube?

Is it possible to slice a cube to make a square cross section? What about a triangular cross section? a rectangular cross section? See how many different types of two-dimensional figures you can make by taking cross sections of a cube. Use sketches or words to describe how you can make each shape.

Cross Sections A cross section is a two-dimensional slice of a three-dimensional figure. This cross section of a box is a rectangle. Plane of slice

This cross section of a sphere is a circle. Plane of slice

Cross section Cross section

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Draw Cross Sections of Different Solids When a medical technician takes a CAT scan or an MRI, all the “slices” are taken parallel to each other, as shown in this series of scans of a head.

How can you use cross sections to help identify solids?

In this investigation, your work is similar to a medical technician’s. You will be drawing MATHscans—parallel cross sections of a solid—and then providing information about the solid you chose. 1

Choose any figure except the cube from your set of solids.

2

Draw a series of three MATHscans of your solid. Be sure you take parallel slices—in other words, do not take one cross section from the top and the next from the side.

3

Exchange MATHscans with another group. Try to decide which solid they scanned. If you cannot tell, see if you can narrow down the choices to three or four possibilities. As you investigate, think about how the group might have taken the slices.

4

Join the group you exchanged MATHscans with. Did you identify the solid they used? Were they able to identify yours?

hot words

cross section two-dimensional three-dimensional

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3 EXPLORING SURFACE AREA AND VOLUME

Take One for Good Measure! You probably have had a lot of experience measuring one-dimensional and two-dimensional objects. Now you will gain some experience in taking two different size measurements for three-dimensional objects.

Find Volumes and Surface Areas How can you measure an object’s volume and surface area?

The volume of a solid is the amount of space it takes up. Its surface area is the total area of its exterior surfaces. Work together to develop strategies for finding the volumes and surface areas of different objects. 1

Find the volume of the smaller of the two objects your teacher provides. It may help to use centimeter cubes.

2

Find the surface area of this same object.

3

Write the two size measurements on a sheet of paper, and tape it to the object.

4

Find the volume and surface area of the larger object. Again, tape your answers to the object.

If one object has a greater volume than another, must it also have a greater surface area? How to Use Grid Paper to Find Surface Area 1. Cover one surface of the object with the Centimeter Grid Paper. In pencil, trace the outline of the surface onto the paper.

98


2. Find the area of the outlined region. (Each grid square represents 1 cm2.) 3. Repeat Steps 1 and 2 for the other outside surfaces. 4. Add the individual areas to find the surface area.


Estimate Volume and Surface Area Maybe you once entered a contest in which you guessed how many jelly beans were in a jar. Now you can apply your estimation skills to find out approximately how many centimeter cubes it would take to fill your classroom. 1

Use any strategy you choose to estimate the volume of your classroom in cubic centimeters. Write down your estimate, and explain how you made it.

2

Estimate the surface area of the classroom in square centimeters. Again, write down your estimate and how you found it.

What are some ways to measure the volumes and surface areas of large objects?

You can also measure surface area in square meters and volume in cubic meters. A square meter measures one meter on each side.

Each edge of a cubic meter is one meter long.

1 m (= 100 cm) 1 m2

1m

1m 1m 1m

1m

3

3

How many square centimeters fit in a square meter? How many cubic centimeters fill a cubic meter?

4

Express the volume of the classroom in cubic meters and the surface area in square meters. (Hint: Would your classroom hold more centimeter cubes or meter cubes?)

Write About Volume and Surface Area Write your responses to the following questions: ■

Why is surface area measured in square units, while volume is measured in cubic units?

■

Explain how you would estimate the surface area and volume of a school bus. What units would you use to express your answers?

hot words

surface area volume

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4

RELATING VOLUME AND SURFACE AREA

Move the Cube You have explored volume and surface area. Now you will investigate how a figure’s shape can affect its surface area.

Pull Out a Cube How does removing one centimeter cube from a 3 3 3 cube affect the surface area?

To begin, build a 3 3 3 cube out of centimeter cubes. Use this cube to investigate each of these questions: ■

Is it possible to remove one of the centimeter cubes and increase the surface area? If you can, explain how in writing or with a sketch.

■

Can you remove one cube and decrease the surface area? If so, explain how this can be done.

■

Can you remove one cube and keep the surface area the same? If you can, explain how.

If you build a figure out of centimeter cubes, does the way the cubes are arranged affect the volume? Does the arrangement affect the surface area?

100



Maximize the Surface Area Experimenting with different arrangements of cubes can help you find a relationship between the number of cubes and the greatest possible surface area for that number. In every arrangement, at least one face of each cube must meet the face of another cube.

1

What’s the greatest surface area you can get for a given volume?

You cannot “arrange” just one centimeter cube, and there is only one arrangement for two cubes. Find the surface areas for one and two cubes. Enter your results in a table like this one. Number of Cubes

1

2

3

4

5

6

Maximum Surface Area (cm2 ) 2

You can arrange groups of more than two cubes in different ways. For each number of cubes in the table, experiment until you find an arrangement that gives the maximum (greatest) surface area. Record your results.

3

Look for a pattern in your results. What do you notice? What do you think causes this pattern? Predict the maximum surface area for 7 cubes.

4

Graph your results on grid paper. (Plot the number of cubes on the x-axis.) What do you notice? Use your graph to predict the maximum surface area for 7 cubes. Does this prediction agree with your first prediction?

5

Write an equation to describe the relationship between the number of cubes and the maximum surface area. (Hint: Your table and the graph you made may help you see the equation.)

hot words

coordinate graph equation

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P H A S E TWO

102-103_Unit03 12/18/03 8:09 AM Page 102 Mac113 mac113:122_EDL:G8 Mathscapes Student Guide U3:

102-103_Unit03 12/18/03 8:10 AM Page 103 Mac113 mac113:122_EDL:G8 Mathscapes Student Guide U3:

You begin by building nets for rectangular prisms. Then you explore surface areas and volumes for different prisms and cylinders. You will also apply what you have learned to find the surface areas and volumes of figures made up of prisms and cylinders. Did you know that space efficiency and functional use determine the shapes of most objects? Thinking about the geometry of everyday things will help you to understand why they are shaped the way they are.

Prisms and Cylinders WHAT’S THE MATH? Investigations in this section focus on: GEOMETRY and MEASUREMENT ■

Finding the surface areas and volumes of prisms

■

Finding the surface areas and volumes of cylinders

■

Finding the areas of compound figures

■

Finding the surface area and volume of a compound solid


Developing a formula for the volume of any prism

■

Developing a formula for the volume of any cylinder


SHAPES AND SPACE

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5

All Boxed Up You have experimented with the volume and surface area of a cube. Now you are ready to find the volumes and surface areas of rectangular prisms—boxes whose sides are not necessarily square.

FINDING THE VOLUME AND SURFACE AREA OF A RECTANGULAR PRISM

Determine the Surface Area of a Rectangular Prism How can you find the surface area of a rectangular prism?

You’re familiar with nets for a cube. Now, you will make a net for a rectangular prism and use it to find the prism’s surface area. 1

A rectangular prism has three dimensions. Identify the length, width, and height for the prism you are working with. Measure these dimensions in centimeters, and record your results.

2

Use scissors, paper, and tape to make a net for your prism. Be sure you can fold up the net so it makes a model of the prism.

3

Measure the dimensions of each face of your net. Write the dimensions inside each face. How do these numbers compare to the length, width, and height you found in Step 1?

4

Find the surface area of your rectangular prism. Write down your result and explain how you found it.

How many different-size faces does a rectangular prism have?

Prisms A prism has a pair of opposite faces, called bases, that are polygons with the same size and shape. All of its other faces are rectangles. A prism is named by the shape of its bases.

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Bases

Triangular prism

Rectangular prism

Shapes that are not prisms


Find the Volume of a Rectangular Prism Instead of counting cubes to find the volume of a rectangular prism, you can use the dimensions of a prism to calculate its volume. 1

Find the volume of the rectangular prism shown. Use any method that makes sense. Write down your answer, and explain how you found it.

How can you calculate the volume and surface area of any rectangular prism?

4 cm 9 cm

2

3 cm

Find the volume of each rectangular prism below. Think about units!

10 in. 5.4 cm 5 in.

3

6 in.

7.5 cm

6 ft 5.4 cm

6 ft

6 ft

Explain how you can find the volume of any rectangular prism if you know its length, width, and height.

hot words

prism face

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6

FINDING SURFACE AREAS AND VOLUMES OF OTHER PRISMS

Wrapping Up Prisms You have found ways to measure the surface area and volume of a rectangular prism. Now you will develop strategies for finding the surface areas and volumes of prisms that do not have rectangular bases.

Find the Surface Areas of Different Prisms How can you measure the surface area of a nonrectangular prism?

106

In this investigation, you will find a way to measure the surface areas of two prisms whose bases are not rectangles. 1

Choose two nonrectangular prisms from your set of solids. What are the names of these prisms?

2

With your group, talk about some strategies that you think you could use to find the surface areas of your solids. Also, think about the supplies you need for these methods.

3

Find the surface area of each of your solids. (Ask your teacher for any supplies you want to use.) Keep a written record of all the work you do. This information may be useful later. Write the name and surface area of each solid, and explain how you found the surface area.



Find the Volumes of Different Prisms How can you find the volume of any prism?

Find the volumes of the two prisms you worked with in the previous investigation. Use any strategy that makes sense to you. Then explain in writing how you found these volumes.

Is there a formula for the volume of a prism? If so, what is it?

Write About Surface Areas and Volumes of Prisms Write your responses to the following questions: ■

Describe two methods that you could use to find the surface area of a nonrectangular prism. Explain why these methods work.

■

Write a formula you can use to calculate the volume of any prism. Using words or pictures, tell why this formula makes sense.

hot words

prism formula

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7

Outside and Inside This lesson gives you an opportunity to explore —solids whose bases are circles. You will begin by cylinders— investigating the surface areas of cylinders and finish by calculating the surface area and volume of an astronomically important cylinder.

EXPLORING THE SURFACE AREAS AND VOLUMES OF CYLINDERS

Investigate the Surface Area of a Cylinder How can you use the net for a cylinder to help find its surface area?

You can use what you know about making a net for other solids to make a net for a solid with curved sides. Then you can use the net to find the surface area of the cylinder. 1

Choose a cylinder from your set of solids. Use grid paper, scissors, and tape to make a net for the cylinder. Notice the shapes that make up this net.

2

Find the surface area of the cylinder. (Hint: Think of formulas that you have used to find the area of the different shapes you see on the net.)

Repeat these steps for the other cylinder in your set. Cylinders A cylinder has two parallel, congruent, circular bases. Bases

These are cylinders.

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These are not cylinders.


Find Volumes and Surface Areas of Telescopes Because it orbits the earth at a height of 381 miles, the Hubble space telescope is able to send back pictures that are not affected by the earth’s atmosphere. At first, the Hubble telescope delivered poor images, but since astronauts made repairs during a 1993 mission, the telescope has performed beautifully. This photograph taken by the telescope shows stars being born in Orion’s Great Nebula. The Hubble telescope rests in an aluminum cylinder. This cylinder has a 7-foot radius and is 43 feet long.

How does the volume of the Hubble telescope’s housing compare to the volume of an amateur telescope?

Galileo Galilei made the first astronomical telescope in 1609. Using this telescope, he was the first person to see Jupiter’s moons and the mountains on the earth’s moon. Telescopes made by Galileo were similar in size and shape to telescopes used by many amateur astronomers today. An amateur telescope is usually about 3 feet long with a radius of about 0.2 feet. ■

Find the volume of the cylinder that holds the Hubble telescope. Then calculate the volume of the typical amateur telescope. Record all your work.

■

Calculate the surface areas of the two cylinders. It may help to sketch the cylinders and to think about what the net for a cylinder looks like.

The cylinder containing the Hubble telescope is how many times as long as a typical amateur telescope? how many times as large?

hot words

cylinder area

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8 INVESTIGATING FIGURES WITH CYLINDERS AND PRISMS

Putting It Together Although many buildings such as office towers and stores are rectangular prisms, some buildings are combinations of different solids. In this lesson, you conclude your work with prisms and cylinders by investigating a building that combines both of these figures.

Find the Areas of Compound Figures What strategies can you use to find the areas of complicated figures?

Before you investigate how to find the surface area and volume of a complicated three-dimensional figure, you will find it helpful to review how to find the areas of complicated two-dimensional figures. Sketch each figure shown below. Then use any method you wish to find its area. Record your answer below the sketch. 2 cm 3 cm

5 cm 7 cm

110


6 cm

6 in. 8 in. 16 in.

6 cm


Size the Building Many aircraft hangars look like a rectangular prism topped by half of a cylinder. To plan the construction of a hangar like the one shown here, a contractor needs to know its surface area (to determine the amount of materials needed) and its volume, which affects the size and number of the ventilation fans the contractor will need. Choose any methods to calculate the surface area and volume of the hangar. (Do not include the area of the bottom of the hangar, since this is not an “outside” surface.)

How can you find the surface area and the volume of an aircraft hangar?

45 m

45 m 90 m

120 m

Write About Cylinders and Prisms Write a paragraph in which you tell everything you know about prisms and cylinders. Make sure you include information about their nets and how you can find their surface areas and volumes.

hot words

surface area volume

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PHASE THREE



You will investigate pyramids and cones and discover how their volumes relate to those of prisms and cylinders. You will determine appropriate units of measure for surface areas and volumes of everyday objects. You will use estimation skills and construct scale models of buildings. Thinking three-dimensionally about the objects that surround you may inspire you to investigate some fascinating careers.Which ones would you like to explore?

Solids with Points WHAT’S THE MATH? Investigations in this section focus on: GEOMETRY and MEASUREMENT ■

Finding the surface areas and volumes of pyramids

■

Finding the cross sections of a cone

■

Choosing appropriate units of measurement for surface areas and volumes

■

Estimating or calculating the volume of a large object


Developing a formula for the volume of a pyramid

■

Developing a formula for the volume of a cone

SC ALE and PROPORTION ■

Creating scale models of buildings


SHAPES AND SPACE

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9 FINDING THE SURFACE AREAS AND VOLUMES OF PYRAMIDS

Pyramid Tips In previous lessons, you found the surface areas and volumes of prisms and cylinders, shapes with two bases. Now you will examine a shape that comes to a point—the pyramid—and discover an important relationship between the volume of a prism and the volume of a pyramid.

Use a Net to Find Surface Area of a Pyramid How can you use the net for a pyramid to help you find the surface area?

Your goal in this investigation is to find the surface area of the square pyramid in your set of solids. 1

Make a net for your solid.

2

Find the surface area of the pyramid. Record your answer, and explain how you found it.

How can you predict the number of triangles in the net for a pyramid?

Pyramids A pyramid has one polygonal base. Each corner of the base connects to the pyramid’s vertex. The height of the pyramid is the perpendicular distance from its vertex to its base. A pyramid is named by the shape of its base. Vertex Height Base

Square pyramid

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Triangular pyramid

Hexagonal pyramid


Find a Formula for the Volume of a Pyramid In this investigation, you will compare the amount of rice a pyramid can hold to the amount that a prism with the same base and height can hold. To avoid spills, use plenty of tape when you build your pyramid and prism. 1

In the previous investigation, you made a net for a square pyramid. Use this net to make a pyramid. To do this, tape together the unattached triangular faces. Since you will be filling the pyramid with rice, do not attach the base to any other faces.

2

Build a square prism that your pyramid fits into perfectly.

How is the volume formula for a pyramid related to the volume formula for a prism?

a. Using grid paper, cut two square bases that are identical to the base of your pyramid. b. Measure the height of your pyramid and the width of its base. (Remember, the height goes straight down from the vertex to the base.) c. Cut out four rectangles that have the height and width you measured in the previous step. d. Use these rectangles and squares to build a square prism. Tape all sides securely, except for the top flap. 3

Fill your pyramid with rice. Pour the rice from the pyramid into the prism. How many times can you do this before the prism is full?

4

Calculate the volume of the prism.

5

What is the volume of your pyramid? (Hint: Your answers to Steps 3 and 4 can help you.)

6

Write a formula for the volume of a pyramid. Explain why your formula makes sense.

hot words

pyramid vertex

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10 INVESTIGATING CONES

Cutting the Cone After investigating prisms, pyramids, and cylinders, it’s —the cone. time to focus on the last in our list of solids— In this lesson, you will explore the cross sections and the volume formula for a cone.Then you will summarize what you have learned about all four types of solids.

Find the Cross Sections of a Cone What twodimensional shapes can you make by slicing a cone?

Earlier, you saw how many different cross sections of a cube you could find. Now you will do the same thing for a cone. These cross sections, called conic sections, are important in the study of algebra. 1

Is it possible to slice a cone to make a circular cross section? Could you get a cross section in the shape of a V? What about a single point?

2

See how many other cross sections of a cone you can find. Using sketches or words, describe how to slice the cone to get each figure.

Cones A cone has one circular base. The tip of the cone is its vertex. The height of the cone is the distance from its vertex to its base.

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Vertex Height Base


Discover the Formula for the Volume of a Cone In an earlier investigation, you made an important discovery about the relationship between the volume formula for a pyramid and the volume formula for a prism. Your goal now is to find a similar relationship between a cone and another solid you studied and then to find a likely formula. Discuss the following with the members of your group. Be prepared to share your insights in a class discussion. ■

How are a cone and a pyramid alike? How are they different?

■

What other solid is a cone similar to? What is the volume formula for this solid?

■

Complete this analogy: a pyramid is to a prism as a cone is to a ____.

■

What do you think the volume formula for a cone might be? Why?

How can knowing the volume formula for a pyramid help you find the volume formula for a cone?

Write About Prisms, Pyramids, Cylinders, and Cones Prepare a report that summarizes what you have learned about prisms, pyramids, cylinders, and cones. Include anything you think is important or interesting. Think of this report as something you or another student could use to study from. You will want to find good ways to organize and display your information so that it is easy to understand. Include pictures or charts where you think they are helpful.

hot words

cross section conic section

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11 USING DIFFERENT AREA AND VOLUME UNITS

Does the Unit Fit? You can express a volume in many different ways, including cubic millimeters, cubic centimeters, and cubic meters. How do you decide which units to use?

Choose Appropriate Units How do you decide which units to use when describing an area or a volume?

You may want to make sketches as you investigate these questions. 1. How many square inches are there in a square foot? How many cubic inches are there in a cubic foot? 2. What is the volume of the Chillaire in cubic meters?

3. What is the area of the home in square inches?

4. What is the volume of the 5. How many square centimeters can of Burble in cubic inches? will the paint in the jar cover?

6. Determine whether the units in each advertisement or the units you worked with are a better way to express the area or volume. Write a short explanation justifying your choice. 118



Fill City Hall with Chicken Soup You know how to find the volume of many different solids. Now, it’s time to find the volume of a real object—the bigger, the better! 1

Think of a large object that would be interesting to consider. For instance, you may want to investigate an elephant, a whale, a cruise ship, or City Hall.

2

Calculate or estimate the volume of this object in cubic inches, feet, centimeters, and meters. Record your answers, and explain how you found them. (You may have to use some clever strategies to find the volume of your object!)

3

Find how many of each of the following would fit inside your object. (Assume that there is no empty space.)

How many tablespoons of chicken soup would it take to fill City Hall? How many wombats could fit inside a whale?

a. tablespoons of chicken soup (1 tablespoon is about 15 cm3) b. cans of soda (355 cm3) c. wombats (250,000 cm3) What other units can you use to measure volume?

hot words

square inch cubic centimeter

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12 USING SOLIDS TO DESIGN BUILDINGS

New Heights in Architecture Although many skyscrapers are rectangular prisms, some famous ones, like San Francisco’s Transamerica Pyramid, resemble other solids. Your project in this last lesson is to design scale models of different buildings and to decide which one you would prefer to build.

Build Scale Models What are some different ways you can design buildings that have about the same base area and height?

Using grid paper, build scale models of three different buildings. One of your buildings should be a prism, one should be a pyramid, and one should be a cylinder. Use your imagination— for instance, pyramids and prisms don’t have to have square bases! Use a scale of 1 cm:1 story (4 m). Your buildings should have almost equal heights and base areas. To make sure your buildings are stable, limit their heights to 15 stories.

Tall Buildings in the United States

120

Building

Year Completed

Height (m)

Number of Stories

City

Woolworth Building

1914

241

60

New York

Trump Building

1930

283

70

New York

Chrysler Building

1930

319

77

New York

Aon Centre

1973

346

80

Chicago

Empire State Building

1931

381

102

New York

Sears Tower

1974

443

110

Chicago



Compare the Buildings Use any method you choose to find the surface area and volume of each of your designs. Record your answers, and explain in writing how you got your results.

How do surface areas and volumes of your buildings compare?

Decide Which Design Is Best If you were actually designing an office building, would you use a cylinder, a prism, or a pyramid? Write an explanation of why you think your choice is the best of the three designs. Of course, you don’t have to choose the shape that is most practical—an unusual design can make a building stand out. When you explain your choice, however, you should show that you have thought about things like construction, maintenance costs, and potential rental income.

hot words

surface area volume

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1 ork w e om

HW

Nets That Catch Cubes 4. The three-dimensional figure shown is a

Applying Skills

1. Which of the nets below can be folded

up into a cube? a.

b.

c.

d.

cube with one corner cut off. How many vertices, edges, and faces does it have? Draw a net for the solid.

Writing

2. Which nets below can be folded

up into a three-dimensional figure like this one? a.

b.

c.

d.


Dear Dr. Math, If I have a net for a solid, I know how I can predict the number of vertices the solid will have. First I draw all the twodimensional shapes that make up the net separately:

Extending Concepts

3. a. Describe and sketch the solid that you

could get by folding each net.

b. For each solid draw two more nets,

both different from the one shown.

122

SHAPES AND SPACE • HOMEWORK 1

Then I count the number of vertices for each shape and add them up. Here, for example, we have a total of 24 vertices. Then I divide that number by 3, because when you fold up a net, three faces always meet at each vertex of the solid. So here I predict 234 8 vertices, which is the number of vertices of a cube. Isn’t this a great method? Manny Faces


2

Any Way You Slice It

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HW

Applying Skills

Extending Concepts

1. Tell whether each two-dimensional

figure below is a possible cross section of this three-dimensional figure.

a.

b.

c.

figure you would obtain by slicing the three-dimensional figure with a plane passing through the following points. O A

H

L N

K

B F E C Cube

a. rectangle

b. hexagon

c. obtuse triangle

d. scalene triangle

can you make by taking cross sections of a cube with one corner cut off? Sketch as many different figures as possible.

2. Sketch and name the two-dimensional

G

you would slice a cube to make each cross section. If it is not possible, write “not possible.”

5. What different two-dimensional figures

d.

D

4. Explain using words and pictures how

6. The picture shows a series of parallel

cross sections for a three-dimensional figure. Sketch two different possibilities for the three-dimensional figure. Explain how you would slice each threedimensional figure to obtain the cross sections. Can you tell the size of the solid? Why or why not?

M Pyramid with square base

a. A, B, C, D

b. D, E, F

c. A, E, F, G, H

d. M, N, O

e. K, L, M, N 3. For the pyramid in item 2, draw a series

of cross sections parallel to the base.

Making Connections

7. A hogan is a house of the Navajo

Indians. Hogans are usually built of logs and earth. They are often dome-shaped as shown in the sketch, with a single door facing east. For the hogan shown, sketch a series of cross sections parallel to the base.


123


3 ork w e om

HW

Take One for Good Measure! 11. Cubes with side lengths 1, 2, and 3 have

Applying Skills

Find the surface area and volume of each object. Each grid square represents 1 cm2. 1.

volumes 1, 8, and 27, respectively. What pattern do you notice? Write a formula for the volume of a cube. 12. Use your formulas to make a table

2.

showing the surface area and volume for cubes with side lengths 1 cm to 8 cm. 3.

2

4.

13. If the surface area of a cube (in cm ) is

5. How many square centimeters are in a

numerically larger than its volume (in cm3), what can you say about the side length of the cube?

square meter? 6. How many square millimeters are in a

Writing

square centimeter? 7. How many cubic millimeters are in a


cubic centimeter? 8. a. Which of the items listed in i–iv could 2

have a surface area of 1,000 cm ? 54 m2? 10,000 cm2? b. Which could have a volume of

900 m3? 1,000 cm3? 27 m3? i. a book iii. a kitchen

ii. a television set iv. a house

Extending Concepts

9. Find an arrangement of centimeter

cubes that has a smaller volume but a greater surface area than the object in item 1. Make a sketch and find the volume and surface area of your object. 10. What is the area of each face of a cube

with side length s? How many faces does a cube have? Write a formula for the surface area of a cube. 124


Dear Dr. Math, I wanted to find the volume of a cube with side length 6 feet. So I converted 6 feet to 2 yards and found the volume by multiplying 2 yd 2 yd 2 yd 8 yd3. But then I tried it a different way: I multiplied 6 ft 6 ft 6 ft 216 ft3. To convert that to cubic yards I reasoned like this: Since there are 3 feet in a yard, there must be 9 cubic feet in a cubic yard. So I divided 216 by 9 and got 24 yd3. But why did these two methods give different answers? Can you help? George


4

Move the Cube

ork w e om

HW

Applying Skills

Extending Concepts

Suppose that you arrange centimeter cubes so that the faces overlap perfectly. Find the maximum surface area for each of the following numbers of cubes. 1. 2

2. 4

3. 6

4. 9

5. 25

6. Which arrangement of cubes in items

10. Find the surface area of each

arrangement of 8 centimeter cubes. Which arrangement has the largest surface area? the smallest surface area? a.

b.

c.

d.

1–5 gives the maximum surface area? 7. For this 3 3 3 cube, would the

surface area increase, decrease, or stay the same if you removed cube A? cube B? cube C? cube D? C

D B A

11. If you move a corner cube of a 3 3 3

cube as shown, does the surface area increase or decrease? Is it possible to move one cube of a 3 3 3 cube so that the surface area decreases? If so, how?

8. What is the surface area of a 4 4 4

cube after you remove: a. a corner cube? b. a center-face cube?

12. How could you arrange 27 centimeter

cubes so that the surface area is as small as possible? Explain your thinking.

c. a middle-edge cube? 9. If you remove one centimeter cube

from a 5 5 5 cube, what are the possibilities for the surface area of the new solid? Describe how you could create each one. What is the volume of the new solid?

Making Connections

13. A car’s cooling system removes excess

heat from the engine. Coolant flows from the engine to the radiator. As it flows through the radiator, it loses heat to the air. For efficient heat transfer, the surface area of the radiator must be large. Explain why this shape for a radiator is more efficient than a cube. SHAPES AND SPACE • HOMEWORK 4

125


5

All

ork w e om

HW

Boxed Up

Applying Skills

Extending Concepts

Tell whether each solid is a prism. If it is a prism, name the prism. 1.

2.

3.

4.

13. Suppose you use the letters l, w, and h to

represent the length, width, and height of a rectangular prism. How could you use these letters to describe the dimensions and area of each of its faces? Assuming that l, w, and h are all different, how many different face sizes are there? h

Find the volume of each rectangular prism. 5.

6. 3m

2 ft

7.

14. Using your results from item 13, write a

6 ft

6m

6m

formula for the surface area of a prism with length l, width w, and height h.

3 ft

15. What is the surface area of a prism if

8.

2.8 cm

9 in.

7.2 cm

10 in. 11 in.

4.5 cm

–8, draw a net, 9. For each prism in items 5– label its dimensions, and find the surface area. Find the volume of a rectangular prism with the specified length, width, and height. 10. 6 cm, 4 cm, 3 cm 11. 9 ft, 2 ft, 1 ft 12. 8.3 in., 6.1 in., 3.8 in.

126

w l


l 22 cm, w 15 cm, and h 12 cm?

16. A rectangular prism must have a volume

of 64 cm3. What dimensions could you choose that give the smallest possible surface area? 17. A rectangular prism must have a volume

of 64 cm3, and its side lengths must be whole numbers. What dimensions give the greatest possible surface area? Making Connections

18. The Maya are Central American Indians.

The Maya are famous for their stelae— stone slabs with pictures of their gods and events from their history. One of the stelae, inscribed in A.D. 771, is 35 feet high, 5 feet wide, and 4.2 feet thick. If the stela is a rectangular prism, what are its volume and surface area?


6

Wrapping Up Prisms

ork w e om

HW

Applying Skills

Extending Concepts

Assume that the bases of all prisms shown are regular polygons. Find the volume and surface area of each prism. 1.

2. 20 m

6 cm 3 cm Base area = 3.9 cm 2

3.

8m Base area = 110 m 2

4. 6.1 m

9 in. 6 in. Base area = 174 in.2

6.2 m Base area = 100 m 2

The apothem (a) is a perpendicular line from the center of a regular polygon to any one of its sides. If the perimeter is p, use the formula A 12 ap to find the area of each regular polygon. 5.

6. 10 cm

the formula V Bh for the volume of a prism makes sense.

13. Faces A and B of the solid shown are

parallel, but face A is smaller than face B. Do you think that the formula V Bh would work for this solid? Explain your thinking. Is this solid a prism? Why? A

B

14. The honeybee carries nectar from

8. 19.05 m

9.9 in.

22 m

flowers to the hive, where it is stored in cells and converted to honey. The cells are hexagonal prisms.

8.2 in.

For each prism find the area of each base, the volume, the area of each rectangular face, and the surface area.

0.5 cm 0.3 cm Area of base = 0.23 cm 2

a. What is the volume of the cell shown?

10. 6 ft 9 cm 1.15 cm 4 cm

12. Use words and pictures to explain why

6.88 cm

6 ft

9.

Its surface area, excluding the bases, is 240 cm2. Its base is a regular polygon with side length 5 cm. How many sides does the base have? How do you know? What is the total surface area of the prism?

Making Connections

1.73 ft

7.

11. Suppose that a prism has height 12 cm.

1.73 ft 2 ft

b. Is it possible to tessellate a plane with

a regular hexagon? Why do hexagonal prism cells use space more efficiently than cylindrical cells would?


127


7 ork w e om

HW

Outside and Inside

Applying Skills

Extending Concepts

Tell whether each solid is a cylinder. If it is not, explain why. 1.

2.

3.

4.

ellipse

For each cylinder below, find the area of each base and the volume. 5.

3 ft

cylinder is S 2πrh 2πr2. Draw a net for a cylinder with height h and radius r. Label its dimensions. Explain why the formula for surface area makes sense. Use the formula to find the surface area of a cylinder with height 10 m and radius 5 m.

15. a. Find the volume of cylinder A

having base radius 1 and height 2. b. Find the volume of a cylinder whose

6.

5 ft

14. The formula for the surface area of a

6m 14 m

dimensions are i. twice those of cylinder A. ii. 3 times those of cylinder A.

7.

8.

54 cm

7.2 cm 21 cm 3.6 cm

9. For each of the cylinders in items 5–6: a. Find the circumference of the base. b. Draw a net and label its dimensions. c. Find the surface area.

Tell whether each figure below is a possible cross section of a cylinder. If it is, explain how you would slice the cylinder to get the cross section.

128

10.

11.

12.

13.


iii. 4 times those of cylinder A. c. How many times the volume of cylinder

A is the volume of each cylinder in item 15b? What pattern do you notice? 16. If the dimensions of one cylinder are

8 times those of another, the volume of the larger cylinder will be how many times the volume of the smaller cylinder?

Making Connections

17. The Skylab space station was first

launched in 1973. It was intended that it would orbit Earth at a height of 170 miles, but the first launch was unsuccessful. The main section of the space station was a cylinder 48 feet long and 22 feet in diameter. Find its volume and surface area.


8

Putting It Together

ork w e om

HW

Applying Skills

Extending Concepts

Find the area of each compound figure. 1.

4m

12 ft

5m

3.

half cylinder. The base of the prism is an equilateral triangle. Draw separate nets for each and label the dimensions. Find the volume and surface area of the solid.

2.

8m 15 m

7. The solid is a triangular prism with a

10 ft

4.

9 cm 7 cm

6 cm 24 m 18 m

20 cm

15 m

4 cm

20 m

40 m

5. Follow the steps to find the volume of

this solid.

12 cm Base area of prism = 62.35 cm 2

Writing

8. Explain how the method for finding the

3 ft

volume of a cylinder is similar to and different from the method for finding the volume of a prism.

4 ft 6 ft

8 ft

a. If the half cylinder were a complete

cylinder, what would the area of each base be? What would its volume be? b. What is the volume of the half

cylinder? c. What is the volume of the prism?

What is the total volume of the solid? 6. Follow the steps to find the surface area

of the solid in item 5. a. Find one-half of the circumference of

a circle with radius 3 ft. b. Copy the net and label its dimensions.

Find the area of your net. What is the surface area of the ? half cylinder?

? ?


Dear Dr. Math, We had to find the surface area of the curved part of this solid. If I made a net, the curved 4 m part would be a 5m rectangle with 8m 10 m length 10 m. Its width would be half the circumference of a circle with radius 4, about 12.6 m. So the area would be 126 m2. Since we only have half a cylinder, the surface area of the curved part must be 12 6 2 2 63 m . What do you think? Angie

c. What is the total surface area of the

five exposed faces of the prism? d. What is the solid’s total surface area?


