expressions formulas grade 7 Bộ Sách Toán THCS Của Mỹ

Expressions
and Formulas
Algebra

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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.

Gravemeijer, K.; Roodhardt, A.; Wijers, M.; Kindt, M., Cole, B. R.; and Burrill, G. (2006)
Expressions and formulas. In Wisconsin Center for Education Research & Freudenthal
Institute (Eds.), Mathematics in Context. Chicago: Encyclopædia Britannica.

2006 Printed by Holt, Rinehart and Winston
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 0-03-039617-4
3 4 5 6 073 09 08 07 06

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The Mathematics in Context Development Team
Development 1991–1997
The initial version of Expressions and Formulas was developed by Koeno Gravemeijer,
Anton Roodhardt, and Monica Wijers. It was adapted for use in American schools by
Beth R. Cole and Gail Burrill.

Wisconsin Center for Education

Freudenthal Institute Staff

Research Staff
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Director

Assistant to the Director

Director

Gail Burrill

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Els Feijs

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Coordinator

Coordinator

Coordinator

Coordinator

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James A, Middleton
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Stephanie Z. Smith

Mary S. Spence

Mieke Abels
Nina Boswinkel
Frans van Galen
Koeno Gravemeijer
Marja van den
Heuvel-Panhuizen
Jan Auke de Jong
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Anton Roodhardt
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Astrid de Wild

Project Staff
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James Browne
Jack Burrill
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Barbara Clarke
Doug Clarke

Beth R. Cole
Fae Dremock
Mary Ann Fix

Revision 2003–2005
The revised version of Expressions and Formulas was developed by Monica Wijers and Martin Kindt.
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
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Els Feijs
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Huub Nilwik
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Martin van Reeuwijk

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

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(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) © PhotoDisc/Getty Images; © Corbis;
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Illustrations
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Inc.; 8 Christine McCabe/© Encyclopædia Britannica, Inc.; 13 (top)
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25 (top right) Thomas Spanos/© Encyclopædia Britannica, Inc.;
29 Holly Cooper-Olds; 32, 36, 40, 41 Christine McCabe/© Encyclopædia

Britannica, Inc.
Photographs
3 © PhotoDisc/Getty Images; 14 © Corbis; 15 John Foxx/Alamy;
26, 32, 33 © PhotoDisc/Getty Images; 34 SuperStock/Alamy;
43 © PhotoDisc/Getty Images

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Contents
Letter to the Student
Section A

Arrow Language
Bus Riddle
Wandering Island
Summary
Check Your Work

Section B

12
14
16
18
22
22

Reverse Operations

Distances
Going Backwards
Beech Trees
Summary
Check Your Work

Section E

6
8
10
10

Formulas
Supermarket
Taxi Fares
Stacking Cups
Bike Sizes
Summary
Check Your Work

Section D

1
3
4
4

Smart Calculations
Making Change

Skillful Computations
Summary
Check Your Work

Section C

vi

25
28
29
30
30

Order of Operations
Home Repairs
Arithmetic Trees
Flexible Computation
Return to the Supermarket
What Comes First?
Summary
Check Your Work

32
34
39
40
42
44
45

Additional Practice

46

Answers to Check Your Work

50

Contents v

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Dear Student,
Welcome to Expressions and Formulas.
Imagine you are shopping for a new bike. How do you determine
the size frame that fits your body best? Bicycle manufacturers have a
formula that uses leg length to find the right size bike for each rider.
In this unit, you will use this formula as well as many others. You will
devise your own formulas by studying the data and processes in the
story. Then you will apply your own formula to solve new problems.
In this unit, you will also learn new forms of mathematical writing.
You will use arrow strings, arithmetic trees, and parentheses. These
new tools will help you interpret problems as well as apply formulas
to find problem solutions.
As you study this unit, look for additional formulas in your daily life
outside the mathematics classroom, such as the formula for sales tax
or cab rates. Formulas are all around us!

Sincerely,

The Mathematics in Context Development Team

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A

Arrow Language

Bus Riddle
Imagine you are a bus driver. Early one morning you start the empty
bus and leave the garage to drive your route. At the first stop, 10 people
get on the bus. At the second stop, six more people get on. At the
third stop, four people get off the bus and seven more get on. At the
fourth stop, five people get on and two people get off. At the fifth stop,
four people get off the bus.

