MathMatters 1: An Integrated Program, Reteaching Workbook

Reteaching Workbook Contents Include: 88 worksheets— one for each lesson To The Student: This Reteaching Workbook giv...

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Reteaching Workbook

Contents Include: 88 worksheets— one for each lesson

To The Student: This Reteaching Workbook gives you additional examples and problems for the concept exercises in each lesson. The exercises are designed to aid your study of mathematics by reinforcing important mathematical skills needed to succeed in the everyday world. The material is organized by chapter and lesson, with one skills practice worksheet for every lesson in MathMatters 1.

To the Teacher: Answers to each worksheet are found in MathMatters 1 Chapter Resource Masters.

Copyright © 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 book may be reproduced in any form, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Send all inquiries to: The McGraw-Hill Companies 8787 Orion Place Columbus, OH 43240-4027 ISBN: 0-07-869307-1 1 2 3 4 5 6 7 8 9 10 XXX

MathMatters 1 Reteaching Workbook 12 11 10 09 08 07 06 05 04

CONTENTS Lesson

1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 4-1 4-2 4-3 4-4 4-5 4-6 4-7

©

Title

Page

Lesson

Collect and Display Data . . . . . . . 1 Measures of Central Tendency and Range . . . . . . . . . . . . . . . . . 2 Stem-and-Leaf Plots . . . . . . . . . . . 3 Problem Solving Skills: Circle Graphs . . . . . . . . . . . . . . 4 Frequency Tables and Pictographs . . . . . . . . . . . . . 5 Bar Graphs and Line Graphs . . . . 6 Scatter Plots and Lines of Best Fit . . . . . . . . . . . . . . . . . 7 Box-and-Whisker Plots . . . . . . . . . 8 Units of Measure . . . . . . . . . . . . . 9 Work with Measurements . . . . . . 10 Linear Measure and Perimeter . . 11 Area . . . . . . . . . . . . . . . . . . . . . . . 12 Problem Solving Skills: Use a Formula . . . . . . . . . . . . . 13 Ratio . . . . . . . . . . . . . . . . . . . . . . 14 Circumference and Area of a Circle . . . . . . . . . . . . . . . . 15 Proportion and Scale Drawings . . . . . . . . . . . . . . . . . 16 Area of Irregular Shapes . . . . . . . 17 Add and Subtract Signed Numbers . . . . . . . . . . . . . . . . . 18 Multiply and Divide Signed Numbers . . . . . . . . . . . . . . . . . 19 Order of Operations . . . . . . . . . . 20 Real Number Properties . . . . . . . 21 Variables and Expressions . . . . . 22 Problem Solving Skills: Find a Pattern . . . . . . . . . . . . . 23 Exponents and Scientific Notation . . . . . . . . . . . . . . . . . 24 Laws of Exponents . . . . . . . . . . . 25 Squares and Square Roots . . . . . 26 Language of Geometry . . . . . . . . 27 Polygons and Polyhedra . . . . . . . 28 Visualize and Name Solids . . . . . 29 Problem Solving Skills: Nets . . . 30 Isometric Drawings . . . . . . . . . . . 31 Perspective and Orthogonal Drawings . . . . . . . . . . . . . . . . . 32 Volume of Prisms and Cylinders . . . . . . . . . . . . . 33

Glencoe/McGraw-Hill

4-8 4-9 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8 7-1 7-2 7-3 7-4 7-5 7-6 7-7 7-8 8-1 8-2 8-3 8-4 8-5

iii

Title

Page

Volume of Pyramids and Cones . . . . . . . . . . . . . . . . 34 Surface Area of Prisms and Cylinders . . . . . . . . . . . . . 35 Introduction to Equations . . . . . . 36 Add or Subtract to Solve Equations . . . . . . . . . . . . . . . . 37 Multiply or Divide to Solve Equations . . . . . . . . . . . . . . . . 38 Solve Two-Step Equations . . . . . 39 Combining Like Terms . . . . . . . . 40 Formulas . . . . . . . . . . . . . . . . . . . 41 Problem Solving Skills: Work Backward . . . . . . . . . . . 42 Graph Open Sentences . . . . . . . . 43 Solving Inequalities . . . . . . . . . . 44 Percents and Proportions . . . . . . 45 Write Equations for Percents . . . 46 Discount and Sale Price . . . . . . . 47 Tax Rates . . . . . . . . . . . . . . . . . . . 48 Simple Interest . . . . . . . . . . . . . . 49 Commission . . . . . . . . . . . . . . . . 50 Percent of Increase and Decrease . . . . . . . . . . . . . . 51 Problem Solving Skills: Make a Table . . . . . . . . . . . . . . 52 Problem Solving Skills: Qualitative Graphing . . . . . . . . 53 Coordinate Plane . . . . . . . . . . . . . 54 Relations and Functions . . . . . . . 55 Linear Graphs . . . . . . . . . . . . . . . 56 Slope of a Line . . . . . . . . . . . . . . 57 Slope-Intercept Form of a Line . . . . . . . . . . . . . . . . . . . 58 Distance and the Pythagorean Theorem . . . . . . . . . . . . . . . . . 59 Solutions of Linear and Nonlinear Equations . . . . . . . . 60 Angles and Transversals . . . . . . . 61 Beginning Constructions . . . . . . 62 Diagonals and Angles of Polygons . . . . . . . . . . . . . . . 63 Problem Solving Skills: Modeling Problems . . . . . . . . . 64 Translations in the Coordinate Plane . . . . . . . . . . . 65 MathMatters 1

Lesson

8-6 8-7 9-1 9-2 9-3 9-4 9-5 9-6 9-7 10-1 10-2 10-3

©

Title

Page

Lesson

Reflections and Line Symmetry . . . . . . . . . . . . . . . . 66 Rotations and Tesselations . . . . . 67 Monomials and Polynomials . . . 68 Add and Subtract Polynomials . . . . . . . . . . . . . . . 69 Multiply Monomials . . . . . . . . . . 70 Multiply a Polynomial by a Monomial . . . . . . . . . . . . . . . 71 Factor Using GCF . . . . . . . . . . . . 72 Divide by a Monomial . . . . . . . . 73 Problem Solving Skills: Use a Model . . . . . . . . . . . . . . 74 Probability . . . . . . . . . . . . . . . . . . 75 Experimental Probability . . . . . . 76 Sample Spaces and Tree Diagrams . . . . . . . . . . . . . . . . . 77

Glencoe/McGraw-Hill

10-4 10-5 10-6 10-7 11-1 11-2 11-3 11-4 11-5 11-6 11-7

iv

Title

Page

Counting Principle . . . . . . . . . . . 78 Independent and Dependent Events . . . . . . . . . . . . . . . . . . . 79 Problem Solving Skills: Make Predictions . . . . . . . . . . . . . . . 80 Expected Value and Fair Games . . . . . . . . . . . . . . . 81 Optical Illusions . . . . . . . . . . . . . 82 Inductive Reasoning . . . . . . . . . . 83 Deductive Reasoning . . . . . . . . . 84 Venn Diagrams . . . . . . . . . . . . . . 85 Logical Reasoning . . . . . . . . . . . 86 Problem Solving Skills: Reasonable Answers . . . . . . . . 87 Non-Routine Problem Solving . . . . . . . . . . . . . . . . . . 88

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

1-1

COLLECT AND DISPLAY DATA Taking a sample is a means of collecting information, or data, from a large group of people. Some ways of sampling an entire group, or population, are random sampling, cluster sampling, systematic sampling, and convenience sampling. E x a m p l e To determine public opinion concerning an Arizona state senator, every tenth person visiting the Grand Canyon in Arizona was interviewed. What kind of sampling does this represent? Name a disadvantage of this kind of sampling for this situation. Solution convenience sampling—Many visitors to the Grand Canyon in Arizona are not Arizona residents. It would be better to interview Arizona residents whose names were chosen from voter lists in several Arizona cities.

EXERCISES Is the sample appropriate for each situation? If not, what is its disadvantage? Suggest a better way of choosing the sample. 1. To determine whether people like a new brand of bran muffin, ask every person who tastes a free sample at a health-food exposition.

2. To determine the quality of ballpoint pens delivered to a schoolsupply store, refer to company records showing 4 out of every 500 pens were found defective.

3. To determine the average amount spent at a store in a day, poll everyone who leaves the store between 5:00 and 7:00 p.m.

4. To determine the proportion of students in your school who own cars, ask everyone in your math class.

6. To determine whether town residents think there is too much traffic on local Route 44, ask all the homeowners who live along the route.

5. To determine the “hottest” fall fashion color, count the number of garments of each color on racks in department stores in different parts of the city.

© Glencoe/McGraw-Hill

1

MathMatters 1

Name _________________________________________________________

Date ____________________________

1-2

RETEACHING

MEASURES OF CENTRAL TENDENCY AND RANGE Baseball statistics are in the newspaper every day during the baseball season. E x a m p l e Find the mean, median, mode, and range for the number of lifetime stolen bases for these home run leaders. Solution Mode is the most frequent number or numbers. The modes are 37, 40, 47, and 49. Median is the middle number. Write the numbers in order from highest to lowest or lowest to highest. Then find the middle number. For 15 numbers, the median is the 8th number. The median is 40. Arithmetic mean is the sum of the data divided by the number of data. The sum of the home runs, 653, divided by the number of players, 15, is 43.53, or about 44. The mean is 44. Subtract the lowest number from the highest to get the range. 70 35 35. The range is 35.

National League Most Home Runs From 1984–1998 Player Fred McGriff Mike Schmidt Dale Murphy Mike Schmidt Howard Johnson Daryl Strawberry Ryne Sandberg Dante Bichette Matt Williams Barry Bonds Kevin Mitchell Andres Galarraga Andre Dawson Larry Walker Mark McGuire

Number of Home Runs 35 36 37 37 38 39 40 40 43 46 47 47 49 49 70

(All data from The World Almanac 1999)

EXERCISES Find the mean, median, mode, and range for each set of baseball data. 1.

2.

American League Runs Batted In (RBI) Leader Year 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

Player Ruben Sierra Cecil Fielder Cecil Fielder Cecil Fielder Albert Belle Kirby Puckett Albert Belle Albert Belle and Mo Vaughn Ken Griffey Jr. Juan Gonzalez

mean: median: mode: range: © Glencoe/McGraw-Hill

RBI 119 132 133 124 129 112 126 148 147 157

Leaders in Lifetime Stolen Bases at End of 1998 Season Player Rickey Henderson Lou Brock Billy Hamilton Ty Cobb Tim Raines Vince Coleman Eddie Collins Arlie Latham Max Carey Honus Wagner

Number of Stolen Bases 1,297 938 912 892 803 752 744 739 738 722

3.

American League Batting Champions Year 1990 1991 1992 1993 1994 1995 1996 1997 1998

Player Average George Brett .329 Julio Franco .341 Edgar Martinez .343 John Olerud .363 Paul O’Neill .359 Edgar Martinez .356 Alex Rodriguez .358 Frank Thomas .347 Bernie Williams .339

mean:

mean:

median:

median:

mode:

mode:

range:

range: 2

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

1-3

STEM-AND-LEAF PLOTS A stem-and-leaf plot can help you organize data so it can be easily analyzed. E x a m p l e Make a stem-and-leaf plot for the high temperatures listed in the chart. Then write a description of the data. Solution Step 1: Form the stem. The high temperatures range from 70 to 99. Use the digits in the tens place, 7, 8, and 9, as the stems. Write them in a column. Draw a vertical line to the right.

Step 2: Form the leaves. Show the first high temperature, 70° for Anchorage, by writing a “leaf,” 0, next to the “stem,” 7. To show 79° for Boston, write 9 next to the 0. Enter the rest of the leaves in the same way.

Step 3: The 70 is an outlier (extremely high or low value). Clusters (groups of values close to one another) appear in the high 80s and high 90s. There is a large gap (space between values) between 89 and 97.

EXERCISES 1. On a separate sheet of paper, make a stem-and-leaf plot for the low temperatures. 2. Write a description of the data noting any outliers, clusters, or gaps.

3. Complete the stem-and-leaf plot for advertised monthly rents for one-bedroom apartments. Use the numbers in the hundreds as the stem. 2 3 4 5

25 80 80 95 00 00 20 50 60 60

4. On a separate sheet of paper, describe the data to a person who is looking for a one-bedroom apartment.


3

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

1-4

PROBLEM SOLVING SKILLS: CIRCLE GRAPHS A circle graph can be used to show data that is expressed as parts of a whole. Favorite Ice Cream Flavors E x a m p l e A survey was taken of 50 cutomers at an ice cream parlor concerning their favorite flavor of ice cream. Vanilla Peach 40% The circle graph at the right shows the results. Use 10% the graph to answer the questions. a. What percent of the customers surveyed prefer Strawberry peach ice cream? Chocolate 20% b. How many of the customers surveyed prefer 30% peach ice cream? c. What percent of the customers surveyed prefer chocolate ice cream? d. How many of the customers surveyed prefer chocolate ice cream? Solution a. 10% of the customers surveyed prefer peach ice cream. b. 10% of the 50 customers surveyed prefer peach ice cream. ↓ ↓ ↓ 0.10 • 50 5 of the customers surveyed prefer peach ice cream. c. 30% of the customers surveyed prefer peach ice cream. d. 30% of the 50 customers surveyed prefer peach ice cream. ↓ ↓ ↓ 0.30 • 50 15 of the customers surveyed prefer peach ice cream.

EXERCISES For Exercises 1–4, use the circle graph from the example. 1. What percent of the customers surveyed prefer vanilla ice cream? 2. How many of the customers surveyed prefer vanilla ice cream? 3. What percent of the customers surveyed prefer strawberry ice cream? 4. How many of the customers surveyed prefer strawberry ice cream? Abi made the circle graph to show how she spends her $2000 a month salary. Use the circle graph for Exercises 5 and 6.

Abi's Monthly Budget Miscellaneous Housing 5% and Utilities Entertainment 45% 10%

5. How much does she spend on clothes each month?

Clothes15%

6. How much does she spend on housing and utilities each month?

Transportation 10% © Glencoe/McGraw-Hill

4

Food 25%

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

1-5

FREQUENCY TABLES AND PICTOGRAPHS A pictograph displays data using symbols and a key. The key tells the number of data items represented by each symbol. E x a m p l e Use the pictograph to answer the questions. a. What does the symbol $ represent? b. What chore do the greatest number of students get paid to do? How many students is that? c. How many more students get paid for yardwork than for housework? Solution a. The symbol represents 10 students. b. Forty students get paid to take care of pets. c. yardwork, 25; housework, 15; 25 15 10. Ten more students get paid for yardwork.

