Mathematics in Context: Building Formulas: Algebra

Building Formulas Algebra

Mathematics in Context is a comprehensive curriculum for the middle grades. It was developed in 1991 through 1997 in collaboration with the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison and the Freudenthal Institute at the University of Utrecht, The Netherlands, with the support of the National Science Foundation Grant No. 9054928. The revision of the curriculum was carried out in 2003 through 2005, with the support of the National Science Foundation Grant No. ESI 0137414.

National Science Foundation Opinions expressed are those of the authors and not necessarily those of the Foundation.

Wijers, M., Roodhardt, A., van Reeuwijk, M., Dekker, T., Burrill, G., Cole, B.R.,& Pligge, M .A. (2006). Building Formulas. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context. Chicago: Encyclopædia Britannica, Inc.

Copyright © 2006 Encyclopædia Britannica, Inc. All rights reserved. Printed in the United States of America. This work is protected under current U.S. copyright laws, and the performance, display, and other applicable uses of it are governed by those laws. Any uses not in conformity with the U.S. copyright statute are prohibited without our express written permission, including but not limited to duplication, adaptation, and transmission by television or other devices or processes. For more information regarding a license, write Encyclopædia Britannica, Inc., 331 North LaSalle Street, Chicago, Illinois 60610. ISBN 0-03-038559-8 1 2 3 4 5 6 073 09 08 07 06 05

The Mathematics in Context Development Team Development 1991–1997 The initial version of Building Formulas was developed by Monica Wijers, Anton Roodhardt, and Martin van Reeuwijk. It was adapted for use in American schools by Gail Burrill, Beth R. Cole, and Margaret A. Pligge.

Wisconsin Center for Education

Freudenthal Institute Staff

Research Staff Thomas A. Romberg

Joan Daniels Pedro

Jan de Lange

Director

Assistant to the Director

Director

Gail Burrill

Margaret R. Meyer

Els Feijs

Martin van Reeuwijk

Coordinator

Coordinator

Coordinator

Coordinator

Sherian Foster James A, Middleton Jasmina Milinkovic Margaret A. Pligge Mary C. Shafer Julia A. Shew Aaron N. Simon Marvin Smith Stephanie Z. Smith Mary S. Spence

Mieke Abels Jansie Niehaus Nina Boswinkel Nanda Querelle Frans van Galen Anton Roodhardt Koeno Gravemeijer Leen Streefland Marja van den Heuvel-Panhuizen Jan Auke de Jong Adri Treffers Vincent Jonker Monica Wijers Ronald Keijzer Astrid de Wild Martin Kindt

Project Staff Jonathan Brendefur Laura Brinker James Browne Jack Burrill Rose Byrd Peter Christiansen Barbara Clarke Doug Clarke Beth R. Cole Fae Dremock Mary Ann Fix

Revision 2003–2005 The revised version of Building Formulas was developed by Truus Dekker. It was adapted for use in American schools by Gail Burrill.

Wisconsin Center for Education

Freudenthal Institute Staff

Research Staff Thomas A. Romberg

David C. Webb

Jan de Lange

Truus Dekker

Director

Coordinator

Director

Coordinator

Gail Burrill

Margaret A. Pligge

Mieke Abels

Monica Wijers

Editorial Coordinator

Editorial Coordinator

Content Coordinator

Content Coordinator

Margaret R. Meyer Anne Park Bryna Rappaport Kathleen A. Steele Ana C. Stephens Candace Ulmer Jill Vettrus

Arthur Bakker Peter Boon Els Feijs Dédé de Haan Martin Kindt

Nathalie Kuijpers Huub Nilwik Sonia Palha Nanda Querelle Martin van Reeuwijk

Project Staff Sarah Ailts Beth R. Cole Erin Hazlett Teri Hedges Karen Hoiberg Carrie Johnson Jean Krusi Elaine McGrath

(c) 2006 Encyclopædia Britannica, Inc. Mathematics in Context and the Mathematics in Context Logo are registered trademarks of Encyclopædia Britannica, Inc. Cover photo credits: (left to right) © Getty Images; © John McAnulty/Corbis; © Corbis Illustrations 6, 7 (bottom) Holly Cooper-Olds; 8 Christine McCabe/© Encyclopædia Britannica, Inc.; 18, 22, 26 Holly Cooper-Olds; 34, 39 Christine McCabe/© Encyclopædia Britannica, Inc.; 40, 41 Holly Cooper-Olds; 44 © Encyclopædia Britannica, Inc.; 48 Christine McCabe/© Encyclopædia Britannica, Inc. Photographs 1 (top right) Expuesto—Nicolas Randall/Alamy; (bottom left) Sam Dudgeon/ HRW; 4 Sam Dudgeon/HRW; 5 Expuesto—Nicolas Randall/Alamy; 6 Sam Dudgeon/HRW; 13 (top) © Corbis; (bottom) © ImageState; 15 © Corbis; 17 Sam Dudgeon/HRW; 30 Brand X Pictures/Alamy; 34 Letraset Phototone; 36 © Pat O'Hara/Corbis; 38 Sam Dudgeon/HRW; 45 Karl Weatherly/Getty Images/PhotoDisc; 47 PhotoDisc/Getty Images; 50 © Corbis

Contents Letter to the Student Section A

Patterns Tiles Beams Summary Check Your Work

Section B

22 26 32 32

Formulas and Geometry Lichens Circles and Solids Pyramids Summary Check Your Work

Section E

13 15 17 20 20

Using Formulas Temperature Building Stairs Summary Check Your Work

Section D

1 6 10 11

Brick Patterns Bricks The Classic More Brick Rows Summary Check Your Work

Section C

vi

34 38 40 42 42

Problem Solving Heavy Training Crickets Egyptian Art Summary Check Your Work

44 47 48 50 50

Additional Practice

52

Answers to Check Your Work

57

Contents v

Dear Student, Welcome to Building Formulas. Throughout this unit, you will study many kinds of formulas. You will learn to identify the parts that make up a formula and create your own formulas. You will help with the construction of a movie set by creating a formula for determining the numbers of blue and white tiles and metal rods needed.

Do you know how to find the area of a lichen, a fungus that grows nearly everywhere? Scientists use the size of a lichen to calculate how long ago a glacier disappeared. You will also work with formulas that someone else created, such as a formula for converting from degrees Celsius to degrees Fahrenheit and formulas that archaeologists use to re-create ancient Egyptian drawings.

C

F

60

140

50

120

40 30

100 80

20

Upon completing this unit, you should understand the meanings of the different parts of a formula, how to graph formulas, how to rewrite formulas, and how to use formulas that you find.

60 10 40 0 10 20

20 0

Sincerely,

The Mathematics in Context Development Team

vi Building Formulas

A

Patterns

Tiles Urvashi is designing a set for a movie scene that takes place outside a mansion. The mansion is surrounded by a large garden decorated with plants, tiled paths, sculptures, and patios.

She wants to design tile patterns for different lengths of garden paths. She decides to use square tiles in two different colors.

white

blue

Section A: Patterns 1

A Patterns

Using one pattern, Urvashi draws paths that have different lengths.

Path Number 3

To make it easy to refer to a path, Urvashi assigns each path a number. 1. a. What do the numbers represent? Path Number 4

b. Using paper or plastic squares of different colors, lay out some other lengths of paths that have this pattern. For each example, write down the path number. Describe the patterns in the numbers of tiles for different paths.

Path Number 5

Urvashi wants to make the pattern more interesting and decides to add a column of tiles at the beginning and another column at the end. The pictures below show how path number 3 is changed. Path Number 3

New Path Number 3

New path number 3 now has a length of five tiles. 2. a. Compare new path number 7 to old path number 7. b. Urvashi has a new path that is now 53 tiles long. What is the path number? 3. Make some examples of Urvashi’s new design.

2 Building Formulas

Patterns A Urvashi makes a table so she can easily find the number of tiles needed for each new path design. Path Number

Number of Blue Tiles

Number of White Tiles

Total Number of Tiles

8

7

15

1 2 3 4 5 6 7

4. Copy and fill in the table. 5. a. How does the number of blue tiles change when you go from one path number to the next? b. Describe at least two other patterns in the table. Use drawings in your explanation. You can also describe a pattern using a NEXT-CURRENT formula. The NEXT-CURRENT formula describing the number of blue tiles is: Number of blue tiles in the NEXT path  Number of blue tiles in the CURRENT path  ? 6. a. Finish the NEXT-CURRENT formula that describes the number of blue tiles. It can be shortened to this: NEXT blue  CURRENT blue  ? b. Write NEXT-CURRENT formulas to describe the number of white tiles and the total number of tiles.