129


9

Pyramid Tips

ork w e om

HW

9. If a pyramid and a prism have the same

Applying Skills

Tell whether each solid is a pyramid. If it is, name the pyramid. 1.

base area and volume, and the height of the prism is 16 cm, what is the height of the pyramid? 10. Suppose that the base of a pyramid is a

2.

regular n-gon with area B and perimeter p and that each triangular face of the pyramid has height s. 3.

a. How many triangles are in the net for

4.

the pyramid? b. The base length of each triangle in the

Assume that the base of each pyramid is a regular polygon. For each pyramid, find its volume, draw a net, find its surface area, and sketch a series of cross sections parallel to the base. 5.

8.54 cm 8 cm

6.

p

net is n . Why does this make sense? What is a formula for the area of each triangle in the net? Why does the formula SA B 12 ps for the total surface area make sense?

7 in. 7.05 in.

Making Connections 6 cm 3 in. Base area = 3.9 in.2

6 cm

7. 10 ft

12.54 ft

11 ft Base area = 208.2 ft2

11. The Maya built pyramids as platforms

for temples. Unlike the Egyptian pyramids, which were true pyramids, the Mayan “pyramids” had flat tops as shown.

Extending Concepts

8. If a pyramid and a prism have the same

base area and the same height, which solid has the greater volume? Why does this make sense? If the volume of the pyramid is 24 m3, what is the volume of the prism?

130


Describe a method you could use to find the volume of such a “pyramid.” Assume that the dimensions of the square base, the dimensions of the square top, the height, and the height of the “missing top” are known.


ork w e om

HW

Cutting the Cone b. What other cross sections can you

Applying Skills

Find the volume of each cone. 1.

2.

4 cm 3 cm

3.

4.

8.9 ft

13 m 16 m

12. a. If the small cone were removed from

the top of the original cone shown, what would the volume of the remaining part (the frustum) be?

26 in. 26 in.

6.3 ft

find of a double cone? Use sketches and words to describe how you would obtain each one.

For which solids (cones, cylinders, prisms, pyramids) is each statement below true?

b. The formula for the surface area of a

cone is πr2 πrs (s is the slant height as shown). Find the surface area of the large cone. 10 m 4m

5. The solid has two parallel bases.

s= 32.3 m

6. The bases of the solids are circular.

30 m

7. Cross sections of the solid parallel to the

base are all the same size.

12 m

8. The solid has at least 3 triangular faces. 9. The solid has at least 3 rectangular faces. 10. Cross sections of the solid parallel to the

Writing

13. How are cones and cylinders alike? How

are they different? How are cones and pyramids alike? How are they different? How is the method for finding the volume of a cone similar to the method for finding the volume of a pyramid? How is it different?

base are polygons.

Extending Concepts

11. a. Tell whether each cross section

below is a possible cross section of this double cone. If it is, explain how you slice the double cone to get the cross section. i.

ii.

iii.

Ellipse Parabola Hyperbola


131


ork w e om

HW

Does the Unit Fit?

Applying Skills

1. How many square inches are in a square

foot? 2. How many cubic centimeters are in a

cubic meter? 3. How many cubic inches are in a cubic

yard? 4. Convert 0.067 square feet to square

inches. 5. Convert 0.00519 square meters to square

centimeters. 6. Convert 5,300,000 cubic centimeters to

cubic meters. Find the volume, in cubic meters and in cubic centimeters, of a rectangular prism with the specified length, width, and height. 7. 350 cm, 200 cm, 100 cm 8. 0.5 m, 0.04 m, 0.03 m

Find the volume, in cubic inches and in cubic feet, of a rectangular prism with the specified length, width, and height. 9. 20 in., 18 in., 35 in. 10. 11 ft, 5 ft, 2 ft 11. 0.4 ft, 0.3 ft, 0.5 ft

Extending Concepts

12. a. Find the surface area (not counting

the ground floor) of a building that is a rectangular prism with length 70 yards, width 62 yards, and height 230 yards. b. There are 1.0936 yards in one meter.

How many square yards are in a square meter? How did you figure it out? What is the surface area of the building in square meters? c. Suppose that one can of a certain

brand of paint covers 300,000 cm2. How many cans would be needed to cover the building? Making Connections

13. Vesuvius, in Italy, is the only active

volcano on the European mainland. Its earliest recorded eruption (A.D. 79) buried the town of Pompeii. Similar to a cone in shape, its height is approximately 1.2 kilometers, and the radius of its base is approximately 11 kilometers. a. Estimate the volume of Vesuvius in

cubic kilometers. Round your answer to the nearest 10 km3. b. What is this volume in cubic meters? c. Which unit is more suitable, a cubic

meter or a cubic kilometer? Why? d. Approximately how many houses of

volume 2,500 m3 would fit inside Vesuvius? (Assume that there would be no empty space.) 132



k wor e m o

HW

New Heights in Architecture b. Draw a net for your prism-shaped

Applying Skills

1. Find the volume and surface area of each

building, including its ground floor. a.

b.

450.7 m 450 m

50 m

c.

50 m 50 m

d. 150 m

28 m

c. Which building do you think has the

smallest surface area? the largest? d. If two buildings have the same

150 m 50 m

building and for your cylinder-shaped building using a scale of 1 cm:5 m.

150 m 31 m Base area = 2,497 m2

2. Each of the buildings in item 1 has

approximately the same volume. Rank the buildings from smallest to largest in surface area. 3. Which building design in item 1 would

you choose if you wanted the smallest surface area for a given volume? Which design gives the largest surface area for a given volume?

Extending Concepts

volume, why might the one with the smaller surface area be preferred? If two buildings have the same surface area, why might the one with the larger volume be preferred? 5. If a pyramid and a cone have the same

base area, height, and volume, which do you think has the greater surface area? Explain your thinking.

Making Connections

6. A building is a rectangular prism with

height 1,350 feet and length and width 209 feet. It has 110 stories. a. Find the volume of the building. b. Estimate the total floor space in

square feet of the building. Explain how you obtained your estimate.

4. a. Design three different buildings,

one a square prism, one a cylinder, and one a square pyramid. Each building should have a base area of approximately 1,000 m2, and all three should have approximately the same volume. Make a sketch of each building and label its dimensions. Tell how many stories each building has.


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WHAT COMES

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PHASEONE

PHASETWO

PHASETHREE

Exploring Growth Sequences

Representing Growth Sequences You will learn ways to calculate growth sequences from actual data and build the sequences into mathematical models to use in estimating data for the future or the past. Describing the models using algebra, charts, and graphs will help you decide how well your models fit the actual data in order to refine your predictions.

Modeling with Growth Sequences In this phase you will use what you have learned about mathematical models to examine real-world statistics about population in different geographical regions. By working with other students, you will compile a picture of growth trends all over the world. You will compare the growth patterns of different countries. In a final report you will combine all your data and describe what it suggests about the future.

In this phase you will experiment with growing numbers in addition and multiplication patterns. By comparing spiral patterns, you can learn about the differences between growing by addition and growing by multiplication. You will see how to build a mathematical model to make forecasts and think about when forecasts are useful.

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PHASE ONE

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In this phase, you will use addition and multiplication to make numbers grow, and then you will examine the difference in the growth patterns you have created. You will use your number sense to make a number grow in equal steps to reach a target value. Soon you will be building mathematical models to help predict future growth.

Exploring Growth Sequences WHAT’S THE MATH? Investigations in this section focus on: NUMBER and OPERATIONS ■

Estimating values in growth sequences

■

Finding growth numbers using guess-and-check strategies

■

Using number sense and inverse operations to build growth sequences

PATTERNS and FUNCTIONS ■

Comparing visual patterns created by addition and multiplication growth numbers


WHAT COMES NEXT?

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1

USING ESTIMATION TO MAKE A FORECAST


Predicting World Population In order to make plans for the future, governments and businesses make forecasts. You will start by looking at data from the past, then use mathematical patterns to predict future data values.

Forecast from Data Can you use actual data to make a forecast?

Analyzing the data given for the world’s population over the years will give you a sense of how large the population will be in the future. Think about how you can use this data to make an estimate. ■

Estimate the world’s population for the year you turn 50 years old.

■

Record your forecast on a World Population Forecasts poster. World Population

138

Year

Population (in billions)

1950 1960 1970 1980 1990 2000

2.52 3.02 3.70 4.45 5.29 6.08

WHAT COMES NEXT? • LESSON 1

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Defend Your Forecast Describe the methods you used to predict the world’s population for the year you will be 50. ■

What mathematics did you use to find a pattern in the data?

■

How did you extend the mathematical pattern you found?

Why do you think your forecast makes sense?

Think About Forecasting Select two of the Forecasting Situations and answer these questions: 1

How is the situation changing with time? Is the number increasing, decreasing, staying about the same, or going up and down? Explain your reasoning.

2

Who would be interested in a forecast for this situation?

3

How might a person or an organization use the forecast? Forecasting Situations

The numbers that describe these situations will vary from year to year. ■

Airline passengers

■

Annual rainfall

■

Books sold

■

Color televisions sold

■

Minimum wage in the United States

■

Smog level in our town’s air

■

Visitors to Yellowstone National Park

■

Price of a new car

hot words

forecast pattern

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2

GROWING BY ADDITION AND MULTIPLICATION


Target Practice Addition and multiplication are natural operations to use when you’re trying to make a number larger. Your calculator allows you to create increasing number patterns quickly.

Find a Growth Number How can you grow 10 into 40 in four steps?

Grow 10 into 40 in four steps by adding. 1

Use a guess-and-check approach to find a number that can be added to 10 four times to reach a final goal of 40.

2

Keep track of your guesses for the addition growth number and the goals they reach.

Make Your Calculator Do the Work

Most calculators can be set up to add or multiply by the same number over and over again, but each calculator operates differently. Follow these steps to check how your calculator does these operations. ■

■

140

Enter 3 5 and record the output after each is pressed. Describe what arithmetic the calculator is doing. Enter 3 1 0 and record the output after each is pressed. Describe what arithmetic the calculator is doing.


Grow 10 into 40 in four steps by multiplying. 1

Use a guess-and-check approach to find a number by which you can multiply 10 four times to grow it into a final goal of 40.

2

Keep track of your guesses for the multiplication growth number and the goals they reach.

You can use a chart like this to keep track of your guesses. 10

?

?

?

40

10 ? ? ? ? 40 Growth Number Guessed

Number Reached

Result

2

160

too big

1

10

too small

1.5

50.625

too big

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Grow from Start to Target in Four Steps Find an addition growth number and a multiplication growth number for each pair of numbers given for the start and target. 1

Start 10; Target 30

2

Start 15; Target 45

3

Start 3.8; Target 11.4

4

Start 1; Target 3

What are the growth numbers?

Compare Addition and Multiplication Growth Write about what you learned about growing numbers by addition compared to growing by multiplication. Include your answers to these questions. ■

Were any of your results surprising?

■

What did you do to make the process of guessing the growth number easier?

■

How did the addition and multiplication growth numbers for a particular start and target compare to each other?

■

Was it quicker to find the addition growth number or the multiplication growth number? Four Steps from Start to Target Start

Target

10

?

?

?

?

30

10

?

?

?

?

30

hot words

addition growth number multiplication growth number

W Homework page 167


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3 VISUALIZING GROWTH SEQUENCES

10:27 AM


Growing Spirals The spiral is a basic geometric growth form found throughout nature and the technological world. The pictures on the opposite page give some examples of different forms.The addition and multiplication sequences help us describe and understand two basic kinds of spirals.

Construct Addition and Multiplication Spirals How can number sequences be made into spirals?

In the examples in Drawing Spirals, the lengths of the segments of the right-angled addition spiral correspond to the numbers in the addition sequence of 1.4, 2.8, 4.2, …. Each segment is placed at right angles to the preceding one, making a quarter-turn in a clockwise direction. The curved addition spiral is sketched by joining the midpoints of the segments on the right-angled spiral. A multiplication spiral is constructed in the same way, but uses numbers in a multiplication sequence. You and your group can construct some sample spirals.

142

■

Each member of your group chooses a different number from this set: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7. This will be the starting number and the growth number.

■

Make an addition sequence and a multiplication sequence of numbers based on the number you chose.

■

Use Centimeter Dot Paper and a ruler to construct two right-angled spirals based on your two sequences.

■

Lightly mark an X at the midpoint of each of the segments.

■

Sketch the curved spirals by connecting the Xs.

■

Work with your group to make a poster of all your spirals. Decide on a title and how you will arrange the spirals on the poster.


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Connect Number Sequences to Spirals ■

In your group, write an answer to this question: How could someone decide whether a spiral was an addition spiral or a multiplication spiral just by looking?

■

If each spiral on your poster continued for 30 segments, what would be the length of the thirtieth segment for each spiral? Which spirals would still fit inside the classroom?

Is there a visual difference between the spirals for addition and multiplication sequences?

Write About Number Growth Write about how looking at a spiral can help you see number growth. What would you look for to tell you whether a spiral shows number growth by addition or multiplication? Drawing Spirals Addition sequence: 1.4, 2.8, 4.2, 5.6 . . . Multiplication sequence: 1.4, 1.96, 2.74, 3.84 . . . Add 1.4 Spiral 2.8 x

x

1.4 x etc.

x

4.2x

x

5.6 x

x

Multiply by 1.4 Spiral

hot words

etc.

x 5.38

1.96

x x 1.4

x x 2.74

3.87 x

W Homework

x

x

sequence

x

page 168

x


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4

PREDICTING WITH ADDITION AND MULTIPLICATION MODELS


Make a Model You can create a model for data that is increasing over time using addition or multiplication growth patterns. Then you can extend the model mathematically into the future and the past.

Make Models of World Population Growth How can models help you forecast world population?

Set up a World Population chart like the one shown. Focus on the world population in 1950 and 1990. Find addition and multiplication growth numbers for each decade. Use your calculator to help. ■

Find a number that can be added to 2.52 four times to reach a goal of approximately 5.29. Keep track of your guesses for the addition growth number.

■

Record the population figures your addition growth model predicts for each decade from 1960 through 2030.

■

Find a number by which you can multiply 2.52 four times to reach a goal of approximately 5.29. Keep track of your guesses for the multiplication growth number.

■

Record the population figures your multiplication growth model predicts for each decade from 1960 through 2030.

World Population (in billions)

144

Year

1950

1960

Addition Model

2.52

5.29

Multiplication Model

2.52

5.29


1970

1980

1990

2000

2010

2020

2030

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Compare Predictions The population number predicted in this lesson depends on which method was used to make the forecast. Compare the addition model forecast to the multiplication model forecast. Then compare these models to the world population predictions your class collected in Lesson 1. Consider these questions: ■

How do the two models compare?

■

Were any of the predictions made in Lesson 1 close to the addition model forecast?

■

Were any of the predictions made in Lesson 1 close to the multiplication model forecast?

■

Were there any similarities between the methods you used in Lesson 1 and the models you created by addition or multiplication?

Look into the Past Use the growth numbers you found for your addition and multiplication models of world population to extend the data into the past by decades until you can answer these questions. 1

When does the addition model estimate a world population of one billion?

2

When does the multiplication model estimate a world population of one billion?

3

Explain the arithmetic you used to extend your models into the past.

When was the population of the world only one billion?

Write About What You Have Learned Write a letter to a friend or to someone in your family explaining what is most important to understand about growth by addition and growth by multiplication.

hot words

growth model predict

W Homework page 169


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P H A S E TWO

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You will use the language of algebra to describe growth sequences in this phase. You will find a more efficient strategy than guess-and-check for calculating growth numbers. By graphing and charting your sequences, you can examine your mathematical models to find how well the models fit the actual data numerically and graphically.

Representing Growth Sequences WHAT’S THE MATH? Investigations in this section focus on: ALGEBRA ■

Using variables and equations to tell how to extend growth sequences

■

Calculating the value of the nth term in a sequence

NUMBER and OPERATIONS ■

Calculating sequences by finding one-step and two-step growth numbers

PATTERNS and FUNCTIONS ■

Graphing growth sequences to compare them to actual data

REASONING ■

Deciding which growth model better fits the data


WHAT COMES NEXT?

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DESCRIBING SEQUENCES ALGEBRAICALLY

10:43 AM


Growing in One Step You have developed some organized guess-and-check methods for finding addition and multiplication growth numbers. Now it’s time to think about more efficient mathematical techniques to calculate growth numbers, and to extend increasing or decreasing sequences.

Calculate Growth Numbers and Extend Sequences How can you find when two population numbers will be equal?

Suppose you want to use the data in the chart to predict when the rural population of Brazil will equal the urban population. Whenever you know two consecutive numbers in a sequence, you can use basic arithmetic operations to calculate the addition and multiplication growth numbers. Then you can extend the sequences to make predictions of future population figures. 1

Assume that you can model both populations by addition sequences. Predict when the population living in rural areas will equal the population living in urban areas.

2

Now assume that you can model both populations by multiplication sequences. Using this model, predict when the population living in rural areas will equal the population living in urban areas.

3

Explain the arithmetic you used to find the addition and multiplication growth numbers and to extend the sequences. Population of Brazil (in millions) Year

1980

1990

Rural Population

39.31

36.17

Urban Population 12.13

21.64

Source: Demographic Yearbooks published by United Nations, years 1981, 1988, 1991, 1993.

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Describe a Sequence Algebraically Extend each of these sequences through the tenth term. Then write an expression that gives the value of the nth term. 1

Addition sequence:

2

Multiplication sequence:

Term

Number

Term

Number

1 2 3 4 . . .

7 9 11 13

1 2 3 4 . . .

2 6 18 54

How can you describe a sequence without actually listing the numbers?

Write About Sequences Tell how to extend addition and multiplication sequences when two consecutive numbers in the sequence are known. Give specific examples. Use variables and equations in your directions. Include information on each of the following points: ■

how to calculate the growth number

■

how to extend the sequence into the future

■

how to calculate the value of the nth term in the sequence Expressing the Value of the nth Term Algebraically This identifies the position of the number in the sequence.

The variable n stands for any term in the sequence.

Term

Number

1 2 3 4 5 . . .

5 8 11 14 17

n

3n 2

The sequence of numbers is in this column.

This expression gives the value of the number in the nth position.

hot words

algebra sequence

W Homework page 170


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6 FINDING INTERMEDIATE VALUES


Growing in Two Steps The U.S. Census Bureau takes a census once every decade, but government planners, business leaders, bankers, and many others need to project population trends over a shorter time period. By calculating a number between two known numbers in a sequence, you can make predictions for fewer than ten years.

Find One-step and Two-step Growth Numbers How can you calculate a number between two consecutive numbers in a sequence?

You already know how to find the one-step growth number when you are given two consecutive numbers in a sequence. There is a connection between the two-step growth number and the one-step growth number. Once you find the connection, you will discover an arithmetic rule for quickly calculating the two-step growth number. Calculate the one-step addition growth number. Use guessand-check to find the two-step addition growth number for each pair of numbers given in the Addition Growth Model.

1

Growth Models

2

How does the two-step growth number compare to the one-step growth number?

3

Calculate the one-step multiplication growth number. Use guess-and-check to find the two-step multiplication growth number for each pair of numbers given in the Multiplication Growth Model.

4

How does the two-step growth number compare to the one-step growth number?

Addition Growth Model one-step 3 75 one-step 4 36 one-step 2 72 two-step 3 … 75 two-step 4 … 36 two-step 2 … 72

Multiplication Growth Model one-step 3

75 one-step 4

36 one-step 2

72

two-step 3 … 75 two-step 4 … 36 two-step 2 … 72

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Estimate an Intermediate Value What will the U.S. population be in 2015?

Using the census data for 1990 and 2000, you can estimate the U.S. population in 1995. With that information, you can predict the population in 2005, 2010, and 2015. 1

First assume that the U.S. population follows an addition growth model. Calculate the population in 1995 and predict the U.S. population in 2015.

2

Now assume that the U.S. population follows a multiplication growth model. Calculate the population in 1995 and predict the U.S. population in 2015.

Relate One-step and Two-step Growth Numbers Write about how you found the two-step growth numbers. ■

Describe the arithmetic you could use to calculate the two-step addition growth number very quickly just by knowing the one-step addition growth number.

■

Describe the arithmetic you could use to calculate the two-step multiplication growth number very quickly just by knowing the one-step multiplication growth number.

■

Assume that P and Q are consecutive numbers in an addition growth sequence. Write algebraic expressions for the value of the one-step and the two-step growth numbers.

■

Assume that P and Q are consecutive numbers in a multiplication growth sequence. Write algebraic expressions for the value of the one-step and the two-step growth numbers. U.S. Population Year

Population (in millions)

1990

250.0

1995

?

2000

282.3

hot words

growth model sequence

W Homework page 171


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What Do You See? In Phase One you drew spirals based on sequences. Now you will analyze the shapes of the graphs made by plotting the terms of addition and multiplication sequences.

GRAPHING THE SEQUENCES

Create Growth Sequences How many different sequences can be built around 36?

The number 36 could appear in many different addition and multiplication sequences. Each person in your group will make up one addition sequence and one multiplication sequence in which 36 is the third term in the sequence. Be sure that everyone uses different growth numbers. ■

Choose an addition growth number. It can be any number you want: a whole number, a decimal, a negative number.

■

Make a table like the example, filling in the numbers in your addition sequence. ■

Next choose a multiplication growth number. It can be a decimal or a whole number, but keep it positive.

■

Make a table like the example, filling in the numbers in your multiplication sequence.

Examples with 36 as the Value of the Third Term Addition Growth Sequence Term

Term

1st

1st

2nd

2nd

3rd

152

Number

Multiplication Growth Sequence

36

3rd

4th

4th

5th

5th

6th

6th

7th . . .

7th . . .


Number

36

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Graph Growth Sequences By graphing a scatter plot of the sequences you created, you will be able to visualize the two growth patterns. Use two pieces of grid paper, one for each sequence. ■

Draw axes. Choose appropriate scales for “Term” on the horizontal axis and for “Number” on the vertical axis.

■

Plot the data on your graph.

■

Connect the points with a dotted line or smooth curve in order to emphasize the visual pattern.

■

On each graph, write your name, the type of growth sequence, and the growth number.

What do your sequences look like on a graph?

Compare Growth-Sequence Graphs What do the graphs have in common? How are they different? Spread out all the graphs your group has made. Decide on at least two different ways to separate the graphs into groups that have something in common. Describe what you learned when you compared the addition and multiplication sequence graphs.

hot words

ordered pair scatter plot

W Homework page 172


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8 COMPARING MODELS TO ACTUAL DATA

10:43 AM


Which Model Fits the Data Better? The predictions made by an addition model are often quite different from those made by a multiplication model. In this lesson, you will learn two ways to help judge which model might provide the more reasonable prediction.

Make Two Growth Models Based on figures for 1910 and 1990, what will the U.S. population be in 2030?

You can create addition and multiplication models for the U.S. population and use them to make two forecasts of the population in 2010 and 2030. ■

Use the data for 1910 and 1990 to find the intermediate values for 1950. Record those values on a copy of the chart.

■

Find and record the intermediate values for 1930 and 1970.

■

Extend the models to 2010 and 2030.

■

Make a graph of the models. Place the year on the horizontal axis and the population on the vertical axis, choosing an appropriate scale for each axis. Using different symbols or colors for each model, graph your addition and multiplication models on the same scatter plot. Draw in a dotted line or curve to emphasize the trend in each graph. Build Two Models of U.S. Population (in millions) Year

Addition Model


1910

92

92

250

250

1930 1950 1970 1990 2010 2030

154


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Compare the Models to Actual Data Comparing the models to the actual population data between 1910 and 1990 tells you which model might predict the future population more closely. First, you’ll make a visual comparison using a scatter plot. 1

Add the following actual U.S. population data (in millions) to your scatter plot, using a third symbol or color: 1930: 122.8; 1950: 152.3; 1970: 204.9.

2

Connect the actual data points with a broken line, and sketch in an extension of that graph until 2030.

3

Which model’s graph appears to be closer to your extended actual graph?

How well do the models fit the data?

You can also make a numerical comparison of the models to the actual data. 4

Calculate the discrepancy between the actual data and model. Do this by finding how different each model is from the actual data for 1930, 1950, and 1970.

5

Find the total amount of discrepancy for each model.

6

Which model’s total discrepancy is less?

Make a Forecast Which model provides a better forecast of the U.S. population for the year 2030? No one really knows what the future holds, but mathematical modeling helps us find some reasonable population forecasts. One indication of how well a model forecasts the future is how well it fits the past. 1

Which model do you think fits the actual data for 1930, 1950, and 1970 better? How did you decide?

2

Which model do you think gives a better population forecast for the year 2030? Why?

3

What is your own prediction for the U.S. population in 2030? Explain your reasons for choosing that number.

hot words

predict

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PHASE THREE

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Now you have an opportunity to build mathematical models of world trends, to make predictions and interpret them. You will also reason about how forecasts that are high or low can have an effect in the real world. You will think about who can make use of your predictions and write a letter to them to explain your findings. You will make some comparisons of rates of growth and amounts of growth. Your final report will sum up what you have learned about mathematical modeling.

Modeling with Growth Sequences WHAT’S THE MATH? Investigations in this section focus on: ALGEBRA ■

Building growth sequences and describing them with graphs and algebraic language

NUMBER and OPERATIONS ■

Comparing rates and amounts of growth

MATH REASONING ■

Comparing models and reasoning about how to use them for forecasting

MATH COMMUNIC ATION ■

Writing about implications of forecasts


WHAT COMES NEXT?

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9 INTERPRETING AND USING MATHEMATICAL MODELS

10:57 AM


Making Plans Building a mathematical model can help plan for the future. Suppose you have been manufacturing and selling Squiggles toys for five months. Sales have grown each month. Now you must think about how many toys you will be selling by the end of a year.

Build a Model How many toys are you likely to sell in future months?

You have learned to project actual data into the future. Here you will use sales figures from five months to predict sales for months 6 and 12. 1

Create addition and multiplication models based on months 1 and 5 for the number of Squiggles toys sold.

2

Use the models to make two forecasts of the sales numbers in Month 6 and Month 12.

3

Draw a graph and write an algebraic description for each model.

4

Use visual and numerical comparisons to decide which model fits the actual data better. Number of Squiggles Sold (per month) Month

158


Sales

1

100

2

300

3

800

4

1,400

5

2,000

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Make a Prediction Consider your models and how well they fit the actual data. In what month do you predict that the company will need to produce 10,000 toys? How did you find your answer?

Can you use the model to predict a target sales figure?

Use the Prediction to Make Decisions Your factory manager says that in order to produce more than 10,000 Squiggles a month, you will need to hire more employees, buy more equipment, and find a larger building. You can use the two mathematical models to estimate how soon you will need to produce more than 10,000 Squiggles a month, but you should consider other factors that might affect future sales. ■

Do you think either one of your models is close enough to use for planning?

■

What other information might be helpful to you in making a reasonable prediction?

■

What advice will you give your factory manager about hiring employees, buying equipment, and locating more space?

■

What might happen to your company if the actual sales in Month 12 turn out to be much greater than your prediction? much less than your prediction?

hot words

growth model project (verb)

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10 MODELING WORLD POPULATION GROWTH

10:58 AM


Looking Ahead You began this unit by guessing about population growth. You have learned how to use mathematical modeling to make forecasts instead of guessing, and thought about the impact the forecasts might have on planning. Now you will begin to prepare a report that analyzes the population growth of a particular country.

Forecast Population How will your country’s population grow over the next several decades?

160

Your group will analyze one region of the world using information from Regional Data Sheets that your teacher will distribute. Each person in the group will work independently on one country in that region. ■

Select one country from the regional list for your report.

■

Create an addition and a multiplication model of population growth based on 1950 and 1990. Make charts, draw scatter plots, and write algebraic descriptions of your models.

■

Decide which model fits the data better by using graphical and numerical tests.

■

Make a population forecast for your country in the year 2040. Explain how you used your models to make the forecast.


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Check for Accuracy Work with a partner to check each other’s work. Write comments and questions about your partner’s work, and then discuss them together. Here are some questions that you should have in mind. ■

Are the calculations of the growth numbers and forecasted values accurate?

■

Are the scales on the graph appropriate?

■

Is the graph accurate and easy to read?

■

Are the algebraic descriptions of the models correct?

■

Is it clear why one of the models was judged to be a better fit than the other?

■

Do you understand why the final forecast was selected?

Are the population forecasts clear and reasonable?

Collect Regional Forecasts After you have reviewed and revised your forecast as needed, meet with your group to copy the population forecasts for the year 2040 for all the countries your group studied. Each of you should keep a copy of the forecasts in order to make some comparisons of the data in the next lesson.

hot words

forecast

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11 COMPARING MODELS IN DIFFERENT WAYS


Rates and Relationships Now that you have looked at the population growth in your country, you can compare it to other countries in your region and in the world. Using mathematics to calculate rates and proportions can produce revealing comparisons of data.

Compare Growth Two Ways What is the difference between growing the most and growing the fastest?

There are two ways to compare the growth of countries. By looking at the addition growth numbers, we can say which country grew the most. Comparing the multiplication growth numbers tells us which country grew the fastest. The growth rate of a country commonly refers to the multiplication growth number. 1

Which of the three countries in the table grew the most? Explain your reasoning.

2

Which of the three countries in the table grew the fastest? Explain your reasoning. Population to the Nearest Million Year

162

1950

1990

Argentina

17

32

Kenya

6

16

U.S.A.

152

250


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Make More Comparisons Mathematics can be used to compare population, population growth, and population density. Work with your group to prepare the following information about your country and region. Show the calculations you performed. This will become part of your country report. ■

Rank the population growth of countries in your region from fastest to slowest and from most to least according to the population figures for 1950 and 1990.

■

Which of the countries in your region have grown at a rate faster than the world’s growth rate from 1950 to 1990?

■

Express the population of your country as a percent of the population of its region and of the population of the world for the years 1950 and 1990.

■

Express the area of your country as a percent of the area of its region and of the area of the world.

■

Express the population density of your country in people per square mile for the years 1950, 1990, and as predicted in 2040.

What do ranking and percentages show about the data?

Draw Conclusions from the Comparisons What did you learn through making these comparisons? Give some of your conclusions in a paragraph at the end of your paper. Include this material in your report.

hot words

rate ratio

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12 INTERPRETING THE PREDICTIONS

10:58 AM


Focus on the Future You have built mathematical models, forecast your country’s population, and compared your country’s growth to the rest of your region and the world. Now you are ready to describe and interpret your results to those who could use this information in planning for the future.

Write a News Release or Letter How can you communicate the significance of your forecast to a target audience?

Explain how your forecast can be of value to a particular individual, a business, a community, a governmental agency, a neighboring country, or the United Nations. One way to do this would be to write a letter summarizing the details of your forecast and explaining what impact the forecast might have. If your forecast is of general interest, you could write your summary and predictions in a short article suitable for a newspaper or magazine. When it is finished, you are ready to organize all of the work you have done into your Country Report. Put your analysis at the beginning of your report to introduce and summarize the information. Include all of your work on making mathematical models and how you used them to make forecasts. In the next step you will evaluate your report, fill in the Cover Sheet, and fasten all the sheets together.

164


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Evaluate Your Report Fill out the Cover Sheet for your Country Report, using the checklist to be sure you have included all the materials. Earlier, your class reviewed a list of criteria for evaluating your work on building mathematical models. Using the list as a guide, write a review of your work including the ideas below and add it to your final report. ■

Does your report include the items on the Criteria List?

■

How well does the work on your report reflect what you have learned in the unit?

■

Are there some things you learned in this unit that your Country Report doesn’t show? If so, what are they?

■

What do you think are the most important things to understand about using mathematical models to make forecasts?

How well does your report meet the criteria?

Your completed, bound report will show what you have learned about mathematical modeling.

hot words

forecast

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1


Predicting World Population

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HW

Applying Skills

Writing

Find the next two terms in each sequence.

9. For each of items a–d below, explain

5. 1, 1.5, 2.25, 3.375, …

why it may be important to make a forecast. Who might be interested in a forecast for the item? How might they use the forecast?

6. 1, 2.1, 3.2, 4.3, …

a. number of 90-year-olds in the

1. 2, 5, 8, 11, …

2. 2, 4, 8, 16, …

3. 5, 11, 17, 23, …

4. 4, 12, 36, 108, …

7. Tell what pattern you found in each

sequence in items 1–6.

United States b. temperature in Phoenix, Arizona c. number of cars sold in a year in the

United States Extending Concepts

d. number of animal species in the world

8. Use the data in the chart to estimate the

population of each country in the year 2030. Use any method you choose. Population (in millions) Year

Yugoslavia

Japan

Kenya

1950 1960 1970 1980 1990

16.35 18.40 20.37 22.30 23.81

82.90 93.22 104.34 116.81 123.54

6.02 8.12 11.23 16.67 24.03

Explain how you found a pattern in each data set and how you extended the pattern. Describe any differences that you notice between the pattern for Yugoslavia and the pattern for Kenya.

166

WHAT COMES NEXT? • HOMEWORK 1

10. Think of three new examples of real-

world items for which forecasting might be important. Describe how each item changes with time. Explain why it may be important to forecast each item.