1. How old is the bus driver?
2. Did you expect the first question to ask about the number of
passengers on the bus after the fifth stop?
3. How could you determine the number of passengers on the bus
after the fifth stop?
Section A: Arrow Language 1

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A Arrow Language

When four people get off the bus and seven get on, the number of
people on the bus changes. There are three more people on the bus
than there were before the bus stopped.
4. Here is a record of people getting on and off the bus at six bus
stops. Copy the table into your notebook. Then complete the table.
Number of Passengers
Getting off the Bus

Number of Passengers
Getting on the Bus

Change

5

8

3 more

9

13

16

16

15

8

9

3
5 fewer

5. Study the last row in the table. What can you say about the
number of passengers getting on and off the bus when you
know that there are five fewer people on the bus?
For the story on page 1, you might have kept track of the number of
passengers on the bus by writing:
10 ؉ 6 ‫ ؍‬16 ؉ 3 ‫ ؍‬19 ؉ 3 ‫ ؍‬22 ؊ 4 ‫ ؍‬18
6. Reflect Do you think that representing the numbers in this
format is acceptable mathematically? Why or why not?
To avoid using the equal sign to compare amounts that are not equal,
you can represent the calculation using an arrow symbol.
؉ 6 16 ⎯⎯→
؉ 3 19 ⎯⎯→
؉ 3 22 ⎯⎯→
؊ 4 18
10 ⎯⎯→

Each change is represented by an arrow. This way of writing a string
of calculations is called arrow language. You can use arrow language
to describe any sequence of additions and subtractions, whether it is

about passengers, money, or any other quantities that change.
7. Why is arrow language a good way to keep track of a changing
total?
Ms. Moss has $1,235 in her bank account. She withdraws $357.
Two days later, she withdraws $275 from the account.
8. Use arrow language to represent the changes in Ms. Moss’s
account. Include the amount of money she has in her account
at the end of the story.
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Arrow Language A
Kate has $37. She earns $10 delivering newspapers on Monday.
She spends $2.00 for a cup of frozen yogurt. On Tuesday, she visits
her grandmother and earns $5.00 washing her car. On Wednesday,
she earns $5.00 for baby-sitting. On Friday, she buys a sandwich
for $2.75 and spends $3.00 for a magazine.
9. a. Use arrow language to show how much money Kate
has left.
b. Suppose Kate wants to buy a radio that costs $53. Does she
have enough money to buy the radio at any time during the
week? If so, which day?
Monday

It snowed 20.25 inches.

Tuesday

It warmed up, and 18.5 inches
of snow melted.

Wednesday

Two inches of snow melted.

Thursday

It snowed 14.5 inches.

Friday

It snowed 11.5 inches in the
morning and then stopped.

Ski Spectacular had 42 inches of snow on
the ground on Sunday. This table records
the weather during the week.
10. How deep was the snow on Friday
afternoon? Explain your answer.

Wandering Island
Wandering Island constantly changes shape. On one side of the island,
the sand washes away. On the other side, sand washes onto shore.
The islanders wonder whether their island is actually getting larger or
smaller. In 1998, the area of the island was 210 square kilometers (km2).
Since then, the islanders have recorded the area that washes away
and the area that is added to the island.

Year

Area Washed Away (in km2)

Area Added (in km2)

1999

5.5

6.0

2000

6.0

3.5

2001

4.0

5.0

2002

6.5

7.5

2003

7.0

6.0

11. What was the area of the island at the end of 2001?
12. a. Was the island larger or smaller at the end of 2003 than it was
in 1998?
b. Explain or show how you got your answer.
Section A: Arrow Language 3

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A Arrow Language

Arrow language can be helpful to represent calculations.
Each calculation can be described with an arrow.
starting number action
⎯ ⎯→ resulting number
A series of calculations can be described by an arrow string.
؉ 6 16 ⎯⎯→
؉ 3 19 ⎯⎯→
؉ 3 22 ⎯⎯→
؊ 4 18
10 ⎯⎯→

Airline Reservations

There are 375 seats on a flight to Atlanta, Georgia, that departs on
March 16. By March 11, 233 of the seats were reserved. The airline
continues to take reservations and cancellations until the plane
departs. If the number of reserved seats is higher than the number
of actual seats on the plane, the airline places the passenger names
on a waiting list.
The table shows the changes over the five days before the flight.
Date