Section A: Patterns 3

A Patterns Urvashi makes a model for path number 10 and finds that she needs 14 white tiles. 7. a. Use a NEXT-CURRENT formula to find out how many white tiles she would need for path number 15. Explain how you did this. b. Find the numbers of white and blue tiles Urvashi would need for path number 30. Explain how you found your answers. To find the number of blue tiles for path number 30, Urvashi did not want to use the NEXT-CURRENT formula because she did not want to calculate the numbers of tiles, step-by-step, for all the paths before it. Stu, one of her coworkers, says that figuring the number of blue tiles for path number 30 is easy. All you have to do is calculate 4  29  2. 8. a. Check Stu’s calculation of the number of blue tiles for path number 30. b. Explain how you think Stu came up with his method. c. Would the same process work for path numbers other than 30? If so, how? By looking at examples of different path numbers, Urvashi discovers a relationship to find the number of white tiles if she knows the path number. She uses P to stand for the path number and W to stand for the number of white tiles. She then writes an arrow string. 4

P ⎯⎯→ W

9. a. Use a drawing to help explain the arrow string. b. Rewrite the arrow string as a formula. The formula you wrote in problem 9b is called a direct formula in contrast to a NEXT-CURRENT formula. 10. a. Write a direct formula to find the number of blue tiles for each path number. b. Explain your formula with an arrow string or a drawing. c. Compare your formula or arrow string to Stu’s formula. 4 Building Formulas

Patterns A 11. Use the direct formulas to check your answers to problems 7a and 7b on page 4. 12. Reflect Compare the use of NEXT-CURRENT formulas to the use of direct formulas. Include advantages and disadvantages of each.

Jim Shew, the budget director for the film, wants to find the cost of the tiles Urvashi will use. Because white and blue tiles are the same price, he needs to know only the total number of tiles for each path number.

13. a. Write at least two ways to find the total number of tiles needed for path number 13. b. Would your methods work for different path numbers? Jim wrote a direct formula to calculate the total number of tiles (T ) for each path number (P). 14. a. Write a formula you think Jim might have used. Explain the formula. b. Use the formula to calculate the total number of tiles the movie company would need for path numbers 15, 23, 92, and 93. Show your calculations. Urvashi and Jim wonder whether any other formulas could be used to find the total number of tiles (T) for each path number (P). Urvashi suggests this formula: T  (P  2)  (P  2)  (P  2) 15. Will this formula also give the total number of tiles for each path number? Use a drawing to explain your answer. Section A: Patterns 5

A Patterns

Jim says this is the same formula as T  3  (P  2), which can also be written as T  3(P  2). 16. a. What did Jim do to get this formula? b. Find another way to write this formula.

After meeting with the film’s director, Urvashi tells Jim they are going to use the new design for the paths. They order 100 tiles. Jim wonders, “Does a path exist that would require exactly 100 tiles?” 17. a. What is the answer? Explain. b. What path number contains exactly 54 tiles?

Beams Construction work has begun on a large building that will be used for part of the movie set. The framework consists of metal beams on concrete columns. Each beam is made from small rods. Beams can have different lengths. The length of a beam is the number of rods along the underside.

6 Building Formulas

Patterns A 18. Why is this beam considered to be of length 6?

To make the building for the movie set, three beams are put together as shown in the picture.

19. a. Look at the drawing. How long are the beams? b. Is there more than one correct answer? Jared works in the factory where these beams are made. He is interested in finding the total number of rods needed for different lengths of beams.

Section A: Patterns 7

A Patterns

Length Number of Beam of Rods (L) (R) 1

Jared started making a table for the number of rods in different length of beams.

3

2

3

4

20. a. Copy and complete Jared’s table in your notebook. b. Explain how you found the numbers to fill in the table. c. Make a drawing or use toothpicks to build a beam of length 5. How many rods would you need for a beam of length 5? d. Think about how you built or drew the beam of length 5. How can you find the total number of rods for a beam of length 6? Length 10? e. How would Jared find the total number of rods for a beam of length 50? Finding a direct formula will give Jared the total number of rods for a beam of any length. 21. a. Try to find a direct formula that gives the number of rods (R) needed to build a beam of any length (L). b. Check your formula by using the entries in the table in problem 20.

Angelina: R  L  3  (L  1)

The people who work in the factory also figure out how many rods are needed for beams of different lengths. They find three different formulas that they think will work.

David: R  L  (L  1)  2L

22. a. Check all of the formulas for lengths 7, 15, and 68 to see if they give the same numbers of rods.

Maria: R  3  (L  1)  4 8 Building Formulas

b. Are all of the formulas the same? Explain. c. Compare these formulas to the one you found for problem 21a.

Patterns A Jared is not convinced that all three formulas will give the correct results for all lengths. He decides that if he can figure out where the formulas came from, he will have a better idea about whether they will work for all lengths. David used this picture to explain his formula, R  L  (L  1)  2L.

23. What is a possible explanation for David’s formula?

I’ll break the beam into parts.

Maria broke the beam into parts and used this picture to describe her formula, R  3  (L  1)  4.

24. Finish Maria’s explanation. Explain how the drawing relates to her formula. Angelina’s formula is R  L  3  (L  1). She tells Jared that she also found the formula by breaking the beam into parts. 25. Write an explanation for Angelina’s formula. You may use drawings. Sara and Josh wrote formulas that are nearly the same as the formulas Angelina and Maria wrote. 26. Sara’s formula is R  3L  (L  1). Josh’s formula is R  3  4 (L  1). Do you prefer one of these formulas? Why? At the rod factory, many of the orders come in by fax. An order came in for rods to make a building with seven beams of equal length. Unfortunately, the fax was hard to read, and no one could tell whether 525 or 532 rods had been ordered. 27. Find the number of rods that were ordered.

Section A: Patterns 9

A Patterns

When solving problems about designs, you can:

• • •

draw some examples of the design; make a table and look for patterns; or express the patterns as formulas.

You explored two different types of formulas that can be used to describe a pattern:

• •

a NEXT-CURRENT formula, going step-by-step a direct formula, working directly with the pattern number

Different direct formulas can be used to describe the same rule or pattern. One way to check to see whether different formulas give the same result is by using a drawing to connect each formula to the same pattern. For example, R  3L  (L  1) is the same as R  4L  1 because they both describe the same pattern.

R  3L  (L  1) can be connected to the pattern with the following drawing.

R  4L  1 represents the same pattern, as shown in the following drawing.

If the meaning is clear, you can leave out the multiplication sign. T  4  L  1 is the same as T  4L  1. When you are adding or multiplying, you can change the order. T  (P  1)  3  2 is the same as T  3(P  1)  2 or T  2  3(P  1).

10 Building Formulas

Here are three formulas: TP2P T  2P  2 T  2(P  1) 1. Show that these formulas describe the same pattern. Terry is designing a tile patio. Her design has an orange square in the middle and a white border around it. These patios can be different sizes. Four sizes are shown. Patio Number 2

Patio Number 1

Patio Number 4 Patio Number 3

2. a. Study the design. Make a table for the number of orange and white tiles for several patio numbers. b. Find a direct formula to calculate the number of orange tiles (O) needed for any patio number (P ). c. Find a direct formula for the number of white tiles (W ) in any patio number. d. How many orange and white tiles are needed for the patio number 10?

Section A: Patterns 11

Here are two formulas for the total number of tiles (T ) in each patio. T  P  P  4P  4 and T  (P  2)  (P  2) 3. a. Add a column to the table you made for problem 2 to show the total number of tiles in each patio. b. Do the formulas both give the same result? You may use drawings to support your answer. Terry has a total of 196 tiles that she is going to use to build one of these patios. 4. a. What patio number is she going to build? Explain. b. How many orange and how many white tiles does she have? Explain.

Find an example of a pattern from the classroom, your trip to school, or your house that could be described by a NEXT-CURRENT formula and one that could be described by a direct formula. Make a problem from one of the patterns.