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2


Target Practice

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HW

Applying Skills

Extending Concepts

Use a calculator to solve each problem. 1. Use a guess-and-check method to find

each growth number. Keep track of your guesses for each growth number and the goals they reach. a. a number that can be added to 5 four

50

b. a number by which you can multiply

5 four times to reach a final goal of 50. 5

Cal begins with 10 and adds 1,000 repeatedly. Tor begins with 10 and multiplies by 2 repeatedly. a. Who do you think will be the first to

pass 5,000? 20,000? b. Make a chart like this showing the

times to reach a final goal of 50. 5

7. Cal and Tor perform an experiment.

50

2. Use the methods you applied in item 1

to find the addition and multiplication growth numbers for each of the following. a. Start 8; Target 23

results of the first twelve steps. Cal: Add 1,000

Tor: Multiply by 2

1,010 2,010 3,010

20 40 80

c. Who passed 5,000 first? 20,000? Who

will pass 100,000 first? d. How does addition growth compare

to multiplication growth?

b. Start 1.6; Target 6.8 c. Start 9; Target 60 d. Start 2; Target 4 3. What do you get if you start with 12 and

add 23 eleven times? 4. What do you get if you start with 8 and

multiply by 2 seven times? 5. If you start with 14 and repeatedly add

7.5, how many steps will it take to reach (or pass) 100? 6. If you start with 14 and repeatedly

Making Connections

8. If you invest $1,000 at 15% simple

interest, $150 (15% of $1,000) is added to the amount at the end of each year. If you invest $1,000 at 9% compound interest (compounded annually), the amount is multiplied by 1.09 at the end of each year. Calculate how much you would have under each plan after 20 years. Which type of interest would you prefer? Why?

multiply by 1.7, how many steps will it take to reach (or pass) 1,000?


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Growing Spirals

Applying Skills

Use a calculator, Centimeter Dot Paper, and a ruler to complete steps A–D for –2. items 1– A. Write a sequence using 1.2 as the starting number and growth number. B. Use your sequence to construct a right-angled spiral. C. Mark an X at the midpoint of each of the segments. D. Sketch the curved spiral by joining the midpoints. 1. Follow steps A–D for an addition

sequence of ten numbers. 2. Follow steps A–D for a multiplication

sequence of ten numbers. 3. How do your spirals in 1 and 2 differ? 4. How many segments of each of the

following spirals would fit on a square measuring 100 cm by 100 cm? a. an addition spiral with a starting

number and growth number of 2 b. a multiplication spiral with a starting

number and growth number of 2

Extending Concepts

5. For spirals a–c, what is the length of the

tenth segment? a. a multiplication spiral with starting

number 2 and growth number 0.5

168


b. a multiplication spiral with starting

number 2; the length of fifth segment is 78.125. What growth number did you use? c. an addition spiral with starting

number 2; the length of fifth segment is 4. What growth number did you use? 6. What does a multiplication spiral look

like if its growth number is between 0 and 1, as in item 5a?

Making Connections

7. The first two terms in the Fibonacci

sequence are 1 and 1. Beginning with the third term, each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13,…. Make a right-angled spiral using this sequence. Does the spiral look more like an addition or a multiplication spiral? 8. Many civilizations through the ages have

seen the spiral as a symbol of life and have used spiral patterns. The spiral below is a two-coil Celtic pattern. Are the spirals addition or multiplication spirals? Why? Describe the visual differences between addition and multiplication spirals.

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12:44 PM


Make a Model

Applying Skills

Find the missing numbers in each growth sequence. 1. addition sequence; growth number is 6: __, __, __, 19, 25, 31, __, __, __ 2. addition sequence; growth number is 2.5: __, __, __, 10.1, 12.6, 15.1, __, __, __ 3. multiplication sequence; growth number is 3: __, __, __, 54, 162, 486, __, __, __ 4. multiplication sequence; growth number is 1.5: __, __, __, 108, 162, 243, __, __, __

Use a guess-and-check method to find the growth number for each sequence. Then find the missing numbers. 5. addition sequence: __, __, 3, __, __, __, __, 4.8, __, __, __ 6. multiplication sequence: __, __, 25, __, __, __, __, 6,553, __, __, __ 7. addition sequence: __, __, 46, __, __, __, __, 67, __, __, __ 8. multiplication sequence: __, __, 7, __, __, __, __, 364, __, __, __

Extending Concepts

9. The population of Kenya in 1950 and 1990 is recorded in the chart. Year

1950

1960

1970

1980

1990

Addition Model

6.02

24.03


6.02

24.03

2000

2010

2020

2030

a. Find a number that can be added to 6.02 four times to reach 24.03. Use the number to

build an addition growth model to predict Kenya’s population every decade from 1960 to 2030. Record your results in the chart. b. Find a number by which you can multiply 6.02 four times to reach 24.03. Use the

number to build a multiplication growth model to predict Kenya’s population every decade from 1960 to 2030. Record your results in the chart. 10. Use each model in item 9 to estimate when Kenya’s population was 1 million. Explain how

you figured it out.


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Growing in One Step

Applying Skills

The first two terms of a growth sequence are given for items 1– 4. Find the growth number and the tenth term of the sequence.

Extending Concepts

13. Write an algebraic description for this

addition sequence.

1. addition sequence: 36, 42 2. multiplication sequence: 64, 72

1

2

3

Number

5

11

17

14. Write an algebraic description for this

multiplication sequence.

3. addition sequence: 28.1, 25.6 4. multiplication sequence: 100, 90

For items 5–7, write the next two terms of each sequence.

Term

1

2

3

Number

3

12

48

Median Age at First Marriage

15.

5. 31, 27, 23, … 6. 100, 80, 64, … 7. 9.3, 8.2, 7.1, … 8. Suppose you have an addition sequence

with a first term of 6 and a growth number of 7. a. How many growth numbers would

you add to 6 to get the tenth term? 20th term? 50th term? 200th term? b. Use your answers to a to find the

values of the 10th, 20th, 50th, and 200th terms. Find terms 5, 6, and 7 of the sequences in items 9–12. 9. nth term is 1 2(n 1) 10. nth term is 5n 4 (n 1)

11. nth term is 7 2

(n 1)

12. nth term is 5 3

170

Term


Year

1980

1990

2000

Women

22.0

23.9

?

Men

24.7

26.1

?

etc.

The table shows the median age at first marriage for women and men. Make forecasts in ten-year intervals. Make two predictions of when the marrying ages for women and men will be the same. Use an addition sequence and a multiplication sequence to model the marrying ages. Show your work. How did you find the growth numbers and extend the sequences? Making Connections

16. The area of the world’s tropical forest was

estimated to be 1,884 million hectares in 1980 and 1,715 million hectares in 1990. Predict this area for the year 2020 using an addition model and a multiplication model.

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6


Growing in Two Steps

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HW

Applying Skills

Extending Concepts

Find the one-step and two-step addition growth numbers. 1.

2 30 2 … 30

2.

3 51 3 … 51

3

48

6

4.

3

multiplication growth number is 8. 13. Find the three-step growth number

between the numbers 5 and 625 for both addition and multiplication sequences. Explain how you found them. 14. In a growth sequence, Q follows P. One

Find the one-step and two-step multiplication growth numbers. 3.

12. Find a pair of numbers whose two-step

other sequence number is between them. Which expression below represents the two-step growth number for an addition sequence? for a multiplication sequence? Explain.

54

…

48

6

…

54

1 Q 2 P Q–P 2

5. What is the two-step addition growth

number if the one-step addition growth number is 84? 109? 26.8? 35.1?

(Q P)2

Q – P

QP 2

6. What is the two-step multiplication

growth number if the one-step multiplication growth number is 121? 49? 289? 729? In items 7–11, find the missing numbers in two ways for each sequence. First, use an addition sequence, and then use a multiplication sequence. 7. 5, _, 125, _, _

8. 4, _, 324, _, _

11.

Making Connections

15.

Women in State Legislatures (out of 7,500 seats)

1971

1981

1991

2001

2011

362

?

1,369

?

?

etc.

a. Use an addition sequence to complete

the table and forecast the decade when women will hold about half the seats.

9. 10, _, 1,000, _, _ 10.

Q P

1985

1990

1995

2000

80.6

?

100.8

?

1985

1990

1995

2000

120.0

?

216.9

?

b. Next use a multiplication sequence to

model the trend and make a different forecast for the decade when women will hold about half the seats. c. In 1981, women held 904 seats. Which

model is closer to the actual data?


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What Do You See? 7.

8. Number

Applying Skills

Follow these steps for items 1–4. A. For each sequence, copy and complete a

Number

166-177_6037a_Unit03

Term

Term

Term/Number table like the one shown. Number

Term

Extending Concepts

1st 2nd

Tell whether each statement is true or false. Explain your thinking.

3rd 4th

9. If the graph of a decreasing growth

5th

sequence hits the horizontal axis, it must represent an addition sequence.

B. Draw axes for a graph. Label the

horizontal axis Term and the vertical axis Number. Choose appropriate scales. C. Plot the data from your table.

10. On the graph of a multiplication

sequence, the vertical change always gets bigger as the values increase. 11. If the graph of a sequence is a straight

D. Connect the points with a dotted line. 1. an addition sequence with third term 8

and growth number 3 2. an addition sequence with third term 12

and growth number –2

3. a multiplication sequence with third

line sloping down from left to right, it represents an addition sequence with a growth number between 0 and 1. 12. If the growth number of a multiplication

sequence is 1, its graph is a horizontal line.

term 4 and growth number 2 4. a multiplication sequence with third

Making Connections

term 4 and growth number 0.4 Tell whether each graph in items 5–8 represents an addition or a multiplication growth sequence. What can you say about the growth number? Number

6. Number

5.

Term

172

Term


13. Radioactive materials such as strontium

decay very slowly. If 48 g of strontium are present to start with, the amount remaining after 29, 58, 87, 116, and 145 years, respectively, will be 24 g, 12 g, 6 g, 3 g, and 1.5 g. What does the graph of this sequence look like? Is it an addition or a multiplication sequence? How can you tell?

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8


Which Model Fits the Data Better?

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HW

Applying Skills

Extending Concepts

–4, find term 3. Then For each sequence 1– find terms 2 and 4. 1. 2, _, _, _, 50; addition sequence 2. 28, _, _, _, 2; addition sequence 3. 8, _, _, _, 312.5; multiplication sequence 4. 512, _, _, _, 2; multiplication sequence 5. Table A shows an addition model and

a multiplication model created from the first and last terms of a data set. Compare the models numerically. Table A 1 2

Terms

3

4

5

Actual Data

32 50

Addition Model Multiplication Model

32 64.5 97 129.5 162 32 48 72 108 162

Addition Model Value 64.5 97 129.50

Table B Actual Data

86 112

162

numbers a and b is half their sum, or a 2 b. For each sequence shown below, calculate the second term as the average of the first and third terms. What type of growth sequence have you created in each case? What is the growth number? a. 20, ____, 30

b. 18, _____, 26

7. The geometric mean of two numbers a

and b is the square root of their product, or ab. Calculate the second term in each sequence below as the geometric mean of the first and third terms. What type of growth sequence did you create in each case? What is the growth number? a. 20, _____, 180

b. 16, _____, 256

Making Connections

How Far Off?

50 86 112 Total discrepancy

a. Copy and use Table B to calculate the

discrepancy between the addition model and the actual data. b. Make another table to calculate the

discrepancy between the results from the multiplication model and the actual data. c. Which model seems to fit the data

better? Why?

6. The average or arithmetic mean of two

8. In the multiplication sequence 3, 6, 12,

the middle term, 6, is the geometric mean of 3 and 12. This sequence can be used to draw a special geometric figure. On a piece of grid paper, draw segment AB 15 units long. Mark point D on line AB so that it is 3 units from point A, making DB 12 units long. Draw segment DC perpendicular to AB and 6 units long. Now draw segments AC and BC. What is the shape ABC? What is special about angle ACB? Would you have the same result using a different multiplication sequence?


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Making Plans

Applying Skills

Extending Concepts

This table shows sales of a new computer in the first few months of manufacture. Sales of New Computer

5. Which of the sales models in items 1 and

2 do you think gives better predictions of future sales? Why? 6. Tell when each model predicts that sales

Month

Actual Sales

Addition Model


1

200

200

200

2

600

3

1,300

7. List some reasons why the company

4

2,100

might need a forecast of future sales.

5

3,000

3,000

3,000

6 . . .

will reach 15,000. When do you think sales will reach 15,000? How did you decide?

8. Think about your reasons in item 7.

How will it affect the company if the forecast for sales in month 12 turns out to be too high? too low?

1. Create an addition model for sales. 2. Create a multiplication model for sales. 3. Graph the actual data and the two

models on the same grid. Which model seems to fit the data better visually? 4. Make charts to find how far off each

model is from the actual data. Which model has the smaller total discrepancy?

Writing

9. Suppose that, based on population data

from 1950 and 1990, a forecast has been made of the number of 65-year-olds in the United States in 2050. a. Why might it be important to have

such a forecast? Who might use it? b. Do you think that a pattern of growth

observed in 1950–1990 is likely to continue until the year 2050? What future factors could affect the trend? Do you think the forecast is likely to be too low or too high? Why? c. What impact could it have if the

forecast is too high? too low?

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Looking Ahead will rise more quickly to start with? later on?

Applying Skills

–2, give the first six terms. Use For items 1– a shortcut to find the 100th term. Write an expression for the nth term.

b. For each sequence, make a graph

showing the first eight terms. Use the same grid for both graphs. Where do the graphs cross? What is special about the point where the graphs cross?

1. addition sequence: 1st term 8, growth

number 7 2. multiplication sequence: 1st term 5,

growth number 1.1 –6, write terms 1–8 and an For items 3– expression for the nth term. 3. 4, 7, 10, 13, …

4. 3, 6, 12, 24, …

5. addition sequence: __, __, 9, 13, … 6. multiplication sequence: __, __, 18, __,

162, … Extending Concepts

Describe in words and symbols how to build sequences in items 7–12. Example: For 5, 15, 45, …, starting number 5; next number current number 3. 7. 2, 5, 7, 10, … 9. 90, 82, 74, 66, …

8. 5, 20, 80, 320, … 10. 64, 16, 4, 1, …

11. 4, __, 10.24, … (multiply) 12. 3, __, 21, … (add) 13. An addition sequence has a first term

of 2 and a growth number of 21. A multiplication sequence has a first term of 2 and a growth number of 2. a. Predict what the graphs of the two

sequences will look like. Which one

Making Connections

14. In 1900, the winning time for the Boston

Marathon was about 160 minutes. In 1980, it was about 132 minutes. Year

1900 1920 1940 1960 1980

Addition Model

160

132


160

132

a. Use an addition model and a

multiplication model to find intermediate values for 1920, 1940, and 1960. Why can’t you check which model fits better? b. Write an algebraic description of each

of your models. c. What does each model predict for the

winning time in 2040? d. If the models are extended farther

into the future, will either model eventually predict a winning time of 0 minutes? Explain your thinking. Is it reasonable to use the models to make predictions for the twenty-third century? WHAT COMES NEXT? • HOMEWORK 10

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Applying Skills

Extending Concepts

What percentage is:

12. a. For each country, use a multiplication

1. 5 of 8?

2. 28 of 250?

3. 29,000 of 40,000?

4. 0.2 of 25?

model to estimate the population in 1970. Use the model to predict the population in 2010.

Use the table below to answer items 5–12.

Country

b. Using the models in part a, predict the

population density, based on the multiplication model, of each country in 2010. Which country do you think will have the highest population density in 2010?

1950 Area 1990 Population Population (thousands (millions) (millions) of km2 )

Peru

8

22

1,285

Mexico

26

86

1,958

Indonesia

76

179

1,905

5. By how much did each population grow

from 1950 to 1990? 6. Rank the countries from most to least

population growth. 7. What was the growth rate for each

population from 1950 to 1990? 8. Rank the countries from fastest to

slowest population growth. 9. Find each country’s 1990 population

Making Connections

13. Starlings were introduced into North

America in 1890, and their numbers increased very rapidly. Use the table to answer the questions. Approximate North America Populations (millions) Year

1890

1950

Starlings

0.00008

50

Humans

80

193

2

density in people per km . 10. Rank the countries from highest to

lowest 1990 population density. 11. Express the 1990 population of each

country as a percent of the world population. The world population in 1990 was about 5,292 million.

a. Which population, the human or the

starling population, increased by more from 1890 to 1950? Which increased faster? Explain your thinking. b. Use an addition model to find an

intermediate value for 1920 for the starling population. Use the model to predict the population in 2040. c. Using your result from part b, predict

176


the population density, based on the addition model, of starlings in 2040. The area of North America is about 21,744,000 km2. How did you figure it out?

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Focus on the Future

Applying Skills

Tell whether each statement is true or false. If it is false, change the underlined words so that it is true. 1. In an addition sequence, the difference between any pair of consecutive numbers is the

same. 2. In a multiplication sequence, the ratio of pairs of consecutive numbers increases steadily. 3. In a decreasing multiplication sequence, the growth number is negative. 4. The two-step multiplication growth number for a pair of numbers can be found by

subtracting and taking the square root. 5. In a decreasing addition sequence, the growth number lies between 0 and 1.

Extending Concepts

6. Would an addition or a multiplication sequence be a good model for the data shown?

Explain why or why not. Time Temperature in BoomTown (°F)

7 A.M.

1 P.M.

7 P.M.

1 A.M.

7 A.M.

1 P.M.

7 P.M.

1 A.M.

58

67

60

52

59

70

64

50

7. Estimated World Population (millions) Year

1500

1650

1800

1950

Population

460

500

954

2,516

a. Find the difference between each pair of consecutive numbers in the table. b. Find the ratio of each pair of consecutive numbers. c. Do you think that either an addition or a multiplication sequence would be a good

model for the data? Explain your thinking. d. Use the 1500 and 1800 data to make two estimates of the 1950 world population. Does

either an addition or a multiplication model give a good prediction? e. Do you think that in the real world there are many data sets that follow neither an

addition nor a multiplication model? Explain your thinking.


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How can algebra help you analyze and solve problems?

EXPLORINGT

HE

U NKNOWN

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PHASEONE

PHASETWO

PHASETHREE

Working with Expressions This phase introduces you to Lab Gear®, a tool for working with and understanding algebra. Lab Gear will help you simplify and evaluate algebraic expressions. You will also investigate ways to multiply expressions. At the end of the phase, you will use everything you have learned to analyze some number tricks.

Minuses and Parentheses In algebra, there are three different uses of the minus sign. In this phase, you will explore these different uses and see how they are related. You will also work with expressions that involve parentheses. All of these ideas will come together at the conclusion of the phase when you describe a geometric pattern.

Solving Equations In Phase Three you will use Lab Gear to model equations. Experimenting with Lab Gear will lead you to develop some algebra rules for solving equations. A final problem involving two different CD offers will give you a chance to apply everything you have learned.

PHASE ONE

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In this phase you will learn to use Lab Gear, a tool for understanding and modeling algebra. By using Lab Gear, you will be able to simplify, evaluate, and multiply algebraic expressions. Algebraic expressions are useful in describing everything from geometric patterns to scientific formulas. What are some ways in which you have used expressions in your previous mathematical work?

Working with Expressions WHAT’S THE MATH? Investigations in this section focus on: ALGEBRA ■

Understanding variables and using them to write expressions

■

Evaluating expressions

■

Simplifying expressions by combining like terms

■

Multiplying two expressions

PATTERN SEEKING ■

Writing expressions to describe number patterns


EXPLORING THE UNKNOWN

181


1

REPRESENTING EXPRESSIONS WITH LAB GEAR

Gearing Up In the same way that a protractor is a tool for working with and understanding angles, Lab Gear is a tool for working with and understanding algebra. You will experiment with Lab Gear blocks to see how they are named. Then you will use them to represent algebraic expressions.

Sort and Name Lab Gear Blocks What are the names for the various Lab Gear blocks?

Your teacher will provide you with a set of Lab Gear blocks. 1

Sort the blocks in a way that makes sense to you.

2

The names of the blocks are based on what they represent. a. What do you think yellow blocks represent? b. What do you think blue blocks represent? c. What name could you give to each different type of block? The names of two blocks that represent variables are shown here to get you started. x

Measuring Lab Gear Blocks You can use the corner piece to measure the length and width of Lab Gear blocks.

y

Be ready to discuss your ideas with the class.

Top View y

Corner piece

x

Lab Gear block

182

EXPLORING THE UNKNOWN • LESSON 1

Length is x, width is y, so this block represents xy.


Name Collections of Blocks For this investigation, you will need a partner and a bag of Lab Gear. 1

Without looking, reach into the bag and pull out a collection of 6 blocks. Sketch a top view of the collection. Write an expression for this collection and then combine like terms to write the expression the short way. Repeat the process four more times, using collections of 7 blocks, 8 blocks, 9 blocks, and 10 blocks.

2

Write each of your expressions on a clean sheet of paper and trade this paper with another pair of students.

3

Put out blocks to match each of the expressions you are given. Then sketch the blocks next to the corresponding expression.

4

Trade back papers. Check to see if the other pair of students correctly sketched blocks to match each expression.

How can writing expressions for collections of blocks help you combine like terms?

How to Write an Expression and Combine Like Terms

The value of this collection is written x2 5 x x2 x x2 OR

We can combine like terms: 3x2 2x 5

hot words

expression like terms

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2 EVALUATING EXPRESSIONS

What’s in the Bag? Blue Lab Gear blocks represent variables. What happens if you know the value of these variables? You will use Lab Gear to explore this question.Then you will be ready to use what you know about evaluating expressions, as well as some logical reasoning, to figure out which Lab Gear blocks are hidden in a bag.

Evaluate Expressions Using Lab Gear How can you evaluate expressions when you know the value of variables?

Write an expression for each collection of blocks. Then use Lab Gear to help evaluate each expression for the given values of the variables. Keep a written record of your work. 1

1 2

a. x 1

b. x 4

c. x

a. x 0

b. x 6

c. x 1

a. y 1

b. y 3

c. y 5

a. x 1, y 2

b. x 3, y 1

1 c. x , y 3 4

2

3

4

Were you always able to replace the variable blocks with constant blocks? If not, how did you evaluate the expressions? Write a short summary of your process.

184



Play “What’s in the Bag?” Follow the directions below to play several rounds of “What’s in the Bag?” Different pairs of students in your group should take turns being the bagkeepers. The other students try to guess which blocks are in the bag. As you play, keep track of any strategies that you use to guess what is in the bag.

What strategies can you use to guess which blocks are in the bag?

Rules for “What’s in the Bag?” ■

Two players, the bagkeepers, secretly place no more than four Lab Gear blocks in a bag.

■

Other players take turns naming values for the variables x and/or y.

■

The bagkeepers carefully evaluate the contents of the bag using these values.Then they tell the final result to the other players.

■

The players continue to name values for the variables until they are able to guess correctly which blocks are in the bag.

Write About Strategies Write a paragraph describing any strategies you used to play “What’s in the Bag?” Include a discussion of how you and your partners figured out the contents of the bag in one round.

hot words

expression

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3 MULTIPLYING EXPRESSIONS

Cornered! You know how to multiply numbers, but can you multiply algebraic expressions? As you will see, the corner piece is very helpful when you need to multiply two expressions. In fact, knowing how to multiply with the corner piece will help you identify some common errors that algebra students make.

Multiply Two Expressions How can you use the corner piece to multiply two algebraic expressions?

186

Work with a partner for the following. 1

What multiplication problem is shown here? Write this algebraically.

2

Use Lab Gear blocks to make a rectangle that fits inside the corner piece. The length and width of your rectangle should exactly match the length and width shown by the blocks on the outside of the corner piece. Sketch your rectangle.

3

What collection of blocks did you use to make your rectangle? What expression do they represent?

4

Write an equation that relates the original multiplication problem and the final product.



Identify Common Errors These equations show some multiplication problems that were done by a student. There are at least one or two common errors here. Can you find them?

How can Lab Gear help you identify some common errors in multiplication?

Use Lab Gear to do each multiplication problem. Decide whether each equation is true or false. If an equation is false, rewrite the equation correctly. Keep a written record of your work. Include a sketch of the Lab Gear blocks for each problem.

hot words

multiplication rectangle

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4 ANALYZING NUMBER TRICKS

Lab Problem No. 1 This is your chance to be a number magician. You will start by exploring some number tricks.Then you will see how to use what you know about Lab Gear and algebra to help explain why the tricks work !

Explore a Number Trick What do you notice about the results of a number trick?

Choose one of the number tricks below. Experiment with different sets of numbers until you see the trick. Keep a written record of your work. Be ready to describe your trick to a partner.

Number Tricks 1. Inner and Outer Products Inner product is 6 • 7

Choose any four consecutive whole numbers. Find the product of the inner numbers and the product of the outer numbers. What do you notice? Does this always work? 2. Calendar Squares Choose any four numbers that form a square on a calendar. Find the product of the two diagonals. What do you notice? Does this always work?

5

6 7

8

Outer product is 5 • 8

Sun

Mon

Tue

Wed

Thu

Fri

Sat

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

The diagonal products are 3 • 11 and 10 • 4

188



Investigate Why It Works Use algebra to show that your number trick always works. 1

Think about the four numbers in your number trick. Suppose the smallest number is represented by x. What are the values of the other numbers?

2

How could you write the products in your number trick using algebra?

3

Use Lab Gear, or other methods, to help you do the multiplication.

4

How do your results show that the number trick always works?

How can you use algebra to show that your number trick always works?

Write About the Results Write a summary of your results. Include the following: ■

a description of the number trick

■

a summary of how you used algebra to show that the trick always works

■

a summary of how you used Lab Gear or other techniques along the way

hot words

product

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P H A S E TWO



Minus signs are used in several different ways in mathematics. In this phase, you will use Lab Gear to help explore these different meanings of minus signs. You will also see how expressions can be written with and without the use of parentheses. At the end of this phase, you will apply everything you have learned to describe a geometric pattern.

Minuses and Parentheses WHAT’S THE MATH? Investigations in this section focus on: NUMBER ■

Working with models for signed-number operations

ALGEBRA ■

Simplifying polynomials

■

Writing expressions with and without parentheses

■

Using the distributive property to simplify expressions

PATTERN SEEKING ■

Describing and analyzing a geometric pattern



191


5

WORKING WITH MINUSES AND NEGATIVES

Extending the Lab Gear Model Do you think you can use Lab Gear to show the expression 5 4? So far, you have only used Lab Gear to represent positive quantities. Now you will explore two ways to show minuses and negatives.

Use the Workmat to Add and Subtract How can you model addition and subtraction of integers?

Work with a partner for each of the following. Record all of your steps on the Workmat Recording Sheet. 1

Use Lab Gear to model each addition problem. a. –6 4 b. –8 10 c. –6 (–3)

2

Create and solve addition and subtraction problems of your own. Be ready to share them with the class.

3

Use Lab Gear to model each subtraction problem. a. –7 – (–3)

b. 5 – 9

c. –4 – (–9)

Adding and Subtracting with Lab Gear To add –5 2...

Put down the first number

Put down the second number

Cancel what you can

Count what remains

5 2 3

To subtract –8 – (–2)...

192


Put down the first number

Take off the second number

Count what remains

8

(2)

6


Show the Number 3 in Different Ways Use Lab Gear to show (or “model”) the number 3 in as many ways as possible. ■

Use the minus area in some of your models.

■

Use upstairs blocks in some of your models.

■

Keep a record of your work using the Workmat Recording Sheet. Also, write an expression that corresponds to each model.

How many different ways can you show a number using Lab Gear?

Showing Numbers in Different Ways Another way to “show minus” with Lab Gear is by putting blocks “upstairs” (on top of other blocks). You can use the upstairs blocks and the minus area to show the same number in a variety of ways. Here are three ways to show –4.

Negative 5 plus 1 5 1 4

1 minus 5 1 5 4

The opposite of 5 1 (5 1) 4

hot words

integers negative integers

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6 SIMPLIFYING POLYNOMIALS

Making Long Expressions Shorter How can you write a long expression in a simpler way? You have already seen how an expression like x x x x can be written as 4x. Now you will use Lab Gear to help simplify special types of expressions called polynomials.

Create and Simplify Polynomials What rules can you find for simplifying polynomials?


Choose as many blocks as you like of each type shown below. Place some blocks inside the minus area of your workmat and some outside the minus area. Do not stack the blocks.

1

5

x

5x

x2

2

Sketch your collection on the Workmat Recording Sheet. Then write a polynomial for the collection of blocks.

3

Simplify the polynomial by canceling and clustering matching blocks.

4

Sketch the simplified collection on the Workmat Recording Sheet. Then write an expression for the simplified polynomial.

Repeat at least five times. Compare your original polynomials to the simplified ones. Write down any algebra rules that you could use in the future.

194



Simplify More Complex Polynomials Do the following steps for each collection of blocks shown here. A.

C.

B.

1

Put out blocks to match each collection. Write the polynomial for the collection.

2

Simplify the collection of blocks. Sketch the result on a Workmat Recording Sheet.

3

Write the simplified polynomial.

What additional rules can you find for simplifying polynomials?

Write down any algebra rules that you could use in the future to simplify polynomials.

Polynomials

A product of numbers and variables is called a term. Examples: 3 2 3x 2xy y 17 5x 5 8 A polynomial is an expression made up of the sum or difference of terms. Examples: 1 3x 5 14x 2 5x 2xy y 2 6y 2

hot words

term expression

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7 WORKING WITH PARENTHESES

Grouping and Ungrouping Is 3(y 4) the same as 3y 4? Parentheses can be used to group numbers and expressions, but you have to be careful if you want to remove the parentheses. You will be exploring some ways to write expressions with and without parentheses.

Decide Whether Equations Are True or False How can you write an expression without parentheses?

The student who wrote these equations wasn’t sure whether the two sides of each equation were really equal.

1

For each equation, show both sides using Lab Gear.

2

Record your setup on the Workmat Recording Sheet.

3

Decide if the two sides of the equation are equal.

4

If an equation is false, rewrite the right side so that it is true.

When can you remove parentheses without changing the value of an expression? What rules can you state for writing expressions without parentheses?

196



Explore the Distributive Property Work with a partner to write each of the following expressions without using parentheses.

1

Use Lab Gear to show each expression. For some expressions, you may need to use the corner piece; for others, you may need to use the minus area of the workmat.

2

Use Lab Gear to help you find a way to write each expression without using parentheses.

How can you use the distributive property to write expressions without parentheses?

The expressions that you worked with all involve examples of the distributive property. How would you complete the following? “The distributive property says that a(b c) ____.”

Write About Parentheses Write a summary that you can refer to later if you need to remember how to work with parentheses. Include the following: ■

examples of expressions that are written with and without parentheses

■

a description of the distributive property

■

any other rules that are useful in working with parentheses

hot words

distributive property

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8 DESCRIBING A GEOMETRIC PATTERN

Lab Problem No. 2 What does algebra have to do with geometric patterns? Here is a chance to find out. First you will describe a pattern of blue and white tiles. Then you will use everything you have learned about writing and simplifying expressions to check your description.

Describe a Patio Pattern How can you use algebra to describe a geometric pattern?

Consider the pattern below. Suppose the length of the patio is x. Write an algebraic expression for each of the following: 1

the total number of tiles needed

2

the number of blue tiles needed

3

the number of white tiles needed

A Pattern of Tiles Small white tiles are placed together to make a large square patio. Then the white tiles are surrounded by a border of blue tiles.

Patio of length 1

198


Patio of length 2

Patio of length 3


Check the Expressions If the expressions you wrote to describe the patio pattern are correct, then when you take the expression for the total number of tiles and subtract the expression for the number of blue tiles, the result should equal the expression for the number of white tiles.

How can you check the expressions you wrote to describe the tile pattern?

As you work, be sure to keep a written record of your process. Use the Workmat Recording Sheet to record any Lab Gear work. 1

Write an expression for the total number of tiles minus the number of blue tiles.

2

Simplify the expression. a. You may want to start by using the corner piece to multiply some expressions. b. You may want to use the workmat to help represent and simplify the resulting expression.

3

What is the final resulting expression? Is it equal to your expression for the number of white tiles?

Write About the Process Write a summary of your work that includes the following: ■

a description of how you used algebra to show that your patio tile expressions were correct

■

a summary of how you simplified expressions and a description of how you used the distributive property

■

a summary of how you used Lab Gear or other techniques along the way

hot words

square expression

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199

PHASE THREE



At the beginning of this phase, you will explore ways to add and subtract polynomials. These ideas will be helpful in the rest of the phase when you develop rules for solving equations. The phase ends with an opportunity to compare two CD offers. This will give you a chance to use everything you have learned about expressions and solving equations.

Solving Equations WHAT’S THE MATH? Investigations in this section focus on: ALGEBRA ■

Adding and subtracting polynomials

■

Solving linear equations

■

Writing expressions and equations to help solve problems

PATTERN SEEKING ■

Using tables and patterns to help analyze a real-world problem



201


9 ADDING AND SUBTRACTING POLYNOMIALS

Polynomial Arithmetic You have already seen how to use Lab Gear to add and subtract integers and simple expressions. Now you will use it to help add and subtract polynomials.

Model a Subtraction Problem How is subtracting a polynomial related to addition?

Use Lab Gear blocks to model the following subtraction problem: (x2 4x 4) (5x 3). Try to find two different ways to show this with Lab Gear. When you have found the result, complete this sentence: “To subtract (5x 3), we ended up adding ____.”