Seats Requested

Seats Cancellations

3/11

Total Seats Reserved
233

3/12

47

0

3/13

51

1

3/14

53

0

3/15

5

12

3/16

16

2

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1. Copy and complete the table.
2. Write an arrow string to represent the calculations you made to
complete the table.
3. On which date does the airline need to form a waiting list?
4. To find the total number of reserved seats, Toni, a reservations
agent, suggests adding all of the new reservations and then

subtracting all of the cancellations at one time instead of using
arrow strings. What are the advantages and disadvantages of
her suggestion?
5. a. Find the result of this arrow string.
؉ 1.40

12.30 ⎯⎯⎯→

⎯⎯

؊ 0.62

⎯⎯⎯→

⎯⎯

؉ 5.83

⎯⎯⎯→

⎯⎯

؊ 1.40

⎯⎯⎯→

⎯⎯

b. Write a story that could be represented by the arrow string.
6. Write a problem that you can solve using arrow language. Then

solve the problem.
7. Why is arrow language useful?

Juan says that it is easier to write 15 ؉ 3 ‫ ؍‬18 ؊ 6 ‫ ؍‬12 ؉ 2 ‫؍‬14 than
to make an arrow string. Tell what is wrong with the string that Juan
wrote and show the arrow string he has tried to represent.

Section A: Arrow Language 5

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B

Smart Calculations

Making Change
At most stores today, making
change is easy. The clerk just
enters the amount of the purchase
and the amount received. Then the
computerized register shows the
amount of change due. Before
computerized registers, however,
making change was not quite so
simple. People invented strategies
for making change using mental
calculation. These strategies are
still useful to make sure you get

the right change.

1. a. When you make a purchase, how do you know if you are
given the correct change?
b. Reflect How might you make change without using a calculator
or computerized register?
A customer’s purchase is $3.70. The customer gives the clerk a $20 bill.
2. Explain how to calculate the correct change without using a pencil
and paper or a calculator.
It is useful to have strategies that work in any situation, with or without
a calculator. Rachel suggests that estimating is a good way to begin.
“In the example,” she explains, “it is easy to tell that the change will be
more than $15.” She says that the first step is to give the customer $15.
Rachel explains that once the $15 is given as change to the customer,
you can work as though the customer has paid only the remaining $5.
“Now the difference between $5.00 and $3.70 must be found. The
difference is $1.30—or one dollar, one quarter, and one nickel.”
3. Do you think that Rachel has proposed a good strategy? Why or
why not?
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Smart Calculations B
Many people use a different strategy that gives small coins and bills
first. Remember, the total cost is $3.70, and the customer pays with
a $20 bill.
The clerk first gives the customer a nickel

and says, “$3.75.”
Next, the clerk gives the customer a quarter
and says, “That’s $4.”
Then the clerk gives the customer a dollar
and says, “That’s $5.”
The clerk then gives the customer a $5 bill
and says, “That makes $10.”
Finally, the clerk gives the customer a $10 bill
and says, “That makes $20.”
This method could be called “making
change to twenty dollars.”
This method could also be called “the smallcoins-and-bills-first method.”
4. a. Does this method give the fewest possible coins and bills in
change? Explain.
b. Why do you think it might also be called the small-coins-andbills-first method?
These methods illustrate strategies to make change without a computer
or calculator. They are strategies for mental calculations, and they can
be illustrated with arrow strings.
Another customer’s bill totals $7.17, and the customer pays with a
$10 bill.
5. a. Describe how you would make change using the small-coinsand-bills-first method.
b. Does your solution give the customer the fewest coins and
bills possible?

Section B: Smart Calculations 7

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B Smart Calculations

Arrow language can be used to illustrate the small-coins-and-bills-first
method. This arrow string shows the change for the $3.70 purchase.
؉ $0.05

؉ $0.25

؉ $1.00

؉ $5.00

؉ $10.00

$3.70 ⎯ ⎯⎯→ $3.75 ⎯ ⎯⎯→ $4.00 ⎯ ⎯⎯→ $5.00 ⎯ ⎯⎯→ $10.00 ⎯ ⎯⎯→ $20.00
6. a. What might the clerk say to the customer when giving the
customer the amount in the third arrow?
b. What is the total amount of change?
c. Write a new arrow string with the same beginning and end but
with only one arrow. Explain your reasoning.
Now try some shopping problems. For each problem, write an arrow
string using the small-coins-and-bills-first method. Then write another
arrow string with only one arrow to show the total change.
7. a. You give $10.00 for a $5.85 purchase of some cat food.
b. You give $20.00 for a $7.89 purchase of a desk fan.
c. You give $10.00 for a $6.86 purchase of a bottle of car polish.
d. You give $5.00 for a $1.76 purchase of pencils.
A customer gives a clerk $2.00 for a $1.85 purchase. The clerk is
about to give the customer change, but she realizes she does not
have a nickel. So the clerk asks the customer for a dime.