12 Building Formulas

B

Brick Patterns

Bricks Many buildings and some pavements are made out of bricks. Bricks can be laid in different patterns.

The windows, doors, arches, or even edges of a building may have different brick designs. You can see different brick patterns on the pavements in these photographs. Section B: Brick Patterns 13

B Brick Patterns

Here are some diagrams of brick patterns. Row 1

Row 2

Row 3

1. a. How many bricks are in each of the three rows? Explain how you got your answers. b. Copy each row and make it longer by drawing some extra bricks. Compare your drawings to those of your classmates. Did everyone end up with the same rows? Study the designs carefully. 2. a. Does any row have a basic pattern that repeats a number of times? If so, make a drawing of the basic pattern for that row. b. How can you use the basic pattern to find the number of bricks in a row? Bricklayers often use a variety of basic patterns in pavements or buildings. One way of creating a pattern is to use standing and lying bricks. A bricklayer could describe such patterns using the letter S for bricks standing upright and the letter L for bricks lying flat.

Name Classic

Exotic

Modern

14 Building Formulas

Drawing

String SLSLS

S

L

The table was made by a new bricklayer to help him remember some basic patterns. 3. Use the copy of the table on Student Activity Sheet 1 to fill in the missing information.

Brick Patterns B 4. a. On the last line of Student Activity Sheet 1, fill in a basic pattern of your own and give it a name. b. If you were to repeat your basic pattern four times, how many bricks would you need? How did you find this number? c. Draw or describe what this row would look like. If a person is building a brick row, it may be important to know not only how many bricks are needed, but also how long (in centimeters or meters) the row will be. Even though you do not know the sizes of the bricks, you can still make some true statements about the lengths of the basic patterns. 5. Reflect Write down two true statements comparing the lengths of the basic patterns shown in the table.

The Classic Ms. Fix saw this brick border and decided she would like to have a brick border for one side of her garden. She has chosen the Classic pattern from the table on page 14. To make the border long enough, she repeats the pattern four times.

6. a. Describe or draw what Ms. Fix’s brick border will look like. b. How many bricks does she need? Write down your calculations. Section B: Brick Patterns 15

B Brick Patterns

Ms. Fix is thinking about using some bricks she has left over from another project. She has to choose between yellow and gray bricks. They have the following measurements:

8 cm

Yellow

6 cm

Gray 12 cm

15 cm

7. a. How long would her brick border be if she used only yellow bricks? Only gray? Write down your calculations. b. Did everyone in your class make the calculations in the same way? Explain. Ms. Fix visits a brickyard to look at some other bricks for her border. She decides to use a formula for finding the length of the basic Classic pattern. To use the formula, Ms. Fix needs to know the length of the lying (long) side of the brick (L) and the length of the standing (short) side of the brick (S). This is the formula she uses: Length of Classic  3S  2L 8. a. Explain the formula. b. Write formulas for the other basic patterns shown in the table on Student Activity Sheet 1. Ms. Fix is interested in a formula that will give the total length for the brick row. (Remember that she wants to repeat the basic Classic pattern four times.) She writes: Total Length  4  Length of Classic Later she realizes she can also use this formula: Total Length  4(3S  2L) 9. Explain how both of the formulas work. Ms. Fix wants a formula that shows how many short sides of a brick and how many long sides are used in the total length. 10. Write such a formula for Ms. Fix. (Hint: This formula should have no parentheses.)

16 Building Formulas

Brick Patterns B

More Brick Rows The Yun family has designed a brick row for their garden using a basic pattern of their own. On the left you see what the row will look like. 11. a. Describe the basic pattern of this row. b. Use the letters L and S to write a formula to calculate the length of the basic pattern. c. Write two formulas you can use to calculate the length of the row shown above. Make one formula with parentheses and one without. d. Describe how you think the formula with parentheses can be rewritten as a formula without parentheses.

Dad, I think the pattern might look nicer if the standing and lying bricks are switched.

It might look nicer, Sueng, but I wonder if switching them would make the row longer.

The picture shows Sueng and her Dad talking about the pattern in the row they are making. 12. a. Does the length of the row change if Sueng’s idea is carried out? How can you tell? b. Write formulas for the length of the new basic pattern and for the length of the whole row. Section B: Brick Patterns 17

B Brick Patterns

Ms. Peterson is fixing up her house. She goes to a hardware store to buy some supplies.

Good morning. How can I help you?

I’m trying to restore the brick row above my front door.

I wrote down a formula that can be used to calculate the length of the row for different-sized bricks.

I left the sketch at home, but I did bring a note.

I can also use this formula to find the total number of bricks I need to buy.

13. a. How can Ms. Peterson use the formula to find the total number of bricks she needs? b. Draw one possible design for the brick row that Ms. Peterson is repairing.

I remember that my row has a basic pattern that repeats a number of times. I also wrote down another formula, which has parentheses.

In that one, I can see the number of times the basic pattern is repeated, but I left that one at home too.

18 Building Formulas

Well, never mind. With what you have told me, we can probably figure out what the pattern is.

In any case, let’s try to figure out the formula you left at home.

Brick Patterns B 14. a. Could Ms. Peterson’s brick row have a basic pattern that occurs four times? Explain why or why not. You may support your explanation with drawings. b. Draw a basic pattern that fits Ms. Peterson’s row. c. Are you sure how many times the basic pattern is repeated? Explain your answer. d. Using the basic pattern and parentheses, write a formula to calculate the length of the row Ms. Peterson is restoring. e. Explain how this formula is related to the formula Length  15S  10L. Another formula for a brick row is: Length  3(2S  4L) 15. a. Draw a brick row that fits this formula. b. Write a formula without parentheses for the length of the row you drew. Explain how you found the numbers for the formula. You may have noticed that there is a rule you can use to change a formula with parentheses into one without parentheses. The formula Length  4(2S  5L) can be rewritten as: Length  8S  20L. 16. a. Where do the 8 and the 20 come from in the second formula? b. If you start with the formula Length  8S  20L, what do you have to do to rewrite it as Length  4(2S  5L)? c. Can all formulas be rewritten with parentheses? Explain why or why not. 17. Rewrite the formula Length  16S  12L using parentheses.

Section B: Brick Patterns 19

B Brick Patterns

If you have to repeat a calculation over and over, using a formula can be helpful. In this section, you have seen different formulas for calculating the lengths of brick rows. Some of the formulas had parentheses and some did not. Most formulas with parentheses can be rewritten without parentheses. For example, Length  2(4S  3L) is the same as Length  8S  6L.

A formula without parentheses can sometimes be rewritten with parentheses. For example, Length  6S  3L is the same as Length  3(2S  L). (Note: 1L is usually written as L.)

Janet is designing a pattern for a patchwork quilt. The pattern is made of upward (U) and downward (D) trapezoids in different colors. Here is the basic pattern Janet uses.

U

20 Building Formulas

D

1. a. Write a formula for the length of a basic pattern. Each row Janet uses for the quilt consists of 30 upward trapezoids and 29 downward trapezoids. b. Write a formula for the length of one row. c. Is it possible to write the formula you made in problem 1b with parentheses? Why or why not? 2. a. Describe or draw a brick row that would fit the formula Length  4(2L  3S). b. Rewrite this as a formula without parentheses. The following formulas are used to calculate the lengths of different brick rows. All of the brick rows except one have a repeating basic pattern. Row 1: Length  12L  8S Row 2: Length  10L  15S Row 3: Length  21L  27S Row 4: Length  9L  13S Row 5: Length  18L  12S Row 6: Length  18L  9S 3. a. Which of the six rows cannot have a repeating basic pattern? Explain. b. Row 5 can have three different repeating basic patterns. What are the formulas for the lengths of these basic patterns?

Do you think it is easier to use a formula with parentheses or without parentheses? Explain.

Section B: Brick Patterns 21

C

Using Formulas

Temperature Kim has a pen pal in Bolivia named Lucrecia. Lucrecia is planning a visit to the United States, and she will stay with Kim’s family. Lucrecia sent this letter from Bolivia.