Adding and Subtracting Polynomials To add (2x2 3x 5) (5x 4)... Put down the first polynomial

Put down the second polynomial

Cancel what you can and count what remains

The result is 2x2 2x 1

202


To subtract (3x 1) 5x... Put down the first polynomial

Add zero if necessary

Take off the second polynomial and count what remains

The result is 2x 1


Add and Subtract Polynomials Use Lab Gear to help you do each of the following addition or subtraction problems. Keep a written record of your work. One of the six expressions cannot be simplified. Which expression is it and why?

How can you use polynomial addition and subtraction to simplify expressions?

Write Polynomial Arithmetic Rules Write a summary of any algebra rules you could use in the future to add and subtract polynomials. Include some specific examples in your summary.

hot words

expression subtraction

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10 SOLVING LINEAR EQUATIONS

Simplify and Solve Now you are ready to use everything you have learned to help you solve equations.You will start by setting up an equation with Lab Gear and simplifying both sides to find a solution. Then you will begin to develop your own rules for solving equations.

Set Up and Solve an Equation How can you model an equation to find a solution?

Work with a partner for this investigation. Record each step of your process on a Workmat/Equation Recording Sheet. 1

Use Lab Gear to show this equation on a workmat. 3x – 2 – (x – 2) 2x 6 – (2x 2)

204

2

Simplify each side of the equation. Write the resulting equation.

3

What value of x makes the two sides equal? How do you know?

4

Check that this value of x is a solution by substituting it into the original equation.



Solve Some Equations Work with classmates to solve each of the following equations. A.

B.

C.

D.

1

Write the starting equation on your Workmat/Equation Recording Sheet.

2

Use Lab Gear to solve the equation. Use the Workmat/ Equation Recording Sheet to keep track of some of the equations and Lab Gear positions along the way.

3

Check your solution in the original equation.

How can simplifying each side of an equation help you find a solution?

Did you and your classmates find any rules or shortcuts for working with Lab Gear? Be ready to discuss these with the class.

hot words

equation solution

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205


11 SOLVING LINEAR EQUATIONS

Staying Balanced How do Lab Gear rules correspond to algebra rules? You will see how thinking of an equation as a balanced scale can help you develop new Lab Gear methods. Then you will translate these methods into algebra rules and solve some equations without using Lab Gear.

Solve Equations with Lab Gear What rules can you use to keep an equation balanced and find its solution?

206

To solve the following equations, it may be helpful to think of each equation as a balanced scale. You can add or remove the same Lab Gear blocks from both sides to keep the equation balanced. A.

B.

C.

D.



1

Write the starting equation on your Workmat/Equation Recording Sheet.

2

Use Lab Gear to solve the equation. Use the Workmat/ Equation Recording Sheet to keep track of some of the equations and Lab Gear positions along the way.

3

Check your solution in the original equation.

Did you and your classmates find any rules or shortcuts for working with Lab Gear? Be ready to discuss these with the class.

Solve Equations Without Lab Gear Work with a partner to solve the following equations: 1

Solve each equation without using Lab Gear. Keep a careful record of each step of your process.

2

Check your solutions by substituting them into the original equations.

How can you solve equations using only paper and pencil?

What algebra rules did you use to help solve the equations? Be ready to share your methods with the class.

Write About the Process Write a summary of everything you know about solving equations. Include the following: ■

algebraic rules or methods you could use in the future to help solve equations

■

specific examples of equations and how you can solve them

■

a discussion of how you can check a solution

hot words

equation solution

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207


12 SOLVING A PROBLEM BY SOLVING AN EQUATION

Lab Problem No. 3 Have you ever seen an advertisement offering 15 CDs for one dollar? Sometimes these deals are not quite what they seem. You will have a chance to compare two different CD offers to find out which is the better deal. Your equation-solving skills will come in handy!

Write Expressions to Describe the Offers What expressions can you write to describe each CD offer?

208

Work with a partner to describe and compare the two magazine advertisements shown below. Let x be the total number of CDs you buy. 1

Write an expression that describes your total cost if you get the CDs from MusicMania.

2

Write an expression that describes your total cost if you get the CDs from DiscZone.



Find the “Break-Even” Point How many CDs would you have to buy to make the cost the same under both plans? This is the “break-even” point. You can find the break-even point by setting your two expressions equal to each other. 1

What equation do you get by setting the two expressions equal?

2

Solve the equation. Keep a step-by-step record of your work.

3

Check your solution.

How can solving an equation help you compare the two CD offers?

What does your result tell you about the two CD offers?

Compare the Two Offers Write a comparison of the two CD offers. Be sure to answer the following questions and include a discussion of how you arrived at your conclusions. ■

When is MusicMania a better deal? When is DiscZone a better deal?

■

How many CDs do you need to buy to make the cost the same under both plans?

■

Describe how you set up and solved an equation. Discuss the methods you used to solve the equation.

hot words

equation solution

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1

Gearing Up

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HW

Applying Skills

Extending Concepts

Write an expression for each collection of Lab Gear blocks. Then combine like terms to write the expression the short way.

Simplify each expression by combining like terms. Explain how you figured out each answer. 13. 7.3x x 2.4x 4.3x 1.1x 2

1.

14. 21 x xy 14 x 2

2.

2

2

2

15. Marta pulled out a collection of Lab

Simplify each algebraic expression by combining like terms. 3. x y x

What could the collection of blocks have been? List as many different possibilities as you can. Assume that the bag contained lots of blocks of each type.

4. y y x 3 5. y y y y 2

2

2

6. x y 1 x 2

2

Gear blocks from a bag. She wrote an expression for the collection. Then she combined like terms and wrote the expression the short way. The short expression was 7y 6.

2

7. x x x x 2

8. x y y x 2

2

2

Writing

2


9. 5 x y 6 10. 2y xy x 3y

Tell whether the two expressions in each pair are equivalent. 11. x x y x and 3x y 12. x y y and x 2y 2

210

2

2

2

EXPLORING THE UNKNOWN • HOMEWORK 1

Dear Dr. Math: I want to simplify the expression 3y2 xy y 4y 2 by combining like terms. How do I know which terms I can combine? Once I know which terms I can combine, how do I combine them? Can you explain the method so that I can figure it out myself in the future? Pete


2

What’s in the Bag?

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HW

Applying Skills

Extending Concepts

Evaluate each expression for the given values of the variables. 1. Evaluate 3x 1 for: a. x 1

b. x 3

c. x 5

2. Evaluate y 2 for: 2

a. y 0

b. y 1

c. y 4

3. Evaluate x 4x for: 2

a. x 1

b. x 2 b. y 4

c. x 6 c. y 12

5. Evaluate x x 3 for: 2

a. x 0

b. x 1

Lab Gear blocks below. Then evaluate the expression for the given values of the variables.

b. x 5

x0

1

x1

7

x2

13

x3

19

c. x 2

6. Write an expression for the collection of

a. x 2

puts Lab Gear blocks in a bag. Julia names values for the variable, and José evaluates the contents of the bag. Figure out which blocks are in the bag. (Hint: There are 3 blocks in the bag and only 1, 5, x, and 5x blocks are allowed.) Values Result

4. Evaluate 4y – 6 for: a. y 0

7. In a game of “What’s in the Bag?” José

c. x 0.2

Making Connections

8. The formula for the volume of a cylinder

is V πr2h, where r represents the radius and h the height. What is the volume of a cylinder if its radius is 3 cm and its height is 1.4 cm? How did you figure it out? How is this similar to evaluating expressions? h r


211


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HW

Cornered!

Applying Skills

Extending Concepts

1. a. What is the multiplication problem

shown below? Write this algebraically.

6. Suppose that you have the Lab Gear

blocks shown here.

a. Write an expression to represent the

collection of blocks. b. Write an expression for the collection

of blocks used to make the rectangle inside the corner piece. c. Write an equation that relates the

multiplication problem and the final product. For each multiplication problem, make a sketch of the Lab Gear setup. Then write an equation that shows the result of the multiplication. 2. x(x 3)

3. (x 2)

4. (x 1)(x 4)

5. (x 2)(x 5)

b. Find a way of arranging these blocks

to form a rectangle. Make a sketch of the arrangement. c. What multiplication problem does

this arrangement show? Write an equation that relates the multiplication problem and the expression in item 6a.

Making Connections

2

7. The sum of the first x even numbers is

equal to x(x 1).

a. How can you write this expression without using parentheses? b. What is the sum of the first 50 even

numbers? How did you find it?

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4

Lab Problem No. 1

ork w e om

HW

Applying Skills

Simplify each algebraic expression by combining like terms. 1. x y x x

2. y y y xy 2

2

Jody: “I called the numbers x, x 2, x 4, x 6. The two products are x(x 6) and (x 2)(x 4). These can be written as x2 6x and x2 6x 8. This shows that whatever the numbers are, the products differ by 8.”

Evaluate each expression for the given values of the variables. 3. Evaluate y 5 for: 2

a. y 1

Making Connections

b. y 3

c. y 7

4. Evaluate x 3x 2 for: 2

a. x 0

b. x 2

c. x 6

Make a sketch of Lab Gear to do each multiplication problem. 5. (x 1)

6. (x 3)(x 4)

2

Extending Concepts

7. Leanne and Jody each gave an argument

to support this statement: “If you pick any four consecutive even numbers, the inner and outer products will differ by 8.” Which argument do you find more convincing? Why?

8. In the old Chinese calendar, each month

was made up of three 10-day periods similar to our week. 1

2

3

4

5

6

7

8

9 10

11

12 13

14 15 16

17 18

19 20

21

22 23

24 25 26

27 28

29 30

The figure shows a calendar with 10 days in each row. Choose any four numbers that form a square. Find the products of the two diagonals. Repeat this for other blocks of four numbers. Describe what you notice.

Inner product is 4 • 6

2

4 6

8

Outer product is 2 • 8

Leanne: “I tried lots of different sets of four consecutive even numbers, and the outer product and the inner product always differed by 8.”


213


5 ork w e om

HW

Extending the Lab Gear Model

Applying Skills

Extending Concepts

What number is shown by each collection of Lab Gear blocks? 1.

2.

3.

4.

13. a. Sketch three different Lab Gear

models for the number 4. In one use the minus area, in the second model use upstairs blocks, and in the third use both the minus area and upstairs blocks. b. What number could you subtract from 4 to end up with 7? How did you figure this out?

Making Connections

14. The highest temperature ever recorded

Solve each addition or subtraction problem.

214

5. 8 3

6. 5 9

7. 8 (6)

8. 4 11

9. 3 (9)

10. (10 3)

11. (7 1)

12. (5 7)


in the United States was 134, recorded in Death Valley, California. The lowest temperature was 80, recorded in Alaska. Write and solve a subtraction problem to find the difference between the highest and lowest recorded temperatures. 15. Mauna Kea is a volcano in Hawaii. It is

the highest island mountain in the world. Its peak is about 13,800 ft above sea level. Its base is about 18,200 ft below sea level. Write and solve a subtraction problem to find the approximate height of Mauna Kea from its base to its peak.


6

Making Long Expressions Shorter

ork w e om

HW

Applying Skills

Extending Concepts

Tell whether each expression below is a polynomial. Explain your answers. 1. 2x 8

of Lab Gear blocks.

2. x 2

2

3. 7x 1 x

10. a. Write a polynomial for this collection

1

3

4. x

Simplify each polynomial. 5. 5x 7x 2x 2

6. 4x 3x 1 2x 2

7. 8y 5 y 3y

2

2

8. 7x x 4 – 6x 3 2

2

9. Write a polynomial for this collection

of Lab Gear blocks. Then write the simplified polynomial.

b. Sketch the simplified collection of

blocks. c. Write the simplified polynomial.

Making Connections

11. The number of diagonals in a regular

polygon with x sides is 0.5x2 1.5x.

a. Is this expression a polynomial? How

can you tell? b. How many diagonals does a regular

polygon have if it has 8 sides? 12 sides? How did you figure out your answers?


215


7 ork w e om

HW

Grouping and Ungrouping

Applying Skills

Extending Concepts

Tell whether each equation is true or false. If it is false, rewrite the right side so that it is true. ? 1. 8 (x 2) 8 x 2 ? 2. 6 (y 7) y 13 ? 2 2 3. x (x 6) x x 6 ? 2 2 4. 2y (y 3) 2y y 3 ? 5. (x 6) x 6 ? 6. (y 1) y 1 Write each expression without using parentheses. 7. 1 (x 7)

8. y (y 3)

9. x (x 2)

10. 5y (y 2)

11. 6x (5x 1)

12. 9y (2y 3)

13. (x 9)

14. (y 6)

2

2

15. Two students, Vinh and Danielle,

explained how they would write the expression 5x3 (2x2 3x 1) without parentheses. Vinh: “Remove the parentheses and make all the terms that were inside the parentheses negative.” Danielle: “Remove the parentheses and switch the signs of all the terms that were inside the parentheses.” Which method do you think is correct? Why? What is the result when the expression is written without parentheses?

2

Making Connections

16. When a manufacturer sells x units of a

product, its income is 12x. Its expenses for the x units are x2 8x 40. The profit can be found by subtracting the expenses from the income as follows: 12x (x2 8x 40). a. Why does it make sense that profit is

found by subtracting expenses from income? b. Write the expression for profit

without parentheses and simplify.

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8

Lab Problem No. 2

ork w e om

HW

Applying Skills

Extending Concepts

Simplify each polynomial.

square piece of paper measuring 10 in. by 10 in. The length of the rectangle is one inch more than its width.

1. 6x x 2x 3 2

15. A rectangle is cut from the corner of a

2

2. 4x 3x 1 8x 2

3. 3x 2x 5 x 2

10 in.

4. 5x x 4 3x 5 2

2

6. 2x 3 6x x

2

Write each expression without using parentheses. 12. x (x 8) 2

13. 5(x 3) 14. (y 7)

x

2

Tell whether each equation is true or false. If it is false, rewrite the right side correctly. ? 7. 4y (5y 2) 9y 2 ? 2 2 8. 3x (x 6) 3x x 6 ? 2 2 9. 6y (y 4) 6y y 4 ? 2 2 2 10. 6y (2y y) 4y y

11. 3x (9x 1)

x1

10 in.

5. 2x 4x 1 x x 2

a. What is the area of the original

square? b. Use x to represent the width of the

rectangle. Write an expression for the area of the rectangle that is removed. c. Write the expression in item 15b

without parentheses. What rule did you use? d. Write an expression for the area of

the paper after the rectangle has been removed. (Hint: Subtract your answer in item 15c from your answer in item 15a). Give your answer both with and without parentheses. Writing

16. Write a summary of the rules you have

learned for removing parentheses from expressions. Be sure to describe how to remove parentheses that are used with subtraction and those that are used with multiplication. Give an example of how to use each rule.


217


9

Polynomial Arithmetic

ork w e om

HW

Applying Skills

Writing

Simplify each addition or subtraction problem. 1. (2x 3) (6x 2)


Dear Dr Math:

2. (5y 3) (4y 1) 3. 2x 3x 2

2

4. 2x 3 (x 5) 5. 2x 1 (x 3) 6. 3y 1 (y 2) 2

2

Complete each sentence. 7. Subtracting (2x 1) is equivalent to

I was trying to simplify this subtraction problem: (4x 3) (2x 1). I figured that I could change it to an addition problem like this: (4x 3) (2x 1), which comes out to 6x 2. My friend got 6x 2 for the answer. Where do you think she went wrong?

adding ____. 8. Subtracting (4y 2) is equivalent to

adding ____. Extending Concepts

9. What addition problem is modeled by

the sequence of Lab Gear moves shown here? Make a sketch of the Lab Gear that will remain after you cancel what you can. What is the result? Step 1: Put down the first polynomial

218

Step 2: Put down the second polynomial


Polly Nomial


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HW

Simplify and Solve

Applying Skills

Extending Concepts

Determine whether the given value of x is a solution to the equation by substituting the value into the equation. Show your work.

10. a. Write the starting equation for this

Lab Gear setup.

1. 3x 15; x 5 2. 2x 6 20; x 7 3. 3x 3 8 (x 1); x 1 4. 4x 5 (x 3) 8 (2x 1); x 2 5. a. Write the starting equation for the Lab

Gear setup shown below.

b. Solve the equation and check your

solution in the original equation. How did you solve this problem?

Making Connections

11. The speed of light is about 11 million

b. Find the value of x that makes the two

sides of the equation equal. c. Check your solution in the original

equation. Solve each equation. Check your solution in the original equation. 6. 2x 14 7. y 9 12

miles per minute. The distance of the earth from the sun is about 93 million miles. Let x represent the time (in minutes) that it takes for light from the sun to reach the earth. Then x satisfies the equation 11x 93. a. Why does this equation make sense? b. What value of x makes both sides

approximately equal? To the nearest minute, how long does it take light from the sun to reach the earth?

8. 6x 24 9. 2x 3 (x 3) 7 5


219


ork w e om

HW

Staying Balanced

Applying Skills

1. a. Write the starting equation for the Lab

Gear setup shown below.

Extending Concepts

12. a. Sketch a sequence of Lab Gear setups

to solve the equation 2x (x 4) 4 (x 2).

b. Solve the equation using algebra rules.

Keep a record of each step of the process. c. Describe each of the Lab Gear rules

that you used in item 12a. Give the corresponding algebra rule from item 12b. b. Solve the equation. c. Check your solution in the original

equation. Solve each equation. Keep a record of each step of the process. Check your solution in the original equation. 2. y 4 8 3. 18 4x 2 4. 3y 4 20 5. 6x 2 16 6. 5y 3 2y 9 7. x 3 2x 4 8. y 3 2y 6 9. 2x 10 2x 2 10. 4x (x 1) x (x 5) 11. 2(y 3) 16

220


Making Connections

13. If a ball is thrown straight up at a speed

of 64 feet per second, its speed (in ft/sec) at x seconds after the toss will be 64 32x. At what instant will the ball’s speed be 0? (Hint: Solve 0 64 32x.) 14. Celsius and Fahrenheit temperatures are

related by the equation F 32 1.8C. If the Fahrenheit temperature is 50, what is the Celsius temperature? Show your work.


k wor e m o

HW

Lab Problem No. 3 13. If you buy 10 books, which book club

Applying Skills

will charge the least? the most?

Solve each equation. Keep a record of each step. Check your solution. 1. 22 3x 1

2. 5x 2 13

3. y 3 4y 9

4. 6x 3 4x 1

14. Write expressions for the total cost of x

books at each book club. 15. How many books do you have to buy for

the cost to be the same at Clubs A and C? Explain how you figured out your answer.

5. x 14 2x 2 6. 2(y 4) 10

16. What equation do you get if you set the

For items 7–12, compare schools A and B. 7. Which school

would charge more if you took 10 lessons? 20 lessons?

expressions for Clubs A and B equal? Solve the equation. How would you interpret the answer? Is there any number of books for which Club B charges less than Club A? Explain your thinking.

School A a $5 registration fee plus $10 per lesson School B $11 per lesson, the first lesson is free.

Writing

8. Write an expression showing the total

cost for x lessons at School A.


9. Write an expression showing the total

cost for x lessons at School B. 10. What equation do you get by setting the

two expressions equal? 11. Solve the equation. Keep a record of each

step. Check your solution. 12. For how many lessons will the cost be

the same at Schools A and B?

Dear Dr. Math: My teacher is always going on about expressions and equations. What’s the difference anyway? And what does she mean when she asks us to solve an equation? Are there any rules that I can use to help me solve an equation? X. Pression

Extending Concepts

Club A First book is free, each additional book costs $8. Club B $5 registration fee, each book costs $9. Club C First two books cost $10 each, remaining books $6 each. EXPLORING THE UNKNOWN • HOMEWORK 12

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PHASEONE Slope as an Angle In this phase, you will focus on tools used to measure the slope angles of slanted objects. You will make your own slopemeasuring device and use it to measure slopes inside and outside your classroom. You will also make scale drawings of the objects you measure.

What math is used in designing and building roads and ramps?

ROADS AND

RAMPS

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PHASETWO

PHASETHREE PHASEFOUR

Right-Triangle Relationships You will measure and calculate the side lengths and slope angles of right triangles. As you study the side length relationships of right triangles, you will compare two different methods for calculating side lengths—scale drawings and the Pythagorean Theorem.

Slope as a Ratio As you measure the vertical rise and horizontal run of some stairs and ramps, you will look for relationships between slope ratio, slope angle, and percent grade. To wrap up the phase, you will have a chance to design a set of stairs.

Road Design In this final phase, you will create two- and threedimensional road models. You will learn about a helpful mathematical tool—the tangent ratio—and then build a hillside model that you can analyze and measure. Then it’s your turn to design and build your own model.

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PHASE ONE

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In this phase, you will measure the slope angles of slanted objects using a Slope-o-meter™ and a protractor. Every profession has specific tools that are used by the people in that field. The Slope-o-meter that you will work with is similar to a protractor, a tool used in many occupations. Besides math students, who else do you think uses a protractor?

Slope as an Angle WHAT’S THE MATH? Investigations in this section focus on: SC ALE and PROPORTION ■

Recognizing the difference between a sketch and a scale drawing.

■

Relating slope angles sighted with the Slope-ometer to angles in scale drawings that are drawn with a protractor.

■

Using ratios to relate lengths in scale drawings to actual lengths.

ACCURACY and MEASUREMENT ■

Estimating and measuring the slopes of lines as angles (from the horizontal) to the nearest degree.

■

Recognizing horizontal and vertical benchmarks for slope.

■

Examining the relationships among the accuracy of measurements, the quality of the measuring tool, and the limitations of human perception.


ROADS AND RAMPS

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1

MEASURING SLOPE AS AN ANGLE


Slopes and Slope-o-meters Most roofs, roads, and ramps in our world have a slope. To begin your investigation of these, you will make and use a tool for measuring slopes.

Making a Slope-o-meter To make a Slope-o-meter, you will need two pieces of printed cardstock. One piece has the rotating protractor, and the other has the backing for the Slope-o-meter. You will also need a heavy washer and a pierced-earring post and back. Rotating protractor 1. Cut out the rotating protractor and tape a heavy washer to the bottom of it. Be sure to center the washer Washer over the vertical line, so that this part of the Slope-o-meter will hang vertically.

2. On the other piece of cardstock, score the dotted lines Backing with a pen or paper clip. Then fold back the edges along the dotted lines and tape them to the back of the cardstock. This will make the backing of your Slope-o-meter sturdy.

226

ROADS AND RAMPS • LESSON 1

3. Use transparent tape to reinforce the swivel point of the rotating protractor with an “X.” Do the same at the center of the backing.

Tape

Tape

4. Use the earring post to pin the swivel point of the rotating protractor to Back the center of the backing.Your Front completed Slope-ometer should Earring back look like this. Earring post

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Measure Slopes Use your Slope-o-meter to measure the slopes of lines posted around the classroom. 1

Before you measure each slope, make an estimate and record it in a table.

2

After you make an estimate, use the Slope-o-meter to measure the slope of the line. Record the measurement in your table. As you record the slopes, think about the accuracy of your Slopeo-meter and consider reasons why different students might find different slopes for the same line.

3

How do your estimates and actual measurements compare?

4

How do your measurements compare with those of your classmates?

How can you use your Slope-o-meter to measure slopes?

How to Use a Slope-o-meter You can rest your Slope-o-meter directly on a surface to measure its slope.You can also align the bottom edge of the Slope-o-meter with a line drawn on the chalkboard.

Read this number on the protractor to find the slope of the surface or line.

ter

me -o-

pe

Slo

me

Na

hot words

slope angle

W Homework page 256


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2 MAKING SCALE DRAWINGS

1:40 PM


Working for Scale If you were going to construct a ramp or road, you might begin by making an accurate scale drawing. You will get a chance to make your own scale drawings after first using a protractor to measure some slopes.

Measure Slopes with a Protractor How can a protractor help you measure slopes?

Consider the slopes in these pictures below and then record this information in your lab journal. 1

Order these slopes from least to greatest.

2

Estimate each slope in degrees measured from the horizontal.

3

Use a protractor to measure the slope in each picture.

4

Make a table comparing your estimated slope measurements to the actual slopes.

Roof

Ladder

Stairs Ramp

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Read a Scale Drawing What information can you get from a scale drawing?

A scale drawing is an accurate drawing of a life-size object that is proportionally smaller. Architects, physicists, and engineers use such drawings to help study specific parts of a situation or problem. Scale drawing Scale : 1 cm 10 m 10 8.5 cm

Use the scale drawing shown here, as well as a protractor and ruler, to answer the following. 1

How tall is the drawing of the ramp in centimeters?

2

What is the height of the actual ramp in meters?

3

What is the measure of the third angle in the drawing? How do you know you are correct?

4

How would the slope of the ramp change if its height increased? decreased?

Make a Scale Drawing Choose a scale and make an accurate scale drawing using a ruler and protractor to solve the following problem. ■

A ramp with a 20° slope is 32 ft long. What horizontal distance does the ramp cover? What vertical distance does it rise?

How can scale drawings be used to find unknown lengths?

hot words

scale scale drawing

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3

USING A LINE OF SIGHT IN SCALE DRAWINGS

1:41 PM


Sighting and Angle of Elevation You can use your Slope-o-meter to measure the slopes of inaccessible objects, such as a faraway roof or a mountain, with a method called sighting.Your Slope-o-meter can also be used to find the slope of your line of sight to an object.

Use a Slope-o-meter for Sighting How can you use sighting to find the slope of a distant object?

Use sighting to measure the slope of the roof line that your teacher has drawn. Sketch the slope and record the measurement in your lab journal. As you work, think about sources of error and why different students might get different readings.

The Sighting Method To find the slope of an object by sighting, hold out your Slope-o-meter. Line up the bottom edge of the Slope-o-meter with the slope line you are measuring.

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Slope-o-meter

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Make a Scale Drawing of a Flagpole How can you use an angle of elevation and a scale drawing to find the height of a flagpole?

Measure 20 m out from the base of a flagpole at school. Stand at that point and use your Slope-o-meter to find the angle of elevation from your eye level to the top of the flagpole. Then create a scale drawing of the situation in your lab journal. Don’t forget to include the distance from the Slope-o-meter to the ground in your scale drawing. 1

What is the height of the flagpole?

2

How does your result compare with those of other students?

3

How accurate do you think your result is? Is it accurate to the nearest 1 cm? the nearest 5 mm?

How to Measure an Angle of Elevation You can use your Slope-o-meter to find an angle of elevation. The angle of elevation is the angle up to an object, measured from the horizontal.

Angle of elevation

To measure an angle of elevation, you must sight along the top (or bottom) edge of your Slope-o-meter, holding the Slope-o-meter close to your eye.

hot words

angle of elevation

W Homework page 258


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P H A S E TWO

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In this phase, you will find the lengths and slope angles of some right triangles using scale drawings and the Pythagorean Theorem. Understanding scale drawings and the Pythagorean Theorem gives you more than one way to find the unknown lengths. Being able to solve problems in different ways is an important skill. How might an architect or a surgeon solve problems in more than one way?

Right-Triangle Relationships WHAT’S THE MATH? Investigations in this section focus on: GEOMETRY ■

Visualizing horizontal, vertical, and slope lines as right triangles.

■

Understanding the relationship between the acute angles of any right triangle.

■

Using vocabulary associated with right triangles.

■

Applying the Pythagorean Theorem in problemsolving situations.

ACCURACY and C ALCULATIONS ■

Making scale drawings to improve accuracy.

■

Using the Pythagorean Theorem to calculate unknown side lengths in right triangles.

■

Comparing measurements found with scale drawings to those calculated with the Pythagorean Theorem.


ROADS AND RAMPS

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4

EXPLORING ANGLE RELATIONSHIPS IN RIGHT TRIANGLES

2:51 PM


Right Triangles Imagine you are climbing up a ladder that is leaning against a building. Is the ladder leaning at a stable and safe angle? In this lesson, you will experiment with the slope angles of ladders to learn about angle relationships in right triangles.

Explore Ladder Stability Set up the experiment you see here. 1

2

3

Tape a paper cup to a meterstick. The cup should be placed near the top, as this is the critical position when a person is on a ladder. To simulate the weight of a person on the ladder, put something into the cup such as wooden blocks or keys.

Simulated “person” (weighted paper cup taped near upper end)

Wall

How can you figure out the maximum and minimum safe slopes for a ladder?

Meterstick Floor Measuring with Slope-o-meter

Lean the stick against a wall.

Use a Slope-o-meter to measure each of these slope angles.

■

For each slope, use a protractor to measure the angles the ladder makes with the floor and with the wall.

■

Which of the protractor measures is the same as your Slope-o-meter measure? Why?

o-m

■

pe-

Find the maximum and minimum stable slopes for the ladder.

Slo

■

ete r

Record your work as you investigate the following:

Measuring with protractors

Wall angle Floor angle

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Find Ladder Angles Find the wall angle for each of the ladder’s floor angles below. You may want to use a protractor or Slope-o-meter first to help position the meterstick so that it forms each angle with the floor. 1. 30°

2. 50°

3. 70°

4. 63°

5. x°

What is the relationship between floor angles and wall angles?

Analyze Right Triangles The ladder experiments that you have done can be represented by drawing right triangles. These diagrams can help you to better understand the relationship between horizontal, vertical, and sloped lines. Horizontal

Or:

e

e lin

Vertical

Vertical

e

e lin

Slop

What can you say about how the floor angle and wall angle are related?

Slop

Horizontal

e

e lin

Slop

Vertical

Or, putting the two triangles together:

Horizontal

Use a protractor and ruler to draw these diagrams in your lab journal. Then write answers to the following questions. 1

How do these diagrams help explain the angle relationships in the ladder experiments you just did?

2

What can you say about how the acute angles in a right triangle are related?

3

Is this relationship true for triangles that are not right triangles? Why or why not?

hot words

right angle right triangle

W Homework page 259


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5

EXPLORING SIDE RELATIONSHIPS IN RIGHT TRIANGLES

2:51 PM


Exploring the Pythagorean Theorem As you’ve seen, a ladder leaning against a building forms a right triangle.Thousands of years ago, people discovered something very special and remarkable about the relationships among the sides of a right triangle. Here you will create some right triangles and explore the relationships among the sides of each triangle.

Measure the Sides of Right Triangles

9 8

7.8 cm 7

Use a metric ruler to measure across the inside of each L-shaped strip. Measure this distance to the nearest 1 10 cm. Record this length and the inner leg lengths in a table.

6

2

5

Cut out four different L-shaped strips of centimeter graph paper. Each strip should be one centimeter wide. A sample strip is shown here.

4

1

10

Follow the steps below to make right triangles, measure the lengths of the sides, and record your data in a table.

6 cm

1

2

3

What relationship can you find among the sides of a right triangle?

3

5 cm

Draw and label the right triangle that corresponds to each L-shaped strip. 7.8 cm

5 cm

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6 cm

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Write About Right Triangles Discuss each of these questions with a partner and then record your answers in your lab journal. 1

The angles of the right triangle have a specific relationship. Describe that relationship.

2

Write a general rule that describes the relationship among the sides of a right triangle. Include a drawing with your rule.

Explore Pythagorean Triples What can you say about whole numbers that satisfy the Pythagorean Theorem?

The whole numbers 3, 4, and 5 are known as a Pythagorean triple. 1

Write the Pythagorean Theorem equation that corresponds to the triangle shown.

2

What do you think is meant by the term Pythagorean triple?

3

The numbers 6 and 8 are two members of another Pythagorean triple. That is, 62 82 c 2.

5

4

3

a. Solve for c. b. Compare the side lengths of this triangle with those of the 3-4-5 triangle. 4

Find the missing values in each set of Pythagorean triples. a. 30, 40, ? b. 15, ?, 25 c. ?, 44, 55

5

Given one Pythagorean triple, how can you find new ones? a. Here is another Pythagorean triple: 5-12-13. Verify that this is a Pythagorean triple. b. Use what you have discovered above to find some new Pythagorean triples.

hot words

Pythagorean Theorem Pythagorean triples

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6

APPLYING SIDE RELATIONSHIPS IN RIGHT TRIANGLES


Using the Pythagorean Theorem Buildings are designed and constructed with a specific degree of accuracy. Boards or beams that are an inch too short or too long can cause major problems in construction. Here you will use the Pythagorean Theorem to help find the length of a beam for a building.

Compare Scale Drawings with the Pythagorean Theorem How can you accurately determine the length of a ramp?

Suppose you are going to make a ramp for your school. It will start at the sidewalk and end at the top of a stairway that leads to the front of the school. The ramp is to cover a horizontal distance of 40 ft and a vertical distance of 5 ft.

5 ft 40 ft

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1

Choose a scale and make an accurate scale drawing using a ruler and protractor to solve this ramp problem. What is the length of the ramp?

2

Use the Pythagorean Theorem to solve the same problem.

3

How do your two results compare? How do the two methods compare? Do you think one of the methods is more accurate? Explain your thinking.

4

How long would a banister for the ramp be if it needs to extend past each end of the ramp by 1 ft?


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Apply the Pythagorean Theorem Imagine that you work for Envirotec Builders. Your boss asks you to order a set of steel beams that will support the sloped roof of a building and overhang each end by 1 ft. The length of the beams must be very accurate because there is no way to cut the beams on-site after they arrive. Here is a side view of the building (not drawn to scale).