EXERCISES Use the pictograph to answer the questions. 1. How many pairs of size-10 shoes are in stock? 2. How many pairs of size-81⁄2 shoes are in stock? 3. Based on this store’s stock, what do you think is the most popular shoe size? 5

51⁄2 6 61⁄2

7 71⁄2

8 81⁄2

9 91⁄2 10

4. If the store sells 17 pairs of size-6 ⁄2 shoes, how many pairs of shoes will be left in that size? 1

5. If the store sells 16 pairs of size-8 shoes, which shoe size will it then have the most of in stock? 6. This chart shows the number of minutes for each device to use 1 kilowatt of electricity. On another piece of paper, construct a pictograph for these data. clothes dryer

shower

iron

dishwasher

vacuum cleaner

30

15

50

20

90


5

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

1-6

BAR GRAPHS AND LINE GRAPHS

E x a m p l e 1 Use the bar graph to answer the following questions. a. In which years did more than 120 banks fail? b. About how many more banks failed in 1993 than in 1994? Solution a. 1991 and 1992 b. 40 10 30

120 90

Number

A bar graph can be used to display data that you find in a list, see in a report, or read in a newspaper or book.

U.S. Bank Failures 150

60 30 0

'91

'92

To display data on a line graph points are plotted and connected in order.

'94 '95 Year

'93

'96

'97

Imported Car Sales

Number of Cars (in millions)

4

E x a m p l e 2 Use the line graph to answer the questions. a. In which year were the most imported cars sold? b. About how many were sold in 1993? Solution a. 1991 b. about 1.8 million

3

2

1

0

'91 '92 '93 '94 '95 '96 '97

EXERCISES

Year

2. How many students were there per computer in ’92–’93?

Number of Students per Computer in Public Schools 20 Number of Students

Use the bar graph for Exercises 1–3. 1. Which two school years had about the same number of students per computer?

16 12 8

Use the weather graph for July to answer Exercises 4–6. 4. What was the lowest high temperature of the month? 5. On how many days was the high temperature above 90°? 6. What is the difference between the highest and the lowest high temperature?


6

'96-'97

'95-'96

'94-'95

'93-'94

'92-'93

'91-'92

0

'90-'91

4

3. How many more students were there per computer in ’95–’96 than in ’96–’97?

School Year

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

1-7

SCATTER PLOTS AND LINES OF BEST FIT Two sets of related data can be shown on a scatter plot. The points on a scatter plot are not connected. A line of best fit can be drawn on some scatter plots. If the line slopes up and to the right, the two sets of data are said to have a positive correlation. If the line slopes down and to the right, the two sets of data are said to have a negative correlation. Average Height of Children

E x a m p l e The scatter plot displays the heights of children ages 1 through 7. a. Estimate a line of best fit. b. Is there a positive or negative correlation between a child’s age and his or her height?

Height (in inches)

80

c. About how tall would an 8-year-old be from this data?

60

40

20

0

Solution

2

3

4 Age

5

6

7

Average Height of Children

b. The correlation is positive because the height increase as the age increases. The line of best fit slopes up and to the right. c. An 8-year-old child would be about 53 inches.

80 Height (in inches)

a. Estimation can cause the line of best fit to vary. A reasonable line of best fit is shown at the right.

60

40

20

0

EXERCISES

1. Estimate a line of best fit. 2. Is there a positive or negative correlation between the number of weeks and the amount of sales? 3. According to this data, what should the sales in week 8 be?

7

1

2

3

4 Age

5

6

7

6

7

Weekly Sales Sales (in thousands of dollars)

The manager of a store that is having an end-of-the-year clearance sale recorded the number of weeks the sale has been in progress and the sales from clearance items.


1

8

6

4

2

0

1

2

3

4 5 Week

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

1-8

BOX-AND-WHISKER PLOTS When you want to identify trends and summarize the information given in a set of data, you can use a box-and-whisker plot. The data shown in a box-and-whisker plot is divided into four equal parts. The first quartile, median (also called the second quartile), and the third quartile are the numbers that separate the data. The middle 50% of the data is represented by the box. The rest of the data is represented by segments on either side of the box, called whiskers.

First quartile

Least value

1

2

3

Third quartile

Median

4

5

6

7

8

9

Greatest value

10

E x a m p l e The box-and-whisker plot above shows ages of children enrolled in a summer day care program. a. How old is the youngest child enrolled? b. How old is the oldest child enrolled? c. What is the median age? d. What is the range of the middle 50%? Solution a. The least value is 1, so the youngest child enrolled is 1 year old. b. The greatest value is 10, so the oldest child enrolled is 10 years old. c. The median is represented by the vertical bar inside the box. The median age is 5. d. The range of the middle 50% is the difference of the third quartile, 8, and the first quartile, 3. Since 8 3 5, the range is 5.

EXERCISES

Amount Spent Per Person at the Movie Theater

The manager of a movie theater gathers data about the amount spent by each person when attending a movie in the evening.

6

7

8

9

10 11 12

13 14 15

1. What is the least amount spent?

2. What is the greatest amount spent?

3. What is the median amount spent?

4. What is the range of the middle 50%?


8

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

2-1

UNITS OF MEASURE The process of finding size, quantities, or amounts is called measurement. The smaller the unit of measure you use, the more precise the measurement. E x a m p l e

1

Which unit of measure gives a more precise measurement, inch or foot? Why? Solution Recall that there are 12 inches in 1 foot. So an inch is more precise that a foot since it is a smaller unit of measure. E x a m p l e

2

Which unit of measure gives a more precise measurement, millimeter or centimeter? Why? Solution Recall that there are 10 millimeters in 1 centimeter. So a millimeter is more precise than a centimeter since it is a smaller unit of measure.

EXERCISES Which unit of measure gives a more precise measurement? 1. milliliter, liter (HINT: There are 1000 milliliters in 1 liter.) 2. inch, yard (HINT: There are 36 inches in 1 yard.) 3. gram, kilogram (HINT: There are 1000 grams in 1 kilogram.) 4. pint, gallon (HINT: There are 8 pints in 1 gallon.) Tell whether each statement is true or false. Write true or false. 5. A pound gives a more precise measurement than an ounce. 6. A foot gives a more precise measurement than a mile. 7. A milligram gives a more precise measurement than a gram. 8. A kilometer gives a more precise measurement than a meter. 9. A cup gives a more precise measurement than a pint.


9

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

2-2

WORK WITH MEASUREMENTS When adding or multiplying measurements, you may need to change the units of the sum or the product. E x a m p l e a.

1

4 lb 6 oz 2 lb 11 oz

b.

Solution 4 lb 6 oz 2 lb 11 oz 6 lb 17 oz ← 17 oz is more than 1 lb. Simplify. 6 lb 17 oz 7 lb 1 oz

3 ft 8 in. 3

Solution 3 ft 8 in. 3 9 ft 24 in. 9 ft 24 in. 11 ft

When subtracting or dividing measurements, you may need to change a measurement before you can perform the operation. E x a m p l e

2

a. Write the answer in simplified form.

b. Complete. 6 L 8

9 gal 2 qt 2 gal 5 qt

Solution

?

mL

Solution Change 6 L to mL. 1 L 1000 mL 6 L 6000 mL 6000 mL 8 750 mL

Rename 9 gal 2 qt as 8 gal 6 qt. 8 gal 6 qt 2 gal 5 qt 6 gal 1 qt

EXERCISES Write each answer in simplified form. 1.

5.

7 lb 13 oz 3 lb 7 oz

3 c 12 fl oz 4

2.

6.

10 ft 7 in. 4 ft 11 in.

3.

10 yd 7 ft 4 yd 9 ft

4.

12 gal 3 qt 8 gal 5 qt

7. 7 yd 1 ft 2

9 lb 7 oz 5

8. 7 c 7 fl oz 3

Complete. 9. 3.5 kg 500 g 11. 2.75 cm • 12


10. 1.8 L 5

kg

12. 6 m 350 cm

mm

10

mL cm

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

2-3

LINEAR MEASURE AND PERIMETER The perimeter of a plane figure is the distance around it. To find the perimeter of any plane figure, find the sum of the lengths of the sides. E x a m p l e Find the perimeter of the figure shown below. Solution The measures of the sides of the figure are 7 cm, 4 cm, 10 cm, and 8 cm. Add these measures. 7 4 10 8 29 So, the perimeter of the figure is 29 cm.

EXERCISES Find the perimeter of each plane figure. 1. 2.

4.

3.

5.

6.

7. Find the perimeter of a rectangular garden that has a width of 16 ft and a length of 35 ft. 8. A triangle has sides that measure 40 cm, 75 cm, and 85 cm. What is its perimeter?


11

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

2-4

AREA To find the area of a parallelogram or a triangle, you use the measure of the base and the height. The height is always perpendicular to the base. Here are some possible positions of the height and base. Triangles

Parallelograms

Area of a parallelogram: E x a m p l e

Ab•h

1

Find the area of the parallelogram.

Solution Ab•h The area is 595 m2.

Area of a triangle:

A 1(b • h) 2

E x a m p l e

2

Find the area of the triangle.

Solution A 12(b • h) The area is 30 cm2.

A 35 • 17 595

A 12(12)(5) 30

EXERCISES Find the area of each figure to the nearest whole number. 1.

2.

3.

4.

5.

6.

7. Find the area of a hair ribbon 18 inches long with its ends cut on the diagonal. The width of the ribbon is 1.5 inches. 8. Find the area of the top surface of a napkin folded into a triangle. The folded edge is 13 cm long, and the height is 7 cm. © Glencoe/McGraw-Hill

12

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

2-5

PROBLEM SOLVING SKILLS: USE A FORMULA Before you can solve a geometric problem of measure, you must recognize which formula you need to use. Sometimes you may need a formula for perimeter. Other times you may need a formula for area. E x a m p l e A rectangular flower garden is 9 ft wide and 12 ft long. How much wire fencing will be needed to enclose it? Solution The fencing needed is to enclose, or go around, the garden. You need to know “distance around.” Find the perimeter of the garden plot. P 2l 2w 2 • 12 2 • 9 24 18 42 So, 42 ft of fencing will be needed to enclose the garden.

EXERCISES Which would you need, a perimeter or an area formula? 1. amount of lace strip to decorate the edges of a handkerchief 2. amount of material needed for a square tarpaulin boat cover Use one of these formulas to solve each problem P 2l 2w Al•w Ab•h

A 12 • b • h

3. The dimensions of one rectangular side of a tent are 3 m by 4 m. How much canvas would be required for both sides? 4. Marna’s backyard measures 80 yd by 90 yd. If sod costs $1.35/yd2, how much will Marna pay for sod to cover the entire yard? 5. Rhonda’s club is making pennants to sell at basketball games. A model for a pennant is shown. How many square feet of felt is needed for 300 pennants? 6. How much wrapping paper will an art dealer need to cover both sides of a large rectangular piece of stained glass? The glass is 7 feet high and 4 feet wide.


13

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

2-6

RATIO When ratios represent the same comparison, they are called equivalent ratios. Finding the equivalent ratios is similar to finding equivalent fractions. E x a m p l e

1

E x a m p l e

Write three ratios equivalent to 4:15.

2

Is this pair of ratios equivalent? 1 2 6 18 9

Solution Write the ratio as a fraction. Then, multiply or divide the numerator and denominator by the same nonzero number. 4 4•2 8 15 15 • 2 30 •3 1 2 4 4 • 45 15 15 3 •4 1 6 4 4 • 60 15 15 4 So, the ratios 8:30, 12:45, and 16:60 are equivalent to 4:15.

Solution 1 2 12 6 2 18 18 6 3 6 3 2 6 9 93 3 2 2 3 3 1 2 and 6 are Yes, the ratios 18 9 equivalent.

EXERCISES Write three equivalent ratios for each of the following.

1. 2:7

2. 11:12

3. 48:36

4. 5 8

1 6 5. 3

6. 9 10

Are the ratios equivalent? Write yes or no. 1 0 7. 2, 5 25 1 0 10. 4, 12 30


8. 3, 4 20 21 11. 5:14, 20:70

14

9. 10:12, 15:18 1 6 , 4 12. 24 6

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

2-7

CIRCUMFERENCE AND AREA OF A CIRCLE The circumference of a circle is the distance around it. The circumference of any circle, divided by the diameter, is always equal to the number 2 . You can find the (pi), which has the approximate value 3.14 or 2 7 circumference of a circle when you know its diameter, using the formula C d. Because the diameter is twice the radius, if you know the radius of a circle, you can find the circumference of the circle using the formula C 2r. To find the area of a circle, use the formula A r2, where r is the radius of the circle. If you are given the diameter of the circle, first find the radius by taking one half of the diameter. E x a m p l e

1

E x a m p l e

2

Find the circumference of the circle shown.

Find the area of the circle. Use 3.14. Round to the nearest tenth.

Solution C d C 3.14 • 10 C 31.4

Solution A r2 A • (2.1)2 A 3.14 • 4.41 A 13.8474

So, the circumference is 31.4 m.

The area of the circle is approximately 13.8 cm2.

1.

2.

3.

4.

5.

6.

7.

8.

9.


15

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

2-8

PROPORTION AND SCALE DRAWINGS A proportion is an equation that states that two ratios are equivalent. A proportion can be written in two ways: 5:15 3:9 or 5 3 15 9 The cross-products of the terms of a proportion are equal. You can use this fact to determine whether a statement is a proportion or to find the unknown term in a proportion. E x a m p l e

1

E x a m p l e

Tell whether the statement is a proportion. 8 2 0 2 2 55 Solution Find the cross-products.

2

Use mental math to solve the proportion. 4 ? 1 2 36 Solution Write the cross-product.

8 • 55 440

4 • 36 12 • ?

20 • 22 440

144 12 • 12 12 ?

So the statement 2 0 8 55 2 2 is a proportion.

Check by substituting 12 for ?. 4 4 4 4 1 3 1 2 12 1 2 12 12 1 36 36 12 3 So ? 12.

EXERCISES Tell whether each statement is a proportion. Write or .

1500 4646

1 5 1. 55 4. 2 3

2. 4 18 2 6 5. 52

1762 5705

460 7950

6 0 3. 9 6. 5 6

Use mental math to solve each proportion. 7. 3:4 9:? 1 4 24 10. 21 ?


8. 6:? 4:20 10 0 11. ? 20 125

16

9. 35:15 ?:3 1 8 ? 12. 45 25

MathMatters 1

Name _________________________________________________________

2-9

2.5 cm

To find the area of a shape like the one at the right, first separate it into smaller figures. Then find and add the area of the smaller figures.