Dear Kim,

en beautiful. We The weather has be th temperatures of have had a week wi nesday, it was even about 25°C. On Wed o hot for me. 30°C. This is a bit to g in the lake. The We went swimmin warm, only 18°C, water was not very cool off. It’s hard but it was great to ly week ago it was on to imagine that a r d to wear a sweate around 16°C. I ha all day. r like in your city? What is the weathe e a sweater? At hom Do I have to bring e evenings. Last it cools down in th understorm, and night we had a th s! opped by 10 degree the temperature dr eing you. I look forward to se Lucrecia

1. Estimate the temperatures in degrees Fahrenheit for the Celsius temperatures Lucrecia mentions in her letter. 22 Building Formulas

Using Formulas C The thermometer shows both Fahrenheit and Celsius temperatures. 2. a. How can you use this thermometer to find the answers to problem 1? b. Check the estimates you made for problem 1. Were they close to what the thermometer tells you? 3. a. Look carefully at the thermometer. An increase of 10°C corresponds to an increase of how many °F? b. Use your answer to part a to answer the following question. An increase of 1°C corresponds to an increase of how many °F? c. Could you have answered part b just by looking at the thermometer? Explain. When you use a thermometer to convert temperatures, you sometimes have to estimate the degrees because of the way the scale lines are drawn.

C 60

F 140

50

120

40

100

30

80

20 60 10 40 0 10 20

20 0

4. Reflect Do you think it is possible to calculate a Fahrenheit temperature for each Celsius temperature? Why or why not? Here is a formula, written different ways, for converting temperatures from degrees Celsius (C) into degrees Fahrenheit (F). 1.8  C  32  F 1.8C  32  F F  1.8C  32 5. a. Explain where the numbers in the formula come from. (Hint: Use the thermometer and your answer for problem 3b.) b. Write the formula using an arrow string.

Section C: Using Formulas 23

C Using Formulas

The formula and the thermometer tell you the relationship between C (the temperature in degrees Celsius) and F (the temperature in degrees Fahrenheit). You can also make a graph to show the relationship. 6. a. First, fill in the table on the top of Student Activity Sheet 2. (Add some temperatures of your own choice, too.) C

20

15

10

5

0

5

F

b. Describe any patterns you see in the table. c. Graph the information from the table at the top of Student Activity Sheet 2. (Notice that C is on the horizontal axis and F is on the vertical axis.) F 60 50 40 30 20 10 40 30 60 60 50 50 40 30 20 20 10 10 0

10

20

30

40

50

60

C

10 10 20 20

40

30 30 40 40 50 50 60 60

Your graph in problem 6 should be a straight line. 7. How could you tell that the graph would be a straight line? 8. There is only one temperature that has the same value in degrees Celsius and degrees Fahrenheit. What temperature is this? Describe how you found your answer. 24 Building Formulas

Using Formulas C To convert temperatures, you can use a thermometer, a graph, a table, or a formula. There are many ways to write a formula that converts between Fahrenheit and Celsius. 9. a. Write a reverse arrow string to convert temperatures from Fahrenheit to Celsius. (Hint: Use the answer to problem 5b.) b. Write a formula that converts temperatures from Fahrenheit to Celsius. Most formulas for converting between the two types of degrees are not easy to use if you are trying to do the calculation mentally. Sometimes people use estimation formulas for converting in their heads. Here is an estimation formula to change Celsius to Fahrenheit. Double the Celsius value and add 30. 10. Make up an estimation formula to convert Fahrenheit to Celsius. The freezing point of water is 0° in Celsius and 32° in Fahrenheit. Check your formula, using the temperatures for freezing. 11. a. Convert two temperatures from Celsius to Fahrenheit and two temperatures from Fahrenheit to Celsius, using the estimation formulas. b. Do the same using the direct formulas. c. Reflect Compare the results. Why would people use estimation formulas if the results are not very accurate? Dale remembered a rule he learned in school last year for converting temperatures. He uses an arrow string to write it on the chalkboard.  40

 1.8

 40

⎯ ⎯⎯→ ⎯⎯⎯→ ⎯ ⎯⎯→ Kim wonders if this rule could be correct. She says: “First you add 40, and then you subtract 40, so nothing happens. You can skip those two arrows.” 12. a. Do you agree with Kim? Why or why not? b. Does Dale’s rule work?

Section C: Using Formulas 25

C Using Formulas

Here are some pictures of Kim and Lucrecia at various places. 13. What is a likely temperature in degrees Celsius for each picture?

Building Stairs The picture shows a staircase. All of the steps are the same size. Each step has two main parts: the riser and the tread. The vertical measure, or the height, of a step is called the rise (R). The horizontal measure, or depth, of a step is called the tread (T).

26 Building Formulas

tread

riser

You are going to use stiff paper to build a model staircase like the one shown. D

•

In the center of a piece of stiff paper, draw a rectangle that is exactly 20 centimeters (cm) long and 10 cm wide. Label the corners of the rectangle A, B, C, and D as shown in the diagram.

C 8 cm

12 cm

A

•

B

Across your paper, draw a dotted line that is 8 cm below the top of the rectangle. D

•

Fill the rectangle with lines that are alternately 3 cm and 2 cm apart, as shown in the next diagram. (It is easy to keep your lines parallel, using a ruler and a triangle.)

•

Fold the paper along the dotted line and cut along the long sides of the rectangle. Do not cut along the short sides.

•

Fold the solid lines like an accordion so that you end up with a staircase. The first fold should be on side DC folded toward you (out). Fold the next line away from you (in). Continue alternating the fold direction until the staircase is finished.

C 3 cm 2 cm 3 cm 2 cm 3 cm 2 cm 3 cm 2 cm

A

B

You now have a model staircase. • Label the wall and the floor on your model as shown.

Section C: Using Formulas 27

C Using Formulas

The stairs you made fit nicely with the floor and the wall. In other words, the treads and the floor are perfectly horizontal and the risers and the wall are perfectly vertical. 14. Reflect Do you think this is a coincidence, or were they designed that way? Why do you think so? 15. a. Measure and record the height and depth of the whole staircase (depth is measured along the floor). b. What are the values for T and R in the steps you made? c. How are the height and depth of the whole staircase related to the rise and tread of each step? Explain. 16. On your model staircase, make the fold between the floor and the wall in a different place. Is the tread of each step in your model still perfectly horizontal? Is the rise of each step exactly vertical? Explain why or why not. 17. a. Would the designs shown here make good staircases? Explain.

3 cm 2 cm 3 cm 2 cm 3 cm 2 cm

Design 1

3 cm 2 cm 3 cm 2 cm 3 cm 2 cm 3 cm 2 cm

Design 2

b. Copy the drawing on the right into your notebook. Draw a fold line where it will create a good model staircase. c. Reflect What are some rules for making good model staircases?

3 cm 2 cm 3 cm 2 cm 3 cm 2 cm 3 cm 2 cm 3 cm 2 cm

28 Building Formulas

Using Formulas C Not all stairs are easy to climb. 18. Order the staircases shown below according to how easy you think they would be to climb. Give reasons to support your choices.

Staircase A

Staircase B

Staircase C Staircase D

Staircase E Section C: Using Formulas 29

C Using Formulas

For problem 18, you may have listed the steepness of the stairs as one factor that affects how easy they are to climb. 19. What are some advantages and disadvantages of steep stairs? If you are not careful about choosing the measurements for the rise and tread of a set of stairs, you can end up with stairs that are difficult to climb. 20. What could you do to make the stairs steeper? Stairs that are easy to climb usually fit the following rule: 2  Rise  Tread  Length of one pace or 2R  T  P An adult’s pace is about 63 cm. So the rule can be written as follows: 2R  T  63 21. a. A contractor wants to build a set of stairs with a rise of 19 cm for each step. What size will the tread be if she follows the rule? Explain. b. For another set of stairs, the contractor knows that the tread must be 23 cm. How high will each rise be if the contractor uses the rule? Explain. You have now found two combinations of rise and tread measurements that fit the rule based on an adult pace of 63 cm. 22. a. Find a few more pairs of numbers that fit the rule. b. On Student Activity Sheet 3, graph all of the pairs of rise and tread measurements that fit the rule. You can make stairs that are difficult to climb even when you use the formula. 23. Which points on the graph would represent stairs that are difficult to climb? 24. What happens to the tread (T ) if you add 1 cm to the rise (R) and are using the rule? Can you see this on your graph? 30 Building Formulas

Using Formulas C 25. Using the rule, when do R and T have the same value? Here is another rule that helps in designing stairs that are easy to climb. Rise  20 cm This means that the rise is less than or equal to 20 cm. 26. a. Why would this rule make stairs easier to climb? b. Find a way to show this rule on your graph. c. Find measurements for some stairs that fit both rules. d. There are some situations that do not allow for stairs that are easy to climb. What could be some reasons for having stairs that are not easy to climb? Think about the dimensions of the paper stairs you made earlier. Suppose the paper stairs are a model that uses the rule 2R  T  63 cm for an actual set of stairs. 27. a. What are the measurements for the rise and tread in the actual flight of stairs? b. What are the height and depth measurements of the whole flight of stairs? Here are some other rules used for building stairs in different kinds of buildings.