How can the Pythagorean Theorem be used to solve real-world problems?

Steel beam

17 ft 9 ft 26 ft

■

How long should the beams be? (Remember, the steel beams must extend past each wall by 1 ft.) Show all of your work.

■

Describe in writing the method you used to find the length of the beams. Did you make a scale drawing or use the Pythagorean Theorem? How accurate do you think your answer is?

The Pythagorean Theorem

For any right triangle, the sum of the squares of the legs equals the square of the hypotenuse.

a2 b2 c2

Leg b

Hypotenuse c

Leg a

hot words

scale drawing Pythagorean Theorem

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PHASE THREE

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In this phase, you will find the slope angles of stairs using ratios. You will have a chance to design your own staircase. You will also do some experiments to investigate the relationship between slope angle, slope ratio, and percent grade. Understanding how to use ratios is very important in many occupations. Architects use ratios to help make models of buildings they are designing. What other professions do you think use ratios? How do you think they use them?

Slope as a Ratio WHAT’S THE MATH? Investigations in this section focus on: RATIOS and MEASUREMENT ■

Understanding the definition of slope as a ratio of vertical rise to horizontal run.

■

Calculating the slope ratios of stairs and other objects.

■

Understanding how percent grade is used to describe slope ratios.

■

Finding the slope angle for a given percent grade.

PROPORTIONAL REASONING ■

Setting up proportions to solve real-world problems.

■

Solving proportions to find equivalent ratios.


ROADS AND RAMPS

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WRITING SLOPES AS RATIOS

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Stairs and Ratios You’ve already seen how to measure a slope as an angle in Phase One of this unit. In these next few lessons you will explore how carpenters, scientists, and mathematicians measure slope as a ratio.

Find the Slope Ratio of Stairs How can you determine the slope ratios for different sets of stairs?

C A

B

Horizontal run Vertical rise

Slope ratio

Vertical rise Horizontal run

Answer these questions for each set of stairs, A, B, and C. Record your findings in a table like the one shown.

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1

What is the slope ratio?

2

Estimate the slope angle.

3

What is the actual slope angle? Measure it with a protractor.


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Design Steps for Different Slope Ratios and Angles Use centimeter dot paper to help answer the following questions. 1

Draw a three-step stairway for each of the following slope ratios. 2 5 8 f. 5 a.

5 2 5 g. 6 b.

3 4 6 h. 5 c.

4 3 2 i. 2 d.

How can you draw stairways with different slope ratios?

5 8 3 j. 3 e.

2

What is the slope angle of each of the stairways you drew in item 1? Measure your drawings with a protractor.

3

a. For each set of stairs you drew in item 1, find the total vertical rise and total horizontal run. b. In each case, how are the total rise and total run for the three steps related to the rise and run for one step?

4

If more steps with the same slope ratio are added to a stairway, will the slope angle of the stairway change? Draw a sketch to explain your answer.

5

Building codes say that handicapped-access ramps can have a 1 . Which of these slopes would be slope ratio no greater than 12 okay? Why? a.

6

3 35

b.

4 50

c. 20°

d. 10°

Do you think the following statement is true? If you have two slope ratios that are reciprocals, then the corresponding slope angles for those ratios will add up to 90°. Test the statement on the stairs with reciprocal slope ratios that appear in item 1, such as 25 and 52 . Can you explain why the statement might or might not be true?

hot words

slope ratio

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USING PROPORTIONAL REASONING TO SOLVE PROBLEMS

3:14 PM


The 12-Inch Tread Most lumber yards sell precut boards called 12-inch steps, which are used for making stair treads. The boards are Douglas fir and are 1 in. thick, 12 in. wide, and can be cut in various lengths. In this lesson, you will investigate how to use 12-in.-wide boards to create some specific stairways.

Explore Stairs with a 12-Inch Tread How can you determine the rise of a step, given the tread and slope of a staircase?

A carpenter needs your help in making a stairway. Each step is to have a 12-in. horizontal tread. The slope of the stairway is to be 25 . ■

■

Each 12 inches

Given the 12-in. tread, what should the rise for each step be? Work with classmates to solve this problem in at least two different ways. Record your work in your lab journal and be prepared to share your methods with the class.

Tread

Rise

Slope is 25 2 5

12 in.

Share and Compare Solution Methods Present your findings to the whole class.

244

■

Take notes in your lab journal while other groups are making their presentations.

■

After all groups have shared their findings, choose a method you can use quickly to calculate the rise for a 12-in. tread for any slope ratio. Explain how the method works.


?

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Design a Stairway The guidelines shown are from a stair-building manual. Imagine that you are a master carpenter and need to design a stairway that goes from the basement to the first floor of a house. The total rise of the stairway must be exactly 10 ft and must follow the guidelines. Your stairway may have a different number of steps from the one shown in the diagram, and it is not necessary for each step to have a 12-in. tread.

How can you design a stairway, given guidelines for the rise and run?

Write up your design recommendations for the stairway. Be sure to show your work and include the following information. 1

What are the maximum and minimum possible values for the total run of the stairway?

2

What is your recommended design for the stairway? a. How many steps would you have? b. What is the total run of your recommended staircase? c. What is the unit run? d. What is the unit rise? Stair-Building Design Guidelines Upper floor Unit run Unit rise

Total rise

Bottom floor Total run

Step 1

Step 2

Calculate unit rise by first choosing the number of rises for your design. When divided into total rise, the result should be a unit rise of 6–8 in. (As an example, the diagram above shows 9 rises.)

For comfortable stairs, calculate unit run with this formula: unit rise unit run between 17 in. and 18 in.

hot words

tread slope

Step 3 Add up the unit runs to find total run.

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9

CONNECTING SLOPE RATIOS AND PERCENT GRADE


7% Grade Ahead! The 7% Grade Ahead sign is a truck warning sign at the start of a steep hill on Highway 92 in northern California. In this lesson, you will explore percent grades.

Suppose you work for a survey crew that needs a chart that relates the slope angle for a hillside to its percent grade. Here is a sketch that shows how you can get the data for the chart.

246

Slope-o-meter taped to string

r

ete

o-m

pe-

Slo

Floor Exactly 200 cm

1

Your teacher will hand out an Angle and Percent Grade Chart. Leave the far right column blank for now—it will be used in a later lesson.

2

Tape the backing of your Slope-o-meter to a piece of string, as shown in the figure.

3

Hold or tape one end of the Slope-o-meter/string device to the floor so that it is 200 cm away from a wall.

4

Hold the other end of the string against the wall and move it up or down until the Slope-o-meter shows the desired slope angle.

5

Use the tape measure to find the resulting wall height. Record the wall height in your chart. This tells you how high a hillside with that slope angle rises for every 200 cm of horizontal distance.

6

Find the slope ratio and convert it to a percent grade. Record the value in your chart.

Percent Grades

A percent grade of 7% means that for every 100 units of horizontal distance traveled, the road ascends (or descends) 7 units.

Thumb holding string

Wall

What is the relationship between slope angles and percent grade?

Measuring tape

Experiment with Slope Angles and Percent Grades


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Design a Skateboard Ramp How can you design a ramp with a specific percent grade?

Ramps-R-Us makes skateboard ramps at whatever percent grade the customer orders. The boss wants you to help design the vertical braces for the ramps. “All our ramps have the same horizontal length (18 ft) and six braces, each 3 ft apart,” she says. “I’m giving each of you a different percent grade to work with. Figure out the heights of all the braces and the total length of a ramp with that percent grade. Then when we get an order, all we will have to do is pull up the specs for that percent grade and build the ramp.” Prepare a report for your boss that includes the following. 1

A labeled diagram of your ramp that shows: a. the percent grade of your ramp b. the height of each of the six braces c. the total length of the ramp

2

A description of how you figured out the height of the braces and the total length of the ramp. Ramp with % grade Total length Braces

3 ft

3 ft

3 ft

3 ft

3 ft

3 ft

hot words

percent percent grade

18 ft

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PHASEFOUR

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In this phase, you will explore the tangent ratio. The tangent ratio ties together the ideas of slope as an angle and slope as a ratio. Then you will construct a threedimensional model of a road on a hill. People solve all types of problems in art, math, science, and engineering by using models.What are some advantages of making and analyzing models?

Road Design WHAT’S THE MATH? Investigations in this section focus on: TRIGONOMETRY ■

Finding the tangent ratio for a given slope angle.

■

Finding the slope angle that corresponds to a given tangent ratio.

■

Calculating unknown side lengths and angle measures in right triangles.

SPATIAL RELATIONSHIPS ■

Visualizing the spatial relationships among horizontal, vertical, and slanted planes.

■

Relating slopes in three dimensions to their twodimensional representations.

SC ALE MODELS ■

Indentifying and calculating the essential lengths and angle measurements needed to construct a three-dimensional model.

■

Devising a strategy for calculating unknown side lengths and angle measures.

■

Building an accurate three-dimensional scale model.


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10 LINKING SLOPE RATIOS AND SLOPE ANGLES


The Tangent Ratio In previous lessons, you investigated slope angle, slope ratio, and percent grade. In this lesson, you will investigate the tangent ratio and see how it ties all of these ideas together. This will help you fill in the last column of your chart from Lesson 9.

Calculate the Tangent Ratio How are slope angles and slope ratios related?

Here is a way to explore the relationship between slope angles and slope ratios. 1

Each member of your group should draw a right triangle (of any size) with a slope angle of 53°.

2

Use a ruler to find the slope ratio of your triangle and convert it to a decimal.

3

Compare your decimal value with the other decimal values in your group.

4

What do you notice about each value?

c

b

53 a

The ratio you calculated is known as the tangent ratio for 53°. The tangent ratio for a given slope angle is constant, no matter how large or small the right triangle is that contains it. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side b (or a in the figure).

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Find Angles, Grades, and Rises Create a table that shows percent grades from 0% to 100% in increments of 10%. Include a column for rises for a set of stairs with a 12-in. tread. Here is an example. Only the first few rows are shown, with one row completely filled in as an example. Slope Angle

Slope Ratio as Decimal (tangent ratio)

Slope Ratio Rise for as Percentage 100 Feet (% grade) of Run

Rise for 12-Inch Tread

How can you determine the rise for a set of stairs, given the slope ratio of the stairs as a percentage?

0% 10% 20% 16.7

0.3

30 ft

30%

3.6 in.

1

What is the rise for a 12-in. tread with a slope angle of 60°? 75°?

2

What is the percent grade of stairs with a rise of 8.5 in.?

3

What are some easy ways to find the rises for a 12-in. tread?

Solve Right Triangle Problems Your goal is to find all the unknown side lengths and angles for the triangles shown below and write an explanation of your work. Note that the triangles are not drawn to scale! 1

Use the tangent ratio, the Pythagorean Theorem, and any other methods you know to find all of the unknown sides and angles for each triangle.

2

Write a step-by-step explanation of how you solved these problems.

3

Have a classmate look at your explanations. Can he or she use your method to solve other such right triangle problems?

hot words

G T

tangent ratio

N

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20% grade

15 cm

6 cm

page 265

28 C

A

D

8.5 cm

O

R

75 yd

U


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11

A Mathematical Hill

ANALYZING A THREEDIMENSIONAL MODEL

When building a road on a steep hill, most engineers design the road to be less steep than if it went straight up the hill. In this lesson, you will make a three-dimensional model of a hill and analyze different roads that travel up it. We will “cut out” the rectangular portion of the hillside shown and look at it more closely: B

B

Steep hill Steep hill

This part comes from inside the hill! A A

Explore a Road Model How can you compare different roads that go up a steep hill?

An engineer looked at Road Design 1 and decided it was too steep for a truck to go straight up along line AB. The engineer began working on a new road design for the hillside and came up with Road Design 2. B

B

Steep hill Steep hill A

A

Road Design 1

252

C

Road Design 2

C

1

How does line AB of Road 1 compare to line CB of Road 2?

2

Why is the slope of Road 2 a better design than that of Road 1?

3

If you were asked to make a three-dimensional scale model of this situation, what measurements would you need?


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Analyze the Hillside How can you use a model to analyze a steep hill?

Use your model and a Slope-o-meter to analyze the hillside. 1

Measure all of the hillside model’s angles and lengths with a protractor and metric ruler. Record the measurements for each triangle on the Model Hillside Recording Sheet.

2

What do you think is the steepest road up the model hill? Use your Slope-o-meter to find and measure the slope angle of that road. Which of the slope angles that you measured with a protractor corresponds to the slope angle you just measured with the Slope-o-meter? How close are the two measurements?

3

Use your Slope-o-meter to measure the slope angle of the road going from C to B. Which slope angle measured with a protractor matches this slope angle? How close are the two?

How to Assemble the Model Hillside B

These faces fit together to form the complete model hillside.

B

D Part 2

D

Part 1

E

A F C C

Work with a partner to make a model of a hillside, as shown in the diagram. This will help you see the mathematics underlying a road design. 1. Carefully cut out Model Hillside Part 1 and Model Hillside Part 2.

4

For road AB, what is the vertical rise? the horizontal run? the percent grade?

2. Use a ballpoint pen and ruler to score the dotted lines. This will help make the folds clean and precise.

5

For road CB, what is the vertical rise? the horizontal run? the percent grade?

6

What are some reasons that a truck might use road CB instead of road AB?

3. Fold along the dotted lines. As you fold, be sure the printed lines of each part are on the outside of the model. Tape the edges of each part of the model together. 4. Place the two pieces together on a flat surface.

hot words

steepness slope angle

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12 CREATING A SCALE DRAWING AND A 3-D MODEL


The Road Project In this lesson, you will be given information about a new hill model. It will be up to you to figure out the size and shape of the triangles that make up the model.

Explore a Sample Road Model How can you find the unknown lengths and angles of the four triangles in this model?

A hillside has a slope angle of 38° and a vertical rise of 800 ft. A road with a 30% grade is to go up this hill. Your design team needs to make scale drawings of the four triangles that make up the model (shown here with heavy lines). B

Grade: 30%

B

B

Vertical rise: 800 ft D B

D

A

D

C D

A Slope angle: 38 C A

C

A

C

Here are some steps to follow to help you and your team make a scale drawing of the four triangles in this model. (The triangles shown here are not to scale, but you may want to copy them into your lab journal and record your results on them.)

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1

To build a model for this hill, do you need to find all the lengths and angles? Record the known lengths and angles. Then make a list of lengths and angles that you still need to find.

2

Why must a length in one triangle match a length in another ––– triangle? For example, why must BC in ABC be the same ––– length as BC in DBC?

3

Find the unknown lengths and angles.

4

Use a ruler and protractor to make a scale drawing of each of the triangles.


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Calculate the Dimensions for the Model How can you make scale drawings of the triangles for your road model?

Your teacher will give you the following three starting measurements for your model. B

Angle BD Percent grade

the slope angle of the hillside the vertical rise of the hill in feet the percent grade of the road BC

Road with given percent grade

Vertical height D A

Hill’s slope angle

C

Use this information to figure out the unknown lengths and angles of each triangle.

B

Record the starting measurements on the Road Model Record and Evaluation Sheet.

D

1

2

On the back of the project sheet, draw and label the four right triangles you see here. (Note: The drawings are not to scale.)

3

Find the angles and lengths needed to draw each triangle of the model. Show your work.

4

Write the measurements you found for each triangle on the drawings you made in item 2.

A

B

D B

C

A

C

D

A

C

Build the Model Use the measurements you just made to draw, cut out, and assemble a scale model of your road. 1

Use a protractor and ruler to make accurate scale drawings of your four triangles. Use heavy paper, or cardboard if it’s available.

2

Cut out the triangles and tape them together to form the model.

3

Do the sides fit together accurately? If they don’t, go back and check your calculations.

How can you construct a model of your road and hill?

hot words

scale percent grade

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Slopes and Slope-o-meters

Applying Skills

Estimate the slope angle of each line. 1.

2.

5. a. Use your Slope-o-meter to measure

the slope angle of each line in items 1–4. Hold this page against a wall to use the Slope-o-meter. b. Make a table comparing your

3.

4.

estimates in items 1–4 with your Slope-o-meter measurements.

Extending Concepts

6. Explain how you could use your Slope-

o-meter to draw a line with a slope angle of 32° on a chalkboard. 7. The three lines below have slope angles

of 20°, 35°, and 60°, respectively.

b. For each line, find the sum of the

actual slope angle and the slope angle that you measured in item a. What do you notice? If you used the incorrect method as described in item a, what would you get for a line with slope angle 42°? Explain your answer. c. If correctly used, does a Slope-o-

a. Suppose you measure the slope angles

of the three lines incorrectly by placing the vertical edge of the Slopeo-meter on the line instead of the horizontal edge, as shown. What do you get for the slope angles of the three lines?

meter measure slope angles from the horizontal or from the vertical? If used incorrectly as described in item a, does it measure slope angles from the horizontal or vertical? Making Connections

8. The Leaning Tower of Pisa was built in

1174 in Tuscany, Italy. It is 180 ft tall. What would you expect the slope angle of the tower to be if it were not leaning? Use your Slope-o-meter and the illustration shown here to measure the slope angle of the tower. 256

ROADS AND RAMPS • HOMEWORK 1

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Working for Scale

ork w e om

HW

Applying Skills

Extending Concepts

1. a. Estimate the slope angle of this roof. Roof

Choose a scale and make an accurate scale drawing using a ruler and a protractor to find the height of each ramp. 7. slope angle 30°, length 7 m 8. slope angle 30°, length 14 m 9. slope angle 60°, length 14 m

b. Use your protractor to measure the

slope angle. How did your estimate in item a compare to the actual slope? Using a scale of 1 cm 1 m, find the length of a ramp on a drawing if its actual length is: 2. 48 m

3. 13 m

4. 7 m

5. Use the scale drawing shown and a ruler

to answer these questions. Scale Drawing 1 cm 10 m

10. Use your results in items 7–9 to answer

these questions. a. If two ramps have the same slope but

the second ramp is twice as long as the first, do you think that the second ramp must also be twice as high? Explain your thinking. b. If two ramps have the same length but

the slope of the second ramp (in degrees) is twice the slope of the first, do you think that the second ramp must also be twice as high as the first? Explain your thinking.

5

a. What are the height and length of the

ramp in the drawing in centimeters?

Writing

11. Answer this letter to Dr. Math.

b. What are the height and length of the

actual ramp in meters? 6. Using a scale of 1 cm 1 m, make an

accurate scale drawing of a ramp that has a slope angle of 25° and a length of 14 m.

Dear Dr. Math: What’s so special about making a scale drawing? I know the lines all have to be straight but I’m pretty good at drawing straight lines freehand. So why do I need to use a ruler and protractor? —“Steady Hand” Luke


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Sighting and Angle of Elevation

Applying Skills

c. Do you think the angle of elevation to

the top of the flagpole could ever be equal to 0°? Why or why not? Make a sketch to explain your answer.

1. Use your Slope-o-meter to measure five

slopes by sighting. Include at least two angles of elevation and two slanted roofs. For each item, make a sketch showing the angle you measured. For angles of elevation, sketch the invisible “line of sight” with a dotted line.

4. This drawing of a lake and hill is not

drawn to scale. In the drawing, AB 600 m. D

2. Maria measured the angle of elevation to

the top of a redwood tree from four different distances: 10 ft, 25 ft, 48 ft, and 60 ft. She obtained angles of elevation of 84°, 59°, 64°, and 76°, but the angles are not listed in the correct order. Match each distance with the correct angle of elevation.

A

16

B

41

C

The angles of elevation are: BAD 16° and CBD 41°. What is the height, CD, of the hill? Explain how you found your answer.

Making Connections Extending Concepts

For items 3 and 4, choose a scale and make an accurate scale drawing using a ruler and a protractor to solve the problem. 3. a. From a point 12 m from the base of a

flagpole, the angle of elevation to the top of the pole is 42°. How tall is the flagpole? b. What is the angle of elevation to the

top of the flagpole from a point 18 m from the base? Explain how you figured out the answer.

258


5. Angel Falls in Venezuela is the highest

waterfall in South America. From a point 650 ft from its base, the angle of elevation to the top of the falls is 72°. How high is the waterfall? Make an accurate scale drawing and explain how you found your answer. Give your answer to the nearest foot.

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Right Triangles a. What is the maximum safe wall angle

Applying Skills

on carpet?

For each ladder’s floor angle, find the wall angle and the slope angle of the ladder. 1. 27°

2. 81°

3. 49°

4. x°

Find the measure of the second acute angle in each triangle. 5.

6. 20

7.

68

b. What is the minimum safe wall angle

on linoleum? c. At what height will the ladder touch

the wall if it stands on carpet with the maximum stable slope angle? with the minimum stable slope angle? Make scale drawings to answer the questions.

46

Making Connections

14. A tessellation is a repeated geometric

Find the measures of the other angles in a right triangle if the measure of one of the acute angles is: 8. 21°

9. 89°

10. 5°

design that covers a plane without gaps or overlaps. The triangle at right can be used to form the beginning of a tessellation as shown.

11. 77°

Extending Concepts

12. An angle in a right triangle is 24° less

than one of the other angles. What could the measures of the other angles be? Explain how you solved this problem. 13. The manufacturer of Reach Higher

ladders states that on linoleum the maximum stable slope angle is 80° and the minimum is 68°. On carpet the maximum is 85° and the minimum is 30°. Their ladders are 25 ft long.

What is the sum of the measures of angles that meet at the center of this pattern? Use what you learned in this lesson about the acute angles of a right triangle to explain how you know that your answer is correct.


259

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5

3:32 PM


Exploring the Pythagorean Theorem

ork w e om

HW

c. Do you agree with this statement: “If

Applying Skills

Write the Pythagorean Theorem equation corresponding to each triangle. 1.

2.

52 20

25 7

48

d. If you add a constant to each number

24

Find the unknown side length in each triangle. 3.

4.

you divide each number in a Pythagorean triple by a constant, you will always get another Pythagorean triple”? If the statement is false, can you modify it to make it true? in a Pythagorean triple will you get another Pythagorean triple? How did you decide?

10

5 12

5.

6

6. 8.5

9.6


4.0

5.1

Find the missing value in each set of Pythagorean triples. 7. 10, 24, ?

8. 21, ?, 35

9. ?, 52, 65

10. 15, 36, ?

Extending Concepts

11. a. Verify that the set 20, 48, 52 is a

Pythagorean triple. b. Do you get another Pythagorean

triple if you divide each number in the set 20, 48, 52 by 4? by 8?

260

Writing


Dear Dr. Math: In a right triangle, doesn’t a 2 b 2 c 2 ? I used this to find the length of the missing side in this triangle. 12 8

x Here is my work: 82 122 x 2 So, x 2 64 144 208 208 14.42 . . . But I know this is wrong because the missing side must be shorter than 12. What did I do wrong? —Emile

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6 ork w e om

HW

3:32 PM


Using the Pythagorean Theorem

Applying Skills

Extending Concepts

Use the Pythagorean Theorem to find each unknown side length. Give each answer as a decimal rounded to the nearest hundredth. 1.

2.

5

2 8 2

3.

legs 17.3 cm and 19.8 cm long. Measure the hypotenuse. Then use the Pythagorean Theorem to calculate the length of the hypotenuse. Can the Pythagorean Theorem still be true even if your calculated answer and the measured length are different? 9. A steel beam spans a building. The wall

4. 3.5

1.7 6.5

8. Carefully draw a right triangle having

3.4

In items 5–7, draw a sketch and use the Pythagorean Theorem to answer the question. Round answers to the nearest hundredth. 5. Find the length of a ramp that covers a

on one side of the building is 25 ft tall and the wall on the other side is 32 ft tall. The beam is 38 ft long and extends 2 ft past each wall. How far apart are the 38 ft walls? Explain 32 ft how you found 25 ft your answer. ?

horizontal distance of 33 ft and a vertical distance of 7 ft. 6. A 25-ft-long ladder is leaning against a

wall of a building. The foot of the ladder is 8 ft out from the bottom of the wall. How high up the wall does the ladder reach? 7. Find the horizontal distance covered by a

ramp if the length of the ramp is 45 ft and the vertical distance covered is 10 ft.

Making Connections

10. The Great Pyramid of Khufu (2680 B.C.)

in Egypt is one of the Seven Wonders of the World and is the largest pyramid ever built. It originally measured 756 ft along each side of its base and 482 ft high. Use the Pythagorean Theorem to find the slant height of the original pyramid. Round your answer to the nearest foot. Slant height 482 ft

756 ft


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7

3:33 PM


Stairs and Ratios

ork w e om

HW

b. What are the total horizontal run and

Applying Skills

Find the slope ratio for each set of stairs. 1.

the total vertical rise of the stairway? How did you find them? c. If the stairway were replaced with a

2.

ramp from point A to point B, what would the slope ratio of the ramp be? Is this the same as the average slope ratio that you calculated in item a? 3.

8. If the slope angle corresponding to a

Use a ruler (or centimeter dot paper) and a protractor for items 4–6.

slope ratio of 43 is 37°, what slope angle would correspond to a slope ratio of 43 ? Make a sketch and explain how you found your answer.

4. Order the slopes below from least to

greatest. 5 9

1 4

35°

3 7

Writing

20°


5. Draw a three-step stairway for each slope

ratio. 2 5 5 a. b. c. 4 7 3 6. Use your protractor to measure the slope angle of each stairway in item 5.

Extending Concepts

7. a. Find the slope ratio for each step in

the two-step stairway shown. What is the average of the slope ratios? 20 4

B

5

3 A

262


Dear Dr. Math: My town has a problem. People in wheelchairs can’t get into the library because there are 8 steps to the door. The slope ratio of the steps is 37. Architects tried to design a ramp but they said there isn’t enough space and that the ramp would end up in the middle of the street. If there’s enough space for the stairs, wouldn’t there be enough space for a wheelchair access ramp? —Left out in Lewiston

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8


The 12-Inch Tread

ork w e om

HW

Applying Skills

a. What are the maximum and

Solve each proportion.

b. What are the maximum and

1.

4 8 7 x

2.

minimum number of steps?

x 2 18 3

642 x 4. 700 100

3.

7 x 8 3.5

2.4 10 5. x 3.6

For each slope ratio, find the rise that goes with a 12-inch tread. 3 6. 5

7 7. 6

7 8. 3

For each slope ratio, find the rise that goes with a 10-inch tread. 9.

4 5

10.

7 2

11.

9 4

minimum possible values for the total run? c. How many steps would you

recommend? d. What is the unit rise of your

recommended stairway? e. What value would you recommend

for the unit run? f. What is the total run for your

recommended stairway?

Making Connections

13. A ziggurat was a type of temple Extending Concepts

12. You have been asked to design a stairway

with a total rise of 16 ft. You must follow these guidelines: ■

■

The unit rise must be between 6 in. and 8 in. unit rise unit run between 17 in. and 18 in.

The stairway may have a different number of steps from the one shown.

common to the Sumerians, Babylonians, and Assyrians. It was a pyramid-like structure, built in receding steps on a rectangular platform. Suppose that a ziggurat has the dimensions shown in this cross-sectional view. Assume all steps have the same rise and the same run. a. What are the rise and run of each

step? How did you find them? b. What is the slope ratio of each step? 30 ft

Unit run 84 ft

Unit rise

Total rise

Total run

150 ft


263

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9

3:33 PM


7% Grade Ahead!

ork w e om

HW

Applying Skills

Extending Concepts

Convert each percent to a ratio or each ratio to a percent. 1. 75% 3.

2.

7 20

2 5

4. 55%

For each slope angle, make a scale drawing to find the rise per 200 cm of run. Round your answer to the nearest centimeter. Then complete the table by finding the slope ratio and percent grade. Slope Angle

Rise per 200 cm

Slope Ratio

Percent Grade

and have been asked to design a skateboard ramp with a percent grade of 25%. The ramp must have a vertical rise of 3 ft, and 6 equally spaced vertical braces. Total length 3 ft Horizontal length

a. Find the horizontal length of the

ramp. How did you figure it out? b. Use the Pythagorean Theorem to find

the total length of the ramp.

5.

23

6.

47

c. How far apart should the braces be?

7.

72

d. Find the height of each brace. Explain

Find the indicated missing length in each triangle. 8.

e. Make a labeled diagram of your ramp.

ramp be if you removed the longest brace?

?

9.

how you found your answers. f. What would the total length of the

40% grade 200 ft

35% grade ? 318 ft

10. Find the horizontal run of a ramp

having a vertical rise of 45 ft and a percent grade of 18%. 11. Find the vertical rise of a ramp having a

horizontal run of 71 ft and a percent grade of 32%.

264

12. You are an employee of Ramps Galore


Making Connections

13. As of 1996, the steepest street in the

world is Baldwin Street in Dunedin, New Zealand, with a maximum grade of 79%. On a road with a 79% grade, what would be the horizontal run corresponding to a vertical rise of 200 ft? How far would you have to travel along the road to gain 200 ft in elevation? Make a sketch and explain your reasoning.

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3:33 PM


The Tangent Ratio

ork w e om

HW

12. You are in a boat in the ocean with your

Applying Skills

Slope-o-meter.

Solve each equation. Round angle measures to the nearest degree and other answers to the nearest hundredth. 1. tan 53° x 3. tan x

9 12

2. tan x 0.15838

10

4. 0.6 x

10 6. tan 63° x

For each triangle, find all unknown side lengths and angles. T

8.

W 6 in.

10 m 33

N

9. H

A

I

11 in.

N

30% grade

O

50 ft

P

10. Use the Tangent Table to complete the

table below. Slope Angle of Stairs

cliff is 35°. You also know the cliff is 1,200 ft straight up above sea level. How far is the boat from the base of the cliff? Make a sketch and calculate the answer using the tangent ratio. b. What do you notice about the tangent

x 5. tan 25° 100

7.

a. The angle of elevation to the top of a

Tangent Percent Grade Ratio

of the angle of elevation as the boat gets very close to the cliff? Why does this make sense?

Making Connections

13. The perpendicular distance from the

center of a regular polygon to a side is the apothem. The length of each side of the regular hexagon shown is 6. Each interior angle of a regular hexagon measures 120°. Calculate the apothem of the hexagon by using the tangent ratio. Explain how you found your answer.

Rise for 100 ft Run

120 Apothem

34 65

60 6

Extending Concepts

11. A 25-ft ladder is leaning up against the

outside wall of a building. The foot of the ladder is 7 ft out from the bottom of the wall. What is the slope angle of the ladder? Make a sketch and explain how you found your answer. ROADS AND RAMPS • HOMEWORK 10

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A Mathematical Hill

ork w e om

HW

c. How much further would you drive

Applying Skills

Use the figure below to answer the questions. B

d. Design a road that starts at point A

D

50 ft

A 100 ft

along road CB than along road AB? Why might you prefer to drive along road CB?

C

30 ft

1. For road AB, what is the horizontal run?

the vertical rise?

and ends at point B that is less steep than road CB. Make a sketch of your road. Do you think that the distance along your road would be greater or less than the distance along road CB?

2. For road AB, find the slope ratio and

percent grade and use the Tangent Table to find the slope angle. 3. Make a sketch of triangle ADC and label

the lengths of the legs. Use the Pythagorean Theorem to find the length ––– of CD to the nearest foot. 4. For road CB, what is the vertical rise? the

horizontal run? 5. For road CB, find the slope ratio and

percent grade, and use the Tangent Table to find the slope angle. Extending Concepts

6. a. For road AB and for road CB, find the

horizontal run, vertical rise, and the percent grade. B 0.4 mi A 1.2 mi

C

0.5 mi

b. What distance would you travel in

driving along each road? How did you find your answers? 266


Writing


Dear Dr. Math: Last Saturday, my family drove to the top of Mount Washington. The road was so twisty that I got carsick. Why did the road zig-zag? I thought that the shortest distance between two points is always a straight line. The road builders would have had much less work to do if they had just built a straight road from the bottom of the mountain to the top. —Nauseous in New Hampshire

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3:34 PM


The Road Project

k wor e m o

HW

Vertical rise: 660 ft

Applying Skills

Grade: 27%

B D

For each triangle, find all unknown side lengths and angles. 1. 40

250 ft

2.

Grade: 28%

Slope angle: 36°

C

a. Sketch triangles ADB, DBC, ADC, and

670 ft

3.

A

ABC. Label any angle measures, side lengths, or percent grades that are known.

240 ft 42 ft

b. Make a list of lengths and angles that

The picture shows a model of a hill with a slope angle of 32° and a vertical rise of 300 ft. The road CB has a grade of 24%. B Vertical rise: 300 ft Grade: 24% D

c. Find the lengths and angles that you

listed in item b and write them on the drawings you made in item a. Explain how you found them. d. Use a protractor and ruler to make

A Slope angle: 32

need to be found in order to build your model.

C

4. Make a sketch of triangle ADB. Label any

angle measures or side lengths that are known. Use the tangent ratio to find the length of leg AD. 5. Make a sketch of triangle DBC. Label

any known side lengths or angles. Use ––– the 24% grade to find the length of DC .