8. Reflect What does the clerk give as change? Explain your strategy.

Skillful Computations
In problem 7, you wrote two arrow strings for the same problem. One
arrow string had several arrows and the other had only one arrow.
9. Shorten the following arrow strings so each has only one arrow.
؉ 50

a. 375 ⎯⎯→
؊1

b. 158 ⎯⎯→

⎯⎯→

?

⎯⎯→

⎯⎯
⎯⎯

؊ 1,000

؉ 50

?

c. 1,274 ⎯⎯
⎯ ⎯⎯→

؉ 100

?

⎯⎯

is the same as 375 ⎯⎯→

?

is the same as 158 ⎯⎯→

⎯⎯
⎯⎯

؉2

?

?

⎯⎯→

?

?

⎯⎯

?

⎯⎯
?

is the same as 1,274 ⎯⎯→

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⎯⎯

/>
?

⎯⎯


Smart Calculations B
You can use arrow strings to represent your thought processes when
making change using mental calculations. Some arrow strings, such
as the one in problem 9a, are easier to calculate when they have
fewer steps. You can make some arrow strings easier to use by
making them shorter or longer.
10. For each of the arrow strings, make a longer string that is easier
to use to do the calculation. Then use the new arrow string to find
the result.

؉ 99

a. 527 ⎯⎯→

؊ 98

b. 274 ⎯⎯→

?

⎯⎯

?

⎯⎯

Change each of the following calculations into an arrow string with
one arrow. Then make a longer arrow string that is simpler to solve
using mental calculation.
11. a. 1,003 – 999

b. 423 + 104

c. 1,793 – 1,010

12. a. Guess the result of this arrow string and then copy and
complete it in your notebook.
؊ 100

273 ⎯⎯→

?

⎯⎯

؉ 99

⎯⎯→

?

⎯⎯

b. If the 273 in part a is replaced by 500, what is the new result?
c. What if 1,453 is substituted for 273?
d. What if 76 is substituted for 273?
e. What if 112 is substituted for 273?
f. Use one arrow to show the result for any first number.
Numbers can be written using different combinations of sums and
differences. Some of the ways make it easier to perform mental
calculations. To calculate 129 ؉ 521, you can write 521 as 500 ؉ 21
and use an arrow string.
؉ 21

؉ 500

129 ⎯⎯→ 150 ⎯⎯→ 650
⎯⎯
Sarah computed 129 ؉ 521 as follows:
؉ 500

129 ⎯⎯→

⎯⎯

؉ 20

⎯⎯→

⎯⎯

؉1

⎯⎯→

⎯⎯

13. Is Sarah’s method correct?
14. How could you rewrite 267 – 28 to make it easier to calculate
using mental computation?
Section B: Smart Calculations 9

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B Smart Calculations

Sometimes an arrow string can be replaced by a shorter string that is
easier to calculate mentally.

⎯⎯

؉ 64

⎯⎯→

⎯⎯

؉ 36

⎯⎯→

⎯⎯

becomes

⎯⎯

؉ 100

⎯⎯→

⎯⎯

Sometimes an arrow string can be replaced by a longer string that
makes the calculation easier to calculate mentally without changing
the result.
⎯⎯

؊ 99

⎯⎯→

⎯⎯

becomes

⎯⎯

؉1

⎯⎯→

⎯⎯

؊ 100

⎯⎯→

or
⎯⎯

؊ 99

⎯⎯→

⎯⎯

becomes

؊ 100

؉1

⎯⎯→
⎯⎯→
⎯⎯
⎯⎯

The small-coins-and-bills-first method is an easy way to make change.

