Building Formulas

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Building Formulas


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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.

© 2010 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, the thistle logo, Mathematics in Context, and the Mathematics in Context logo are registered trademarks of Encyclopædia Britannica, Inc.

All rights reserved.

No part of this work may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage or retrieval system, without permission in writing from the publisher.

International Standard Book Number 978-1-59339-937-5 Printed in the United States of America

Wijers, M., Roodhardt, A., van Reeuwijk, M., Dekker, T., Burrill, G., Cole, B.R., &

Pligge, M. A. (2010). Building formulas. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in context. Chicago:

Encyclopædia Britannica, Inc.

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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

Project Staff

Jonathan Brendefur Sherian Foster Mieke Abels Jansie Niehaus

Laura Brinker James A, Middleton Nina Boswinkel Nanda Querelle James Browne Jasmina Milinkovic Frans van Galen Anton Roodhardt Jack Burrill Margaret A. Pligge Koeno Gravemeijer Leen Streefland

Rose Byrd Mary C. Shafer Marja van den Heuvel-Panhuizen

Peter Christiansen Julia A. Shew Jan Auke de Jong Adri Treffers

Barbara Clarke Aaron N. Simon Vincent Jonker Monica Wijers

Doug Clarke Marvin Smith Ronald Keijzer Astrid de Wild

Beth R. Cole Stephanie Z. Smith Martin Kindt

Fae Dremock Mary S. Spence

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

Project Staff

Sarah Ailts Margaret R. Meyer Arthur Bakker Nathalie Kuijpers

Beth R. Cole Anne Park Peter Boon Huub Nilwik

Erin Hazlett Bryna Rappaport Els Feijs Sonia Palha

Teri Hedges Kathleen A. Steele Dédé de Haan Nanda Querelle

Karen Hoiberg Ana C. Stephens Martin Kindt Martin van Reeuwijk

Carrie Johnson Candace Ulmer

Jean Krusi Jill Vettrus

Elaine McGrath 05/13/2005 07:14 Page iii


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.


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

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Letter to the Student vi

Section A Patterns

Tiles 1

Beams 6

Summary 10

Check Your Work 11

Section B Brick Patterns

Bricks 13

The Classic 15

More Brick Rows 17

Summary 20

Check Your Work 20

Section C Using Formulas

Temperature 22

Building Stairs 26

Summary 32

Check Your Work 32

Section D Formulas and Geometry

Lichens 34

Circles and Solids 38

Pyramids 40

Summary 42

Check Your Work 42

Section E Problem Solving

Heavy Training 44

Crickets 47

Egyptian Art 48

Summary 50

Check Your Work 50

Additional Practice 52 Answers to Check Your Work 57 05/13/2005 07:14 Page 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.

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.

Sincerely, T

Thhee MMaatthheemmaattiiccss iinn CCoonntteexxtt DDeevveellooppmmeenntt TTeeaamm

0 0

10 10

20 20 30 40 50 60


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A Patterns


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 05/13/2005 07:14 Page 1


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

To make it easy to refer to a path, Urvashi assigns each path a number.

1. a. What do the numbers represent?

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.

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.

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.



Path Number 3

Path Number 4

Path Number 5

Path Number 3 New Path Number 3 05/13/2005 07:14 Page 2


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

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.

Patterns A

Path Number of Number of Total Number

Number Blue Tiles White Tiles of Tiles 1


3 8 7 15

4 5 6 7 05/13/2005 07:14 Page 3




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.

P→ W 4

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. 05/13/2005 07:14 Page 4


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.

Patterns A

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 formulato 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.

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. 05/13/2005 07:14 Page 5


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?

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. 05/13/2005 07:14 Page 6


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.

Patterns A 05/13/2005 07:15 Page 7


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



Length of Beam


Number of Rods

(R) 1





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.

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.

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

b. Are all of the formulas the same? Explain.

c. Compare these formulas to the one you found Angelina:

R L  3  (L  1) David:

R L  (L  1)  2L Maria:

R 3  (L  1)  4 05/13/2005 07:15 Page 8


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?

Patterns A

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 formu- las 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.

I’ll break the beam into parts. 05/13/2005 07:15 Page 9



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).

A 05/13/2005 07:15 Page 10


Here are three formulas:

T P  2  P 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.

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?

Patio Number 1

Patio Number 3

Patio Number 2

Patio Number 4 05/13/2005 07:15 Page 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?


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.

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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.

B Brick Patterns

Bricks 05/13/2005 07:15 Page 13


Here are some diagrams of brick patterns.

Brick Patterns


Name Drawing String

Classic SLSLS



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 patternthat 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.

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. 05/13/2005 07:15 Page 14


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.

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.

Brick Patterns B

The Classic

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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:

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.)

Brick Patterns


Yellow 8 cm Gray

6 cm

12 cm 15 cm 05/13/2005 07:15 Page 16


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.

Brick Patterns B

More Brick Rows

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.

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.

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. 05/13/2005 07:15 Page 17


Ms. Peterson is fixing up her house.

She goes to a hardware store to buy some supplies.

Brick Patterns


Good morning.

How can I help you?

I’m trying to restore the brick row

above my front door.

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.

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

I remember that my row has a basic pattern that repeats

a number of times. I also wrote down another formula, which

has parentheses.

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.

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.

In that one, I can see the number of times the basic

pattern is repeated, but I left that one at home too.