Extending Concepts

6. Follow the guidelines to build a model

for a road going up a hill. The slope angle of the hill is 36° and its vertical rise is 660 ft. The road CB is to have a grade of 27%.

accurate scale drawings of your four triangles. Explain how you would build the model of the hill. Making Connections

7. Stonehenge, on Salisbury Plain, England,

was built between 1900 B.C. and 1600 B.C. It consists of concentric rings of standing stones. Each stone in the outer circle has approximately the shape of a rectangular prism with height 18 ft, length 7 ft, and A width 4 ft. Find the approximate length 18 ft of a diagonal (AB) of one of these C stones. Explain how 7 ft you figured it out. 4 ft B


267

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PHASEONE Direct and Inverse Variation At the start of this phase, you will look at what it means for a mathematical relationship to be a function. Then you will collect and graph data on heelto-toe pacing and data on the apparent size of an object. You will see how these sets of data relate to direct and inverse variation.

How can you describe mathematical relationships?

FAMILY

PORTRAITS

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PHASETWO

PHASETHREE PHASEFOUR

Linear Functions In this phase you will look at functions whose graphs are straight lines. You will start by using a ratio to describe a line’s slope. Next you will see how a line’s equation can give you information about the line’s slope and y-intercept. Finally, you will see how you can write the equation of a line that goes through two given points.

Quadratic Functions How is the area of a slide’s projected image related to the projector’s distance from the screen? You will graph data to help you answer this question and learn how it connects to quadratic functions. You will also graph other quadratic functions and explore the relationship between their equations and their graphs.

Exponential Functions You will begin by thinking about the number of regions formed when a sheet of paper is folded in half over and over. You will graph this exponential function and explore some rules of exponent arithmetic. Finally, you will work with a calculator to see how scientific notation can help you write very large and very small numbers.

PHASE ONE



In this phase you will see what it means for a mathematical relationship to be a function. You will also do some experiments to help you investigate two types of functions—direct and inverse variation. You will be collecting data on the relationship between the number of heel-to-toe paces you take and the distance you cover. What other mathematical relationships have you worked with in the past?

Direct and Inverse Variation WHAT’S THE MATH? Investigations in this section focus on: ALGEBRA ■

Understanding what makes a mathematical relationship a function

■

Recognizing functions from their graphs

■

Graphing and describing direct variation functions

■

Graphing and describing inverse variation functions

GEOMETRY and MEASUREMENT ■

Collecting measurement data

■

Displaying and analyzing measurement data


FAMILY PORTRAITS

271


1

UNDERSTANDING FUNCTIONS

Inputs and Outputs Take any number, double it, and add 1. This rule tells you how to take any input value and get an output value. Can you think of other rules that relate inputs and outputs? You will look at examples of such rules, and explore a special type of rule called a function.

Find Each Machine’s Rule How can you write rules to describe the relationship between an input and an output?

In each picture, a machine takes input values and gives you output values. Each machine has a printout showing some sample inputs and outputs. A.

B.

2

Input (x) Output (y)


7


4

20

5

14

0

0

0

1

2.1

10.5

2

7

1

4

3

15

3

10

2.5

4

D.

10

E.

4


272

C.

10

31

1

4

0

4

F.

10


3


5

25

5

2

4

25

5

6

36

3

153

36

6

7

49

3

42.7

4

2

25

2

4

100

3

1

3

8.45

6.1

1

For each machine, describe the relationship between the input and the output. Whenever possible, write an equation that describes the machine’s rule.

2

Write a brief description of how Machines A, B, D, and E are different from Machines C and F.

FAMILY PORTRAITS • LESSON 1


Decide If It’s a Function

Distance from Home

A mathematical rule that assigns exactly one output value to each input value is a function. Each of the following graphs shows a relationship. The input values are along the horizontal axis; the output values are along the vertical axis. B.

y

C. Weight

A.

Time

x

Age of Baby E.

y

F. Car’s Speed

Postage

D.

0

How can you tell whether or not a graph represents a function?

Weight of Envelope

0

x

Time

1

Which graphs show functions? Why?

2

How were you able to tell which graphs show functions? State a general rule you can use to tell whether or not a graph is a function.

Set Up a Function Album In this unit you will be investigating different types of functions and recording facts in a function album. For the first page of your album, write a summary of what you know so far about functions. Include the following: ■

the definition of a function, in your own words

■

examples of functions

■

how you can tell whether a graph shows a function

hot words

function rule

W Homework page 302


273


2

EXPLORING DIRECT VARIATION

Keeping a Steady Pace How long is your classroom? You could measure it directly with a tape measure, but it is often easier to measure long distances indirectly. You will be finding a way to measure this length by pacing. You will also see how finding distances by pacing is related to a type of function known as direct variation.

Make a Conversion Graph How can you make a graph to show the relationship between paces and distance?

Your teacher has set up some marks on the ground that are one meter apart. 1

Starting at the first mark, begin pacing “heel-to-toe.” Record various numbers of paces and their distances in meters.

2

Plot your data to make a conversion graph that relates the number of paces to the distance covered.

3

Write an equation for the graph. Let x be the number of paces and y be the distance in meters. Your equation should help you convert a given number of paces into distance covered.

Find the Length of Your Classroom Carefully pace off the length of your classroom. Then use your graph and/or equation to find the length of the classroom. Write a summary of your findings that includes the following:

274

■

The length of your classroom

■

A complete description of your method



Explore the y kx Family Equations that have the form y kx (where k is not equal to 0) are called direct variation functions. What do the graphs of these equations have in common? 1

Work with a partner to write down at least 8 equations of the form y kx. Use both positive and negative values for k. Also, choose some values of k that are between 0 and 1.

2

Graph each of your equations on the same coordinate plane.

3

What do you notice about the graphs? Write a list of as many generalizations as possible. Here are some things to consider.

What can you say about the graphs of equations that have the form y kx?

a. What do all the graphs have in common? b. Where do the graphs intersect? c. Which graphs slant upward as you move from left to right? downward? d. Which graphs are steepest? flattest?

hot words

variation, direct

W Homework page 303


275


3

EXPLORING INVERSE VARIATION

How Long Is a Meter? Have you ever noticed how objects appear to get smaller as you move farther away from them? You will collect data involving the “apparent size” of a meter. Then you will see how this is related to inverse variation.

Graph Apparent Size Versus Distance How is the apparent size of an object related to your distance from it?

Work with a partner for this investigation. Decide which of you will be Partner A and which will be Partner B. 1

Partner A: stand with your back to the meter-long strip of paper that your teacher has hung. Then take at least 8 heel-totoe paces in a straight line away from the strip.

2

Partner A: turn around and use a ruler to measure the apparent size of the strip to the nearest tenth of a centimeter.

3

Partner B: record the number of paces, x, and the apparent size, y, in the table on the Apparent Size Recording Sheet.

4

Repeat steps 1–3 with different numbers of paces. Collect at least 10 data points.

Measuring Apparent Size

Plot your points using the recording sheet’s axes. What is your graph’s shape? Find x • y for each pair of values. What do you notice?

To measure the apparent size of an object:

276

■

Stand facing the object.

■

Hold your arm straight out.

■

Hold a ruler vertically aligned with the object.

■

Read the apparent length of the object.



Explore the y xk Family Equations that have the form y xk or xy k (where k is not equal to 0) are called inverse variation functions. What do the graphs of these equations have in common? Your teacher will assign your group a value of k to work with. 1

Make a table of values for your equation.

What can you say about the graphs of equations that have the form y xk ?

a. Include at least 10 pairs of points with positive values of x. b. Include at least 10 pairs of points with negative values of x. c. Include some values of x that are very close to 0. 2

Plot these points to help you graph the equation.

Compare your graph to those of other groups.

Update the Function Album Add two pages to your function album—one that summarizes what you know about direct variation functions, and one that summarizes what you know about inverse variation functions. Include the following on each page: ■

a verbal description of the function family

■

a typical graph from the function family

■

a sample equation for the function family

■

an example of how the function occurs in a real-world situation

■

anything else that might be a useful reference about the function family

hot words

variation, inverse

W Homework page 304


277

P H A S E TWO



Linear functions have graphs that are straight lines. In this phase, you will take a close look at the ideas of slope and y-intercept. You will see how these concepts are connected to the equation of a linear function. A train moving at a constant speed has a distance-versustime graph that is a straight line. A graph showing the number of paces you take versus the distance you cover is also a straight line. What other relationships have you seen that have linear graphs?

Linear Functions WHAT’S THE MATH? Investigations in this section focus on: ALGEBRA ■

Finding the slope of a line

■

Finding the y-intercept of a line

■

Connecting the slope and y-intercept of a line to its equation

■

Writing the equations of lines based on their graphs

RATIO and PROPORTION ■

Using ratios to describe slopes


FAMILY PORTRAITS

279


4

FINDING THE SLOPE OF A LINE

A New Slant on Linear Functions A linear function has a graph that is a straight line. Now it’s time to think about how you can describe the slant, or slope, of a line. You will explore this idea and then see what it means for a line to have a negative slope.

Explore the Slope of a Line How can you use math to describe the steepness, or slope, of a line?


Start at any point on Line A and move horizontally as directed. y

y

Line A 0

Line B

x

0

x

y

a. Move 1 unit to the right. How many units does the line rise? What result do you get if you start at any other point on the line?

? 1 unit 0

x

b. Move 2 units to the right. How many units does the line rise? What result do you get if you start at any other point on the line? c. Move 3 units to the right. How many units does the line rise? What result do you get if you start at any other point on the line?

280

2

Repeat the same moves with Line B. What results do you get?

3

Write a brief statement about what you have noticed. Be ready to share your results with the class.



Find Slopes of Some Lines For each pair of points, plot the points and draw the line through them. Then calculate the slope of the line. 1

(1, 4) and (2, 1)

2

(1, 6) and (6, 4)

3

(4, 2) and (2, 3)

4

(3, 2) and (7, 2)

5

(1, 1) and (5, 1)

What can you say about the slope of lines that slant downward from left to right?

What do all of the lines in 1–3 have in common? What do you notice about their slopes? What do the lines in 4 and 5 have in common? What do you notice about their slopes? How to Find the Slope of a Line Choose two points along the line. The rise is the difference in the y-coordinates. The run is the difference in the x-coordinates. The slope of the line is the ratio of the rise to the run: y (4, 5)

Rise 5 3 2

Rise Slope Run

(1, 3) Run 4 1 3 Slope

0

2 Rise 3 Run x

hot words

rise run slope

W Homework page 305


281


5

WORKING WITH SLOPE AND INTERCEPT

What’s in an Equation? Did you know that the equation of a line is like a coded message? You will be experimenting with the graphs of some lines. This will help you see how information about the slope and y-intercept of a line is contained in its equation.

Connect Equations, Slopes, and Intercepts How are the slope and y-intercept of a line related to its equation?

Your teacher will give you and a partner two equations to work with. For each equation, do the following. 1

Make a table of values that satisfy the equation.

2

Plot the points from your table to help you graph the equation.

3

Find the slope of your graph.

4

Find the y-intercept of your graph.

For each equation, how are the slope and y-intercept of the graph related to the equation? What connections do you see? The y-Intercept of a Line The y-intercept of a line is the point where it crosses the y-axis. y

This line crosses the y-axis at (0, 3), so its y-intercept is (0, 3). 0

x

The y-intercept of this line is (0, 2).

282



Analyze a Set of Equations What can you say about lines just by looking at their equations?

Consider the set of six equations shown below. A. y = 5x

B. y = 31 x – 2

C. y = –x

D. y = 3x + 4

E. y = 0.5x + 1

F. y = –2x – 421

Work with classmates to write a response for each of the following. Be prepared to discuss your responses with the class. 1

Which equation has the flattest graph? How do you know?

2

Which equations have graphs that slant downward as you go from left to right? How do you know?

3

Which equation has a graph that crosses the y-axis at the highest point? Why?

4

Write an equation of your own whose graph would be steeper than the graph of any of the six given equations.

5

Write an equation of your own whose graph would be flatter than those of the six given equations.

6

Write an equation of your own whose graph would cross the y-axis at a lower point than would any graph of the six given equations.

hot words

y-intercept slope

W Homework page 306


283


6 WRITING THE EQUATION OF A LINE

The Shortest Distance Between Two Points . . . Until now, you have usually started with the equation of a line and then created its graph. Now suppose you have the graph of a line and want to find its equation. You will be starting with two points, drawing the line through the points, and finding a way to write the line’s equation.

Write the Equation of a Line How can you write the equation of a line that goes through two points?

For this investigation, you will need a copy of the reproducible Starting Points. 1

Choose any two of the five given points on the coordinate plane. Using a straightedge, carefully draw a line through the points. y

C D

B E A

0

284

x

2

Is the slope of your line positive, negative, or zero? Is the y-intercept positive, negative, or zero? Based on your answers, what can you say about the equation of your line? What do you think your equation will look like?

3

Calculate the slope of your line.

4

Find the y-intercept of your line.

5

Use the information you have gathered to write the equation of your line.



Check the Equation Refer to your line and the equation you wrote for it. Write a brief response to each of the following items. 1

According to your graph, give the coordinates of at least three points that lie on your line.

2

Since these points lie on the line, how do their coordinates relate to the line’s equation?

3

Use this fact to see if your equation is correct. If your equation is not correct, look for errors in your work and revise your equation.

How can you be sure the equation you wrote is correct?

Update the Function Album Add a page to your function album that summarizes what you know about linear functions. Include the following: ■

a verbal description of this family of functions

■


■

a sample equation for the function family

■

a description of how to calculate slope

■

a description of how to find the equation of a line through two points

■


hot words

y-intercept slope

W Homework page 307


285

PHASE THREE



In this phase, you will look at functions whose graphs are parabolas. These are called quadratic functions. The first situation you will explore involves the area of a slide projector’s image. By collecting data, graphing it, and writing an equation, you will see how this situation relates to quadratic functions. You will also see how a real-world problem about rectangular pens for animals results in a quadratic function.

Quadratic Functions WHAT’S THE MATH? Investigations in this section focus on: ALGEBRA ■

Writing equations for quadratic functions

■

Graphing parabolas

■

Understanding how the constant a in the equation y ax2 affects the graph of the equation

■

Using quadratic functions to solve problems

PATTERNS ■

Describing and extending patterns that result from quadratic functions


FAMILY PORTRAITS

287


7 EXPLORING A QUADRATIC RELATIONSHIP

The Area of a Projected Image How is the area of a slide’s projected image related to the projector’s distance from the screen? You will see how this relationship—called a quadratic function—can be described with a graph and an equation.

Make a Graph of Area Versus Distance What does the graph of this area-versusdistance relationship look like?

The figure below shows the relationship between the area of a projected image and the projector’s distance from the screen. 1

2

Make a table of values that relates the area of the image to the distance of the projector from the screen.

Distance from screen (m)

Area of image (cm2)

0 1 2 3 4 5

0

Plot the points from your table to help make a graph of the relationship.

How is the graph of this relationship different from the graph of a linear function? Area Versus Distance When a square slide is projected onto a screen, the area of the image depends upon the distance of the projector from the screen.

3m 20 cm

2m 1m

0m

288



Write an Equation for the Relationship Use your table to help you write an equation that describes the relationship between the area of a projected image and the distance of the projector from the screen. Given any distance, d, your equation should tell how to calculate the area, A.

What equation can you write to describe the area-versusdistance relationship?

Write About the Results Write a brief summary of your findings. Include the following: ■

a description of how your equation is different from the equation of a linear function

■

a description of how your graph is different from the graph of a linear function

■

a discussion of how the area of the image changes as the distance changes

hot words

quadratic equation

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289


8 GRAPHING PARABOLAS

Going Around a Curve What does the graph of a quadratic function look like? You have already seen one quadratic relationship: the area of a slide’s projected image as a function of the projector’s distance from the screen. Now you will explore the graphs of other equations that contain an x2 term.

Make and Describe a Graph What does the graph of the function y x2 look like?

Equations of the form y ax2 bx c, where a 0, are called quadratic functions. The simplest of these is y x2. 1

Make a table of values for the equation y x2. Be sure to include at least four negative values of x.

2

Use your table to plot points and make a graph.

The graph you made is called a parabola. Write a description of your graph. Include the ideas of symmetry and axis of symmetry in your description.

Symmetry If you can fold a figure in half so that the two halves match perfectly, then the figure has symmetry. The fold line is called the line of symmetry or the axis of symmetry.

290



Experiment with Parabolas Work with classmates to make a graph for each of the six equations shown. Then be ready to discuss the questions below with the rest of the class. Column A

Column B

y = x2

y = –x2

y = 2x2

y = –2x2

y = 21 x2

y = – 21 x2

1

How do the graphs of the equations in Column A compare to those in Column B? What is similar? What is different? Include a discussion of symmetry in your answer.

2

What does the constant a in the equation y ax2 tell you about the graph?

3

Answer this question without actually making any graphs: How does the graph of y 3x2 compare to the graph of y 5x2?

How is the shape of a parabola related to its equation?

hot words

parabola symmetry

W Homework page 309


291


9 SOLVING A PROBLEM THAT INVOLVES A PARABOLA

Fenced In Suppose you have 32 m of fencing material. What is the largest rectangle you can fence off? And what does this problem have to do with parabolas? You will explore this situation and make a graph to describe it. This will help you see the connection between perimeters, areas, and parabolas.

Explore Rectangular Pens How can you find a rectangle with the greatest area for a given perimeter?

A farmer has 32 m of fencing material and wants to fence off a rectangular pen for animals. One side of the pen must lie along a creek. What length along the creek results in a pen with the greatest area for the animals? 1

What is the perimeter of any pen the farmer can make?

2

Use a sheet of graph paper to help sketch all of the possible pens that have whole-number lengths. One possible pen is shown here.

Area 60 m2 Width 10 m Length 6 m Creek

3

Find the area of each pen.

Which length along the creek results in the pen with the greatest area?

292



Plot the Pens Make a graph that shows the relationship between the length of the fence along the creek and the area of the rectangular pen.

How can a graph of the situation help solve the problem?

100 90 80 70 60 2 Area of Pen (m ) 50 40 30 20 10 0

1 2 3 4 5 6 7 8 9 Length Along the Creek (m)

1

Describe your graph.

2

What is the line of symmetry of your graph?

3

From your graph, what length along the creek results in a pen with the greatest area? Does this agree with your earlier results?

Update the Function Album Add a page to your function album that summarizes what you know about quadratic functions. Include the following: ■

a verbal description of the function family

■

sample equations for the function family

■

a typical graph from the function family and a discussion of symmetry

■

a discussion of how the constant a affects the graph of y ax2

■


hot words

area function

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293

PHASEFOUR



The more times you fold a sheet of paper in half, the more rectangular regions there are when unfolded. How is the number of regions related to the number of folds? You may be surprised when you collect data to help answer this question. Exponential functions have some special characteristics. In this phase, you will explore laws of exponents. You will also see how your calculator handles very large and very small numbers.

Exponential Functions WHAT’S THE MATH? Investigations in this section focus on: ALGEBRA ■

Graphing exponential functions

■

Writing equations for exponential functions

PATTERNS ■

Describing and extending patterns that result from exponential functions

NUMBER ■

Developing and using laws of exponents

■

Working with scientific notation


FAMILY PORTRAITS

295


10 EXPLORING EXPONENTIAL FUNCTIONS

Folds and Regions Imagine folding a sheet of paper in half 25 times. If you unfolded the sheet of paper, how many rectangular regions would be formed? You will explore this question and see how it is connected to exponential functions. Then you will compare the graphs of several exponential functions.

Calculate the Number of Regions What is the relationship between the number of times a sheet of paper is folded and the number of regions formed?

296

When you fold a sheet of paper in half one time, two regions are formed. If you fold the sheet of paper in half again, four regions are formed.

0 folds 1 region

1 fold 2 regions

2 folds 4 regions

1

Make a table that relates the number of folds to the number of regions. (Save your table for use later in this unit!)

2

Describe the relationship between the number of folds and the number of regions using words, variables, or any other method that makes sense to you.

3

Suppose a sheet of paper could be folded in half 25 times. Find the number of regions that would be formed.



Compare Graphs of Exponential Functions Functions of the form y ax are called exponential functions. Work with classmates to compare the exponential functions y 3x, y 4x, and y 5x. 1

Make a table of values for each function. Include x 0, 1, 2, 3, 4, 5, and 6. One way to organize your tables is shown here. (Save your tables for use later in this unit!) x

3x

4x

0

30 = 1

40 = ?

1

31 = 3

What do graphs of exponential functions look like?

5x

2 3 4 5 2

Make graphs for all three functions on the same set of axes.

Write a brief description of what all three graphs have in common. What are some differences?

hot words

exponent power

W Homework page 311


297


11 DEVELOPING LAWS OF EXPONENTS

Rules of the Road for Exponents How can you evaluate an expression that contains more than one exponent? As you will see, the tables of values you have already developed can be quite useful in exploring laws of exponents. You will also see how one of these laws can help you make more complete graphs of exponential functions.

Evaluate Expressions with Exponents How can tables of values help you evaluate expressions with exponents?

Here is an example of how a table can help you evaluate 23 • 24 and translate the result back to an exponent.

23 • 24 8 • 16 128 27 22 23 24 25 26 27

4 8 16 32 64 128

Use the tables you have already made to help you evaluate each expression. Then use your tables to translate your result back into an exponent. Look for patterns as you work. 1. 32 • 33 215 4. 9 2 7. (25)4

2. 27 • 212 45 5. 3 4 8. (26)3

3. 210 • 26 221 6. 3 2 9. (52)2

Write About Laws of Exponents Write a summary of any patterns you noticed in the previous investigation. ■ ■ ■

298

In general, what is true about an • am? an In general, what is true about m? a In general, what is true about (an)m?



Graph an Exponential Function How can negative exponents help you graph exponential functions?

Negative Exponents

If a is a positive number, then an For example, 43

1 . an

1 1 1 . 3 4 4 4 4 64

Now that you know how to work with negative exponents, you can graph exponential functions for both positive and negative values of x. Your teacher will give you a function to work with. 1

Make a table of values for your function. Be sure to include at least three negative values for x, x 0, and at least three positive values for x.

2

Plot the points to help make a graph of your equation.

Write a brief description of the shape of your graph.

hot words

exponent power

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299


12 WORKING WITH SCIENTIFIC NOTATION

The Very Large and the Very Small You have seen how quickly the function y 2x grows as x increases. What is the largest value of 2x that your calculator can handle? What is the smallest? As you explore scientific notation, you will see how it can help you answer these questions about your calculator.

Make a Powers-of-10 Table What patterns do you notice in a table of powers of 10?

You will need a copy of the Powers-of-10 Table. Fill in as much of the table as you can. Work with students around you to help with parts of the table you are unsure about. Power of Ten

Number

Common Name of Number

Power of Ten

Number (written as a fraction)

Number (written as a decimal)

Common Name of Number

1 100

0.01

one onehundredth

100 101

10 –1

102

10 –2

103

10 –3

Be ready to discuss the following questions with the class. ■

What patterns do you notice in your table?

■

How can you tell how many zeros are in a number based on the power of 10?

Scientific Notation

Numbers written in scientific notation have the form a 10 n, where a is a number between 1 and 10, and n is an integer. Here are some examples: 4,000 4 103 320,000,000 3.2 108 0.0007 7 104

300



Explore a Calculator’s Limits Find the greatest power of 2 you can enter into your calculator without causing it to display an error message. 1

Experiment by entering various powers of 2 into your calculator using the y x or ^ key.

2

How does your calculator display this largest power of 2?

3

How would you write this number using scientific notation?

4

If you wrote out this number in full, how many digits would it have?

How does your calculator display very large and very small numbers?

Repeat the above process to find the smallest (negative) power of 2 you can enter into your calculator.

Update the Function Album Add a page to your function album that summarizes what you know about exponential functions. Include the following: ■

sample equations for the function family

■


■

a summary of laws of exponents

■

a description of how to write numbers in scientific notation

■


hot words

power scientific notation

W Homework page 313


301


1

Inputs and Outputs

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HW

Applying Skills

Extending Concepts

A machine takes input values and gives output values. Some sample inputs and outputs are shown. Describe each machine’s rule using words and/or an equation. 1.

3.

Input (x) 2 0 1 2.2

Output (y) 6 0 3 6.6

2.

Input (x) 3 0 2 6

Output (y) 2 2 2 2

4.

Input (x) 3 0 1 2

Output (y) 9 0 1 4

Input (x) 9 16 81 100

Output (y) 3, 3 4, 4 9, 9 10, 10

5. Which rules in items 1–4 are functions?

Why? Tell whether each graph shows a function and explain your thinking. 6. y

7. y

10. Is the relationship y x a function?

Is the relationship y x 2 a function? Explain your thinking.

11. Describe two real-world examples of

functions. How do you know these relationships are functions? 12. Sketch a graph of your own that

represents a function and a graph that does not represent a function. How can you tell whether a graph represents a function?

Making Connections

13. Suppose that a weight is attached to

the end of a swinging pendulum. As the pendulum swings, the distance of the weight from the center of the pendulum’s arc varies as shown in the graph. Distance from Center

x

9.

Time

Is the relationship between distance and time a function . . .

Salary

Temperature

8.

x

a. if time is the input value and distance Time

Age

from the center is the output value? b. if distance from the center is the input

value and time is the output value? Explain your answers.

302

FAMILY PORTRAITS • HOMEWORK 1


2

Keeping a Steady Pace

ork w e om

HW

Applying Skills

Extending Concepts

Graph each equation on the same coordinate plane. 1. y 2x

2. y 0.4x

3. y 3x

4. y 4x

Tell whether each function is a direct variation function. If it is, write an equation describing the relationship. If it is not, explain why not.

Tell whether each graph represents an equation of the form y kx. If it does, tell whether k is positive or negative. If not, say why not. 5.

6.

y

y

x

12. A person’s salary as a function of the

number of hours she works (assume she makes $15 per hour) 13. The number of legs as a function of the

number of dogs 14. The cost of a rental car as a function of

x

the number of miles driven (the car costs $20 plus 10 cents per mile) 15. The height of a child as a function of his

or her age 7.

8.

y

y

x

16. The revenue for a show as a function of x

9. Where do the graphs of y 8x and

y 4x intersect?

the number of tickets sold (tickets cost $18 each)

Making Connections

17. At maximum speed, a cheetah can run

10. Which equation below has the steepest

graph? the flattest graph? How do you know? y x

y 5x

y 0.5x

y 6x

11. Which equations below have graphs that

slant downward from left to right? How do you know? yx

y 8x

y 0.1x

y 2x

about 100 feet per second. The giant tortoise can cover about 0.25 feet per second. For each animal, write an equation relating distance and time. Assume each animal is moving at its maximum speed. What do the graphs of these equations look like? How are the graphs alike? How are they different?


303


3

How Long Is a Meter?

ork w e om

HW

11. The graph appears only in the first and

Applying Skills

second quadrants.

1. a. Make a table of values for the function

y 12x . Include x 24, 12, 6, 4, 3, 1, 0.5, 0.5, 1, 3, 4, 6, 12, 24.

b. Plot your points and make a graph of

the equation. Tell whether each of the following could be the first-quadrant graph of the equation y xk (k is a positive number). Explain why or why not. 2. y

3. y

12. In the first quadrant, the graph rises

steeply as x gets closer to zero. 13. As x gets very large, the graph will

eventually cross the x-axis.

Extending Concepts

14. Jim makes $10 per hour. His total pay

is a function of the number of hours he works. Is this an inverse variation function? Why or why not? 15. Suppose you drive at constant speed.

x

4. y

x

5. y

The time (in hours) it takes to drive 500 miles and your speed (in mph) are related by the equation t 500 s . a. Make a graph of this equation. Show s

on the horizontal axis. b. What happens to your graph as s gets x

x

Tell whether each equation represents direct or inverse variation. 6. y 80x

80

8. y x

7. xy 80

y

9. x 80

Tell whether each statement about the graph of y xk is true or false. If it is false, change it to make it true. Assume that k is positive. 10. As x increases in the first quadrant, the

graph rises steeply.

304

very large? What happens as s gets very close to zero? Explain why your observations make sense.


Writing

16. Write a short summary about the graphs

of direct variation and inverse variation functions. Describe how the graphs of the two types of functions differ.


4

A New Slant on Linear Functions

ork w e om

HW

Applying Skills

Extending Concepts

14. A line has slope 12 , and the point (1, 4)

Find the slope of each line. 1.

2.

y

0

lies on the line. Find the coordinates of a second point that lies on the line and graph the line.

y

x 0

x

15. Explain why the slope of a vertical line is

undefined. For each pair of points, plot the points and draw the line through them. Then calculate the slope of the line. 3. (1, 2) and (3, 6) 4. (2, 6) and (4, 0) 5. (3, 2) and (1, 4) 6. (2, 5) and (1, 5) 7. (2, 2) and (0, 5) 8. (4, 1) and (6, 3) 9. If a line is horizontal, what is its slope?

Making Connections

16. Apollo 10, which orbited the moon, was

launched in 1969. The graph represents one portion of its journey and shows the distance traveled as a function of time. a. Choose two points on the graph and

write their coordinates. Find the rise and run corresponding to this pair of points. What does the rise represent? What does the run represent?

Why?

100,000

10. If the slope of a line is negative, does it

slope upward or downward as you move from left to right? Why? 11. If the slope of a line is 4, how many units

75,000 Distance Traveled (miles)

50,000 25,000

does the line rise if you move 2 units to the right?

1

12. If the slope of a line is 5, what rise

corresponds to a run of 3?

13. If the slope of a line is 6, what run

2 3 Time (hours)

4

b. What is the slope of the graph? What

does it represent?

corresponds to a rise of 12?


305


5

What’s in an Equation?

ork w e om

HW

Applying Skills

Extending Concepts

Find the y-intercept of each line. 1.

2.

y

0

Find the slope and y-intercept of the graph of each equation.

y 0

x

x

Find the slope and y-intercept of the graph of each equation. 3. y 5x 8

4. y 2x 1

5. y x 4

6. y 0.3x 2.5

7. y 3x 9. y 2 x

1 4 10. y 2

8. y x

2 5

11. Which equation in items 3–10 has the

steepest graph? the flattest graph? 12. Which equations in items 3–10 have

graphs that slant upwards from left to right? Write the equation of a line with the following slope and y-intercept: 13. slope 7, y-intercept (0, 9)

17. 2y 4x 6

18. x y 7

19. y 5(x 1)

20. y 2x 4

In items 21–23, use the equations below. 1 A. y 4x 2 B. y x 3 2 C. y 2x 4 21. Use the slope and y-intercept to graph

each equation. Label the lines A, B, and C. 22. Write an equation whose graph crosses

the y-axis at a lower point than all of the given lines. 23. Write an equation whose graph crosses

the y-axis at a higher point than Line A and that is steeper than all three graphs.

Making Connections

24. Fahrenheit and Celsius are different

15. slope 4.5, y-intercept (0, 0)

temperature scales. The equation F 1.8C 32 describes the relationship between Fahrenheit and Celsius.

16. slope 6, y-intercept (0, 1)

a. What is the slope of the graph of this

14. slope 4, y-intercept (0, 1)

equation? How much does the Fahrenheit temperature increase when the Celsius temperature increases by 1°? by 2°? How do you know? b. What is the y-intercept of the graph?

What is the Fahrenheit temperature when the Celsius temperature is 0°?

306



6 ork w e om

HW

The Shortest Distance Between Two Points . . .

Applying Skills

1. a. Plot the points (1, 4) and (3, 0) on a

Extending Concepts

4. Find the equation of the line shown here.

coordinate plane and draw the line through the points.

y

b. Is the slope of the line positive,

negative, or zero? 0

c. Is the y-intercept of the line positive,

x

negative, or zero? d. What can you say about the equation

of the line? e. Calculate the slope of the line.

Writing

5. Answer the Dr. Math letter.

f. Use your graph to find the y-intercept

of the line. g. Write the equation of the line. h. Use your graph to find the

coordinates of three new points that lie on the line. Check that the coordinates of each point satisfy the equation that you wrote. Show your work. 2. Repeat item 1 using the points (1, 2)

and (2, 5). 3. Repeat item 1 using the points (2, 1)

and (6, 1).

Dear Dr. Math, I wanted to find the equation of the line through the points (1, 0) and (3, 1). I plotted the points and drew the line as I’ve shown. I calculated the slope this way: rise 1 0 1, y run 3 (1) 4, slope rriusne 41. The y-intercept looked 0 x 1 like about 3, so I wrote this equation for the line: y 41x 31. But when I checked, I found that the coordinates (3, 1) didn’t satisfy this equation. I guess the y-intercept wasn’t quite right after all. How can I figure out exactly what the y-intercept is? Mario


307


7

The Area of a Projected Image

ork w e om

HW

c. How is your equation different from a

Applying Skills

linear equation?

Tell whether or not each function is a quadratic function and why. 1. y 3x 10

d. Predict what the graph of your

2. y 5x

2

3 2 4. y 8 x 5. y 0.1x x For each table of values, write an equation that describes the relationship between the two variables. Then complete the table. 3. y

6.

x 0 1 2 3 4

y 0 3 12 27 ?

7.

s 0 1 2 3 4

t 0 8 32 72 ?

8.

p 0 1 2 3 4

q 0 2 8 18 ?

Extending Concepts

equation will look like. How will it differ from the graph of a linear equation? e. Plot the points from your table and

make a graph of the relationship.

Making Connections

10. Suppose that a rock is dropped from a

tall building. Let d represent the distance (in feet) from the point of release and t the time (in seconds) since the rock was released. The variables d and t are related by the equation d 16t 2. a. Is this equation linear or quadratic?