3.5 cm

AREA OF IRREGULAR SHAPES

2.5 cm 3.5 cm

RETEACHING

Date ____________________________

4 cm

8 cm

8 cm

9 cm

E x a m p l e Find the area of the figure shown above.

Notice that the two smallest rectangles, X and Y, have the same area. So find the area of one and multiply by 2.

3.5 cm

2.5 cm X

2.5 cm

4 cm

3.5 cm

Solution The dashed segment shown in the figure at the right separates the larger figure into three smaller rectangles. First find the area of each of the smaller rectangles.

Y

8 cm

8 cm Z

Area of X 2.5 • 3.5 8.75 cm2 Area of X and Y 2 • 8.75 17.5 cm2

9 cm

Then find the area of the larger rectangle, Z. Notice that the length of this rectangle is 9 cm, and that the width is 8 cm 3.5 cm 4.5 cm. Area of Z 9 • 4.5 40.5 cm2 The area of the figure is 17.5 cm2 + 40.5 cm2 58 cm2.

EXERCISES Find the area of each figure. Use the dashed segments in Exercises 1 and 2 to help you. Draw dashed segments in the figure in Exercise 3 to help you. 3.

11 m

3 ft

8

12 in.

14 in.

18 m

25 m

4 in.

5 ft

9.5 ft

9.5 ft

5m 7m

7 in.

3 ft 5 ft

2.

21 in.

5 ft

1.

11 ft

6m


17

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

3-1

ADD AND SUBTRACT SIGNED NUMBERS To add integers on a number line: • start with zero; • move to the right for a positive integer; • move to the left for a negative integer. E x a m p l e 1 Add 4 5. Solution Use a number line.

E x a m p l e 2 Add 2 (1). Solution Use a number line.

The sum is 3. 2 (1) 3 The sum is 1. 4 5 1 You can use integer chips to model the subtraction of integers. E x a m p l e

3

Subtract 7 2. Solution To model 7 2, start with 7

You cannot take away two

Therefore, add two combinations of and chips so that you can subtract two chips.

chips.

When you take away two result is 9 chips. Therefore, 7 2 9.

chips.

chips, the

Notice that the result is the same as when you add the opposite of 2. 7 2 7 (2) 9

EXERCISES Draw a number line and add. 1. 4 (4)

2. 12 (8)

3. 5 (8)

4. 2 (9) Subtract.

5. 6 (3)

6. 3 (7)

8. 8 7 (6)

9. 9 5 17

7. 3 7 10. 15 (8)

11. 21 (8)

12. 19 (7)

13. 14 6

14. 3 (32)

15. 6 6

16. 9 (9)

17. 34 43

18. 16 (1)


18

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

3-2

MULTIPLY AND DIVIDE SIGNED NUMBERS The rules for multiplying integers are as follows: • The product of two integers having the same signs is positive. • The product of two integers having opposite signs is negative. E x a m p l e

1

E x a m p l e

Find the product of 9 • 8. Solution Since the signs of the factors are opposite, the sign of the product is negative. 9 • 8 72

2

Find the product of 7 • (6). Solution Since the signs of the factors are the same, the sign of the product is positive. 7 • (6) 42

The rules for dividing integers are as follows: • The quotient of two integers having the same signs is positive. • The quotient of two integers having opposite signs is negative. E x a m p l e 3 Divide 36 6. Then check by multiplying. Solution The signs of the dividend and divisor are different. So, the quotient will be negative. 36 6 6 Check: 6 • (6) 36

E x a m p l e 4 Divide 60 (12). Then check by multiplying. Solution The signs of the dividend and the divisor are the same. So, the quotient will be positive. 60 (12) 5 Check: 12 • 5 60

EXERCISES Use the rules for multiplying integers to find each product. 1. 8 • 5

40

4. 3 • 12

36

7. 6 • (6)

2. 7 • (11) 5. 8 • (10)

36

8. 8 • (6)

135

10. 3 • 5 • (9)

77 80

17. 54 (9) © Glencoe/McGraw-Hill

2 6

15. 32 (4)

48

9. 2 • (30)

18. 72 8 19

9

60

240

11. 6 • 5 • 8

8

36

6. 4 • 9

150 12. 5 • (6) • (5) 13. 4 • (7) • 2 Find each quotient. Then check by multiplying. 14. 14 (7)

63

3. 9 • (7)

56

16. 63 7 19. 18 (3)

9 6 MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

3-3

ORDER OF OPERATIONS It would be very confusing if people got different answers when evaluating an expression. To avoid that problem, operations are always done in a specified order. 1. Parentheses—perform all calculations within grouping symbols first. 2. Exponents—do all calculations with exponents. 3. Multiplication and division—multiply and divide in order from left to right. 4. Addition and subtraction—add or subtract in order from left to right. E x a m p l e 1 Simplify 21 7 • 2 4. Multiply.

Subtract.

}

}

Solution 21 7 • 2 4 21 14 4 74 ← 11

Add.

E x a m p l e 2 Simplify 32 • (7 1) 8 • 2 4. Solution

}

Parentheses.

3 • (7 1) 8 • 2 4 32 • 8 8 • 2 4 9•88•24 72 16 4 72 4 68 2

← ← ← ←

Exponents. Multiply. Divide. Subtract.

EXERCISES What operation should you do first? 1. 7 • 3 5 2

multiply

3. 56 (4 4) • 3

parentheses

5. 210 7 • 8 4

multiply

2. 74 (2 • 3)2 1 4. 23 5 • 3 6. 25 16 2 8

parentheses exponent subtract

Evaluate each expression. 7. 7 • 3 5 2

18

9. 56 (4 4) • 3

21

11. 2 • 2 0 • (18 56) © Glencoe/McGraw-Hill

8. 74 (2 • 3)2 1

4 20

39

10. 23 5 • 3

23

12. (25 8) 62 15

38 MathMatters 1

Name _________________________________________________________

Date ____________________________

3-4

RETEACHING

REAL NUMBER PROPERTIES Commutative Property

The order of addends or factors does not affect the answer.

abba a•bb•a

3 2 5; 2 3 5 7 • 4 28; 4 • 7 28

Associative Property

The grouping of addends or factors does not affect the answer.

a (b c) (a b) c a • (b • c) (a • b) • c

3 (4 5) (3 4) 5 4 • (2 • 3) (4 • 2) • 3

Distributive Property

A factor outside the parentheses can be used to multiply each term within the parentheses.

a (b c) (a • b) (a • c) a (b c) (a • b) (a • c)

6(2 3) (6 • 2) (6 • 3) 3(12 7) (3 • 12) (3 • 7)

EXERCISES Match each equation with the property illustrated. 1. 5 • (2 • 2) 10 • 2

d

a. commutative property of addition

2. 2 • 7 7 • 2

b

b. commutative property of multiplication

3. 4 (2 7) (4 • 2) (4 • 7)

e

c. associative property of addition

4. 15 (21 7) 15 (7 21)

a

d. associative property of multiplication

5. (8 3) 17 8 (3 17)

c

e. distributive property

Complete. 6.

10

8. 837

•

1.8

73 73 • 10

7. (5.6 8.2)

16

9. 6 • (5 • 12) (6 •

10. 2(3 8) (

16 837

2

12. 4(18 2) (4 •

•

3) (2 • 8)

18

11.

) (4 • 2)

3

5.6 (8.2 1.8)

5

) • 12

(5 2) (3 • 5) (3 • 2)

13. 0.3(4 9) (0.3 • 4) (0.3 •

9

)

Evaluate, using mental math. Name the property or properties you need. 14. 56 • 21 • 0 • 17 15. 36 18 14 16. 7 • 6 • 5 © Glencoe/McGraw-Hill

0; associative 68; commutative, associative 210; associative 21

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

3-5

VARIABLES AND EXPRESSIONS A variable is a letter that is used to represent an unknown number. To evaluate an expression means to find the value of the expression for a particular value of the variable or variables.

Variables a, b

E x a m p l e

E x a m p l e

Variable Expressions 3x 2 variables n5

1

2

Evaluate 14 n, if n 3.

Evaluate 2n 4, if n 8.

Solution 14 n 14 3 11

Solution 2 n4 2 • 8 4 16 4 20 Write 3 in place of n.

Write 8 in place of n.

EXERCISES Match each variable expression with its meaning. 1. n 2 2. 2 n

e

a. 2 more than a number

a

b. 2 divided by a number

3. 2n

d

c. 2 less than a number

d. twice a number c 4. n 2 e. a number divided by 2 b 5. 2 n Write an expression for each situation.

12 n

6. 12 cookies divided among n people

8n d7 7 less than d compact discs 8x 8 times x people

7. 8 more than n dollars 8. 9.

Evaluate each expression. Let n 18. 10. n 5

13

11. 8n

144

12. n 9

2

13. 14 n

32

Complete each table. 14.

16.

15.

0 6 12 18 © Glencoe/McGraw-Hill

0 1 2 6

0 1 2 30 22

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

3-6

PROBLEM SOLVING SKILLS: FIND A PATTERN In solving some problems, sometimes you can organize the information in a table that will show a pattern. E x a m p l e Every day, Maria saves twice as many pennies as she did the day before. If Maria begins the year by putting away one penny, how many pennies will she have by January 10? Solution Make a table.

Add the total number of pennies saved. 1 2 4 8 16 32 64 128 256 512 1023 Maria will have saved 1023 pennies.

EXERCISES Look for a pattern and choose the rule from those given. Then write the unknown numbers. 1. 0.24, 0.48, 0.72, , a. Double the previous number. b. Add 0.24 to the previous number. c. Add 0.024 to the previous number. 2. 12.6, 12.4, , 12.0, a. Add 0.2 to the previous number. b. Subtract 0.2 from the previous number. c. Subtract 0.02 from the previous number. 3. Find each product and look for a pattern. 10 • 10

15 • 15

9 • 11

14 • 16

4. If 45 • 45 is 2025, what is 44 • 46? 5. If 33 • 35 is 1155, what is 34 • 34? 6. Nikia put $10 in her savings account in January, $13 in February, $16 in March, and so on. If the pattern continues, how much money will she put into her account in December? How much will she have put into the account for the entire year?


23

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

3-7

EXPONENTS AND SCIENTIFIC NOTATION Multiples of a number can be written in exponential form. The base tells what factor is being multiplied. The exponent tells how many equal factors there are.

6 6 66 •

•

3

exponent Read “six to the third power.” base

Very large and very small numbers can be written as the product of a number greater than or equal to 1 but less than 10 and a power of 10. A number expressed in this form is in scientific notation. E x a m p l e 1 a. Write 3 • 3 • 3 • 3 in exponential form.

b. Write 75 in standard form.

Solution a. 3 • 3 • 3 • 3 34

4 factors

b. 75 7 • 7 • 7 • 7 • 7 16,807

5 factors

E x a m p l e 2 a. Write 5,600,000 in scientific notation. b. Write 3.8 • 102 in scientific form. Solution a. Move the decimal point to the right to get a number greater than or equal to 1 and less than 10.

5,600,000. ←

6 places

Write 5,600,000 as the product of 5.6 and a power of 10 equal to the number of decimal places you moved the decimal point. 5,600,000 5.6 • 1,000,000 5.6 • 106 b. 3.8 • 102 3.8 100 380

EXERCISES Write in exponential form. 1. 5 • 5 • 5 • 5 • 5 • 5

2. 10 • 10 • 10 • 10

Write in standard form. 3. 73

4. 80

5. 26

6. 302

Write in scientific notation. 7. 3700

8. 24,000,000

Write in standard form. 9. 1.6 • 104


10. 3.088 • 108

24

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

3-8

LAWS OF EXPONENTS You can use the rules of exponents in simplifying expressions containing numbers in exponential form. Product Rule: To multiply numbers with the same base, add the exponents: am • an am n Quotient Rule: To divide numbers with the same base, subtract the exponents: am an am n Power Rule: To raise an exponential number to a power, multiply the exponents: (am)n am • n E x a m p l e 1 Simplify. a. 33 • 32 Solution a. 33 • 32 33 2 35 33 • 32 (3 • 3 • 3) • (3 • 3) 35 E x a m p l e Simplify (42)3.

b. 54 52 b. 54 52 54 2 52 •5 • 5 5 • 5 54 52 52 5•5

2 Solution (42)3 42 • 3 46 (42)3 42 • 42 • 42 (4 • 4) • (4 • 4) • (4 • 4) 46

EXERCISES Use the product rule to simplify. 1. 26 • 28

2. 53 • 56

3. 104 • 107

4. 45 • 4

Use the quotient rule to simplify. 5. 78 7

6. 23 22

7. 46 46

8. 109 106

Use the power rule to simplify. 9. (23)4

10. (52)0

11. (72)10

13. Which of the following are equal to 210? b. (22)5 c. 210 20 a. 22 • 25


12. (103)6

14. Which of the following are equal to twice 216? b. 217 c. 131,072 d. 232 a. 416

25

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

3-9

SQUARES AND SQUARE ROOTS The square of 6 is 36, and the square of 6 is 36. 6 • 6 62 36 and (6) • (6) (6)2 36 6 is the positive square root of 36. 3 6 6

6 is the negative square root of 36. 3 6 6

Any number whose square roots are integers is a perfect square. For example, the numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are perfect squares. The square root of all numbers that are not perfect squares are irrational numbers—nonterminating, nonrepeating decimals. The square roots of these numbers are given as approximations, usually rounded to the nearest thousandth. The square root of 2, for example, is 1.414. E x a m p l e

1

E x a m p l e

2

Find each square. 2 a. 52 b. 3 c. (0.5)2 4

Find each square root a. 0 .8 1 b. 9 c. 3 8 25

Solution a. 52 5 • 5 25 2 b. 3 3 • 3 9 4 4 4 16

Solution a. 0.9 • 0.9 0.81, so 0 .8 1 0.9 b. 3 • 3 9, so, 9 3 5 5 25 25 5

c. (0.5)2 0.5 • 0.5 0.25

c. Use a calculator. 3 8 6.164

EXERCISES Find each square. 2. 222 1. 192 7. (0.19)2

8. (1.6)2

Find each square root. 13. 1 0 2 4 14. 6 7 6 19. 2 7

20. 1 9


3. 252 9. (3.2)2

4. (16)2

10. 5 7

2

5. 392

6. (28)2

11. 8 9

2

12. 3 4

2

15. 2 8 9

16. 0 .8 1

17. 0 .0 0 4 9 18. 2 .2 5

21. 8 9

22. 5 1

23. 7 5

26

24. 1 3

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-1

LANGUAGE OF GEOMETRY Several geometric figures are shown in the table along with their names and symbols. E x a m p l e

1

Name the endpoints of the line segment in the chart. Solution Points O and P. E x a m p l e

2

Name the vertex and sides of the angle in the chart. Solution The vertex is point G. → → The sides are GF and GH.