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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. 05/13/2005 07:15 Page 19


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 D 05/13/2005 07:15 Page 20


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?


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. 05/13/2005 07:15 Page 21


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.

C Using Formulas


Dear Kim,

The weather has been beautiful. We have had a week with temperatures of about 25°C. On Wednesday, it was even

30°C. This is a bit too hot for me.

We went swimming in the lake. The water was not very warm, only 18°C, but it was great to cool off. It’s hard

to imagine that a week ago it was only around 16°C. I had to wear a sweater all day.

What is the weather like in your city?

Do I have to bring a sweater? At home it cools down in the evenings. Last night we had a thunderstorm, and the temperature dropped by 10 degrees!

I look forward to seeing you.


1. Estimate the temperatures in degrees Fahrenheitfor the Celsius 05/13/2005 07:15 Page 22


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.

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.

Using Formulas C

0 0

10 10

20 20 30 40 50 60


20 40 60 80 100 120 140 05/13/2005 07:15 Page 23


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.)

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.)

Using Formulas


6060 5050 303020201010 5050

40 40 30

30 20

20 4040



404040 30


60 00 101010 20 30 40 50 606060 F






606060 60 60 60

50 50 50

40 40 40

30 30 30

20 20 20

10 10 10


C 20 15 10 5 0 5


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? 05/13/2005 07:15 Page 24


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 stringto 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 stringto write it on the chalkboard.


→ 


→  →


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?

Using Formulas C 05/13/2005 07:15 Page 25


Here are some pictures of Kim and Lucrecia at various places.

13. What is a likely temperature in degrees Celsius for each picture?

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 ).

Using Formulas


Building Stairs


riser 05/13/2005 07:16 Page 26


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

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.

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

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.

You now have a model staircase.

Label the wall and the floor on your model as shown.

8 cm

12 cm D






3 cm 2 cm

3 cm 2 cm 3 cm 2 cm 3 cm 2 cm 05/13/2005 07:16 Page 27


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?


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?

Using Formulas


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

3 cm 2 cm 3 cm 2 cm

3 cm 2 cm 3 cm 2 cm 3 cm 2 cm 05/13/2005 07:16 Page 28


Staircase A

Staircase B

Staircase C

Staircase E

Staircase D 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.

Using Formulas C 05/13/2005 07:16 Page 29


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

Using Formulas


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?


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 measure- ments that fit the rule based on an adult paceof 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 05/13/2005 07:16 Page 30


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.

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?

Private Homes

Rise — maximum 20 cm Tread — minimum 23 cm

Public Buildings

Rise — maximum 18 cm Tread — minimum 28 cm 05/13/2005 07:16 Page 31


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:


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

F95(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?

C 05/13/2005 07:16 Page 32


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?

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?

i. ii.

0.5 meter 5 meters

0.75 meter 2 meters


1 meter

0.2 meter 05/13/2005 07:16 Page 33


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 areacovered by this lichen in square centimeters (cm2).

D Formulas and Geometry

Lichens 05/13/2005 07:16 Page 34


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:



 radius  radius or




Your calculator may have a π key. If it doesn’t, use 3.14 as an approximation for



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


r2can be written as an arrow string.

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.

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.

Formulas and Geometry D

r ⎯⎯→square ...⎯⎯→



Radius (in cm) Area (in cm2)

0 0

0.5 1


1.5 2 2.5 3 4

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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?

Formulas and Geometry

D 05/13/2005 07:16 Page 36


To “unsquare” a number, mathematicians use the symbol . It is usually read as taking the square rootof 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.

number ⎯⎯square⎯→ ……




11. a. What is the answer if the number is 5? If the number is 10?

If the number is 23?

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.

r ⎯⎯square⎯→ …… ⎯ ⎯⎯→ areaπ 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.

Formulas and Geometry D

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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 


 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?

Formulas and Geometry


Circles and Solids

6 cm diameter



overlap 1 cm 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. 05/13/2005 07:17 Page 38


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 volumeof 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 areasof 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?

Formulas and Geometry D 05/13/2005 07:17 Page 39


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

Formulas and Geometry



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

Volume13  area of Base  Height This formula can be rewritten as:

Volume 13 a2h

(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:

a ⎯⎯→ .... ⎯13 square⎯⎯→ .... ⎯ ⎯→ Volumeh 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 is 112 dm.

21. How many cubic decimeters (dm3) of plastic are needed?

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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 212 dm3each.

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.

Formulas and Geometry D

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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 ⴝ


ⴛ radius ⴛ radius or

Area ⴝ



“Unsquaring” a number is called taking the square root of a number.

The square root兹苵苵 of a perfect square has no decimal or a finite number of decimals.

兹苵苵81 ⴝ 9 兹苵苵614ⴝ 21᎑᎑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.

兹苵苵10 ⴝ 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. 兹苵苵16 ⴝ c. 兹苵苵48 ⴝ

b. 兹苵苵121᎑

4苵 ⴝ d. 兹苵苵1000苵苵 ⴝ

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.


SE_ppi_63_ISBN9375_2010.qxd:SE_ppi_63_ISBN9375_2010.qxd 4/7/09 10:33 AM Page 42


Here is an arrow string.

number →4 .... square→ .... →3 .... →7 answer 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? 05/13/2005 07:17 Page 43


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.

E Problem Solving

Heavy Training

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. 05/13/2005 07:17 Page 44


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.

Problem Solving E 05/13/2005 07:17 Page 45




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