Complete each of the following arrow strings.
1

؉ 15

1. a. 20 ⎯⎯→

⎯⎯

؉ 0.03

b. 6.77 ⎯⎯⎯→
؉ 0.20

؊8

⎯⎯→

⎯⎯

c. 12.10 ⎯⎯⎯→

؉–
2

⎯⎯

؉ 0.20

⎯⎯⎯→

⎯⎯

⎯ ⎯→

؉ 0.70

⎯⎯

⎯⎯⎯→

⎯⎯

⎯⎯

؉ 13

⎯
⎯⎯→ 20

؉7

⎯⎯→ 20

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2. For each of these arrow strings, either write a new string that will
make the computation easier to calculate and explain why it is
easier, or explain why the string is already as easy to calculate
as possible.
؉ 237

a. 423 ⎯⎯⎯→
؊ 24

b. 544 ⎯⎯⎯→
؉ 54

c. 29 ⎯⎯⎯→
؉ 34

⎯⎯
⎯⎯

⎯⎯

d. 998 ⎯⎯⎯→

؊ 25

⎯⎯⎯→

⎯⎯

⎯⎯

3. Write two examples in which a shorter string is easier to
calculate mentally. Include both the short and long strings for
each example.
4. Write two examples in which a longer string would be easier to
calculate mentally. Show both the short and long strings for each
example.
5. Explain why knowing how to shorten an arrow string can be
useful in making change.

Write an arrow string that shows how to make change for a $4.15
purchase if you handed the clerk a $20.00 bill. Show how you would
alter this string if the clerk had no quarters or dimes to use in making
change.

Section B: Smart Calculations 11

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C

Formulas

Supermarket
Tomatoes cost $1.50 a pound. Carl buys
2 pounds (lb) of tomatoes.
1. a. Find the total price of Carl’s tomatoes.
b. Write an arrow string that shows how
you found the price.

At Veggies-R-Us, you can weigh fruits
and vegetables yourself and find out how
much your purchase costs. You select the
button on the scale that corresponds to the
fruit or vegetable you are weighing.
APPLES

BANANAS

CABBAGE

CARROTS

CELERY

CORN

CUCUMBERS

GRAPES

GREEN BEANS

LETTUCE

LEMONS

ONIONS

ORANGES

PEAS

PEPPERS

POTATOES

TOMATOES

PRICE &
STICKER

The scale’s built-in calculator computes the purchase
price and prints out a small price tag. The price tag lists
the fruit or vegetable, the price per pound, the weight,
and the total price.

weight

⎯⎯→

⎯⎯→

price

The scale, like an arrow string, takes the
weight as an input and gives the price as
an output.

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Formulas C

2. Find the price for each of the following weights of
tomatoes, using this arrow string:
؋ $1.50

weight ⎯⎯⎯⎯⎯
→ price
؋ $1.50

→
a. 4 lb ⎯⎯⎯⎯⎯
؋ $1.50

b. 3 lb ⎯⎯⎯⎯⎯
→
؋ $1.50

c. 0.5 lb ⎯⎯⎯⎯⎯
→

?

⎯⎯
?

⎯⎯
?

⎯⎯

The prices for other fruits and vegetables are calculated in the same
way. Green beans cost $0.90 per pound.
3. a. Write an arrow string to show the calculation for green beans.
b. Calculate the price for 3 lb of green beans.
The Corner Store does not have a calculating scale. The price
of tomatoes at the Corner Store is $1.20 per pound. Siu bought
tomatoes, and her bill was $6.
4. What was the weight of Siu’s tomatoes? How did you find
your answer?

Section C: Formulas 13

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C Formulas

Taxi Fares
In some cabs, the fare for the
ride is shown on a meter. At
the Rainbow Cab Company,
the fare increases during
the ride depending on the
distance traveled. You pay a
base amount no matter how
far you go, as well as a price
for each mile you ride. The
Rainbow Cab Company
charges these rates.