Private Homes Rise — maximum 20 cm Tread — minimum 23 cm

Public Buildings Rise — maximum 18 cm Tread — minimum 28 cm

28. a. Why do you think that there is a maximum for the rise? b. Why is there a minimum for the tread and not a maximum? Section C: Using Formulas 31

C Using Formulas

In this section, you used formulas in different situations. Formulas can be used to:

•

convert from one measuring system to another, such as how to convert temperatures; and

•

investigate possibilities within certain constraints, such as how to build a staircase that has a total height of 3 meters (m) but is easy to climb.

You will encounter many more situations in which formulas can be used.

Do not believe everything you read! The following was printed on the cover of a notebook:

•

To convert Fahrenheit temperatures to Celsius temperatures, use this formula: 5

C   9 (F 32).

•

To convert Celsius temperatures to Fahrenheit temperatures, use this formula: 9

F   5 (C  32). You know, for example, that 0°C corresponds to 32°F. 1. The second formula is not correct! Write a letter to the company that produced the notebook to explain why. What mistake was made?

32 Building Formulas

The length of time it takes a driver to stop a car is affected by how fast the car is going. Suppose the following formula finds the stopping distance in feet if you know the speed of the car. 7  Speed  74  Stopping Distance 2. a. What is the stopping distance if the car’s speed is 20 miles per hour (mi/h)? 40 mi/h? 60 mi/h? b. Create a graph of the relationship between speed and stopping distance. Place the speed along the horizontal axis and the distance along the vertical axis. c. Are there any restrictions for possible speed and/or distance values? Explain. A rule for building an exit ramp states that the vertical distance of the ramp must be no more than one-eighth of the horizontal distance. 3. a. Which of the following ramps fits this rule? i.

iii.

ii. 5 meters

0.2 meter

0.75 meter

0.5 meter 2 meters

1 meter

b. Write the rule for an exit ramp in mathematical language. 4. Design a staircase for a public building with a total height of 3 m. The staircase should take up as little floor space as possible (that is, it should have the smallest possible depth measurement). Make sure it fits the rule for a public building and the rule 2R  T  63. (R represents rise in centimeters; T represents tread in centimeters. Public Building Rise – maximum 18 cm

Tread – minimum 28 cm

Many formulas have constraints in order for them to make sense in a context. What do you think the constraints would be for the formulas for converting between Fahrenheit and Celsius?

Section C: Using Formulas 33

D

Formulas and Geometry

Lichens

Many formulas are used in geometry. In this section, you will revisit some of the formulas you studied earlier for finding the area and volume of different shapes and solids. A lichen (pronounced LIKE-en) is a type of fungus that grows on rocks, on walls, on trees, and in the tundra. Lichens are virtually indestructible. No place is too hot, too cold, or too dry for them to live. Scientists can use lichens to estimate when glaciers disappeared. Lichens are always the first to move into new areas. So as the glacier recedes, lichens will appear very soon. The scientists know how fast lichens grow, so they use the area covered by the lichens to calculate how long ago a glacier disappeared. Many lichens grow more or less in the shape of a circle. 1. Estimate the area covered by this lichen in square centimeters (cm2).

34 Building Formulas

Formulas and Geometry D You can use a circle as a model for the area covered by a lichen. Remember that the formula to find the area of a circle is: Area  π  radius  radius or Area  πr 2 Your calculator may have a π key. If it doesn’t, use 3.14 as an approximation for π. 2. a. Make a drawing of a circle with a radius of 2 cm. Use a compass! b. What is the diameter of your circle? The formula Area  πr 2 can be written as an arrow string. π square r ⎯⎯→ ..... ⎯⎯→ areaF 3. Use the formula or the arrow string to find the area of the circle from problem 2. Round your answer to the nearest tenth and be sure to include the unit measurement. The diameter of the lichen shown on page 34 is about 1 cm. 4. a. What is the radius of a circle with a diameter of 1 cm? b. Use the formula or the arrow string to find the area of the circle. Round your answer to the nearest tenth and be sure to include the unit measurement. c. Was your estimation of the area covered by the lichen close? The table shows the relationship between the radius of a circle and its area. Radius (in cm) Area (in

cm2)

0 0

0.5

1

1.5

2

2.5

3

4

3.1

5. a. Copy the table in your notebook and fill in the empty spaces. b. Use graph paper to draw a graph to represent this relationship. c. Describe the graph. Does it seem to be a straight line? Explain how you can tell. Section D: Formulas and Geometry 35

D Formulas and Geometry

A scientific article reported that a lichen on a glacier covered 34 cm2. 6. If you knew the radius, you could figure out how wide the lichen in the report was. How could you find an estimate for the radius? 7. a. Sammi says the radius would be 17 cm because 34 divided by 2 is 17. What do you think of Sammi’s idea? b. Sammi insists that his idea is a good one. Think of some examples of areas that would either support his idea or show that it is wrong. c. Jorge has a different idea. He says that because the formula for the area uses square numbers, you can “unsquare” the number. What do you think of Jorge’s idea?

36 Building Formulas

Formulas and Geometry D To “unsquare” a number, mathematicians use the symbol . It is usually read as taking the square root of instead of unsquaring. 8. Find the square root of each of the following numbers. Why don’t you need a calculator to do so? a. 25

b. 64

c. 121

d. 14

9. a. Use the  key on your calculator to find the square root of 150. b. Write on a sheet of paper the answer your calculator gives you for  150.  Clear the calculator and calculate the square of this number. If there is a difference, can you explain the difference between this number and the answer you got in part a? For most numbers, it is not possible to find the exact square root because there are an infinite number of decimal places, and the decimals never form a repeating pattern. The only time you get an exact answer is when you start with a square number like 49 or 614. 10. What does the calculator do since it cannot show a decimal that keeps going? Here is an arrow string that makes use of square numbers. square number ⎯⎯⎯→ ……  3 answer

⎯⎯→

11. a. What is the answer if the number is 5? If the number is 10? If the number is 32 ? b. Reverse the arrow string. Use the  sign. c. Find the number for each of the answers: 12, 24 , and 35 . d. Make an arrow string using squares and roots. Find two numbers and two answers and have them checked by a classmate. Here is the arrow string for the area of a circle. square  π area ⎯⎯→ r ⎯⎯⎯→ …… ⎯ 12. a. Reverse the arrow string for the area of a circle. b. Use the reverse arrow string to find a formula for the radius of a circle if you know its area. c. Use the reverse arrow string or the formula to find the radius of a circle with an area of 35 cm2. Give your answer to one decimal place. Section D: Formulas and Geometry 37

D Formulas and Geometry

Circles and Solids Valerie wants to make a mold she can later use to make candles. She decides to use a cylinder-shaped mold. For the base of the mold, she has cut a circle that has a 6-cm diameter.

6 cm diameter

height overlap 1 cm

base 13. a. Make an accurate drawing of a circle that is 6 cm in diameter. Use a compass. b. Use a strip of paper to find the size of the mantle of the mold. Allow at least 1 cm overlap to glue the mantle together. What are the measurements of the mantle without the overlap? Valerie used this formula for the mantle of her mold: circumference of a circle  π  diameter c. Explain why this formula makes sense. Fruit drinks come in cans of different sizes. Some cans are narrow and tall; others are wide and short. 14. a. What shapes are juice cans usually? b. Is it possible for cans in different shapes to contain the same amount of liquid?

38 Building Formulas

Formulas and Geometry D This juice can is made up of two circles and a rectangle.