EXERCISES Write the symbol for each figure. 1.

2.

3.

4.

Draw a figure on your own paper to illustrate each of the following. → 5. TU 8. an angle with vertex Q


6. point R on plane q — 9. LM

27

7. CAT ↔ 10. JK

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-2

POLYGONS AND POLYHEDRA Examples of commonly used polygons are given below. Each polygon is named by the letters at its vertices, listed in order. E x a m p l e

1

Name the sides of the quadrilateral in the chart. Solution The quadrilateral has 4 sides, — — — — MO , OP , PN , and NM . E x a m p l e

2

Identify the polygon. Solution There are 6 sides. It is a hexagon.

EXERCISES Draw a figure on your own paper to illustrate each of the following. 1. an octagon

2. a quadrilateral with no two sides equal in length

3. triangle LMN

4. a pentagon with only two sides equal in length

5. hexagon RSTUVW

6. a quadrilateral with vertices A, B, C, and D

7. a triangle with sides — — — XY , YZ , and ZX

8. octagon MNPQRSTU

9. pentagon DEFGH


28

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-3

VISUALIZE AND NAME SOLIDS

EXERCISES Match each figure with its description or with the pattern that can be folded to form it. Identify the figure. 1.

a. a square base and four triangular faces

2.

b.

3.

c. a curved surface having a center equally distant from all points on the surface

4.

d. two octagons as bases

5.

e. four pairs of opposite faces that are squares

6.

f. a single base and five triangular faces


29

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-4

PROBLEM SOLVING SKILLS: NETS The surface of a three-dimensional figure can be shown as a two-dimensional figure called a net. The surface of the three-dimensional figure is formed by folding the net. E x a m p l e

Solution The net shows two rectangular bases and four rectangular faces. So the net must be a rectangular prism.

Identify the three-dimensional shape formed by the net at the right.

base

face

face

face

face

base

EXERCISES Label the base(s) and faces of each net. Then identify the three-dimensional shape formed by the net. 1.

2. base face face

face

face

face

face

base

face

face

base

3.

4. face

face

face base

face

base face

face

face base

face

face base


30

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-5

ISOMETRIC DRAWINGS Parallel and perpendicular lines are important parts of three-dimensional figures.

a b

Parallel lines lie in a plane and do not intersect. In the figure at the right, line a is parallel to line b. The symbol | | is used to indicate parallel lines. So “line a | | line b” is read “line a is parallel to line b.”

m

Perpendicular lines lie in a plane and intersect to form four right angles. In the figure at the right, line m is perpendicular to line n. The symbol –| is used to indicate perpendicular lines. So “line m –| line n” is read “line m is perpendicular to line n.”

n

EXERCISES Tell whether each pair of lines are parallel or perpendicular. Then use the symbol | | or –| to write a statement about the relationship. 1.

2. x

p

y

q

3.

4.

c

d


j

31

k

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-6

PERSPECTIVE AND ORTHOGONAL DRAWINGS All one-point perspective drawings have one vanishing point which lies on the horizon line. In the perspective drawing at the right, point X is the vanishing point and line m is the horizon line.

X

m

E x a m p l e Locate and label the vanishing point and horizon line.

Solution Draw line segments connecting the vertices front and back faces of the prism. Label the point at which the line segments intersect A. Draw a line through point A that is parallel to the horizontal edges of the back face. Label it b.

A

b

Point A is the vanishing point and line b is the horizon line.

EXERCISES Locate and label the vanishing point and horizon line. 1.


2.

32

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-7

VOLUME OF PRISMS AND CYLINDERS You can find the volume of an object by counting the number of unit cubes that would fill the space. You can also use a formula. The volume of a prism or a cylinder is equal to the area of its base times its height, so you can use these formulas. For a rectangular prism:

(

V (l • w) • h E x a m p l e

For a triangular prism: • • V 1 2 b h h

1

For a cylinder:

)

V r 2 • h

E x a m p l e

2

Find the volume of the rectangular prism.

Find the volume of the cylinder. Round to the nearest cubic centimeter. Use 3.14. Solution The diameter of the base is 14 cm, so the radius is 7 cm. V r 2 • h V 3.14 • 72 • 24 V 3692.64 The volume is approximately 3693 cm3.

Solution V (l • w) • h V (22 • 24) • 12 V 6336 The volume of the prism is 6336 in3.

EXERCISES Find the volume of each rectangular prism. 1. l 2 ft, w 4 ft, h 6 ft ______________________________________________________ 2. l 8 cm, w 5 cm, h 6 cm _________________________________________________ 3. l 10 m, w 5 m, h 4 m ___________________________________________________ 4. l 16 in., w 15 in., h 18 in.________________________________________________ Find the volume of each cylinder. 5. r 2 in., h 5 in.

6. r 4 ft, h 10 ft

7. r 6 m, h 4 m

8. r 12 cm, h 20 cm

Find the volume. 9.


10.

33

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-8

VOLUME OF PYRAMIDS AND CONES Formulas for the volume of a pyramid and a cone are given below. Compare the formula for the volume of a prism with that of a pyramid. Then compare the formula for the volume of a cone with that of a cylinder. Rectangular Prism

You know VB•h V (l • w) • h

Rectangular Pyramid

Cylinder

New V 1 • B • h 3 V 1 • l • w • h 3

You know VB•h

Cone

New V 1 • B • h 3 V 1 • r 2 • h 3

V r 2 • h

E x a m p l e Find the volume to the nearest whole number. Use 3.14.

a.

b.

Solution a. V 1 • l • w • h 3 V 1 • 3 • 4 • 5 3

b. V 1 • r 2 • h 3 V 1 • 3.14 • 42 • 6 3 V 1 • 3.14 • 16 • 6 3 The volume of the cone is approximately 100 in3.

The volume of the pyramid is approximately 20 in3.

EXERCISES Find the volume of each, to the nearest tenth. Use 3.14. 1.


2.

3.

34

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

4-9

SURFACE AREA OF PRISMS AND CYLINDERS To find the surface area, SA, of a prism, find the sum of the areas of all its faces. To find the surface area of a cylinder, add the area of the curved surface to the sum of the areas of the two circular bases. Use these formulas. rectangular prism: SA 2 • (l • w l • h w • h)

cylinder: SA 2rh 2r 2

E x a m p l e Sketch and label the unfolded faces of each figure. Then use a formula to find the surface area. Use 3.14. Round to the nearest whole number. a.

b.

Solution a. 2 • (2 • 3) 12 2 • (2 • 4) 16 2 • (3 • 4) 24 SA 12 16 24 52 SA 52 cm2

b. SA 2rh 2r 2 2rh 2 • 3.14 • 3 • 4 75.36 2 2r 2 • 3.14 • 32 2 • 3.14 • 9 56.52 SA 75.36 56.52 207.24 SA 207 cm2

EXERCISES Sketch and label the unfolded faces of each figure on another sheet of paper. Then use a formula to find the surface area. Use 3.14. Round to the nearest whole number. 1.


2.

35

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-1

INTRODUCTION TO EQUATIONS An equation is a statement that two numbers or expressions are equal. An open sentence is a sentence containing one or more variables. These are open sentences. x14

y23

An open sentence can be either true or false, depending on what value is substituted for the variables. A value of the variable that makes the equation true is called a solution of the equation. E x a m p l e

1

Tell whether the equation is true, false, or is an open sentence. a. 4(3 2) 20 b. 2x 4 6 c. 4 1 2(6 8) Solution a. 4(3 2) 20 4(5) 20 20 20

b. 2x 4 6

The equation is true.

The equation contains a variable.

c. 4 1 2(6 8) 3 2(2) 3 4 means “is not equal to”

This equation is false.

The equation is an open sentence. E x a m p l e

2

Use mental math to solve the equation 2 x 4. Solution Think: What number added to 2 equals 4? You know that 2 6 4, so x 6.

EXERCISES Tell whether the equation is true, false, or an open sentence. 1. 4(9 3) 22

2. 2x 3 5

3. 12 4 2(10 2)

4. 9 3 2(3 6)

5. 3y 4 16

6. 2(4 6) 16 20

Use mental math to solve the following equations. 7. x 4 9 11. 2a 18


8. y 2 7 12. 11 p 8

36

9. c 9 17

10. d 8 2

13. 6 r 1

14. 7 13 z

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-2

ADD OR SUBTRACT TO SOLVE EQUATIONS To solve an equation, find all values of the variable that make the equation true. To do so, you must perform operations on the equation so that you can get the variable alone on one side of the equals sign. Many equations contain addition or subtraction. x 7 10

y38

To solve such equations, you must reverse, or “undo,” the addition or subtraction in the equation. Undo addition by subtracting. Undo subtraction by adding. In doing so, you must keep the equation in balance by adding or subtracting the same number from both sides of the equation. E x a m p l e

1

Solve x 7 10. Solution You want to get x alone on the left side of the equals sign. Undo the addition in the equation by subtracting 7 from → both sides of the equation.

3 7 10 10 10

CHECK: Substitute the value 3 into the original equation. E x a m p l e

x 7 10 x 7 7 10 7 x03 x3

2

Solve y 3 8. Solution Undo the subtraction by adding 3 to both sides of the equation.

→

y38 y3383 y 0 11 y 11 11 3 8 88

CHECK: Substitute the value 11 into the original equation.

EXERCISES Solve each equation. Check the solution. 1. x 5 7

2. y 3 10

3. x 3 10

4. n 4 9

5. t 6 6

6. x 4 3

7. p 7 7

8. r 21 9

9. y 4 1

10. 6 n 5


11. 8 f 8 37

12. 7 y 13

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-3

MULTIPLY OR DIVIDE TO SOLVE EQUATIONS Multiplication and division are inverse operations. Therefore, you can solve an equation involving multiplication by dividing. Likewise, you can solve an equation involving division by multiplying. Remember, both sides of the equation must be multiplied or divided by the same number. E x a m p l e

1

E x a m p l e

2

Solve n 4. 2 Solution Undo the division. 2 n 2(4) Multiply each side by 2 to balance 2 the equation.

Solve 3x 6. Solution Undo the multiplication. 3 x 6 Divide each side by 3 to balance the 3 3 equation.

1x 2 x2

1n 8 n8 CHECK: 8 4 2

CHECK: 3(2) 6

1. 3x 12

2. n 7 3

3. 10m 80

4. t 7 7

5. 16 4y

6. 6b 72

7. 5 x 9

y 8. 0 8

9. c 5 2

10. 8a 64

11. 18y 36

12. x 8 5

y 13. 8 3

14. 12x 48

15. 4n 16

16. 2y 0

17. x 7 8

18. 16c 96

19. 18d 234

20. w 21 5


38

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-4

SOLVE TWO-STEP EQUATIONS Some equations, called two-step equations, require more than one inverse operation to solve. Follow these steps: • First, undo addition or subtraction. • Then undo multiplication or division. E x a m p l e 1 Solve 3x 4 5. Solution Undo the subtraction. Undo the multiplication.

E x a m p l e 2 y Solve 3 13. 5 Solution Undo the addition.

Undo the division.

3x 4 4 5 4 3x 9 3 x 9 3 3

y 3 3 13 3 5 y 10 5

x3

y 50

y 5 5(10) 5

EXERCISES Solve each equation. Check the solution. 1. 5n 4 29 2. m 3 8 7

3. x 6 4 2

4. 8y 7 17

5. 9t 5 14

6. 12 n 18 2

7. 15 5y 20

8. 3 7x 17

y 9. 6 5 4

Write an equation for the following problem and solve. 10. Jeff bought two tapes on sale at the music store. They both cost the same amount. Then Jeff bought a tape-head cleaner for $6. Altogether Jeff spent $22. How much did Jeff pay for one tape? © Glencoe/McGraw-Hill

39

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-5

COMBINING LIKE TERMS Those parts of a variable expression that are separated by addition or subtraction signs are called terms. The expression 3x 5y 2x x2 contains 4 terms. 3x and 2x are like terms. The variable parts are identical. 5y, x 2, and 3x are unlike terms. The variable parts are different. You can simplify expressions by combining like terms. E x a m p l e 1 Simplify 3a 4b 5a. Solution Rewrite using the commutative property.

3a 4b 5a 3a 5a 4b 3a 5a 4b (3 5)a 4b

Use the distribute property.

You can solve equations by combining like terms. E x a m p l e 2 Solve 4x 16 2 2x. Solution Add 2x to each side. Simplify. Add 16 to each side. Divide both sides by 6.

4x 2x 16 2 2x 2x 6x 16 2 6x 16 16 2 16 6x 18 6 x 1 8 6 6 x3

CHECK: 4(3) 16 2 2(3) 12 16 2 6 4 4

EXERCISES Simplify by combining like terms. 2. 5a 8 2a 1 1. x 3y 2x 5 4. m m3 2m m3 Solve and check. 7. 5y 3 4y 5 10. 5x 4 2x 2 2


3. 3x 7y 2 3x 2y 2

5. 7x 3xy 2xy x

6. 5x 2y 2xy 2 4x 2y

8. m 3m 2 2m 14

9. 3t 17 t 1

11. 4y 2y 7 2y y

40

12. 4n 2n 6

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-6

FORMULAS An equation that states a relationship between two quantities is a formula. Formulas are commonly used in many fields, including mathematics, the sciences, statistics, and banking. Here are some of these formulas. Volume of a prism V lwh

Rate/Distance/Time d rt

Perimeter of a rectangle p 2l 2w

Area of a triangle A 1bh 2

V volume l length w width h height

d distance r rate t time

p perimeter l length w width

A area b base h height

E x a m p l e

1

E x a m p l e

2

Find the volume of a rectangular prism 12 ft long, 2 ft wide, and 5 ft high. Solution Substitute the values into the formula for volume. Solve for V. Vl•w•h V 12(2)(5) V 120

How long would it take a car to go 135 mi at an average rate of 45 mi/h? Solution Substitute the values into the formula for distance. Solve for t. d rt 135 45t 13 5 45t 45 45 3t

Volume is expressed in cubic units. So, the volume is 120 ft3.

It would take the car 3 h to go a distance of 135 mi.