9. a. Make a table of values that relates the

side length of a square (measured in yards) and its area (measured in square feet). A table has been started for you. 1 yd 1 ft 1 ft 1 ft 2 yd

Side Length (yd) 0 1 2 3 4 5

Area (ft)2 0 9

b. Write an equation that describes the

relationship between the side length, s, and the area, A.

308


How do you know? b. Make a table of values that satisfy the

equation. Use values 0 through 5 for t. c. Plot your points and make a graph of

the equation. d. How does the distance that the rock

falls during the first second compare with the distance that it falls during the fifth second? Explain your thinking.


8

Going Around a Curve

ork w e om

HW

14. If a is negative, the graph of y ax has 2

Applying Skills

no points above the x-axis. Make a table of values for each equation. Include negative as well as positive values of x. Use your table to plot points and make a graph. 1. y x

2. y 3x

2

2

3. y 0.5x

4. y 2x

2

2

5. y 0.25x

6. y 4x

2

15. The graph of y ax is symmetric about 2

the x-axis. 16. You can get the graph of y 2x by 2

flipping the graph of y 2x2 over the x-axis.

2

The graphs shown here represent the equations y 5x2, y 2x2, y 0.3x2, y x2, and y 3x2. Which graph represents the following equations? C

B A

D E

7. y 0.3x

8. y 5x

9. y x

10. y 2x

2

2

2 2

11. y 3x

2

Extending Concepts

Tell whether each statement is true or false. If it is true, explain why it makes sense. If it is false, change it so that it is true.

Writing


Dear Dr. Math, I’m confused. I was trying to figure out the slope of the graph of y x 2. I know the slope is the amount that the graph rises when you move one unit to the right. When I moved one unit to the right from x 1 to x 2, the rise was 3. So I figured the slope must be 3. But when I moved one unit to the right from x 2 to x 3, the rise was 5. What’s going on? What is the slope? Why did I get two different answers for the rise? Going ’Round The Bend y 5

12. If the point (3, 80) lies on the graph of

an equation of the form y ax , then the point (3, 80) also lies on the graph.

3

2

x 1

2

3

13. The graph of y 10x is wider than the 2

graph of y 9x2.


309


9 ork w e om

HW

Fenced In

Applying Skills

Making Connections

For each table of values, (a) make a graph; (b) describe the graph and find its line of symmetry; (c) find the value of x where the graph reaches its maximum height. 1.

x 0 1 2 3 4 5 6

y 0 10 16 18 16 10 0

2.

x 0 1 2 3 4 5 6 7

y 0 12 20 24 24 20 12 0

Extending Concepts

4. If a ball is thrown straight up at a speed

of 96 feet per second, its height (in feet) after t seconds will be h 96t 16t 2. a. Find the height of the ball after 0, 1, 2,

3, 4, 5, and 6 seconds. Show your results in a table. (For example, the height after 2 seconds is 96 • 2 16 • 22 192 16 • 4 192 64 128 feet). b. Use your table to plot points and

make a graph. Show time on the horizontal axis and height on the vertical axis. c. According to your graph, when does

3. Suppose you want to find two positive

numbers whose sum is 11 and whose product is as large as possible. a. List all the whole-number possibilities

for the two numbers. For each one, find the product. Organize your results in a table like the one shown. 1st Number 1 2 • •

2nd Number 10 9 • •

Product 10 18 • •

b. Make a graph. Show the value of the

first number on the horizontal axis and the product on the vertical axis. c. According to your graph, what value

for the first number results in the largest product?

310


the ball reach its maximum height? d. Use your graph to estimate the times

at which the height of the ball is 100 feet. Why do you think there are two such times? e. How could you have predicted from

the equation that the parabola would open downward?


Folds and Regions

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HW

Applying Skills

Extending Concepts

1. Make a table of values for the function

y 6 . Include x 0, 1, 2, 3, 4, 5. Plot the points and make a graph of the function. x

Suppose that a is greater than 1. Tell whether each graph below could be the graph of the function y ax. Explain why or why not. 2. y

3. y

x

x

20? 50? b. Which graph rises more steeply,

y x 2 or y 2x ? Explain your thinking.

11. On April 1, Kate receives $10,000. At

the end of the month, this amount is squared. On April 1, John receives $1. The next day and each of the following days until the end of the month, his money is doubled. a. Predict who will have the most money

x

4. y

2

10. a. Which is larger, x or 2 if x is 3? 8?

at the end of the month and estimate how much more this person will have.

5. y

b. Using an exponent, write the amount x

of money each person will have at the end of the month.

x

Tell whether each statement is true or false. If it is false, change it so that it is true. 6. The graph of y 9 rises more steeply x

c. Use your calculator to calculate how

much money each person will have at the end of the month. Who will have more money? How much more?

than the graph of y 8x.

7. The y-intercept of the graph of y 7

x

is a higher point on the y-axis than the y-intercept of the graph of y 5x. 8. The y-intercept of the graph of y 9 is x

(0, 9). 9. The graph of y 8 passes through the x

point (1, 8).

Making Connections

12. Under favorable laboratory conditions,

the number of cholera bacteria in a colony can double every half hour. If the colony starts with 1 bacterium, how many bacteria will there be at the end of 12 hours?


311


ork w e om

HW

Rules of the Road for Exponents

Applying Skills

Extending Concepts

1. Make tables of values for the functions

y 2 and y 3 . Include the values x 0, 1, 2, ..., 12. x

x

Use your tables from item 1 to help you evaluate each expression. Then translate your result back into an expression using an exponent. 3

5

2. 2 • 2 3 4

5. (2 )

2

7

3. 3 • 3 2 5

6. (3 )

6

5

4. 2 • 2

22. 2

3

25. 4

312

23. 3

90 1

0

28. 3 , 10 3

20

29. 20 , 3 4

30. 2

, 42

4 2

29 37 9. 6 4 2 3 Use the laws of exponents to simplify each expression. Write your answer using an exponent. 214 3 8 5 3 10. 4 • 4 11. (x ) 12. 2 2 516 4 9 7 9 15. a • a 13. 9 14. (8 ) 5 618 6 10 3 10 16. 5 • 5 17. (a ) 18. 15 6 x21 8 20 7 2 19. x • x 20. (3 ) 21. 15 x Evaluate each expression. Write your answer as a fraction. 2

0

27. 90 , 1

7. (2 )

8.

3

Tell whether the second expression is greater than, less than, or equal to the first expression.

5

24. 2

4

26. 5


Making Connections

31. The half-life of a radioactive substance

is the time it takes for half the amount originally present to decay. If the half-life of a particular substance is 1 year and if 1 gram is originally present, the amount remaining after x years will be 2x grams. a. How much will remain after 5 years?

after 8 years? b. Will the amount remaining ever reach

zero? How do you know?


k wor e m o

HW

The Very Large and the Very Small

Applying Skills

Extending Concepts

Write each number using scientific notation. 1. 8,200

2. 870,000

3. 2,500,000

4. 76,000

5. 0.03

6. 0.00064

7. 0.0014

8. 0.00000007

4

2

11. 6 10

2

13. 7 10

8

15. 3 10

the sun is about 3.7 107 miles. The distance from Pluto to the sun is about 3.7 109 miles. a. Estimate how much greater the

Write each number using standard notation. 9. 4.7 10

19. The distance from the planet Mercury to

7

10. 1.9 10

10

12. 8.5 10

5

14. 3.3 10

distance from Pluto to the sun is than the distance from Mercury to the sun. b. Calculate how much further it is

to the sun from Pluto than from Mercury by writing the distances in standard notation and subtracting. Give your answer in scientific notation and standard notation.

11

16. 9.2 10

Use your calculator to multiply each pair of numbers. Give your answer in scientific notation and in standard notation.

Writing


17. 23,000 and 11,400,000 18. 0.000041 and 0.000006

Dear Dr. Math, I’ve found a great method for converting a number into scientific notation. I just count the number of zeros and that’s my exponent. So 24,000 is 24 103 because it has 3 zeros. 32,800,000 is 328 105 because it has 5 zeros. My method even works for the small numbers: 0.0005 has 3 zeros after the decimal point so it’s 5 103. How do you like my method? Should I tell my teacher about it? Lucy


313

MS_GR8-ENG/SP Glossary 4.0 1/8/04 12:32 PM Page 314 mac85 Pdrive 01:es:glencoe:glmat1-Gr8:G8 Mathscapes Student Guide GLO:

Glossary/Glosario Cómo usar el glosario en español: 1. Busca el término en inglés que desees encontrar. 2. El término en español, junto con la definición, se encuentra debajo del término en inglés.

A addition growth number a number that when added to a given number a certain number of times results in a given goal number número del aumento de la adición número que cuando es sumado a un número dado una cierta cantidad de veces tiene como resultado un número final determinado Example: Grow 8 into 14 in two steps by adding (8 3 3 14). 3 is the addition growth number. Ejemplo: Aumenta el 8 a 14 en dos pasos sumando (8 3 3 14). 3 es el número del aumento de la adición. algebra a branch of mathematics in which symbols are used to represent numbers and express mathematical relationships álgebra rama de las matemáticas en la cual se usan símbolos para representar números y expresar relaciones matemáticas angle of elevation an angle formed by an upward line of sight and the horizontal ángulo de elevación ángulo formado por una recta ascendente de la vista y la horizontal

Example: The average of 3, 4, 7, and 10 is (3 4 7 10) ÷ 4 or 6. Ejemplo: El promedio de 3, 4, 7 y 10 es (3 4 7 10) 4 o 6. average speed the average rate at which an object moves promedio de velocidad el promedio en el que un objeto se mueve

C combination a selection of elements from a larger set in which the order does not matter combinación selección de elementos de un conjunto más grande en el cual el orden no importa Example: 456, 564, and 654 are one combination of three digits from 4,567. Ejemplo: 456, 564 y 654 son una combinación de tres dígitos de 4,567. conic section the curved shape that results when a conical surface is intersected by a plane sección cónica forma curvada que es el resultado de una superficie cónica cuando es intersecada por un plano

Example/Ejemplo:

angle of elevation ángulo de elevación

Example/Ejemplo: horizontal/horizontal

area the size of a surface, usually expressed in square units área el tamaño de una superficie, por lo general se expresa en unidades cuadradas Example/Ejemplo: 2 ft 2 pie

area ⫽ 8 ft2 área ⫽ 8 pie2

4 ft 4 pie

314

average the sum of a set of values divided by the number of values promedio la suma de un conjunto de valores dividido entre el número de valores

GLOSSARY/GLOSARIO

This ellipse is a conic section. Esta elipse es una sección cónica.

coordinate graph the representation of points in space in relation to reference lines—usually, a horizontal x-axis and a vertical y-axis gráficas de coordenadas la representación de puntos en el espacio en relación con las rectas de referencia —generalmente, un eje x horizontal y un eje y vertical


coordinates an ordered pair of numbers that describes a point on a coordinate graph. The first number in the pair represents the point’s distance from the origin (0, 0) along the x-axis, and the second represents its distance from the origin along the y-axis. coordenadas un par de números ordenados que describen un punto en una gráfica de coordenadas. El primer número en el par representa la distancia del punto desde el origen (0, 0) a lo largo del eje x y el segundo representa su distancia desde el origen a lo largo del eje y. Example/Ejemplo:

dependent events a group of happenings, each of which affects the probability of the occurrence of the others sucesos dependientes un grupo de eventos, en el que cada uno afecta la probabilidad de la ocurrencia de los otros distance the length of the shortest line segment between two points, lines, planes, and so forth distancia la longitud del segmento más corto entre dos puntos, rectas, planos, etcétera distance-from graph a coordinate graph that shows distance from a specified point as a function of time gráfica de distancia gráfica de coordenadas que muestra la distancia desde un punto específico como una función de tiempo

y 4 3 2 1

D

P (2, 3)

x 1 2 3 4 Point P has coordinates (2, 3). El punto P tiene las coordenadas (2, 3).

correlation the way in which a change in one variable corresponds to a change in another correlación la forma en la que un cambio en una variable corresponde al cambio en otra cross section the figure formed by the intersection of a solid and a plane sección cruzada la figura formada por la intersección de un sólido y un plano Example/Ejemplo:

distributive property the mathematical rule that states that for any numbers a, b, and c, a(b c) ab ac propiedad distributiva regla matemática que expresa que para cualquier número a, b y c, a(b c) ab ac

E edge a line along which two planes of a solid figure meet arista una línea en donde se unen dos planos de un cuerpo geométrico equation a mathematical sentence stating that two expressions are equal ecuación enunciado matemático que expresa que dos expresiones son iguales Example/Ejemplo: 3 (7 8) 9 5 estimate an approximation or rough calculation estimado una aproximación o calculación aproximada

cross section sección cruzada

cylinder a solid shape with parallel circular bases cilindro un cuerpo geométrico con bases circulares paralelas Example/Ejemplo:

experimental probability a ratio that shows the total number of times the favorable outcome happened to the total number of times the experiment was done probabilidad experimental una razón que muestra el número total de veces que ocurrió un resultado favorable en el número total de veces que se realizó el experimento exponent a numeral that indicates how many times a number or expression is to be multiplied by itself exponente numeral que indica cuántas veces un número o expresión se debe multiplicar por sí mismo

cylinder cilindro

Example: In the equation 23 8, the exponent is 3. Ejemplo: En la ecuación 23 8, el exponente es 3.

GLOSSARY/GLOSARIO

315


expression a mathematical combination of numbers, variables, and operations expresión una combinación matemática de números, variables y operaciones Example/Ejemplo: 6x y 2

growth model a description of the way data change over time modelo de crecimiento descripción de la forma en que cambian los datos con el tiempo

F

I

face a two-dimensional side of a three-dimensional figure cara un lado bidimensional de una figura tridimensional

integers the set of all whole numbers and their additive inverses {. . . 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5 . . .} enteros el conjunto de todos los números enteros y sus inversos aditivos {. . . 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5 . . .}

factorial represented by the symbol !, the product of all the whole numbers between 1 and a given positive whole number factorial representado por el símbolo !, el producto de todos los números enteros entre 1 y un número positivo dado Example/Ejemplo: 5! 1 2 3 4 5 120 forecast to predict a trend, based on statistical data predicción predecir una tendencia, en base a los datos estadísticos formula an equation that shows the relationship between two or more quantities; a calculation performed by spreadsheet fórmula ecuación que muestra la relación entre dos o más cantidades; cálculo realizado en una hoja de cálculo Example: A πr2 is the formula for calculating the area of a circle; A2 B2 is a spreadsheet formula Ejemplo: A πr2 es la fórmula para calcular el área de un círculo; A2 B2 es una fórmula de una hoja de cálculo function assigns exactly one output value to each input value función asigna exactamente un valor independiente a cada valor dependiente Example: You are driving at 50 mi/hr. There is a relationship between the amount of time you drive and the distance you will travel. You say that the distance is a function of the time. Ejemplo: Estás manejando a 50 mi/h. Hay una relación entre la cantidad de tiempo que manejas y la distancia que viajarás. Tú dices que la distancia es una función del tiempo.

316

G

L like terms terms that include the same variables raised to the same powers. Like terms can be combined. términos semejantes términos que incluyen las mismas variables elevadas a las mismas potencias. Los términos semejantes pueden combinarse. Example: 5x2 and 6x2 are like terms; 3xy and 3zy are not like terms. Ejemplo: 5x2 y 6x2 son términos semejantes; 3xy y 3zy no son términos semejantes. line of best fit in a scatter plot, a line drawn as near as possible to the various points so as to best represent the trend being graphed línea de regresión una gráfica de dispersión, una recta dibujada tan cerca como sea posible de los varios puntos para representar mejor la tendencia que se está trazando Example/Ejemplo:

line of best fit línea de regresión

line of symmetry a line along which a figure can be folded so that the two resulting halves match eje de simetría recta por la que se puede doblar una figura de manera que las dos mitades resultantes sean iguales Example/Ejemplo:

S

T ST is a line of symmetry. ST es un eje de simetría.

GLOSSARY/GLOSARIO


M

N

mean the quotient obtained when the sum of the numbers in a set is divided by the number of addends media el cociente que se obtiene cuando la suma de los números de un conjunto se divide entre el número de sumandos

negative integers the set of all integers that are less than zero enteros negativos el conjunto de todos los enteros que son menos de cero

Example: The mean of 3, 4, 7, and 10 is (3 4 7 10) 4 or 6. Ejemplo: La media de 3, 4, 7 y 10 es (3 4 7 10) 4 o 6.

Examples/Ejemplos: 1, 2, 3, 4, 5, . . . net a two-dimensional plan that can be folded to make a three-dimensional model of a solid red un diagrama bidimensional que se puede doblar para hacer un modelo tridimensional de un cuerpo sólido

median the middle number in an ordered set of numbers mediana el número medio en un conjunto de números ordenado

Example/Ejemplo:

Example: 1, 3, 9, 16, 22, 25, 27 16 is the median. Ejemplo: 1, 3, 9, 16, 22, 25, 27 16 es la mediana. mode the number or element that occurs most frequently in a set of data moda el número o elemento que se presenta con más frecuencia en un conjunto de datos Example: 1, 1, 1, 2, 2, 3, 5, 5, 6, 6, 6, 6, 8 6 is the mode. Ejemplo: 1, 1, 1, 2, 2, 3, 5, 5, 6, 6, 6, 6, 8 6 es la moda.

net of a cube red de un cubo

O ordered pair two numbers that tell the x-coordinate and y-coordinate of a point par ordenado dos números que expresan la coordenada x y la coordenada y de un punto Example: The coordinates (3, 4) are an ordered pair. The x-coordinate is 3, and the y-coordinate is 4. Ejemplo: Las coordenadas (3, 4) son un par ordenado. La coordenada x es 3 y la coordenada y es 4.

multiplication one of the four basic arithmetical operations, involving the repeated addition of numbers multiplicación una de las cuatro operaciones aritméticas básicas que comprende la adición repetida de números multiplication growth number a number that when used to multiply a given number a given number of times results in a given goal number número del aumento de la multiplicación número que cuando se usa para multiplicar un número dado una cierta cantidad de veces tiene como resultado un número final determinado Example: grow 10 into 40 in two steps by multiplying (10 2 2 40) 2 is the multiplication growth number. Ejemplo: aumenta el 10 a 40 en dos pasos multiplicando (10 2 2 40) 2 es el número del aumento de la multiplicación

P parabola the curve formed by a quadratic equation such as y x2 parábola la curva formada por una ecuación cuadrática como y x2 Example/Ejemplo: 8

parabola parábola

6

y ⴝ x2

4 2 ⴚ6

ⴚ4

ⴚ2

ⴚ2

2

4

6

GLOSSARY/ GLOSARIO

317


percent a number expressed in relation to 100, represented by the symbol % por ciento un número expresado con relación a 100, representado por el signo % Example: 76 out of 100 students use computers. 76 percent of students use computers. Ejemplo: 76 de 100 estudiantes usan computadoras. El 76 por ciento de los estudiantes usan computadoras.

predict to anticipate a trend by studying statistical data predecir anticipar una tendencia al estudiar los datos estadísticos prism a solid figure that has two parallel, congruent polygonal faces (called bases) prisma figura sólida que tiene dos caras poligonales, congruentes y paralelas (llamadas bases) Example/Ejemplo:

percent grade the ratio of the rise to the run of a hill, ramp, or incline expressed as a percent grado de por ciento la razón de la elevación del trayecto de una colina, rampa o inclinación expresada en por ciento Example/Ejemplo:

do

e

ad

gr

6

a /gr

8 6 percent grade – or 75% 8 6 – grado de por ciento o 75% 8

permutation a possible arrangement of a group of objects. The number of possible arrangements of n objects is expressed by the term n! permutación ordenación posible de un grupo de elementos. El número de ordenaciones posibles de los elementos n se expresa con el término n! picture graph a graph that uses pictures or symbols to represent numbers pictografía una gráfica que usa dibujos o símbolos para representar números point one of four undefined terms in geometry used to define all other terms. A point has no size. punto uno de los cuatro términos indefinidos de la geometría que se utiliza para definir todos los otros términos. Un punto no tiene dimensiones. power represented by the exponent n, to which a number is raised by multiplying itself n times potencia representada por el exponente n, al cual un número es elevado al multiplicarse por sí mismo n veces Example: 7 raised to the fourth power 74 7 7 7 7 2,401 Ejemplo: 7 elevado a la cuarta potencia 74 7 7 7 7 2,401

318

GLOSSARY/GLOSARIO

prisms/prismas

probability the study of likelihood or chance that describes the chances of an event occurring probabilidad el estudio de las probabilidades que describen las posibilidades de que ocurra un suceso product the result obtained by multiplying two numbers or variables producto el resultado obtenido al multiplicar dos números o variables project to extend a numerical model, to either greater or lesser values, in order to guess likely quantities in an unknown situation proyectar extender un modelo numérico a valores mayores o menores para acertar cantidades probables en una situación incógnita pyramid a solid geometrical figure that has a polygonal base and triangular faces that meet at a common vertex pirámide cuerpo geométrico que tiene una base poligonal y caras triangulares que se encuentran en un vértice común Example/Ejemplos:

pyramids/pirámides


Pythagorean Theorem a mathematical idea stating that the sum of the squared lengths of the two shorter sides of a right triangle is equal to the squared length of the hypotenuse Teorema pitagórico concepto matemático que establece que la suma de las longitudes cuadradas de los dos lados más pequeños de un triángulo rectángulo es igual a la longitud cuadrada de la hipotenusa

ratio a comparison of two numbers razón comparación de dos números Example: The ratio of consonants to vowels in the alphabet is 21:5. Ejemplo: La razón entre las consonantes y las vocales en el abecedario es de 21:5. rectangle a parallelogram with four right angles rectángulo paralelogramo con cuatro ángulos rectos

Example/Ejemplo:

Example/Ejemplo: b

c

a For a right triangle, a 2 ⫹ b 2 ⫽ c 2. Para un triángulo recto, a 2 ⫹ b 2 ⫽ c 2.

Pythagorean triple a set of three positive integers a, b, and c, such that a2 b2 c2 triple pitagórico conjunto de tres números enteros positivos a, b, y c, de manera que a2 b2 c2 Example: for the Pythagorean triple {3, 4, 5} 32 42 52 9 16 25 Ejemplo: para el triple pitagórico {3, 4, 5} 32 42 52 9 16 25

Q quadratic equation a polynomial equation of the second degree, generally expressed as ax2 bx c 0, where a, b, and c are real numbers and a is not equal to zero ecuación cuadrática ecuación polinómica de segundo grado, generalmente expresada como ax2 bx c 0, donde a, b, y c son números reales y a no es igual a cero

rectangle/rectángulo

right angle an angle that measures 90° ángulo recto un ángulo que mide 90° Example/Ejemplo:

A

⬔A is a right angle. ⬔A es un ángulo recto. right triangle a triangle with one right angle triángulo rectángulo un triángulo con un ángulo recto Example/Ejemplo: A

C

B

䉭ABC is a right triangle.

䉭ABC es un triángulo rectángulo.

R range in statistics, the difference between the largest and smallest values in a sample rango en estadísticas, la diferencia entre los valores más grandes y menores en una muestra rate [1] fixed ratio between two things; [2] a comparison of two different kinds of units, for example, miles per hour or dollars per hour tasa [1] razón fija entre dos cosas; [2] comparación de dos tipos diferentes de unidades, por ejemplo, millas por hora o dólares por hora

rise the amount of vertical increase between two points elevación el aumento de la longitud vertical entre dos puntos rule a statement that describes a relationship between numbers or objects regla un enunciado que describe a la relación entre números o objetos run the horizontal distance between two points trayecto distancia horizontal entre dos puntos

GLOSSARY/GLOSARIO

319


S scale the ratio between the actual size of an object and a proportional representation escala la razón entre el tamaño real de un objeto y una representación proporcional

slope [1] a way of describing the steepness of a line, ramp, hill, and so on; [2] the ratio of the rise to the run pendiente [1] modo de describir la inclinación de una recta, rampa, colina, etc. [2] la razón entre la elevación y el trayecto.

scale drawing a proportionally correct drawing of an object or area at actual, enlarged, or reduced size dibujo a escala un dibujo proporcionalmente correcto de un objeto o área en su tamaño real, ampliado o reducido

slope angle the angle that a line forms with the x-axis or other horizontal ángulo de la pendiente el ángulo que forma una recta con el eje x o otro eje horizontal

scatter plot (or scatter diagram) a two-dimensional graph in which the points corresponding to two related factors (for example, smoking and life expectancy) are graphed and observed for correlation gráfica de dispersión (diagrama de dispersión) gráfica bidimensional donde los puntos correspondientes a dos factores relacionados (por ejemplo, fumar y la esperanza de vida) se representan gráficamente y se observan para su correlación Example/Ejemplo:

Diameter (inches) Diámetro (pulgadas)

AGE AND DIAMETER OF RED MAPLE TREES EDAD Y DIÁMETRO DE LOS ÁRBOLES ARCE ROJOS 12 10 8

4

speed-time graph a graph used to chart how the speed of an object changes over time gráfica de velocidad y tiempo gráfica que se usa para representar cómo la velocidad de un objeto cambia con el tiempo

Example/Ejemplo:

2

A 10

20

30 40 Age (years) Edad (años)

Examples: 9,572 9.572 103 and _ 0.00042 4.2 10 4 Ejemplos: 9,572 9.572 103_ y 0.00042 4.2 10 4 sequence a set of elements, especially numbers, arranged in order according to some rule secuencia conjunto de elementos, en especial números, ordenados de acuerdo a una regla

GLOSSARY/GLOSARIO

B AB ⴝ CD ⴝ AC ⴝ BD

50

scientific notation a system of writing numbers using exponents and powers of ten. A number in scientific notation is written as a number between 1 and 10 multiplied by a power of ten. notación científica sistema de escritura de números que usa exponentes y potencias de 10. Un número en la notación científica se escribe como un número entre 1 y 10 multiplicado por una potencia de 10.

320

solution the answer to a mathematical problem. In algebra, a solution usually consists of a value or set of values for a variable. solución la respuesta a un problema matemático. En álgebra, una solución generalmente consiste en el valor o conjunto de valores de una variable.

square a rectangle with congruent sides cuadrado un rectángulo con lados congruentes

6

0

slope ratio the slope of a line as a ratio of the rise to the run razón de la pendiente la pendiente de una línea como la razón entre la elevación y el trayecto

C

D

square cuadrado

square centimeter a unit used to measure the size of a surface; the equivalent of a square measuring one centimeter on each side centímetro cuadrado unidad usada para medir el tamaño de una superficie; el equivalente de un cuadrado que mide un centímetro en cada lado square inch a unit used to measure the size of a surface; the equivalent of a square measuring one inch on each side pulgada cuadrada unidad usada para medir el tamaño de una superficie; el equivalente de un cuadrado que mide una pulgada en cada lado statistics the branch of mathematics concerning the collection and analysis of data estadística la rama de las matemáticas que estudia la colección y análisis de datos


steepness a way of describing the amount of incline (or slope) of a ramp, hill, line, and so on inclinación modo de describir el empinamiento (o pendiente) de una rampa, colina, recta, etc. stem-and-leaf plot a method of presenting numerical data between 1 and 99 by separating each number into its tens digit (stem) and its units digit (leaf) and then arranging the data in ascending order of the tens digits diagrama de tallo y hojas método de presentación datos numéricos entre 1 y 99 separando cada número en sus dígitos de diez (tallo) y sus dígitos de unidades (hojas) y luego ordenando los datos en orden ascendente de los dígitos de diez Example/Ejemplo:

stem tallo 0 1 2 3 4 5

leaf hoja 6 18225 61 7 3 8

stem-and-leaf plot for the data set 11, 26, 18, 12, 12, 15, 43, 37, 58, 6, and 21 diagrama de tallo y hojas para el conjunto de datos 11, 26, 18, 12, 12, 15, 43, 37, 58, 6 y 21

symmetry see line of symmetry simetría ver eje de simetría Example/Ejemplo:

This hexagon has symmetry around the dotted line. Este hexágono tiene simetría alrededor de la línea de puntos.

T tangent ratio the ratio of the length of the side opposite a right triangle’s acute angle to the length of the side adjacent to it razón de la tangente la razón de la longitud del lado opuesto al ángulo agudo de un triángulo rectángulo entre la longitud de su lado adyacente Example/Ejemplo: S length of the side opposite to S tan S length of the side adjacent to S tan S

subtraction one of the four basic arithmetical operations, taking one number or quantity away from another substracción una de las cuatro operaciones aritméticas básicas, sacando un número o cantidad de otro surface area the sum of the areas of all the faces of a geometric solid, measured in square units área de la superficie la suma de las áreas de todas las caras de un cuerpo geométrico, medida en unidades cuadradas

4

5

or 0.75

The tangent ratio of S is

3 4

or 0.75.

R

3

tangente S

longitud de un lado opuesto a S longitud de un lado adyacente a S

tangente S

3 4

T

o 0.75

La razón de la tangente de S es

3 4

o 0.75.

term product of numbers and variables; x, ax2, 2x4y2, and 4ab are four examples of a term término producto de números y variables; x, ax2, 2x4y2, y 4ab son cuatro ejemplos de un término

Example/Ejemplo:

h3

3 4

8

w3 The surface area of this prism is 2(3 3) 4(3 8) square units. El área de la superficie de este prisma es 2(3 3) 4(3 8) unidades cuadradas.

theoretical probability the ratio of the number of favorable outcomes to the total number of possible outcomes probabilidad teórica la razón del número de resultados favorables en el número total de resultados posibles three-dimensional having three measurable qualities: length, height, and width tridimensional que tiene tres propiedades de medición; longitud, altura y ancho

GLOSSARY/GLOSARIO

321


total distance the amount of space between a starting point and an endpoint, represented by d in the equation d s (speed) t (time) distancia total la cantidad de espacio entre el punto de partida y el final, representada por d en la ecuación d v (velocidad) t (tiempo)

vertex (pl. vertices) the common point of two rays of an angle, two sides of a polygon, or three or more faces of a polyhedron vértice el punto común de las dos semirrectas de un ángulo, dos lados de un polígono o tres o más caras de un poliedro

total distance graph a coordinate graph that shows cumulative distance traveled as a function of time gráfica de la distancia total una gráfica de coordenadas que muestra la distancia acumulada viajada como una función de tiempo

Examples/Ejemplos:

tread the horizontal depth of one step on a stairway anchura la profundidad horizontal de un escalón en una escalera tree diagram a connected, branching graph used to diagram probabilities or factors diagrama de árbol gráfica de ramas conectadas que se usan para hacer un diagrama de probabilidades o factores

vertex of an angle vértice de un ángulo

vertices of a triangle vértices de un triángulo

vertices of a cube vértices de un cubo

volume the space occupied by a solid, measured in cubic units volumen el espacio que ocupa un cuerpo, medido en unidades cúbicas

Example/Ejemplo:

Example/Ejemplo:

⫽5

h⫽2 w⫽3

tree diagram diagrama de árbol

two-dimensional having two measurable qualities: length and width bidimensional que tiene dos propiedades de medición; longitud y ancho

V variation a relationship between two variables. Direct variation, represented by the equation y kx, exists when the increase in the value of one variable results in an increase in the value of the other. Inverse variation, represented by the equation y xk , exists when an increase in the value of one variable results in a decrease in the value of the other. variación relación entre dos variables. La variación directa, representada por la ecuación y kx, existe cuando el aumento del valor de una variable tiene como resultado el aumento del valor de la otra. La variación inversa, representada por la ecuación y xk , existe cuando el aumento del valor de una variable tiene como resultado la disminución del valor de la otra.