The can shown in the drawing has a height of 15 cm. The diameter of the bottom is 7 cm. 15. a. Calculate the area of the bottom of the can. b. Calculate the volume of the can. Remember that the formula for the volume of any cylinder is: Volume  area of Base  Height c. What are the measurements of the rectangle that makes the sides of the can? d. The can is made of tin. How much tin (in cm2) is needed to make this can? This type of fruit juice is also available in cans that are twice as high. 16. a. Compare the amounts of fruit juice that each can contains. b. How do the surface areas of the cans compare? Be prepared to explain your answer without making calculations. 17. Suppose one can has double the diameter of another can. a. Do you think the amount of liquid that fits in the larger can will double? Give mathematical reasons to support your answer. b. What can you tell about the surface area of the larger can compared to that of the original can? Section D: Formulas and Geometry 39

D Formulas and Geometry

Pyramids Nicolas, an artist, makes clocks in the shape of a pyramid by pouring plastic into a mold. The base is a square.

Remember that the formula for the volume of any pyramid is: 1

Volume   3  area of Base  Height This formula can be rewritten as: 1

2 Volume   3a h

(a is the length of one side of the base; h is the height) 18. Explain why this formula can be used to calculate the volume of a pyramid. 19. Write the formula as an arrow string. Matthew made this arrow string:  13 square h Volume a ⎯ ⎯→ .... ⎯ ⎯⎯→ .... ⎯⎯→ 20. Matthew made a mistake. What was his mistake? Nicolas wants to know how much plastic is needed for 250 of the clock pyramids. The square measures 2 dm by 2 dm, and the height 1 is 1  2 dm. 21. How many cubic decimeters (dm3) of plastic are needed?

40 Building Formulas

Formulas and Geometry D Nicolas thinks the clock pyramids should be a little larger so they will fit in the gift boxes he can buy. He wants the new pyramids to have a volume of 2 12 dm3 each. 22. a. Write the reverse arrow string to find the area of the base of the new pyramid. b. Find the length of the square that is the base of the pyramid. c. To how many decimal places did you round your answer for part b? Explain why you think what you did is reasonable.

Section D: Formulas and Geometry 41

D Formulas and Geometry

Formulas are used to find areas and volumes of geometric figures. In this section, some of the formulas used square numbers, like the formula you used to find the area of a circle: Area  π  radius  radius or Area  πr 2 “Unsquaring” a number is called taking the square root of a number. The square root  of a square number has no decimal or a finite number of decimals.  81  9 1  6 14  2 2

The square root of a number that is not a perfect square has an infinite number of decimals that do not have a repeating pattern.  10  3.1622776… If you need to round an answer, find a reasonable number of decimals according to the situation.

1. Find an answer for each of the following. Round those that are not perfect squares to one decimal place. a.  16  c.  48  b.  1241 

d.  1000  

2. a. Find the area of a circle with a diameter of 30 cm. Show your work. b. Find the radius of a circle with an area of 10,000 cm2. Round your answer to the nearest centimeter.

42 Building Formulas

Here is an arrow string. 4 . . . . ⎯square 3 . . . . ⎯⎯→ 7 answer number ⎯⎯→ ⎯⎯→ . . . . ⎯⎯→ 3. a. Use the arrow string to find the answer for number = 6. b. Use the reverse arrow string to find the number if answer  55. The following problem is about pyramids with a square base. 4. Is there a difference in volume between a pyramid with a side of the square of 4 and height 6 and a pyramid with a side of the square of 6 and height 4? Explain why or why not.

How does the volume of a pyramid with a square base compare to the volume of cube with the same square base?

Section D: Formulas and Geometry 43

E

Problem Solving

Heavy Training Your heart rate when you are lying or sitting is considered your normal or resting heart rate. When you fall asleep, your heart rate slows, and when you exercise or are upset, your heart rate increases.

With a partner, find your resting heart rate. To do this, find your pulse in your neck or wrist and count the beats for 20 seconds. Your partner should watch the clock and tell you when to start counting and when to stop. Heart rate is usually reported in terms of beats per minute (bpm). Use the pulse that you counted in 20 seconds to find your resting heart rate in beats per minute. Switch roles with your partner and repeat the above procedure so that both of you know your resting heart rates.

44 Building Formulas

Problem Solving E

Athletes who take part in endurance sports need to be in very good condition. When they compete, their heart rates increase. Because it is dangerous for a person’s heart rate to be too high for too long, athletes train specifically to increase their endurance. It is important for athletes to determine their maximum heart rate. Finding the exact value of an athlete’s maximum heart rate is difficult. There is, however, a rule that gives a close approximation. Subtract your age (A) in years from 220 to find your maximum heart rate (M) in beats per minute. 1. a. Write this rule as a formula. b. How does your resting heart rate compare to the maximum heart rate you calculated from the formula? 2. Make a graph on Student Activity Sheet 4 that could be placed in a gym to help people find their maximum heart rates. Warning: The M value can vary with your physical condition. You should not use the above formula to gauge your own workouts without consulting a physician.

Section E: Problem Solving 45

E

Problem Solving Use the formula and the graph you made for problem 2 to answer the following questions. 220 216

Maximum Heart Rate (in bpm)

212 208 204 200 196 192 188 184 180 12

14

16

18

20

22

24

26

28

Age (in years)

3. a. Who has a higher maximum heart rate: you or a teacher? Explain. b. What can you say about the relationship between age and maximum heart rate? 4. a. What is the highest possible maximum heart rate according to the formula? The lowest? b. Do you think this rule applies for people of any age? Explain. Jacob is the trainer for John, Anita, and Carmen. He decides that these athletes should be using 75% of their maximum heart rate during their workouts. This heart rate is called the training value (T ). 5. a. Write 75% as a fraction. b. Use the formula M  220  A to create a new formula for the relationship between the training value T and age A. 46 Building Formulas

Problem Solving E

Crickets Some crickets are brown in color and have transparent wings that make them difficult to spot. These insects often form a chorus at night, singing a high-pitched, two-note song. The approximate temperature in Fahrenheit can be estimated by counting the number of cricket chirps made in one minute, dividing this number by 4, and adding 40. You can use the following formula to find the approximate temperature (F) if you know the number of chirps (N) made in one minute: N

— 4  40  F 6. A cricket chirps 100 times per minute. Estimate the air temperature. 7. a. About how cold is it when you hear no cricket chirps? b. Is there a maximum number of chirps you could hear in one minute? Explain your answer. 8. a. Choose at least five different numbers of chirps (N) and estimate the corresponding temperature for each (F). Make a table in which to record your data. b. Use a sheet of graph paper to make a graph of the data you gathered in part a to show how the number of chirps (N) and the temperature (F) are related. c. Tell whether each of the following statements is true or false and explain why. i.

If the number of chirps in one minute doubles, then the temperature gets twice as high.

ii.

If you counted one more chirp in a minute than in a previous minute, then the corresponding temperature would rise about one-half of a degree Fahrenheit.

9. Kim wants a formula so that she can predict the number of chirps she will hear per minute when she knows the outside temperature in degrees Fahrenheit. Write a formula that Kim could use. Section E: Problem Solving 47

E

Problem Solving

Egyptian Art The ancient Egyptians were fascinated by proportions. When they made drawings or sculptures of people, the measurements followed a set of rules. These rules help modern archaeologists reconstruct Egyptian pictures that have been damaged. Archaeologists have written these rules as formulas that are easy to use. The formulas are written in terms of the height of parts of the body from the ground. H  height of the hairline N  height of the neck

E  height of the elbow

W  height of the wrist

K  height of the knee

These are the formulas used to figure out the proportions (each measurement is from the ground up): H  3K

EWWK

E  2K

8 N   9H

10. a. Translate the formula E  2K into an English sentence about the drawing. b. Describe how you can check whether E  W equals W  K in the drawing. c. Show whether the measurements of the Egyptian in the picture fit the given formulas.

48 Building Formulas

Problem Solving E On a recent excavation, archaeologists discovered pieces of an ancient Egyptian drawing. Unfortunately, parts of the drawing were missing. The archaeologists made careful tracings of the pieces they found. 11. Use Student Activity Sheet 5 to reconstruct the original drawing.

•

First cut out the tracings that the archaeologists made.