1. How far would a car go in 4 h at an average speed of 55 mi/h?

2. Find the area of a triangle with a base of 3 cm and a height of 8 cm.

3. Find the perimeter of a rectangle 4 m long and 3.5 m wide.

4. Find the volume of a rectangular prism 4 in. long, 5 in. wide, and 7 in. high.

5. How long would it take you to walk 10 mi if you walk at an average rate of 4 mi/h?

6. What is the height of a rectangular prism if the volume is 210 cm3, the width, 6 cm, and the length, 7 cm?


41

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-7

PROBLEM SOLVING SKILLS: WORK BACKWARD Some problems give you the final result of a series of steps and ask you to find the starting condition. To solve such a problem you can work backward to the beginning. E x a m p l e Kim spent a total of $18 at the movies. She bought 3 tickets, a bucket of popcorn for $3.50, and a drink for $1.00. How much did she spend for each ticket? Solution Everything Kim spent must add up to $18. First, subtract the cost of the drink and the popcorn from that total. $18 $1.00 $17 $17 $3.50 $13.50 3t $13.50

Subtract the cost of the drink. Subtract the cost of the popcorn. The cost of three tickets is $13.50. Let t be the cost of one ticket. Divide by 3 to find the cost of a ticket. Kim spent $4.50 for each ticket.

$13 .50 $4.50 t 3

EXERCISES Solve by working backward. 1. Mario had $5 left after shopping at the mall. He spent $12 at the record shop, $23 at the sports store, and $5 at the food mart. How much did Mario have when he went to the mall?

2. Patty works at a pet store. Last week she earned $45. She makes $5 an hour. Mrs. Sanchez paid her an extra $10 for clipping her garden hedge. How many hours did Patty work at the pet store?

3. David, Chen, and Leslie raised money for the school in a walk-a-thon. Altogether they made $46. Leslie made $10. Chen made twice as much as David. How much did David make?

4. Willie had 55 baseball cards after trading with his friends. He traded 5 cards for one special card. Lee gave him 11 cards for a Mickey Mantel card. How many cards did Willie start with?

5. Classes start at 8:30 A.M. at Lee’s school. It takes him 15 minutes to ride his bike to school, but he likes to arrive 20 minutes early. What time should he leave home?

6. After Julie went to sleep, the temperature rose 6° F before falling twice as far as it had risen. The next morning, the temperature was 14° F. What was the temperature when Julie went to sleep?


42

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-8

GRAPH OPEN SENTENCES An open sentence can be an equation or an inequality and can be graphed on a number line. A solid dot means the number is included. An open circle means the number is not included. E x a m p l e

1

a. Graph x 5. Solution The graph of x 5 is a solid dot on 5 on a number line.

E x a m p l e

b. Graph x 5. Solution The graph of x 5 is an open circle on 5 with an arrow pointing to numbers less than 5.

2

a. Graph x 5. Solution The graph of x 5 is an open circle on 5 with an arrow pointing to the numbers greater than 5.

E x a m p l e

b. Graph x 5. Solution The graph of x 5 is a solid dot on 5 with an arrow pointing to the numbers less than 5.

3

Graph x 5. Solution The graph of x 5 is a solid dot on 5 with an arrow pointing to the numbers greater than 5.

EXERCISES Graph each open sentence. 2. y 4

1. x 4 6

4

2

0

2

4

6

6

4

2

0

2

4

6

6

0

2

4

6

4

2

0

2

4

6

4

2

0

2

4

6

6. y 0

5. x 5 6

2

4. n 2

3. p 3 6

4

4

2


0

2

4

6

6

43

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

5-9

SOLVING INEQUALITIES Inequalities can be solved in almost the same way as equations. However, when you multiply or divide by a negative number, you must reverse the sign.

E x a m p l e

1

E x a m p l e

Solve x 3 5. 3 Solution Subtract 3 from each side.

2

Solve 3y 4 2. Solution Add 4 to each side.

x 3 3 5 3 3 x 2 3 x(3) 2(3) 3 x 6

Multiply each side by 3.

3y 4 4 2 4 3y 6 3 y 6 3 3 y 2

Divide each side by 3. Graph the solution.

Graph the solution.

EXERCISES Solve each inequality and graph the solution. 1. 4s 2 10 8 6 4 2

2. 3t 5 2 0

2

4

6

8 6 4 2

8

3y 3. 4 2 1 8 6 4 2


2

4

6

8

0

2

4

6

8

0

2

4

6

8

x

2 4. 3 0

2

4

6

8 6 4 2

8

5. 5n 3 12 8 6 4 2

0

6. 6m 2 14 0

2

4

6

8 6 4 2

8

44

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

6-1

PERCENTS AND PROPORTIONS You can use a proportion to solve problems involving percents. E x a m p l e

1

E x a m p l e

What number is 20% of 750? Solution Write a proportion. part part wh ole wh ole 2 0 x 10 0 750 Solve the proportion. 20 • 750 100x 150 x

2

What percent is 300 is 270? Solution Write a proportion. Let the unknown percent be x% and write it as x. 100 x 27 0 10 0 300 300x 27,000 3 0 0x 27,0 00 300 300 x 90

E x a m p l e

3

12 is 25% of what number? Solution Write a proportion. Let n the number part → 25 1 2 wh ole 10 0 n Solve the proportion. 12 • 100 25 • n 1200 25n 48 n So, 12 is 25% of 48.

So, 270 is 90% of 300.

EXERCISES Find the percent of each number, using a proportion. 1. 40% of 700

2. 25% of 4800

3. 55% of 165

4. 50% of 492

5. 11% of 300

1 6. 12% of 72 2

7. 7.5% of 2000

1 8. 12% of 64 2

9. 0.4% of 20

Write and solve a proportion. 10. What percent of 80 is 4?

11. What percent of 56 is 7?



14. 36 is what percent of 144?


16. 45% of what number is 135?

17. 33 is 75% of what number?


45

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

6-2

WRITE EQUATIONS FOR PERCENTS You can write equations to solve problems involving percents. E x a m p l e

1

E x a m p l e

What number is 10% of 300? Solution Write an equation. Use a decimal or a fraction for the given percent. x 0.10 • 300 x 30 1 or x • 300 10 x 30

2

What percent of 400 is 320? Solution Let p the percent p • 400 320 400p 320 400 320 400 400 p 0.8 So, 320 is 80% of 400.

E x a m p l e

3

14 is 20% of what number? Solution Let n the number 0.20 • n 14 Solve the equation. 0.20n 14 0.20n 14 0.20 20 n 70 So, 14 is 20% of 70 Check: 70 • 0.20 14

EXERCISES Find the percent of each number, using an equation. 1. 27% of 700

2. 30% of 120

3. 75% of 60

4. 80% of 98

1 5. 2% of 40 2

6. 6.7% of 500

7. 0.5% of 34

8. 14.8% of 2000

1 9. 33% of 360 3

Write and solve an equation. 10. What percent of 72 is 27?









19. 34% of what number is 238?


46

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

6-3

DISCOUNT AND SALE PRICE Many stores offer a discount at one time or another to encourage people to buy their merchandise. The discount is the percent of the regular price by which that price is reduced. The sale price is the regular price less the discount. E x a m p l e

1

E x a m p l e

The regular price of a wool coat at Sterling Clothes is $84.50. Sterling is offering a 15% discount this week. What will the sale price be after the discount? Solution First, find the amount of the discount. $84.50 0.15 12.675 or $12.68 to the nearest cent Then, subtract the discount from the regular price. $84.50 12.68 $71.82 So, the sale price will be $71.82.

2

The employees at Book Market get a 10% discount on any purchase. Dave works at Book Market and wants a book that costs $14.95. How much will he pay for it? Solution First, find the amount of the discount. Use mental math: $14.95 • 0.1 $1.495 or $1.50 Now, subtract the discount from the regular price. $14.95 1.50 $13.45 So, Dave will pay $13.45.

1. Regular price: $225.75 Percent of discount: 30%



4. Regular price: $6000 Percent of discount: 8.5%

5. Regular price: $340 Percent of discount: 20%





47

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

6-4

TAX RATES Sales taxes, property taxes, and income taxes are examples of some of the taxes people pay. A tax is usually a percentage of a certain amount. E x a m p l e The price of a coat is $80. The sales tax is 7%. a. Find the amount of the sales tax. b. Find the total cost of the coat with tax. Solution a. To find the amount of the sales tax, find 7% of $80. 7% of $80 sales tax 0.07 • 80 sales tax 5.60 sales tax The sales tax is $5.60. b. To find the total cost of the coat with tax, add the amount of sales tax to the price of the coat. sales tax price total cost 5.60 80 total cost 85.60 total cost

EXERCISES Find the amount of the sales tax and total cost of each item. 1. Video: $15 Sales tax rate: 5%

2. Sweater: $40 Sales tax rate: 8% Amount of sales tax: Total cost of sweater:

Amount of sales tax: Total cost of video: 3. Book: $25 Sales tax rate: 7% Amount of sales tax: Total cost of book:

4. Calculator: $88 Sales tax rate: 6% Amount of sales tax: Total cost of calculator:

5. Software: $35 Sales tax rate: 9% Amount of sales tax: Total cost of software:

6. CD: $18 Sales tax rate: 5.5% Amount of sales tax: Total cost of CD:


48

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

6-5

SIMPLE INTEREST Interest (I) is paid for the use of money over a period of time. The money earning interest, or that has been borrowed, is the principal (p). The rate (r) is the percent of the principal charged for the use of the money over a period of time. The time (t) is the number of time periods during which the principal earns interest or has not been paid back. The formula for interest earned or to be paid is I prt. E x a m p l e

1

E x a m p l e

Find the interest due on a loan of $500, at a rate of 12% per year for 6 months. Solution Identify the principal, rate and time. p $500, r 12%, t 0.5 (6 months is a half year) Substitute these amounts in the formula. I prt I $500 • 0.12 • 0.5 I $30 The interest due on the loan is $30.

2

Find the total amount in a savings 1 account that pays 62% interest per year after 6 years. The original amount invested was $350. Solution Identify the principal, rate, and time. Then use the interest formula. 1 p $350, r 6%, t 6 2 I $350 • 0.065 • 6 I $136.50 To find the total amount in the savings account, add the amount of interest to the principal. $350 $136.50 $486.50 So, the total amount after 6 years is $486.50.

EXERCISES Find the interest and the total amount. Principal

Rate

Time

1. $640

5%

6 mo

2. $3000

4%

2 yr

3. $475

11%

3 yr

4. $700 6. $200

7.5% 51% 4 9.5%

2 yr 21 yr 2 3 mo

7. $860

12%

4 mo

8. $138

18%

2 mo

9. $620

6.65%

2 yr

5. $58.50


Interest

49

Total Amount

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

6-6

COMMISSION Salespeople who earn a commission receive a percent of their total sales. The percent they earn is called the commission rate. The amount of money they earn is called the commission. E x a m p l e Tyler earns a base salary of $200 a week plus a commission of 5% on all of his sales. Last week his sales were $500. a. Find the commission Tyler earned last week. b. Find Tyler’s total income last week. Solution a. The commission is 5% of $500. 5% of $500 commission 0.05 • 500 commission 25 commission Tyler earned a commission of $25 last week. b. Tyler’s total income is his base salary plus his commission. total income base salary commission total income $200 $25 total income $225 Tyler’s total income last week was $225.

EXERCISES Find the commission and the total income. 1. Base salary: $350/wk Total sales: $600/wk Commission rate: 5% Commission: Total income:

2. Base salary: $600/mo Total sales: $800/mo Sales tax rate: 2% Commission: Total income:

3. Base salary: $400/wk Total sales: $1000/wk Commission rate: 4% Commission: Total income:

4. Base salary: $900/mo Total sales: $1500/mo Sales tax rate: 8% Commission: Total income:

5. Base salary: $550/mo Total sales: $450/mo Commission rate: 5% Commission: Total income:

6. Base salary: $25,000/yr Total sales: $50,000/yr Sales tax rate: 10% Commission: Total income:


50

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

6-7

PERCENT OF INCREASE AND DECREASE To find the percent of increase or decrease, express the ratio of the amount of increase or decrease to the original amount as a percent. E x a m p l e

1

E x a m p l e

2

Find the percent of increase. Original number: 56 New number: 63

Find the percent of decrease. Original number: 240 New number: 18

Solution

Solution

Find the amount of increase. 63 56 7 Write a ratio. amount of increase 7 original number 56 Find the percent for this ratio. 7 0.125 12.5% 56 So, the percent of increase is 12.5%.

First, find the amount of the decrease. 240 18 222 Write a ratio. amount of increase 222 original number 240 Write a percent for this ratio. 222 0.925 92.5% 240 So, the percent of decrease is 92.5%.

EXERCISES Find the percent of increase. Round to the nearest tenth. 2. Original weight: 120 lb 1. Original price: $300 New weight: 140 lb New price: $360 3. Original rent: $500 New rent: $575

4. Original number: 4500 New number: 4590

Find the percent of decrease. Round to the nearest tenth. 5. Original price: $50 New price: $35

6. Original number: 400 New number: 352

7. Original salary: $60 New salary: $45

8. Original size: 80 cm New size: 48 cm


51

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

6-8

PROBLEM SOLVING SKILLS: MAKE A TABLE Some problems ask you to find out in how many different ways items can be arranged. Many of these problems can be solved by making a table. The table must be clearly set up, then carefully filled in. E x a m p l e In how many ways can you make 15¢ using any combination of pennies, nickels and dimes? Solution Make a table that lets you record the number of each kind of coin. Each row must give a total of 15¢. Work in a certain order. For example, begin by using the least amount of pennies and work up to using all pennies. The completed table shows that 6 combinations are possible.

EXERCISES Copy and complete the table to find all possible answers for each problem. 1. How many different 3-digit numbers can you make using the digits 9, 2 and 1? No digit should be used more than once in any number. Hundreds 9 9

Tens 2 1

2. There are 4 runners, A, B, C, and D, in a race. In how many different ways can the runners finish first, second, third, or fourth.

Ones 1 2

First A A

Number of ways:

Second B B

Third C D

Fourth D C

Number of ways:

Make a table to find all possible answers for each problem. 3. How many different 4-digit numbers can you make using the digits 8, 4, 2, and 1? No digit should be used more than once in any number.


4. How many different ways can you make 12¢ using any combination of pennies, nickels, and dimes?

52

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

7-1

PROBLEM SOLVING SKILLS: QUALITATIVE GRAPHING A qualitative graph communicates information about situations such as the speed of a car and distance from home when traveling. E x a m p l e Carl is on a trip. The graph at the right provides information about how far Carl is from home. a. What is represented by the horizotal axis? b. What is represented by the vertical axis? c. What is represented by the segment labeled a? d. What is represented by the segment labeled b? e. What is represented by the segment labeled c? f. What is represented by the segment labeled d? Solution a. The horizontal axis represents the time Carl has been on his trip. b. The vertical axis represents Carl’s distance from home. c. The segment labeled a represents time traveling to the first destination.

distance

d c b a

0

time

d. The segment labeled b represents time spent at the destination. e. The segment labeled c represents traveling to a second destination. f. The segment labeled d represents continued travel to a second destination but at a slower speed.