322

GLOSSARY/GLOSARIO

The volume of this rectangular prism is 30 cubic units. 2 ⫻ 3 ⫻ 5 ⫽ 30 El volumen de este prisma rectangular es 30 unidades cúbicas. 2 ⫻ 3 ⫻ 5 ⫽ 30

Y y-intercept the point at which a line or curve cuts across the y-axis intersección y el punto en el que una recta o curva cruza el eje y

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I N DEX quadratic functions, 286–293, 308–310 writing equations, 55 writing expressions, 182–183, 188–189, 198–199

A Accuracy of scale drawings, 238–239 of Slope-o-meters, 227

Addition grouping symbols and, 196–197, 216 integer, 192–193, 214 of polynomials, 202–203, 218 simplifying polynomials, 194–195, 215

Angle of elevation, 230–231, 258 Angles. See also Slope measuring with a protractor, 228–229, 257 measuring with a slope-o-meter, 226–227, 256 of a right triangle, 234–235, 259 slope, 225–231, 256–258 tangent ratio and, 250–251, 265

Addition sequences algebraic description of, 148–149, 151, 170, 171 comparing models to data, 155, 158–161, 173 graphing, 152–153, 158, 161, 172–174 growth patterns for, 140–141, 167 growth spirals for, 142–143, 168 interpolating intermediate values, 150–151, 154–155, 171, 173 modeling with, 158–165, 174–177 prediction with, 144–145, 154–155, 158–165, 169, 173–177

Apothem, of a regular polygon, 265 Area. See also Formulas; Surface area of compound two-dimensional figures. 110, 129 converting among units of, 118, 132 expressed as percent, 163 of a rectangle, 246

Arithmetic mean. See also Mean, 173 Average. See also Mean; Median; Mode

Additive inverse, 144–145, 148–149, 169, 170

speed, 71, 74–75, 86, 88

Axis of symmetry, 290

Algebra. See also Equations; Expressions; Formulas; Graphing; Polynomials correlations, 20–21, 40 describing patterns, 140–141, 167 direct variation functions, 274–275, 303 distance, time, speed relationship, 52–53, 70–71, 74–75 distributive property, 197, 198–199, 216, 217 evaluating expressions, 184–185, 196–197, 211 exponential functions, 294–301, 311–313 input/output rules, 272–273, 302 integer equations, 192–193, 214 inverse variation functions, 276–277, 304 laws of exponents, 298–299, 312 linear equations, 204–209, 219–221 linear functions, 278–285, 305–307 modeling equations, 202–207, 218–220 modeling expressions, 182–187, 192–197, 210–212, 214–216 modeling growth sequences, 146–149, 158–165, 170, 174–177 plotting variables, 18–19, 39 polynomials, 194–199, 202–203, 215–218 proportions, 244–245, 263 Pythagorean Theorem, 236–239, 260, 261

B Back-to-back stem-and-leaf plots, 16–17, 38

Bar graph, interpreting, 7 Base of of of of

a a a a

cone, 116 cylinder, 108 prism, 104 pyramid, 114

Benchmarks, horizontal and vertical for slope, 227, 231

Break-even point, 209

Combinations systematic list of, 27, 42 tree diagrams and, 28–29, 43

Comparison. See also Graphing; Graphs actual and average speed, 74–75, 88 addition and multiplication growth, 140–141, 167 using back-to-back stem-and-leaf plots, 16–17, 38 of distance-time graphs, 65–67, 85 distance-time and speed-time graphs, 72–73, 87 equations, slopes, and y-intercepts, 282–283, 306 estimated and measured distances, 52–53 of forecasting techniques, 138–139 using goodness of fit, 155, 158–161, 173 of graphs of exponential functions, 297, 311 of growth models, 145, 158–165, 169, 174–177 of growth predictions, 154–155, 173 of growth spirals, 142–143, 168 using linear equations, 208–209, 221 using mean, median, mode, and range, 8–9, 35 measurements and Pythagorean Theorem calculations, 238–239 population growth, growth rate, and density, 163 prism and pyramid volumes, 115, 130 quadratic equations and their graphs, 290–291, 309 quadratic and linear relationships, 288–289, 308 using a rating scale, 6–7, 34 using scatter plots, 18–19, 39 simple and compound interest, 167 slope estimates and measurements, 227, 228, 256, 257 of stairway slopes, 242–243, 262 using stem-and-leaf plots, 12–13, 37 surface area and volume, 98–99, 120–121, 124, 133 using tree diagrams, 29

Compound interest, 167

C Calculator large and small numbers on, 301 repeated operations, 140

Circle graph, interpreting, 7, 34 Classification of growth spirals, 142–143, 168 of prisms, 104–105, 126 of pyramids, 114–115, 130 of three-dimensional figures, 94–95, 122

Concert tour event probabilities and combinations, 26–27, 33 performance order permutations, 30–33, 44 selection tree diagram, 28–29

Concert Tour Game, 26–27, 33 Cone, 116–117, 131 cross sections, 116, 131 surface area formula, 131

INDEX • MATHSCAPE

323


Conic sections, 116, 131

Distributive property, 197, 198–199,

Coordinates in a sequence, 54–55, 80

Exponents, 194–301, 311–313

216, 217

laws of, 298–299, 312 negative, 299, 312 scientific notation and, 300–301, 313

Double bar graph, interpreting, 7

Correlations, types of, 20–21, 40

Expressions. See also Equations, 181–189, 210–213 Polynomials checking, 199 combining like terms, 182–183, 210 distributive property, 197, 198–199, 216, 217 evaluating, 184–185, 196–197, 211 exponential, 298–299, 312 for geometric patterns, 198–199, 217 grouping symbols and, 196–197, 216 for growth sequences, 148–149, 170 integer, 192–193, 214 modeling, 182–187, 192–197, 210–212, 214–216 multiplying, 186–189, 212, 213 simplifying, 182–183, 194–195, 210, 215 writing, 182–183, 188–189, 198–199

Cross sections of a cone, 116, 131 of a cube, 96 of a cylinder, 108–109, 128 of a pyramid, 115, 130 of a sphere, 96 of three-dimensional figures, 96–97, 123

Cube cross sections, 96 nets for, 95 volume and surface area, 100–101, 125

Cylinder cross sections, 108–109, 128 net for, 108, 128 volume and surface area, 108–109, 128

E Equations. See also Expressions; Formulas checking solutions for, 205, 219 of the form y = ax, 296–297, 311 of the form y = k/x, 276–277, 304 of the form y = kx, 274–275, 303 of the form y = ax2, 290–291, 309 integer, 192–193, 214 linear, 204–209, 219–221 for lines between points, 284–285, 307 making comparisons with, 208–209 modeling, 202–207, 218–220 of motion, 55, 74–75, 80, 88 polynomial operations, 202–203, 218 proportions and, 244–245, 263 Pythagorean Theorem, 237, 260 slope and, 280–283, 305, 306 y-intercept and, 282–283, 306

Estimation. See also Prediction

D Decimals as growth numbers, 140–141, 148–149, 167, 170 growth spirals and, 142–143, 168 for tangent ratios, 251–252, 265

Density, population, 163 Designing ramps, 247 roads, 248–255, 265–267 stairs, 244–245, 263

Diagrams growth spirals, 142–143, 168 sequence, 54–57, 80, 81 tree, 28–29, 33, 43, 45

average speed, 72–73 distance, 52–53, 79 using graphs of sequences, 154–155, 173 using growth models, 144–145, 158–165, 169, 174–177 by interpolating an intermediate value, 151, 171 using line of best fit, 22–23, 41 metric length, 16 using patterns, 140–141, 166 slope, 227, 228, 242, 256, 257 speed, 50–51, 54–57, 78, 80, 81 surface area, 99, 124 visual, 16, 22–23, 41, 99, 154–155, 173, 227, 228, 242, 256, 257 volume, 99, 119, 124, 132

Direct correlation, 20–21, 40

Experimental probability, 26, 42

Direct variation, 274–275, 303

Exponential expressions,

Direction sequence diagrams and, 56–57, 81 speed and, 72–73, 87

Discrepancy, model and actual data, 155, 158–161, 173

Distance. See also Distance-time graphs calculating, 52–55, 79, 80 estimating, 52–53, 79 measuring, 52–53, 79 time, speed relationship, 52–55, 70–71, 74–75, 79–80

Distance-time graphs, 58–67, 82–85 interpreting, 66–67, 70–71, 85, 86 making, 60–61, 70–71, 82, 86 types of, 64–65, 83

298–299, 312

Exponential functions, 294–301, 311–313 definition, 297

Exponential growth algebraic description of, 148–149, 151, 170, 171 graphing, 152–155, 158–161, 172–174 interpolating intermediate values, 150–151, 154–155, 171, 173 modeling, 158–165, 174–177 patterns, 140–141, 167 predicting with, 144–145, 158–165, 169, 174–177 spiral models of, 142–143, 168

F Face, of a three-dimensional figure, 95 Factorials, 32 Fibonacci sequence, 168 Forecasting. See Prediction Formulas distance, 55, 74–75, 80, 88 probability, 26 surface area of a cone, 131 surface area of a cube, 124 surface area of a cylinder, 128 surface area of a pyramid, 130 surface area of a rectangular prism, 126 volume of a cone, 117, 131 volume of a cylinder, 109 volume of a prism, 107, 127 volume of a pyramid, 125 volume of a rectangular prism, 105, 126

Functions. See also Formulas algebraic representation of, 148–151, 170, 171 comparing, 140–145, 166–167 definition, 273 direct variation, 274–275, 303 exponential, 294–301, 311–313 graphing, 152–155, 172–173 input/output relationships, 184–185, 211 input/output rules, 272–273, 302 inverse variation, 276–277, 304 linear, 278–285, 305–307 modeling, 158–165, 174–177 quadratic, 286–293, 308–310 tangent, 250–251, 265

Fundamental Counting Principle, 29

324

INDEX • MATHSCAPE


population growth, 154–155, 160–161 quadratic functions, 288–293, 308–310 sales growth, 158–159, 174

G Galilei, Galileo, 109 Generalizations angle relationships in right triangles, 234–235, 259 laws of exponents, 298–299, 312 Pythagorean Theorem, 236–237, 260 relationship among right triangle sides, 236–237, 260 tangent ratio, 250–251, 265

Graphs. See also Models bar, 7 circle, 7, 34 distance-time, 58–67, 82–85 double bar, 7 identifying functions from, 272–273, 302 linear, 278–285, 305–307 scatter plot, 18–23, 39–41, 152–155, 158–161, 172 sequence diagrams and, 54–57, 80, 81 speed-time, 72–73, 87

Geometric mean, 173 Geometric patterns, algebraic expressions for, 198–199, 217

Geometry. See also Measurement angle of elevation, 230–231, 258 classification of prisms, 104–105, 126 classification of pyramids, 114–115, 130 classification of three-dimensional figures, 94–95, 122 cone, 116–117, 131 conic sections, 116, 131 cross sections, 96–97, 108–109, 116, 123, 128, 130, 131 cylinder, 108–109, 128 growth spirals, 142–143, 168 nets, 94–95, 104, 114–115, 122, 126 prism, 104–107, 126, 127 pyramid, 114–115, 130 Pythagorean Theorem, 236–239, 260, 261 Pythagorean triples, 237, 260 rectangular prism, 104–105, 126 right triangle relationships, 232–239, 259–261 symmetry, 290 two-dimensional drawings and threedimensional models, 252–255, 266, 267

for direct variation, 275, 303 distance-time, 60–61, 82 finding line, 280–281, 305 for inverse variation, 277, 304 positive, negative, zero, 280–281, 305 relationship to a linear equation, 282–283, 306 sequence diagrams and, 54–55, 80 speed-time, 72–73, 87

Graphing. See also Modeling change over time, 62–63, 70–75, 86–88 direct variation functions, 274–275, 303 exponential functions, 296–299, 311 growth sequences, 152–153, 172 inverse variation functions, 276–277, 304 line of best fit, 21, 40 linear functions, 280–285, 305–307 maximum surface area, 101 motion, 58–67, 72–73, 82–85, 87

adding, 192–193, 214 on the coordinate plane, 277, 281 as exponents, 300–301, 313 subtracting, 192–193, 214

Inverse correlation, 20–21, 40 Inverse operations addition and subtraction, 144–145, 169 multiplication and division, 144–145, 169 square and square root, 150–151, 171

Inverse variation, 276–277, 304 Irrational numbers, 238–239, 261

Grouping symbols, 196–197, 216 Growth model, 156–165, 174–177

L

Growth rate, 162–163, 176

Lab Gear

Growth sequences algebraic description of, 148–149, 151, 170, 171 comparing addition and multiplication, 140–141, 167 comparing models to data, 155, 173 graphing, 152–155, 172, 173 interpolating intermediate values, 150–151, 154–155, 171, 173 modeling with, 158–165, 174–177 predicting with, 144–145, 154–155, 158–165, 169, 173–177 spirals for, 142–143, 168

Guess-and-check, for finding growth numbers, 140–141, 144–145, 150, 167, 169

Grade. See Percent grade Graph slope

Integers. See also Negative numbers

adding and subtracting integers, 192–193, 214 evaluating expressions, 184–185, 211 evaluating expressions with parentheses, 196–197 multiplying expressions, 186–187, 212 representing expressions, 182–183, 210 simplifying polynomials, 194–195, 215 solving linear equations, 204–207, 219, 220

Leg, of a right triangle, 239 Like terms combining, 182–183, 210 combining to simplify expressions, 202–203, 218

Line of best fit, 21, 40

H

making predictions with 22–23, 41

Height of a cone, 116 of a pyramid, 114

Hill three-dimensional model of, 252–253 three-dimensional scale-model of, 254–255, 267

Homework, 34–45, 78–89, 122–133, 166–177, 210–221, 256–267, 302–313

Hypotenuse, of a right triangle, 239

Line of sight, 231 Line of symmetry, 290 Linear equations, 204–209, 219–221, 278–285, 305–307 of the form y = k/x, 276–277, 304 of the form y = kx, 274–275, 303 for lines between points, 284–285, 307 modeling, 204–207, 219, 220 slope and, 280–283, 305, 306 solving, 204–209, 219–221 y-intercept and, 282–283, 306

Linear functions, 278–285, 305–307 Linear graph

I Indirect measurement using proportion, 247, 264 using the Pythagorean Theorem, 238–241, 261, 262 using tangent, 250–251, 265

slope and, 280–283, 305, 306 writing an equation for, 284–285, 307 y-intercept and, 282–283, 306

INDEX • MATHSCAPE

325


Linear growth algebraic description of, 148–149, 151, 170, 171 graphing, 152–155, 158–161, 172–174 interpolating intermediate values, 150–151, 154–155, 171, 173 modeling, 158–165, 174–177 patterns, 140–141, 167 predicting with, 144–145, 154–155, 158–165, 169, 173–177 spiral models of, 142–143, 168

Logic. See also Number sense, Patterns; Sequences following clues, 10 impossible graphs, 67, 85 inconsistent statements, 85 making a data-based argument, 8–9, 35 making generalizations, 234–237, 250–251, 259, 260, 265 number trick analysis, 188–189, 213 proof using expressions, 188–189, 213 recognizing enough information, 70–71 relative location, 70–71, 86 relative speed, 54–55, 80 tree diagrams, 28–29, 33, 43, 45 working backward, 10–11, 36

maximum and minimum angles, 234 maximum surface area, 100–101, 125 metric length, 16, 142–143, 168, 276 model lengths and angles, 252–253 perimeter, 292 right triangle side lengths, 236 selecting appropriate units for area and volume, 118–119, 132 slope using a protractor, 228–229, 242–243, 257, 262 slope using a Slope-o-meter, 227, 234–235, 246, 256 time and distance, 50–53, 78, 79 volume and surface area of compound figures, 111, 129 volume and surface area of a cylinder, 108–109, 128 volume and surface area of a nonrectangular prism, 106–107, 127 volume and surface area of a pyramid, 114–115, 130 volume and surface area of a rectangular prism, 104–105, 126 volume and surface area relationships, 98–101, 124–125

Measures of central tendency. See also Mean; Median; Mode; Range, 8–9, 35

Median

M Map, distance, 62–63, 83 Mathematical forecasting. See Prediction

Maximum surface area, 100–101, 125 Mean for analyzing athletic performance, 8–9, 35 for analyzing a survey, 6–7, 34 arithmetic, 173 back-to-back stem-and-leaf plot and, 17, 38 geometric, 173 stem-and-leaf plot and, 12–13, 37 working backward to create a data set from, 10–11, 36

Measurement. See also Scale drawing accuracy, 227 angle of elevation, 230–231, 258 area, 288, 292 converting among units of area, 118, 132 converting among units of length, 52–53, 79 converting among units of time, 52–53, 79 converting among units of volume, 118–119, 132 estimating speed, 50–51, 54–57, 78, 80, 81 indirect, 238–241, 247, 250–251, 261, 262, 264, 265, 274

326

INDEX • MATHSCAPE

for analyzing athletic performance, 8–9, 35 back-to-back stem-and-leaf plot and, 17, 38 stem-and-leaf plot and, 12–13, 37 working backward to create a data set from, 10–11, 36

Minus sign grouping symbols and, 196–197, 216 three uses of, 192–193, 214

Mode for analyzing athletic performance, 8–9, 35 working backward to create a data set from, 10–11, 36

Modeling. See also Graphing expressions, 182–187, 210–212 growth sequences, 142–143, 168 integer equations, 192–193, 214 linear equations, 204–207, 219, 220 multiplication equations, 186–187, 212 polynomial addition and subtraction, 202–203, 218 polynomials, 194–197, 215, 216 population growth, 144–145, 154–155, 160–165, 169, 173, 176 sales growth, 158–159, 174

Models. See also Functions; Graphs; Lab Gear; Nets; Scale model goodness of fit comparison of, 155, 158–161, 173 growth, 158–165, 174–177

predicting with, 144–145, 154–155, 158–165, 169, 173–177 prism and pyramid hills, 252–253, 266 spirals, 142–143, 168

Motion distance, time graphs, 58–67, 82–85 distance, time, speed relationship, 52–53, 70–71, 74–75 estimating speed, 50–51, 54–57, 78, 80, 81 measuring, 50–53, 78, 79 representing, 48–57, 78–81 sequences, 54–57, 80, 81 speed-time graphs, 72–73, 87 study, 76–77, 89

Multiple representations of an event, 51 data and statements, 13, 37 data tables and equations, 208–209 descriptions and graphs, 70–71, 86 exponential functions and graphs, 297, 299 graphs and stories, 61, 75, 82, 88 graphs and tables, 60–61 linear equation and graph, 282–283, 306 models and expressions, 182–187, 210–212 models and integer equations, 192–193, 214 models and linear equations, 204–207, 219, 220 models and polynomial operations, 202–203, 218 models and polynomials, 194–195, 215 of motion, 76–77, 89 nets for cylinders, 108, 128 nets for pyramids, 114–115 nets for rectangular prisms, 104–105, 126 nets for three-dimensional figures, 94–95, 122 quadratic equation and graph, 290–291, 309 scale drawings and threedimensional models, 254–255, 267 scatter plots and tables, 19, 23 scientific and standard notation, 300–301, 313 sequences and equations, 55 sequences and expressions, 148–149, 170 sequences and graphs, 152–155, 172–173 sequences and spirals, 142–143, 168 sequences and stories, 56–57, 81 slope ratios and slope angles, 242–243, 262 story, map, and graph of motion, 62–63, 83 table, graph, expression, 158–159, 174 tables and back-to-back stem-andleaf plots, 16–17, 38


Multiplication of algebraic expressions, 186–189, 212, 213 distance calculation, 52–53, 79 Fundamental Counting Principle, 29 grouping properties and, 197, 216

Ordering degrees of slope, 228 exponential expressions, 312 population growth, 163 population growth rates, 163 slope ratios, 262

Multiplication sequences algebraic description of, 148–149, 151, 170, 171 comparing models to data, 155, 173 graphing, 152–155, 172, 173 growth patterns for, 140–141, 167 growth spirals for, 142–143, 168 interpolating intermediate values, 150–151, 154–155, 171, 173 modeling with, 158–165, 174–177 prediction with, 144–145, 154–155, 158–165, 169, 173–177

Multiplicative inverse, 144–145, 148–149, 169, 170

Mystery Data game, 10–11

N Negative correlation, 20–21, 40 Negative exponents, 299, 312 Negative numbers decreasing growth sequences and, 148–149, 170 as exponents, 299, 312

Negative slope, 280–281, 305 Nets for cubes, 95 for cylinders, 108, 128 for pyramids, 114–115 for rectangular prisms, 104–105, 126 for three-dimensional figures, 94–95, 122

Nonrectangular prisms, volume and surface area of, 106–107, 127

Number sense. See also Estimation; Patterns laws of exponents, 298–299, 312 relationship of a to the graph of y = ax2, 291 relationship of b to the graph of equation y = mx + b, 282–283, 306 relationship of m to the graph of equation y = mx + b, 280–283, 305, 306 scientific notation, 300–301, 313

Number theory, numbers between numbers, 140–141, 167

Numeration laws of exponents, 298–299, 312 scientific notation, 300–301, 313 square root, 150–151, 171

Population

O

Outlier, 12–13

P Parabola, graph of a quadratic function, 290–291

Patterns. See also Sequences addition and multiplication growth, 140–145, 167–169 algebraic expressions for, 188–189, 198–199, 213, 217 for angles of right triangles, 234–235, 259 calendar, 188–189, 213 for exponential functions, 296–297, 311 geometric, 198–199 maximum surface area, 101 powers of ten, 300 prediction and, 138–139, 167 for quadratic relationships, 288–289, 308

Percent population comparison with, 163 ratio and, 246–247, 251, 264, 265 survey analysis and, 6–7, 34

Percent grade, 246–247, 264 tangent ratio and, 250–251, 265

Perfect negative correlation, 21 Perfect positive correlation, 21 Perimeter, 292 Permutation, 30–33, 44, 45 Permutation Theorem, 32–33, 45 Plane, cross–sectional, 96–97, 123 Plane figures. See Two-dimensional figures

Plotting points on distance-time graphs, 60–61, 82 positions on sequence diagrams, 54–57, 80, 81

Polynomials adding, 202–203, 218 definition, 194 geometric patterns and, 198–199 grouping properties and, 196–197, 216 simplifying, 194–195, 215 subtracting, 202–203, 218 term of, 194

density, 163 estimating, 138–139, 166 expressions for growth, 148–149, 151, 170, 171 fitting to a model, 154–155, 173 growth rate, 162–163, 176 interpolating intermediate values, 150–151, 171 interpreting predictions, 164–165, 177 modeling growth, 158–161, 174–175 sequence models of, 144–145, 169

Positive correlation, 20–21, 40 Positive slope, 280–281, 305 Powers of ten, 300–301, 313 Prediction by interpolating an intermediate value, 151, 171 using estimation, 138–139, 166 using exponential functions, 311 using growth models, 144–145, 158–165, 169, 174–177 using growth sequence graphs, 154–155, 173 using line of best fit, 22–23, 41 using mathematical models, 158–165, 174–177 maximum surface area, 101 number of triangles in pyramid nets, 114 using quadratic equations, 308, 310 using a scatter plot, 22–23, 41 shape formed by a net, 94–95, 122

Prism hill models, 252–255, 266, 267 net for, 104–105, 126 surface area formula, 126 surface area and volume of nonrectangular, 106–107, 127 surface area and volume of rectangular, 104–105, 126 volume formula, 105, 107

Probability combinations, 27–29, 42, 43 experimental, 26, 42 formula, 26 the Fundamental Counting Principle and, 29 order and, 30–31, 44 Permutation Theorem and, 32–33, 45 permutations, 30–31, 44 systematic list and, 27, 42 theoretical, 26–27, 42 tree diagram and, 28–29, 43

Proof, using algebraic expressions, 188–189, 213

Proportion percent grade and, 246–247, 264 scale and, 120–121, 133 scale drawing and, 229 slope ratio and, 244–245, 263

INDEX • MATHSCAPE

327


volume to surface area, 98–101, 120–121, 124, 125, 133

Protractor measuring a model, 253 measuring right triangles, 234–235 measuring slope, 228–229, 242–243, 257, 262

Reciprocals, of slope ratios, 243 Rectangular prism net for, 104–105, 126 volume and surface area of, 104–105, 126

Puzzles, anagram, 31 Pyramid cross sections, 115, 130 hill models, 252–255, 266, 267 net for, 114–115 surface area formula, 130 volume formula, 115 volume and surface area, 114–115, 130

Pythagorean triples, 237, 260

Quadratic equation

308–310 definition, 290

R percent grade, 246–247, 264 Pythagorean Theorem and, 238–239, 261 Scale drawing of, 229, 257

Range for analyzing athletic performance, 8–9, 35 back-to-back stem-and-leaf plot and, 17, 38 stem-and-leaf plot and, 12–13, 37

Ranking, using mean, median, mode and range, 8–9, 35

Rate. See also Speed graph slope, 60–61, 82 growth, 162–163, 176

Rating scale, for making comparisons, 6–7, 34

Ratio percent and, 246–247, 264 probability and, 26–27, 42 proportion and, 244–245, 263 right triangle relationships and, 232–239, 259–261 road design and, 248–255, 265–267 scale and, 229, 231, 257, 258 slope and, 240–247, 262–264, 280–283, 305, 306 tangent, 250–251, 265

328

INDEX • MATHSCAPE

230–231, 258

Slant height, of a cone, 131

Right triangle

Slope. See also Graph slope

angle relationships, 234–235, 259 Pythagorean Theorem, 238–239, 261 side relationships, 236–237, 260 tangent ratio, 250–251, 265

Rise, slope ratio and, 242, 245, 281 Road

Run, slope ratio and, 242, 245, 281

S Scale. See also Scale drawing; Scale model distance-time, 60–61, 82 ratio and, 229, 231, 257, 258 slope and, 228–229, 257

Scale drawing

Ramp

Sighting, angle of elevation and,

Relative speed, 54–55, 80

graph of, 290–293, 309, 310 for a table of values, 288–289, 308

Quadratic function, 286–293,

236–239, 261

Simple interest, 167

analysis using a model, 252–253, 266 design, 248–255, 265–267 project, 254–255, 267

Q

Side, lengths for a right triangle,

Relative location, 70–71, 86

Pythagorean Theorem, 236–239, 260, 261

modeling with, 158–165, 174–177 prediction with, 144–145, 154–155, 158–165, 169, 173–177

accuracy and, 238–239 angle of elevation and, 230–231, 258 for a hill model, 254–255, 267 line of sight and, 231 reading and making, 229, 257

Scale model. See also Model of a hill and road, 254–255, 267 of a skyscraper, 120–121, 133

Scatter plot correlations and, 20–21, 40 creating, 19, 39 interpreting, 18–19, 39 line of best fit and, 21, 40 making predictions with, 22–23, 41 scale selection, 19

Scientific notation, 300–301, 313 Sequences algebraic description of, 148–149, 151, 170, 171 diagrams of motion over time, 54–57, 80, 81 Fibonacci, 168 graphing, 152–155, 158–161, 172–174 growth patterns for, 140–141, 167 growth spirals for, 142–143, 168 interpolating intermediate values, 150–151, 154–155, 171, 173 interpolating missing data from, 56–57, 81

as an angle, 225–231, 256–258 angle of elevation and, 230–231, 258 estimating, 227, 228, 256, 257 measuring with a protractor, 228–229, 242–243, 257, 262 measuring with a Slope-o-meter, 227, 234–235, 246, 256 percent grade and, 246–247, 251, 264, 265 as a ratio, 240–247, 262–264 tangent ratio and, 250–251, 265

Slope-o-meter, 225 hillside analysis with, 253 making, 226 measuring angles of elevation with, 231, 258 measuring slope with, 227, 234–235, 246, 256 sighting with, 230, 258

Solids. See Three-dimensional figures Solve a simpler problem, to find volume and surface area of compound figures, 110–111, 129

Spatial visualization. See also Graphing; Modeling angle and percent grade, 246–247, 264 conic sections, 116–117, 131 cross sections, 96–97, 116, 123, 128, 130 estimating distance, 52–53, 79 estimating speed, 50–51, 54–57, 78, 80, 81 estimating surface area and volume, 99, 119, 124, 132 growth spirals, 142–143, 168 line of best fit, 21, 40 nets for cylinders, 108, 128 nets for pyramids, 114–115 nets for rectangular prisms, 104–105, 126 nets for three-dimensional figures, 94–95, 122 relating equations and slopes, 282–283, 306 relating equations and y-intercepts, 282–283, 306 relating parabola shapes and quadratic equations, 290–291, 309 roads on hillside models, 252–253, 266 scatter plot correlation, 18–19, 39


of a cone, 117, 131 of a cylinder, 108–109, 128 estimating, 99, 124 maximum, 100–101, 125 measuring, 98–99, 124 of a nonrectangular prism, 106–107, 127 of a pyramid, 114–115, 130 of a rectangular prism, 104–105, 126 relationship to volume, 98–101, 124, 125 of a skyscraper, 120–121, 133

sighting and angle of elevation, 230–231, 252 snapshot sequences, 54–57, 80, 81 stair rise and run, 244–245, 263 stem-and-leaf plots, 12–13, 37 tree diagrams, 28–29, 43 two-dimensional sides of threedimensional figures, 254–255, 267 volume and surface area, 110–111, 129

Speed average, 71, 74–75, 86, 88 calculating, 52–55, 72–75, 79, 80 distance-time graphs, 59–67, 82–85 estimating, 50–51, 54–57, 72–73, 78, 80, 81 relative, 54–55, 80 sequence diagrams and, 55–57, 80, 81 speed-time graphs, 72–73, 87 time, distance relationship, 52–55, 70–71, 74–75 variability, 72–73, 87

V

Symmetry, 290

Variables

combinations, 27, 42

Vertex (vertices) of a cone, 116 of a pyramid, 114 of a three-dimensional figure, 95

Spirals, for growth sequences,

Term, of a polynomial, 194

Statistics. See also Probability analysis using a back-to-back stemand-leaf plot, 16–17, 38 analysis using mean and percent, 6–7, 34 analysis using a scatter plot, 18–23, 39–41 analysis using a stem-and-leaf plot, 12–13, 37 analysis using a systematic list, 27, 42 analysis using a tree diagram, 28–29, 43 correlations, 20–21, 40 line of best fit, 21–23, 40, 41 outliers, 12–13 ranking using mean, median, mode, and range, 8–9, 35 working backward to create data sets, 10–11, 36

Stem-and-leaf plot back-to-back, 16–17, 38 creating and comparing, 12–13, 37

Subtraction grouping symbols and, 196–197, 216 integer, 192–193, 214 of polynomials, 202–203, 218 for simplifying polynomials, 194–195, 215

Surface area

algebraic, 182 correlations and, 20–21, 40 scatter plots and, 18–19, 39

Systematic list, for finding

Tangent ratio, 250–251, 265

designing, 244–245, 263 slope ratios for, 242–243, 262 stair-building design guidelines, 245

area of compound figures, 110, 129 cross sections of three-dimensional figures, 96–97, 123 as faces of three-dimensional figures, 94–95, 108, 122 as prism bases, 104–107, 126, 127

Symbols, factorial, 32

Sphere, cross section of, 96

Stairs

Two-dimensional figures

percent, 6–7, 34

T

Square root, 150–151, 171

right triangle relationships, 232–239, 259–261 tangent ratio, 250–251, 265

Survey, analysis using mean and

Speed-time graphs, 72–73, 87

142–143, 168

Trigonometry

Volume of compound figures, 111, 129 of a cone, 117, 131 converting among units of, 118–119, 132 of a cylinder, 108–109, 128 estimating, 99, 119, 124, 132 measuring, 98–99, 124 of a nonrectangular prism, 106–107, 127 of a pyramid, 114–115, 130 of a rectangular prism, 104–105, 126 relationship to surface area, 98–101, 124, 125

Tessellation, 259 Theorem Permutation, 32–33, 45 Pythagorean, 236–239, 260–261

Theoretical probability, 26–27, 42 combinations and, 27–29, 42, 43 Permutation Theorem and, 32–33, 45 permutations and, 30–31, 44

Three-dimensional figures classification of, 94–95, 122 cones, 116–117, 131 cross sections of, 96–97, 108–109, 116, 123, 128, 130, 131 cylinders, 108–109, 128 measuring surface area and volume, 98–99, 124 nets for, 94–95, 108, 114–115, 122, 128 nonrectangular prisms, 106–107, 127 pyramid and prism hill models, 252–255, 266, 267 pyramids, 114–115, 130 rectangular prisms, 104–105, 126

Time converting among units of, 52–53, 79 distance, speed relationship, 52–55, 70–71, 74–75 distance-time graphs, 58–67, 82–85 estimating, 50–51, 78 speed-time graphs, 72–73, 87

W Weak, positive correlation, 21 Working backward, creating data sets, 10–11, 36

Y y-intercept, 282–283, 306

Z Zero slope, 280–281, 305

Tree-diagram, 28–29, 33, 43, 45 permutations and, 31, 44

Triangle, right triangle relationships, 232–239, 259–261

of a compound figure, 111, 129

INDEX • MATHSCAPE

329

MS_GR8-Photo Credits 1/8/04 12:27 PM Page 1 mac85 Mac 85:1st shift: 1268_tm:6037a:6037a:

PHOTO C RE D I TS Unless otherwise indicated below, all photography by Chris Conroy. 2 M. Tcherevkoff/Getty Images; 3 (tl)Reza Estakhrian/Getty Images, (tr)Peter Zeray/Photonica; 10 Aric Crabb, courtesy San Jose Lasers/American Baseball League; 19 T. Davis and W. Bilenduke/Getty Images; 24 Peter Zeray/Photonica; 46 Frank Herholdt/Getty Images; 47 (tr)David Madison/Getty Images, (tc b)map reproduced with permission of copyright owner, Compass Maps, Inc.; 51 NASA; 53 (bl)Froomer Pictures/Getty Images; 54 Marvin E. Newman/Getty Images; 58–59 map reproduced with permission of copyright owner, Compass Maps, Inc.; 68–69 David Madison/Getty Images; 74 Steven E. Sutton/Duomo Photography; 75 Stephen Wilkes/Getty Images; 77 Tom Raymond/Getty Images; 97 CNRI/Science Photo Library/Photo Researchers, Inc.; 109 111 NASA; 121 Brian Yervin/Photo Researchers, Inc.; 134 (tl)PhotoDisc/Getty Images; 135 (tr)Andrea Moore; 156 Andrea Moore; 161 (tr)Andrea Moore; 179 (tl)Shiniichi Eguchi/Photonica; 180 Shiniichi Eguchi/Photonica; 222 (l)Michael L. Marrella; 269 (tl)Tom Tracy/Getty Images, (tc)Kathleen Campbell/Getty Images; 278–279 Tom Tracy/Getty Images; 286 Kathleen Campbell/Getty Images.

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