•

Use the formulas to place each piece correctly. (Keep track of your calculations so that you can explain your process to someone else.) Glue the pieces onto a sheet of paper.

•

Then sketch the missing portions.

Later, while excavating the site, the team of archaeologists found fragments of another drawing. This drawing had grid lines over it, which the ancient Egyptians may have used to check the proportions. The archaeologists thought that this drawing would be easier to reconstruct than the first one because of the lines. After they looked at the pieces, however, they realized that it would be harder. 12. Reflect Look at Student Activity Sheet 6. Why is this drawing harder to reconstruct? After looking carefully at the formulas, the archaeologists found a way to get around the problem. 13. Use Student Activity Sheet 6 to reconstruct this picture. Cut out the pieces, glue them into the correct positions, and sketch the missing parts. (Remember to keep track of your calculations.)

Section E: Problem Solving 49

E Problem Solving

Formulas are used to describe relationships and solve problems in many situations. Information from pictures, stories, graphs, tables, or other formulas can help you figure out a formula you can use to solve a problem. When solving problems, be sure to choose the best way to organize and represent the information.

A Dutch mathematician invented one of the first pendulum clocks in 1656. Pendulums are used to regulate the movement of clocks because the time interval of each swing of a pendulum is always the same. The time a pendulum needs to make a swing depends only on the length of the pendulum. Here is a formula that can be used to determine the time interval in seconds (T ) of 100 swings if the length (L) in centimeters of the pendulum is known: L T  20   1. a. Calculate how much time it would take a pendulum 100 cm long to make 100 swings. b. How many seconds does one swing of the same pendulum take? LaShonda wants to make a pendulum clock in which the pendulum takes exactly one minute to swing 100 times. 2. How long does the pendulum on LaShonda’s clock need to be? Explain how you found your answer.

50 Building Formulas

3. a. Choose at least five different pendulum lengths between 1 cm and 36 cm and calculate the time needed for each pendulum to make 100 swings. Make a table in which to record your data. b. Use a sheet of graph paper to make a graph of the data you gathered in part a to show how the length (L) and the time (T) are related. c. Tell whether each of the following statements is true or false and explain why: i. The longer the length of the pendulum, the slower it swings. ii. If you double the length of the pendulum, it takes twice as much time to make 100 swings. iii. If you add 1 cm to the length of a pendulum, the time needed to make 100 swings is increased by 20 seconds. 4. Greg wants to make a pendulum clock in which the pendulum takes exactly one second for each swing. Find out how long his pendulum needs to be. 5. Caroline rewrites the formula to determine the time needed for one swing instead of 100 swings of the pendulum. She writes: L . Is Caroline’s formula correct? Time for 1 swing  0.2   Explain why or why not.

You have studied other formulas in mathematics. Name at least two of them and explain how they are used.

Section E: Problem Solving 51

Additional Practice Section A

Patterns The paths shown in the drawing are made from green and white hexagonal tiles. Tiles with a hexagonal shape have six equal sides. The path is said to be of length 5 because it has five green tiles down the center. The second path is of length 6.

Length  6

Length  5

1. How many green and white tiles are in a path of length 4? Length 1? 2. How can you find the total number of tiles in a path of length 10? 3. Copy and complete the table. Path Length (L)

Number of Green Tiles (G )

Number of White Tiles (W )

Total Number of Tiles (T )

5

5

14

19

6

6

16

22

4. a. Describe the pattern of green tiles, using a NEXT-CURRENT formula. b. Write a direct formula to describe the number of green tiles (G) for a path of any length (L). c. Write a direct formula to describe the number of white tiles (W ) for a path of any length (L). d. Write a direct formula to describe the total number of tiles (T ) for a path of any length(L). 52 Building Formulas

5. a. Marsha has 75 white tiles. Can she use them all in one path? Explain why or why not. b. Marsha thinks the number of white tiles is always even. Is she right? Give mathematical reasons to support your answer.

Section B

Brick Patterns Urvashi, the movie scene designer, needs garden walls for the large garden surrounding a mansion. This is the basic pattern she uses for the walls:

1. a. Write a formula for this basic pattern. Length  . . . . b. Make a drawing of her garden wall that consists of two times the basic pattern. For one garden wall, Urvashi thinks she will need 50 times the basic pattern. She uses this formula. Total Number of Bricks  100L  200S 2. a. Explain the letters and the numbers in this formula. b. Write the formula in a different way, using parentheses. c. How many bricks does Urvashi need in total for this garden wall? 3. a. Draw a garden wall that fits this formula. Length  3(2S  2L) b. Write the formula for problem 3a without parentheses. c. Is the basic pattern the same as the one Urvashi used for her garden walls?

Additional Practice 53

Additional Practice

Section C

Using Formulas

“City Taxi” in Europe uses this formula for the total cost in euros (T ) for any number of kilometers (K) T  5  (0.8  K) 1. a. What do the numbers in this formula represent? b. Write the formula as an arrow string; K ⎯⎯→ ....... c. Find the cost for a taxi ride of 12 km. d. Jeanette paid 11 euros for a ride. Her company wants to know how many kilometers she traveled. Write a reverse arrow string and calculate how many kilometers Jeanette traveled. A truck rental agency advertised “$19.95 a day and 69 cents a mile in town” for renting a truck. 2. a. Use the information to write a direct formula for the total cost in dollars (T) for any number of miles (M) for a one-day rental. b. Calculate the total rent if you need the truck for one day and expect to drive no more than 40 miles. The amount of blood in your body depends on, among other things, your weight. A rule to estimate the amount of blood is: 1

  weight (in kilograms) blood (in liters)  13 3. a. Elmar weighs 65 kilograms (kg). Estimate the number of liters of blood in his body. b. Nathan is twice Elmar’s weight. Does he have twice the amount of blood Elmar has? Explain your thinking. c. Do you think you could use this formula for a two-month-old baby? Why or why not?

54 Building Formulas

Additional Practice

Section D

Formulas and Geometry Mark found a circular-shaped lichen on the garden wall. Its diameter is about 3 cm. 1. Find the area of the lichen. Selma says: “I always use 3 for an estimation of pi when I estimate the size of a lichen. It makes calculations much easier.” 2. Use examples to show whether Selma can use 3 for π. Write down any assumptions about the size of the lichen you make. The local yogurt shop serves yogurt by pushing the yogurt into a cone and filling it to the top of the cone. 3. a. How much yogurt will fill a cone that is 12 cm high with a diameter of 6 cm? Remember that the formula for the volume of a cone is: V  13  area of base  height or V  13 πr 2h b. Suppose you could choose a cone that is twice as wide as the cone from 3a. Does this new cone have twice the amount of yogurt? Why or why not?

Section E

Problem Solving How tall will you be as a grown-up? Here are some estimation rules, expressed as a formula. There are about 3 feet (ft) in 1 meter (m). boy: height in inches when grown up  height father height mother 13  4.5 2

(

)

girl: height in inches when grown up  height father height mother 13  4.5 2

(

)

1. a. Why are there different rules for boys and girls? b. Use one of the formulas to find out approximately what height you will be as a grown-up. Use estimations if you do not know the height of your parents. Additional Practice 55

Additional Practice The first formula can be rewritten as: height in inches when grown up  1 2 (height father  height mother)  11

2. a. Show that this formula is equal to the first one. b. Rewrite the second formula. c. Can the formula for girls be described as 2 less than the mean of father’s and mother’s height? Explain why or why not. The higher you stand, the farther you can look! Here is a rule to estimate the relationship between the height (h) in meters and the distance to the horizon (d) in kilometers. The rule holds only for a clear day. 13 h d   3. a. Write this formula as an arrow string. Use the  sign. b. Hank is at the beach and is 1.80 m tall. How many kilometers away can he still see a ship? c. Bert is 1.50 m tall. He used the formula and found the distance to be 4.416 km. Comment on Bert’s answer. d. Frances claims: “If you stand twice as high, you can see twice as far!” Is Frances right? Explain. e. Mohamed claims: “If you stand four times as high, you can see twice as far!” Is Mohamed right? Explain.