EXERCISES Match each situation with one of the A. graphs shown. Each graph should be used only once.

B.

1. Distance from home when returning from a vacation 2. Speed of a bicyclist on a trip that included one stop for lunch

0

0

time

time

D.

C.

3. Heart rate during an aerobics class 0


53

time

0

time

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

7-2

COORDINATE PLANE Two number lines that are perpendicular to one another can be used to locate points on a coordinate plane. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The point at which the lines intersect is called the origin. Its coordinates are (0, 0). E x a m p l e

1

E x a m p l e

2

Locate point F(3, 2) on the coordinate plane.

Give the coordinates of point D. Solution Find D on the grid. The x-coordinate of D is 4; the y-coordinate is 5. So, the coordinate of point D are (4, 5).

Solution Point F is 3 units to the right of the origin and 2 units down.

EXERCISES Refer to the figure. Give the coordinates of the point or points. 1. point A

y

2. point B

5 4

3. a point in the third quadrant

2 1

5 4 3 2 1 0 1 S 2

5. What point has the coordinates (3, 3)? 6. What point has the coordinates (4, 2)?

B

C

3

D

4. a point with a y-coordinate of 4

A

T

3 4

1 2 3 4 5x E

5

7. What point has the coordinates (1, 4)? Graph each point on the grid at the right. 8. X(2, 3) © Glencoe/McGraw-Hill

9. Y(3, 2) 54

10. Z(1, 1) MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

7-3

RELATIONS AND FUNCTIONS Definitions

E x a m p l e s

A relation is a set of ordered pairs.

(2, 3), (4, 1), (6, 2), (5, 3)

The first value in each ordered pair is in the domain of the relation.

domain: 5, 2, 4, 6

The second value in each ordered pair is in the range of the relation.

range: 3, 1, 2, 3

If each value of the domain is paired with exactly one value of the range, then the relation is a function. A mapping is a visual way to show the pairing of the domain and range.

–5 2 4 6

–3 –1 2 3

The mapping shows exactly one arrow from each value of the domain. The relation is a function.

EXERCISES State the domain and range of each relation. Then state whether each relation is a function. 1. (3, 4), (1, 5), (5, 2)

2. (1, 5), (6, 1), (4, 1), (6, 2)

domain:

domain:

range:

range:

Is it a function?

Is it a function?

3. (0, 5), (3, 4), (3, 4), (8, 5)

4. (10, 4), (4, 6), (7, 2), (10, 5)

domain:

domain:

range:

range:

Is it a function?

Is it a function?

5. (3, 5), (2, 4), (2, 2), (3, 9)

6. (6, 4), (8, 3), (6, 4), (8, 3)

domain:

domain:

range:

range:

Is it a function?

Is it a function?


55

MathMatters 1

Name _________________________________________________________

Date ____________________________

7-4

RETEACHING

LINEAR GRAPHS An equation in two variables has an infinite number of solutions. The solution of an equation like x y 7 is an ordered pair of numbers. For instance, (3, 4), (2, 5), and (1, 8) are three solutions of x y 7. E x a m p l e Graph the equation y 2x 1. Solution Make a table of three ordered pairs that are solutions of the equation. x

2x 1

0 1 2

2(0) 1 2(1) 1 2(2) 1

y 1 1 3

(x, y) (0, 1) (1, 1) (2, 3)

Graph the ordered pairs. Draw a line through the points. All points along the line are solutions of y 2x 1.

EXERCISES For each equation, complete the table for three solutions. Then graph the equation. 2. y x 4

1. y 3x x

3x

y

(x, y)

x 4

x

3. y 2x 5 y

(x, y)

x

1

0

0

0

1

1

1

2

2

y

4

2

2x 5

y

y

y

4

4

4

2

2

2

0

2

4

x

4

2

0

2

4

x

4

2

0

2

2

2

4

4

4


56

(x, y)

2

4

x

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

7-5

SLOPE OF A LINE The steepness of rise or descent along a line is called the slope of the line. E x a m p l e

1

Find the slope of the line graphed at the right. Solution Choose two points on the line, say point A (2, 3) and point B (1, 1). Find the number of units of change vertically and horizontally in moving from A to B. Subtract the y-coordinate of A from that of B. Subtract the x-coordinate of A from that of B. ris e difference of y-coordinates 1 3 2 slope run difference of x-coordinates 1 ( 2) 3 The slope of the graphed line is 2 . 3 E x a m p l e 2 Find the slope of the line graphed at the right. The coordinates of point A are (1, 0). The coordinates of point B are (2, 2). Subtract the y-coordinates and the x-coordinates in the same order. 0 2 2 2 slope 1 2 1 The slope is 2.

EXERCISES Find the slope of the line that passes through each pair of points. 1. S(2, 7) and T(4, 13)

2. V(0, 3) and W(2, 8)

4.


3. X(1, 1) and Y(3, 4)

5.

57

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

7-6

SLOPE-INTERCEPT FORM OF A LINE When a linear equation is in the form y mx b, or slopeintercept form, you can read the slope (m) and the y-intercept (b) directly from the equation. With this information you can graph the equation. E x a m p l e Graph the equation y 2x 1. 3 Solution The y-intercept is 1, which means that the graph crosses the y-axis at 1. The ordered pair is (0, 1). Use the slope to find another point on the graph. 2 Start at (0, 1). The slope is 3. Go up 2 units and over to the right 3 units. Mark that point. You may want to mark another point, again going up 2 and over to the right 3 units. Draw a straight line through the points you have marked. Check: Solve the equation for any value of x. Let x 6 y 2(6) 1 3 y41 y3 Is the point (6, 3) on the graph? Yes.

EXERCISES Graph each equation using the slope-intercept method. Check your work. 1. y 3x 3 2. y 2x 2 4 6

6

5

5

4

4

3

3

2 1

2 1

6 5 4 3 2 1 0 1


y

6 5 4 3 2 1 0 1

1 2 3 4 5 6x

2

2

3 4

3 4

5 6

5 6

58

y

1 2 3 4 5 6x

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

7-7

DISTANCE AND THE PYTHAGOREAN THEOREM The distance formula given below can be used to find the distance between two points, P(x1, y1) and Q(x2, y2), on a coordinate plane. 2 PQ (x x1) y (2 y1)2 2

E x a m p l e Given points P(1, 4) and Q(3, 6) use the distance formula to — find the length of PQ to the nearest tenth. Solution In this case, P(x1, y1) is P(1, 4) and Q(x2, y2) is Q(3, 6). So x1 1, x2 3, y1 4 and y2 6. Substitute these values into the distance formula and simplify. 2 PQ (x x1) y (2 y1)2 2

PQ (3 ) 12 6 ( ) 42 2 PQ 2 22

PQ 4 4 PQ 8 2.8

EXERCISES Use the distance formula to find the distance between each pair of points. 1. P(3, 4) and Q(1, 1)

2. M(2, 6) and N(1, 3)

x1

x2

x1

x2

y1

y2

y1

y2

PQ

MN

3. A(2, 1) and B(5, 2)

4. R(0, 5) and S(4, 9)

x1

x2

x1

x2

y1

y2

y1

y2

AB


RS

59

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

7-8

SOLUTIONS OF LINEAR AND NONLINEAR EQUATIONS Values that make an equation true are called solutions of the equation. Some equations have two variables and thus have an infinite number of solutions. Each solution to such an equation is written as an ordered pair. E x a m p l e

1

Determine if (3, 4) is a solution of y 2x 2. Solution Substitute 3 for x and 4 for y and simplify. y 2x 2 ? 4 2(3) 2 ? 462 44 The result is a true statement. So (3, 4) is a solution of y 2x 2. E x a m p l e

2

Determine if (1, 4) is a solution of y 4x 2. Solution Substitute 1 for x and 4 for y and simplify. y 4x 2 ? 4 4(1)2 ? 4 4 4 4 The result is a false statement. So (1, 4) is not a solution of y 4x 2.

EXERCISES Determine if the ordered pair is a solution. 1. (1, 2) y 3x 5 4. (3, 5)

2. (4, 2) y|x|2 5. (1, 3)

yx2

y 3x 2

7. (2, 2)

8. (6, 4)

y x2 4


3. (0, 5) y x2 5 6. (3, 2) y|x|1 9. (2, 6) y2|x|2

y 4

60

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

8-1

ANGLES AND TRANSVERSALS In the figure at the right, lines AB and CD are parallel, and line KL is a transversal. You can use these facts to identify types of angles and find their measures, given the measure of just one angle. In the figure, the four pairs of vertical angles are 1 and 3, 2 and 4, 5 and 7, and 6 and 8. The four pairs of corresponding angles are 1 and 5, 2 and 6, 4 and 8, and 3 and 7. E x a m p l e If m2 80°, find the measures of the other numbered angles. Solution Angles 2 and 3 are supplementary, so m3 180° 80° 100° Since vertical angles have the same measure, m1 m3 100° and m2 m4 80° Since corresponding angles have the same measure, m1 m5 100° and m2 m6 80° m4 m8 80° and m3 m7 100°

EXERCISES Use the figure to find the measure of each angle named. 1.

m1

2.

AB || CD

↔

m3

m3

m4

↔

m2

↔

m2

↔

↔

FK || HJ and LT KH

3.

m2

m5

m6

m4

m1

m3

m7


61

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

8-2

BEGINNING CONSTRUCTIONS You need only compass and straightedge to copy a given angle or to construct the bisector of a given angle. E x a m p l e

1

Copy ABC, shown at the right. Solution

→ Given ABC. Use straightedge to draw DG . With compass point on B, draw an arc intersecting the sides of the angle. Label the intersection points X and Y. Keep the same setting, → and with compass point on D, draw an arc intersecting DG . Label the intersection point E. With compass, measure the distance from X to Y. Keep that setting and place point on E. Draw an arc intersecting the first arc. Label the intersection → point Q. Draw DQ to complete the copy of the angle. E x a m p l e

2

Bisect ABC, shown at the right. Solution Given ABC. With compass point on B, draw an arc that → → intersects both BA and BC . Label the intersection points X and Y, as shown. With compass point on X, draw an arc a little greater than one half of arc XY. With the same setting, place compass point on Y and draw an arc so that it intersects the → first one. Label their intersection point Z. DrawBZ , which bisects ABC.

EXERCISES Copy each angle on your own paper. 1.

2.

Trace each angle. Then bisect the angle. 3. 4.


5.

62

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

8-3

DIAGONALS AND ANGLES OF POLYGONS The number of diagonals in a polygon with n sides is given by the n(n 3) formula: 2. E x a m p l e

1

Find the number of diagonals for a quadrilateral. Solution A quadrilateral has four sides, so applying the formula gives 4(4 3) 2 2 diagonals. To find the sum of the measures of the angles of any polygon, count the number of triangles formed by diagonals and multiply by 180°. E x a m p l e

2

Find the unknown angle measure.

Solution Copy the pentagon and draw all the diagonals from one vertex. The diagonals form three triangles, so the sum of the angles is 3 • 180°, or 540°. Subtract the sum of the known angle measures from 540°. The unknown angle measure is 165°.

EXERCISES Find the number of diagonals in each polygon. Tell the number of triangles that can be formed and the sum of the angles of the polygon. Then find the unknown angle measure. 1.

x°

3.

2.

4.

x° x°


63

x°

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

8-4

PROBLEM SOLVING SKILLS: MODELING PROBLEMS You can solve some problems using objects, people, or drawings to “act it out.” E x a m p l e In how many ways can three pennies be placed on a 3 by 3 grid so there is only one penny in any row or column? Solution Draw a grid like the one shown. Move pennies around to find the solution. Record the solutions on paper. You might find: (1)

(2)

(3)

(4)

But solutions (1) and (2) are the same, and (3) and (4) are the same. Move your grid around to show they are the same, or imagine it rotated 180° in either direction. There are just two solutions.

EXERCISES 1. Move only two coins from the first figure to make the second figure. 2. In how many ways can you make change in the amount of 32¢? Use coins or cardboard cutouts of coins to act out the problem. 3. In how many ways can 4 pennies be placed on a 4 by 4 grid so there is only one penny in any row or column? Use as many of the grids given below as you need to act out the problem.


64

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

8-5

TRANSLATIONS IN THE COORDINATE PLANE A translation, or slide, of a figure results in a new figure exactly like the original. The original figure is called the preimage of the translated figure. The new figure is called the image of the original. When you translate a figure, try to visualize all the points of the figure moving along a plane the same distance and in the same direction. The sides and angles of the new image are equal in measure to the sides and angles of the preimage. Each side of the new figure is parallel to the corresponding side of the original. E x a m p l e Graph the image of XYZ under the translation 2 units right and 4 units down. Solution Find the coordinates of each image point of each vertex of the triangle. Preimage Translation Image X(2, 1) (2 2, 1 4) X (0, 5) Y(0, 3) (0 2, 3 4) Y (2, 1) Z(3, 2) (3 2, 2 4) Z (5, 2) X Y Z is the image of XYZ under this translation.

EXERCISES Graph the image of each figure under the given translation. 1. 3 units left, 4 units up

2. 5 units left

Write the coordinates of the preimage on your own paper. Then graph the image under the given translation, and write the coordinates of the image on your own paper. 3. 3 units right, 3 units up


4. 2 units left, 1 unit down

65

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

8-6

REFLECTIONS AND LINE SYMMETRY A reflection is a transformation under which a figure is flipped, or reflected, over a line called the line of reflection. E x a m p l e Graph the image of ABC with vertices A(2, 3), B(3, 5) and C(4, 1) under a reflection across the y-axis. Solution Find the image point for each vertex of the triangle. In a reflection across the y-axis, the y-coordinates remain the same, but the x-coordinates are opposites. Multiply the x-coordinate of each vertex by 1. Preimage Image A(2, 3) A (2, 3) B(3, 5) B (3, 5) C(4, 1) C (4, 1) The reflection of ABC across the y-axis is A B C .

EXERCISES Graph the image of each point under a reflection across the given axis. 1. A(2, 1); y-axis 2. B(4, 3); x-axis 3. C(1, 2); y-axis 4. D(3, 4); x-axis 5. Graph the image of FGH under a reflection across the y-axis.


6. Graph the image of KLM under a reflection across the x-axis. Write the coordinates of the vertices of the image.

66

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

8-7

ROTATIONS AND TESSELATIONS A rotation is a transformation that produces an image by turning, or rotating, a figure about a point. A rotation is described in terms of • the point, or turn center, about which the figure is rotated; • the amount of turn, expressed as a fractional part of a whole turn by the number of degrees of turn, called the angle of rotation; • the direction of rotation, either clockwise or counterclockwise. E x a m p l e Triangle PQR has been rotated about point S. Give its angle of rotation in a clockwise direction. Solution Draw a line from S to Q. Draw a line from Q to S. Measure the angle between the two lines. The angle of rotation is 90° clockwise.

EXERCISES

Each drawing shows a line segment, LM, and its rotation image L M about turn center T. Give the angle of rotation in a clockwise direction. 1.

2.

3.

A figure has been rotated about turn center T. Describe each angle of rotation clockwise and counterclockwise. 4.

5.

C

B

A T

B' C'


67

A'

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

9-1

MONOMIALS AND POLYNOMIALS A monomial is an expression that is a single number, a variable, or the product of a number and one or more variables. A polynomial is an expression that is the sum of two or more monomials. The monomials are called the terms of the polynomials. The following expressions are polynomials: x (y)

x 2 4x 4

3y 2 z (y) 2

Those terms of a polynomial that are exactly alike or are alike except for their coefficients are called like terms. like terms:

xy 2, 3xy 2, 2xy 2

unlike terms:

xy 2, x 2y, 3z, 4x

You can simplify a polynomial by combining like terms. E x a m p l e Simplify. a. 2y 2 8y 3 4y 3y 2 4

b. 4x 3 3x 2 5x x 3

Solution a. 2y 2 8y 3 4y 3y 2 4 2y 2 3y 2 8y 4y 3 (4) (2 3)y 2 (8 4)y [3 (4)] 5y 2 4y 1

Rearrange, or collect, like terms. Use the distributive property to combine like terms.

b. 4x 3 3x 2 5x x 3 4x 3 x 3 (3x) (5x) 2 (4 1)x 3 [3 (5)]x 2 5x 3 (8)x 2 5x 3 8x 2 1. m2 m 6 3m 2

2. 5s 2 5s 7 4s

3. t 4 3t 3 2t 4 3 2t 3 t 2 1

4. 6x 2 8x 8 2x 2 9x 4

5. 3 16z 2 8z 3 5z 2 11 4z

6. 7a 3a 2 8a 12a 2 9

7. 5x 3 3x 4 3x 2 8x 3 6 6x

8. 3y 4 3y 3y 3 y 2 y y 2

9. n 2 3n 4n 2 n 3 3n 3 1


10. 8r 2 4r 3 3r 2 r 4r 2

68

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

9-2

ADD AND SUBTRACT POLYNOMIALS You can add and subtract polynomials in vertical form. E x a m p l e

1

Simplify (3y 3 4y 2 6) (2y 3 2y 2 3y 6). Solution Put like terms in the same columns. Write 0 as a coefficient where there is no like term.

Combine like terms.

3y 3 4y 2 0y 6 + 2y 3 2y 2 3y 6

3y 3 4y 2 0y 6 + 2y 3 2y 2 3y 6 5y 3 2y 2 3y 12

To subtract a polynomial from another polynomial, you add the opposite. E x a m p l e

2

Simplify (4t 4 t 2 5t 4) (3t 4 3t 8). Solution Change to an addition Arrange the problem in problem by changing all vertical columns. signs in the polynomial being subtracted. 4t 4 t 2 5t 4 (3t 4 0t 2 3t 8)

4t 4 t 2 5t 4 + 3t 4 0t 2 3t 8

Combine like terms.

4t 4 t 2 5t 4 + 3t 4 0t 2 3t 8 t 4 t 2 2t 4

EXERCISES Add. 1. (4y 3 2y 2 y) (y 3 2y 2 4y)

2. (4m 2 m 3) (2m 2 7)

Subtract. 3. (6n 2 4n 3) (3n 2 5n 1)

4. (5p 3 3p 2 p 4) (2p 3 p 2 p 1)

Simplify. 5. (4y 2 y 5) (3y 2 5y 6)

6. (8x 2 4x 2) (5x 2 3x 4)

7. (4m 2 3m 3) (2m 2 3m 3)


8. (5r 3 r 2 5) (6r 3 r 2 3)

69

MathMatters 1

Name _________________________________________________________

Date ____________________________

9-3

RETEACHING

MULTIPLY MONOMIALS The product rule for exponents is as follows: am . an amn So, to multiply two powers having the same base, you add the exponents. E x a m p l e

1

Simplify (4a2b)(5ab3). Solution (4a 2b)(5ab 3) (4)(5)(a 2)(a)(b)(b 3)

Use the commutative and associative properties to regroup the coefficients and variables.

(20)(a 21)(b 13) 20a 3b 4 The power rule for exponents is as follows: (am)n amn To find the power of a monomial that is itself a power, multiply exponents. E x a m p l e

Use the product rule to add exponents for each like base.

The power of a product rule is as follows: (ab)m amb m To find the power of a product, find the power of each factor and multiply.

2

E x a m p l e

Simplify (y 2)3. Solution (y 2)3 y 2 3 y 6 Check: (y 2)3 y 2 • y 2 • y 2 y 222 y6

3

Simplify (4a 2b)3. Solution (4a2b)3 (4)3(a 2)3(b)3 64a 6b 3

•

EXERCISES Simplify. 1. (2a 3)(5a)

2. (4m 2n)(3mn 3)

3. (2xy)(4x 2)

4. (3xy)(2x 3y 4)

5. (7st 4)(2s 2t 2)

6. (3w 2y)(6)

7. (m 5n 2)(mn 4)

8. (3t 2)(3st)

9. (4xyz)(2x 2yz 2)

10. (s 3)3

11. (2x 2)2

12. (4rt)2

13. (3m)2

14. (2p 2)3

15. (5x 2yz 3)2


70

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

9-4

MULTIPLY A POLYNOMIAL BY A MONOMIAL You can set up the multiplication of a polynomial by a monomial in a manner similar to a multiplication with whole numbers. E x a m p l e Simplify. 6xy(4x 2 3y) Solution Write the factors as shown.

Multiply the first term in the polynomial by the monomial.

Multiply the second term in the polynomial by the monomial.

4x 2 3y 6xy

4x 2 3y 6xy 3 24x y

4x 2 3y 6xy 3 24x y 18xy 2

EXERCISES Simplify.

1. 7(2x 3y)

2. s(4s 2 3s)

3. m 2(5m 2)

4. 6m(3mn 2n)

5. 4y(4x 2y 3xy)

6. 3p(8pq 2)

7. 5(2x 2 x 2)

8. 4m(3m 2 2m 3)

9. 3x 2(x 2 3x 5)

10. xy(4x 3y 2z)

11. 3t(3s 2t 5st)

12. mn 2(m 3 mn n 2)

Write an expression for the area of each rectangle. Then simplify the expression. 13. 14.


71

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

9-5

FACTOR USING GCF To find the product of a monomial and a polynomial, multiply each term of the polynomial by the monomial. You can reverse this process to find what factors were multiplied to obtain the product. This reverse process is called factoring. To factor a polynomial, find the greatest monomial factor common to all the terms of the polynomial. Then write the polynomial as the product of that greatest common factor and another polynomial. E x a m p l e Factor the polynomial: 5s 2t 10st. Solution Write the factors of each monomial term and find the greatest common monomial factor of each term. factors of 5s 2t: (5)(s)(s)(t)

factors of 10st: (2 • 5)(s)(t)

GCF: 5st Rewrite the polynomial as the product of the greatest common factor and another polynomial. Use the distributive property. 5s 2t 10st 5st • s 5st • 2 5st(s 2)

EXERCISES Factor each polynomial. 1. 4m 2n

2. 4x 2 3x

3. 9x 2 6x

4. 6p 2 12p

5. 5r 2 10r

6. 12m 2 4m

7. 7s 2t 14s

8. 4x 2y 6xy 2

9. 3p 2q 8pq 2

10. 4m 3 2m 2 6m

11. 3a 2 6a 3

12. 2x 3y 4x 2y 6xy

13. 3r 3s r 2s r

14. 2x 2 4xy 8y 2

15. 8m 3n 4m 2n 24mn


72

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

9-6

DIVIDE BY A MONOMIAL The quotient rule for exponents is as follows: To divide two powers having the same base, subtract the exponent of the denominator from that of the numerator. am mn n a a Use this rule to find the quotient of two monomials with the same base. E x a m p l e 1 6 18x Simplify . 6x 2 Solution 18x 6 18 x 62 3x 4 2 6x 6 ab a b When a, b, and c are real numbers, and c 0, . c c c

You can use this rule to divide a polynomial by a monomial. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. E x a m p l e

2

4x 3y 2 6x 2y 2xy Simplify . 2xy Solution 4x 3y 2 6x 2y 2 2xy 4x 3y 2 6x 2y 2xy 2xy 2xy 2xy 2xy 31 21 21 21 2x y 3x y 1x 11y 11 2x 2 y 3xy 1

EXERCISES Simplify. r 5s 2. r4

w3 1. w

w2 6s 2t 4 5. 6st 2

st 2

5x 2 10x 9. 5

x 2 2x © Glencoe/McGraw-Hill

3xy 2 6. xy

4s 2t 4. 2s

6mn 3. 3n

rs

2m 15a 2b 2 7. 3ab

3y

5ab

6xy 2 9x 2y 11. xy

4m 8n 10. 2

6y 9x

2m 4n 73

2st 1.2m 3n 8. 6m

0.2 m 2n

6m 3 4m 2 8m 12. 2m

3m 2 2m 4 MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

9-7

PROBLEM SOLVING SKILLS: USE A MODEL When a problem involves a linear function, a graph of the function can be a useful model. E x a m p l e The TV repair service charges a flat fee for making a house call and then an additional charge for each hour of work done. They use this chart. Find the flat fee and the hourly rate. Solution Graph the ordered pairs for the information in the table. Use the three points to draw a line. Find the slope and y-intercept of the line. Then write the equation of the line. The equation of this line is y 10x 10. For 0 hours, the fee is $10. This is the flat fee. Each hour (x) is multiplied by 10, so the fee per hour is $10.

EXERCISES Use the information in the chart. Model the problem on a coordinate plane on your own paper and solve. 1. A messenger service charges a fixed fee per delivery and then an additional cost per mile. Find the fixed fee and the fee per mile. Total amount charged for delivery

$7

$9

$13

Number of miles

1

2

4

2. A parking garage charges a fixed fee per vehicle. In addition, an hourly fee is added. Use the chart to find the fixed fee and the amount charged per hour. Amount charged for vehicle

$5

$11

$20

$32

Number of hours parked

1

3

6

10


74

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

10-1

PROBABILITY When you flip a coin, there are two possible equally likely outcomes, H or T. An event is an outcome or combination of outcomes. The probability of an event is found using this formula: number of favorable outcomes P(E ) number of possible outcomes The probability is usually written as a fraction in lowest terms. E x a m p l e

1

E x a m p l e

2

Find the probability. Use the spinner. a. P(1) b. P(2 or 6)

Find the probability of choosing an M from a set of cards, with one letter from the word MATHEMATICS written on each card. You choose a card without looking.

Solution

Solution 2 ← P(M) 1 1 ←

1 ← one favorable outcome a. P(1) 8 ← total number of outcomes ← one 2 and three 6’s b. P(2 or 6) 4 8 ← total number of outcomes 1 2

2 M’s 1 letter in all

EXERCISES List the outcomes. 1. spinning the spinner in Example 1 2. drawing a letter in Example 2 3. tossing a six-sided number cube 4. choosing a digit at random from 55,238,175 Find the probability. Use the spinner in Example 1. 5. P(6)

6. P(1 or 2)

7. P(odd number)

8. Find the probability that a student chosen at random from a class of 8 girls and 12 boys is a girl. 9. Find the probability of selecting a vowel if a letter is chosen at random from the word WINTER. 10. A day of the week is chosen at random. a. Find the probability it has 6 letters. b. Find the probability it has fewer than 6 letters. © Glencoe/McGraw-Hill

75

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

10-2

EXPERIMENTAL PROBABILITY The probability of an event based on the results of an experiment is called experimental probability. For example, if 100 toys are randomly selected from a bin and checked and 2 are found to be defective, the experimental 2 1 probability of finding a defective toy is 100 50. E x a m p l e A spinner divided into 8 equal parts is spun 50 times. The outcomes are shown in the table at the right. Find each experimental probability. a. P(1) b. P(even number) Solution number of favorable outcomes a. P(E ) number of possible outcomes

87 98 b. P(even number) 5 0 3 2 1 6 50 25

5 1 P(1) 5 0 10

EXERCISES Rob selected a disk from a bag, recorded the color, and returned the disk to the bag. He repeated the experiment 60 times. Use the results in the table to find each probability.

1. P(red)

2. P(green)

3. P(blue)

4. P(white)

5. P(red or blue)

6. P(green or red)

7. P(not red)

8. P(not white)

9. One hundred flashlights were selected at random from a group for sale. Of these, 2 did not work. Find the experimental probability of selecting a flashlight that works. 10. On a production line, 14 bottles of carbonated water were found to be over- or underfilled out of 2000 bottles selected randomly. What is the experimental probability of a bottle not being filled properly? © Glencoe/McGraw-Hill

76

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

10-3

SAMPLE SPACES AND TREE DIAGRAMS The set of all possible outcomes of an experiment is called a sample space. One way to picture a sample space is to make a tree diagram. E x a m p l e

1

E x a m p l e

Make a tree diagram to show the sample space if a number cube is rolled and a coin is tossed. Solution There are 6 outcomes for the number cube (1, 2, 3, 4, 5, 6) and 2 outcomes for the coin (H, T).

2

You have an equal chance of winning (W) or losing (L) a game. Make a tree diagram to show the outcomes when playing three games. Find P(winning exactly 1 game). Solution There are two outcomes each time.

There are eight possible outcomes. Three of these outcomes involve winning exactly one game. 3 P(winning exactly one game) 8

There are twelve possible outcomes.

EXERCISES Use a tree diagram to find the possible outcomes in the sample space. 1. Roll a six-sided number cube and choose a letter from the word AND.

Find each probability if you spin Spinner 1 and Spinner 2.