56 Building Formulas

Section A

Patterns 1. Yes, they describe the same pattern. Here are some sample strategies you could use: Strategy 1 A pattern could be: Path 2

Path 3

Path 4

T  P  2  P can be explained as a row of P orange tiles, 2 white tiles, and another row of orange tiles. T  2P  2 can be explained as 2 rows of P orange tiles and 2 white tiles. T  2(P  1) can be explained as 2 rows of P orange tiles and a white tile. Strategy 2 P  2  P is the same as 2P  2, and this can also be written as 2(P  1). 2. a. Sample table: Patio Number (P )

Number of Orange Tiles (O)

Number of White Tiles (W )

1

1

8

2

4

12

3

9

16

4

16

20

5

25

24

6

36

28

Answers to Check Your Work 57

Answers to Check Your Work b. Discuss your formula with a classmate. Sample formulas: The Number of Orange Tiles is the Patio Number times itself. Number of Orange Tiles  Patio Number  Patio Number GPP G  P2 c. Discuss your formula with a classmate. You can find many equivalent formulas. W  (P  2)  (P  2)  P  P W  2(P  2)  2P W  4  4P W  4(P  1) W  4(P  2)  4 W  (P  2)  (P  2)  (P  P) d. You can find the answer by extending the table or by using the formulas you found. Number of Orange Tiles in Patio Number 10 is 100. Number of White Tiles in Patio Number 10 is 44. 3. a. Sample table: Path Length (L)

Number of Orange Tiles (O)

Number of White Tiles (W )

Total Number of Tiles (T )

1

1

8

9

2

4

12

16

3

9

16

25

4

16

20

36

5

25

24

49

6

36

28

64

b. Yes, both formulas give the same result. Sample drawing: The length of the side of the whole square (including orange and white tiles) is P  2, so the total number of tiles is (P  2)  (P  2).

1 PP

P 3

58 Building Formulas

2

4

Answers to Check Your Work 4. a. Terry is building patio number 12. The side of the square Terry is going to build has a length of 14 since 14  14  196, so the patio number is two less. b. Use one of the formulas from 2b to find the number of white tiles. For example, W  4(P  1) will give you 4(12  1)  4  13  52 white tiles.

Section B

Brick Patterns 1. a. Length  1U  1D or, since the 1 in 1U is the same as U, Length  U  D b. Length  30U  29D c. If you think very hard, both yes and no can be your answer! Discuss your explanation with a classmate. Sample explanations: No, it is not possible, because 30 and 29 are not divisible by the same number; there is no number that goes into 29 and 30 evenly. But you might also say it is possible if you first rewrite the formula; for instance: Length (29U  U)  29D, so Length  29U  29D  U Length  29(U  D)  U 2. a. Sample response: A row with a basic pattern with two lying bricks and three standing bricks that is shown four times. b. Length  8L  12S 3. a. Row 4. Sample explanation: A basic pattern cannot be repeated with Row 4 since there is no number that goes into 9 and 13 evenly. All other rows have basic patterns. b. The formulas for the length of the different basic patterns for Row 5 are: Length  3L  2S Length  6L  4S Length  9L  6S

Answers to Check Your Work 59

Answers to Check Your Work

Section C

Using Formulas 1. Here is a sample letter. Your letter will be different, but your explanation should be similar to the one shown in the letter. To Whom It May Concern: Recently I realized that your company has printed a formula incorrectly. It is located on the notebook cover as follows:

• To convert Celsius temperatures to Fahrenheit temperatures, use this formula: 5

C   9 (F  32).

• To convert Fahrenheit temperatures to Celsius temperatures, use this formula: 9

F 5  (C  32). The first formula is correct, but the second is not. The second formula should be the reverse of the first, but it is not. When you use numbers to check out the formulas, you can see that the second one does not work. I know that 0°C is equal to 32°F, so if I put F = 32 in the first formula, I should get C = 0, and when I try it, that happens. When I put C = 0 in the second formula, I should get F = 32, but I get 57.6, which is incorrect. The correct second formula can be found using arrow language to reverse the first formula: 5

Formula I

  32 9 F ⎯⎯ ⎯⎯→ _____ ⎯ ⎯⎯→ C 5

   32 9 Formula II C ⎯ ⎯⎯→ _____ ⎯⎯⎯→ F 5

9

Dividing by   . So the correct 9 is the same as multiplying by 5 9 second formula is F  5  C  32

It was a mistake to use the parentheses. Please correct this error. Sincerely,

60 Building Formulas

Answers to Check Your Work 2. a. Speed (mi/h)

Stopping Distance (ft)

20

66

40

206

60

346

b. Automobile Stopping Distances

Stopping Distance ( in ft)

400

300

200

100

10

20

30

40

50

60

70

80

Speed (in mi/h)

c. Yes. Sample explanation: Restrictions depend on where you are driving. For example, the maximum speed is often 65 mi/h on the interstate and 40 or 35 mi/h in the city. Another restriction is that the formula will not make sense if you use speeds that are too small. If you are traveling at 10 mi/h, the formula would show that it would take 4 seconds to stop.

Answers to Check Your Work 61

Answers to Check Your Work 3. a. The ramp in part i fits the rule. 1

1

  5, which is less than   5 0.5  10 8 3 1 ramp ii. 0.75  8  2, which is more than 8  2 2 1 1   1 or   1, which is more than   1 ramp iii. 0.2  10 5 8 1 b. vertical distance ≤ 8  horizontal distance

4. Designs will vary. Sample design that has 20 steps. 300 cm

660 cm

Make sure you have noted the rise and tread in your design and that they fit the rules. In the sample design, the measurements were: rise is 15 cm tread is 33 cm

Section D

Formulas and Geometry 1. a.  16  4 1 b.  12 14  3 2

c.  48 ≈ 6.9 d.  1000  ≈ 31.6 2. a. A circle with a diameter of 30 cm has a radius of 12  30  15 Area  π  15  15  706.85834… Rounded to one decimal place, area ≈ 706.9 cm2 b. You can use a reverse arrow string for the area formula. π

 area ⎯⎯→ . . . . . ⎯⎯→ radius

When you use your calculator, do not round results before the end of your calculations. π

 56.418958… 10,000 ⎯⎯→ 3183.09886… ⎯⎯→ The radius of the circle is 56 cm, rounded to the nearest whole number.

62 Building Formulas

Answers to Check Your Work 7

square

4  3 12 ⎯⎯→ 19 3. a. 6 ⎯⎯→ 2 ⎯ ⎯⎯→ 4 ⎯⎯→

b. First make the reverse arrow string: 7

4

:3

 answer ⎯⎯→ …. ⎯⎯→ …. ⎯⎯→ …. ⎯⎯→ number

7

4

:3

 55 ⎯⎯→ 48 ⎯⎯→ 16 ⎯⎯→ 4 ⎯⎯→ 8

4. Yes, there is a difference. Discuss your explanation with a classmate. Sample explanation: In one case you square the 4 and in the other you square the 6. If you compute the volume of both pyramids, you would find volume (I)  ( 13 ) 4  4  6 and volume (II)  ( 13 ) 6  6  4

Section E

Problem Solving 1. a. 200 seconds. Strategies will vary. Some students may substitute the given values into the formula: T  20   L T  20   100   T  20  10  200 seconds b. 2 seconds 2. The pendulum needs to be 9 cm long. Strategies will vary. Sample strategy: T  20   L 60  20   L 3   L 9 L 3. a. Answers will vary. Sample response: Pendulum Length

 L

Time

4 cm

2

40 sec.

9 cm

3

60 sec.

16 cm

4

80 sec.

25 cm

5

100 sec.

36 cm

6

120 sec.

Answers to Check Your Work 63

Answers to Check Your Work b. Graphs will vary. Sample graph: Pendulum Movement

30 28

Length ( in cm)

26 24 22 20 18 16 14 12 10 8 6 4 2 10

20

30

40

50

60

70

80

90

100

Time (in sec.)

c.

i. True. The time increases as the pendulum length increases. ii. False. You might use your table or graph to think about your answer. iii. False. One way to show this is to choose a length from your table, add 1 cm and compute the new time to see if the number of seconds is increased by 20.

4. Greg should make the pendulum 25 cm long. 5. Yes, Caroline’s formula is correct. Explanations will vary. You may 1 of the time needed –— reason that the time needed for 1 swing is 100 for 100 swings, so you need to divide the numbers in the formula by 100. So, T  0.2  L .

64 Building Formulas