2

3

1

2. P(2 and A)

A

B

4

8

5 7

6

3. P(3 and vowel) Spinner 1

E

C D Spinner 2

4. P(even number and consonant) © Glencoe/McGraw-Hill

77

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

10-4

COUNTING PRINCIPLE You can use the counting principle to find the number of outcomes of an experiment. Just multiply the number of outcomes at each stage of the activity. Remember, number of favorable outcomes P(E ) number of possible outcomes E x a m p l e

1

E x a m p l e

Veronica has a choice of four recording artists she likes, each on CD, record, or tape. How many choices does she have? Solution There are two stages: choosing an artist and choosing a format. artist format • 3 12 4

2

A player tosses a number cube and chooses one of 26 alphabet cards. Using the counting principle, find P(even number and E, F or G). Solution Find the number of possible outcomes. number cube alphabet cards • 6 26 156 Find the number of favorable outcomes. even number letters • 3 3 9 9 3 P(even number and E, F or G) 156 52

Veronica has 12 choices.

EXERCISES Use the counting principle to find the number of possible outcomes. 1. making a lunch from 3 choices of soup, 5 choices of sandwich, and 4 choices of a drink 2. selecting one of 26 alphabet cards and then tossing a coin twice 3. tossing a coin 7 times 4. rolling a number cube 4 times Use the counting principle to help find each probability. 5. A number cube is rolled 3 times. Find P(all three numbers even). 6. A number cube is rolled and a coin tossed 2 times. Find P(even number and all heads). 7. A coin is tossed 10 times. Find P(all heads or all tails). 8. A card is selected from 26 alphabet cards and a number cube is rolled. Find P(vowel and prime number). Remember: 1 is not a prime number. © Glencoe/McGraw-Hill

78

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

10-5

INDEPENDENT AND DEPENDENT EVENTS Two events are said to be independent if the result of the second event does not depend on the result of the first event. If the result of the first event does affect the result of the second, the two events are said to be dependent. E x a m p l e

1

Eleven cards, each having one of the letters of MATHEMATICS, are placed in a bowl. A letter card is selected at random from the bowl. It is replaced and once again a letter is selected. Find P(M, then M). Solution Since the first letter is replaced, there will be eleven cards in the bowl for both selections. The two events are independent. 2 • 2 4 P(M, then M) 1 1 11 121 E x a m p l e

2

From the bowl with eleven cards, each having one letter of MATHEMATICS, a card is selected at random. Then a second card is selected at random without the first card being replaced. Find P(M, then M). Solution 2 Once that M is You know that the probability of selecting an M is 1. 1 withdrawn, however, there are only 10 letters left. Only one of those 1 letters is an M. So, the probability of selecting an M as the second letter is 1. 0 2 • 1 2 1 P(M, then M) 11 10 110 55

EXERCISES A bag contains 8 red marbles, 3 blue marbles and 5 white marbles. Find each probability if the first marble is replaced before the second one is drawn. 1. P(red, red)

2. P(red, blue)

3. P(red, green)

4. P(red, blue or white)

Find each probability if the first marble is not replaced before the second one is drawn. 5. P(white, then red)

6. P(red, then red)

7. P(black, then red)

8. P(red, then blue or white)


79

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

10-6

PROBLEM SOLVING SKILLS: MAKE PREDICTIONS A random sample is chosen from a population so that every member has an equal chance of being chosen. If you use a random sample to find an experimental probability, you can make predictions about the whole population. To do so, use the formula: probability • number in total population predicted number E x a m p l e Seth and Marla conducted a poll of 50 students selected randomly out of a total of 500 students to see which of the three candidates for president of the student council each student was planning to vote for: Beth Sundfeld, Kim Chung, or Gabriel Melendez. The results of the poll are shown in the table. Predict how many students in the school will vote for Gabriel. Solution 1 8 9 First, find P(Gabriel). P(Gabriel) 50 2 5 Then use the formula to predict the number of all 500 students who will vote for Gabriel. P(Gabriel) • Total number of students Predicted number who will vote for Gabriel 9 • 500 180 2 5 So, 180 students out of a total of 500 will vote for Gabriel.

EXERCISES A radio station asked a random sample of 1000 out of the 250,000 listeners to find out how many preferred one of four types of music. 1. Find P(classical). 2. Find P(popular). 3. Find P(big band). 4. Find P(other). If all the people in the city were polled, how many would you expect to select each type of music? 5. Popular

6. Big band

7. Classical

8. Other


80

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

10-7

EXPECTED VALUE AND FAIR GAMES Expected value is the amount you can expect to win or lose in situations in which the winners are determined randomly. The following formula can be used to find the expected value, E, for different sample spaces. For event A: E P(A) • payoff for A For events A and B: E [P(A) • payoff for A] [P(B) • payoff for B] For events A, B, and C: E [P(A) • payoff for A] [P(B) • payoff for B] [P(C ) • payoff for C] This pattern continues as more events are added. The number of events equals the number of products in the formula. E x a m p l e A charity raffles off a $500 stereo system by selling 1000 tickets for $1 per ticket. What is the expected value? Solution 1 or 0.001. The probability of winning the stereo is 10 00 The expected value is calculated below. E P(A) • payoff for A 0.001 • $500 $0.50 The expected value is $0.50 per ticket sold.

EXERCISES 1. A baseball team raffles off a $300 television by selling 600 tickets for $1 per ticket. What is the expected value?

2. The student council raffles off a season football pass by selling 400 tickets for $1 per ticket. If the season pass is worth $50, what is the expected value?

3. A charity raffles off a $300 bicycle by selling 400 tickets for $2 per ticket. What is the expected value?


81

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

11-1

OPTICAL ILLUSIONS Sometimes you can look at a picture and see something that you think is there, but in reality it is not there. Such a picture, which deceives your eyes, is called an optical illusion. E x a m p l e Refer to the figure at the right. Make a statement about A B , BC , D , and D A in the figure. Then check to see if your statement is C true or false. Solution , BC , C D , and DA appear to bend in toward In the figure AB the circle in the middle. Make a statement: A B , BC , CD , and D A are not straight lines. Test the statement. Lay a ruler down along each line segment. AB, BC, CD, and DA are all straight. All have the same length. So, the statement is false.

EXERCISES Look carefully at the figure. Make a statement about each figure. Then check to see whether the statement is true or false. 1.

2.

3.


82

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

11-2

INDUCTIVE REASONING You are using inductive reasoning when you: • examine several examples • find a pattern or rule that explains those examples • make a trial generalization or conjecture stating the rules that explain those examples. You might find examples that support your conjecture, but you cannot prove it is true by example. In fact, just one example that does not support your conjecture, called a counterexample, proves that your conjecture is not true. E x a m p l e Joann notices the following: 12 21 33 34 43 77 62 26 88 She makes the conjecture: To a 2-digit number, adding the same 2-digit number with the digits reversed gives a sum with repeating digits. Test this conjecture: Can you find a counterexample? Solution Try other examples. Do they support the conjecture? 72 27 99 (yes) 78 87 165 (no) You found a counterexample. Joann’s conjecture is not true.

EXERCISES Make a conjecture. Give at least 3 examples to support it. 1. 2 • 8 16 3 • 10 30 4 • 6 24 5 • 12 60 Any multiple of an even number is

2. Take your age and double it. Add 10 and double the result. Subtract 20 and divide the result by 4. Repeat with someone else’s age.

Here are some conjectures. Test them. Can you find a counterexample? 3. Any multiple of an odd number is odd. 4. The sum of any number and an odd number is an even number. 5. When a number that ends in 2 zeros is multiplied by a number that ends in 3 zeros, their product always ends in exactly 5 zeros.


83

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

11-3

DEDUCTIVE REASONING A conditional statement has two parts, a hypothesis and a conclusion. hypothesis

conclusion

If a person lives in Salt Lake City, then that person lives in Utah. A conditional statement is false if you can find a counterexample that satisfies the hypothesis but does not satisfy the conclusion. E x a m p l e

1

Write two conditional statements, using these two sentences: A figure is a triangle. A figure has three angles. Solution If a figure is a triangle, then that figure has exactly three angles. If a figure has exactly three angles, then that figure is a triangle. E x a m p l e

2

Is the statement true or false? If false, give a counterexample. If the number is divisible by 3, then it is divisible by 6. Solution The number 9 is a counterexample: 9 is divisible by 3 (satisfies the hypothesis) but is not divisible by 6 (does not satisfy the conclusion). So, the statement is false.

EXERCISES 1. Write two conditional statements, using these pairs of sentences: The pond has ice a foot thick. We can skate on the pond.

Is each conditional true or false? If false, give a counterexample. 2. If a number is divisible by 9, then the number is divisible by 3. 3. If a number is divisible by 12, then it is divisible by 18.


84

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

11-4

VENN DIAGRAMS Venn diagrams show relationships of sets. In a Venn diagram there are two or three intersecting circles. Each circle represents one set. The circles are in a rectangle representing the universe, the total number of things being considered. Different parts of the intersecting circles represent those parts of the sets that share one or more characteristics. E x a m p l e Here are the results of a survey of the reading habits of 100 people: 18, only mystery; 20, only romance; 14, only nonfiction; 22, only mystery and romance; 11, only romance and nonfiction; 25, only mystery and nonfiction; 10, all three types. How many readers in all read mystery books? Solution Draw a Venn diagram. Label each circle for one kind of book. Write the total number of items in the universe. In the center section, enter the number who read all three types (10). In the parts of the circles not intersected, enter the numbers who read only mysteries (18), only romance (20), and only nonfiction (14). In the parts of the circles that intersect, enter the numbers as shown. Add all the numbers inside the circle representing mystery books. There are 55 people in all who read mystery books.

EXERCISES Use a Venn diagram to answer the questions. A survey of 170 drivers asked which of these features drivers want in a car: a sunroof, 4-wheel drive, or stereo tape deck. The survey showed that 48 drivers want all three; 8 want only the sunroof; 8 want 4-wheel drive only; 10 want only the stereo tape deck; 32 drivers want only a sunroof and 4-wheel drive; 30 want 4-wheel drive and stereo tape deck only; and 32 want sunroof and stereo tape deck only. 1. How many in all want a stereo tape deck? 2. How many in all look for a sunroof? 3. How many in all look for 4-wheel drive? 4. How many want 4-wheel drive and tape deck but not a sunroof? 5. Out of those surveyed, how many want none of these features? © Glencoe/McGraw-Hill

85

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

11-5

LOGICAL REASONING For logic problems, it is useful to make a table for keeping track of clues. E x a m p l e Lou, Sue, and Drew ran in the same race. Drew finished second. Lou congratulated Sue on doing better than he himself did. Who finished third? Solution Make a table as shown. Put ✔ in a box to indicate that a fact is true. Put ✘ in any box for a fact that is not true. Since Drew finished second, put ✔ in the box where the row labeled “Drew” and the column labeled “2nd” meet. Put ✘’s in the other two boxes in the “Drew” row. Put ✘’s in the other boxes in the “2nd” column, since neither Lou nor Sue came in second. Since Sue did better than Lou, she must have come in first. Put ✔ in the box where the row labeled “Sue” and the column labeled “1st” meet. Put ✘’s in the empty boxes in Sue’s row and in the “1st” column. That leaves only one empty box: Lou must have finished third.

EXERCISES Solve each problem. Complete the table to help solve each problem. 1. The Smiths have three children, Loretta, Tom, and Steve. The middle child is a girl. Tom is not the oldest. Who is the youngest child?

2. Amy, Bud, and José have different pets. One has a bird, one a snake, and one has a kitten José’s pet has fur. Bud’s pet does not have wings. Who has the snake?


86

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

11-6

PROBLEM SOLVING SKILLS: REASONABLE ANSWERS Answers to a problem should make sense. You can check answers for sense by determining whether or not they are reasonable. One strategy that can be used to determine if answers are reasonable is eliminate possibilities. E x a m p l e Five years ago Raul was twice as old as Rachael. The sum of their ages today is 16. How old is Raul? Select the most reasonable answer. a. 2 b. 4 c. 7 d. 9 Solution The fact that five years ago Raul was twice as old as Rachael makes the first two choices unreasonable, so the first two choices should be eliminated. If Raul is 7, he would have been 2 five years ago and Rachael would have been 1. So Rachael would be 6 now and 6 7 13. So the correct answer must be d.

EXERCISES 1. Ten years ago Arlan was five times as old as his granddaughter. The sum of their ages today is 86. How old is his Arlan? Select the most reasonable answer. a. 11

b. 21

c. 55

d. 65

2. A dog pen is twice as long as it is wide. The perimeter of the pen is 48 square feet. How wide is the pen? Select the most reasonable answer. a. 2 feet

b. 4 feet

c. 8 feet

d. 16 feet

3. Trevor has three times as many baseball cards as Rick. Together they have 60 cards. How many cards does Rick have? Select the most reasonable answer. a. 3

b. 15

c. 45

d. 60

4. On Monday, Gregory’s family traveled for 6 hours at a average rate of 50 mph. On Tuesday, they traveled half as many hours at the same average rate. How far did they travel in all? Select the most reasonable answer. a. 150 miles

b. 300 miles

c. 400 miles

d. 450 miles

5. Fran started a walking program. Each week she walks 4 miles further than she walked the previous week. If she started out walking 1 mile, in how many weeks will she be walking 33 miles? Select the most reasonable answer. a. 4 weeks


b. 8 weeks

c. 32 weeks

87

d. 33 weeks

MathMatters 1

Name _________________________________________________________

RETEACHING

Date ____________________________

11-7

NON-ROUTINE PROBLEM SOLVING Mysteries, problems, and brainteasers can often be solved by changing your approach—looking at the problem in a new way. Study the picture at the right. Do you see the stairs from above (side B nearer to you) or from below (side A nearer to you)? Keep trying! You can shift your focus. E x a m p l e Study the arrangement of toothpicks at the right. How can you remove 2 toothpicks so that exactly 2 squares remain? Solution A possible solution is shown. Notice that the 2 squares that remain are not the same size, and one is within the other. Can you find another solution?

EXERCISES Use the toothpick arrangement in the Example for Exercises 1–3. 1. Remove 4 toothpicks to make 1 square. 2. Remove 4 toothpicks to make 2 squares. 3. Move 3 toothpicks and rearrange them so that there are 3 small squares, all the same size and shape. 4. Move two dots so that the arrangement of dots is inverted (points up rather than down). 5. Study the arrangement of toothpicks at the right. How can you remove 2 toothpicks to make exactly 2 triangles? 6. Study the arrangement of 15 toothpicks at the right. Remove 3 toothpicks so that exactly 3 squares are left. 7. Using the arrangement of 15 toothpicks from Exercise 6, remove 2 toothpicks to make 3 squares.


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