The base price is $2.00.
The price per mile is $1.50.
5. What is the price for each of these rides?
a. stadium to the railroad station: 4 miles
b. suburb to the center of the city: 7 miles
c. convention center to the airport: 20 miles
The meter has a built-in calculator to find the fare. The meter
calculation can be described by an arrow string.
6. Which of these strings shows the correct price? Explain
your answer.
؋ $ 1.50

number of miles ⎯⎯⎯→
؉ number of miles

⎯⎯
⎯⎯ ⎯⎯→
$2.00 ⎯⎯⎯
؋ $2.00

number of miles ⎯ ⎯⎯→

⎯⎯
⎯⎯

؉ $2.00

⎯ ⎯⎯→ total price
؋ $ 1.50

⎯⎯⎯→ total price

⎯⎯

؋ $ 1.50

⎯⎯⎯→ total price

14 Expressions and Formulas

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Formulas C
The Rainbow Cab Company changed its fares. The new prices can be
found using an arrow string.
؋ $ 1.30

number of miles ⎯⎯⎯→

⎯⎯

؉ $3.00

⎯⎯⎯→ total price

7. Is a cab ride now more or less expensive than it was before?
8. Use the new rate to find the fare for each trip.
a. from the stadium to the railroad station: 4 miles
b. from a suburb to the center of the city: 7 miles
c. from a convention center to the airport: 20 miles
9. Compare your answers before the rate change (from problem 5)
to those after the rate change.
After the company changed its rates, George
slept through his alarm and had to take a cab
to work. He was surprised at the cost: $18.60!
10. a. Use the new rate to calculate the distance from
George’s home to work.
b. Write an arrow string to show your calculations.
The arrow string for the price of a taxi ride shows how to find the
price for any number of miles.

؋ $ 1.30

number of miles ⎯⎯⎯→

⎯⎯

؉ $3.00

⎯⎯⎯→ total price

Section C: Formulas 15

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Stacking Cups
Materials:

rim

Each group will need a centimeter ruler
and at least four cups of the same size.
Plastic cups from sporting events or
fast-food restaurants work well.

hold

base

Measure and record the following:
• the total height of a cup
• the height of the rim
• the height of the hold
(Note: The hold is the distance from the bottom
of the cup to the bottom of the rim.)
• Stack two cups. Measure the height of the
stack.
• Without measuring, guess the height of a
stack of four cups.
• Write down how you made your guess.
With a partner, share your guess and the
strategy you used.
Make a stack of four cups and measure it. Was
your guess correct?

16 Expressions and Formulas

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Formulas C
11. Calculate the height of a stack of 17 cups. Describe your
calculation with an arrow string.
12. a. There is a space under a counter where cups will be stored.
The space is 50 centimeters (cm) high. How many cups can
be stacked to fit under the counter? Show your work.
b. Use arrow language to explain how you found your answer.
Sometimes a formula can help you solve a problem. You can write a

formula to find the height of a stack of cups if you know the number
of cups.
13. Complete the following arrow string for a formula using the number
of cups as the input and the height of the stack as the output.
?

?

? ⎯⎯→ height of stack
number of cups ⎯⎯→ ⎯⎯

Suppose another class has cups of different sizes. The students use a
formula for finding the height of a stack of their cups.
؊1

number of cups ⎯⎯→

⎯⎯

؋3

⎯⎯→

⎯⎯

؉ 15

⎯⎯→ height of stack

14. a. How tall is a stack of 10 of these cups?

b. How tall is a stack of 5 of these cups?
c. Sketch one of the cups. Label your drawing with the correct
height.
d. Explain what each of the numbers in the formula represents.
Now consider this arrow string.
؋3

number of cups ⎯⎯→

⎯⎯

؉ 12

⎯⎯→ height

15. Could this arrow string be used for the same cup from problem 14?
Explain.
We can write the arrow string as a formula, like this.
number of cups ؋ 3 ؉ 12 ‫ ؍‬height
16. Could the formula in problem 13 also be used to solve the
problem?
17. These cups will be stored in a space 50 cm high. How many cups
can be placed in a stack? Explain how you found your answer.
Section C: Formulas 17

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C Formulas

Bike Sizes
You have discovered some formulas written as arrow strings. On the
next pages, you will use formulas that other people have developed.

saddle height
frame
height

inseam
height

Bike shops use formulas to find the best saddle and frame height for
each customer. One number used in these formulas is the inseam
of the cyclist. This is the length of the cyclist’s leg, measured in
centimeters along the inside seam of the pants.
The saddle height is calculated with this formula.
inseam (in cm) ؋ 1.08 ‫ ؍‬saddle height (in cm)
18. a. Do you think you can use any numbers at all for inseam
length? Why or why not?
b. Write an arrow string for the formula.
c. Use the arrow string to complete this table.
Inseam (in cm)

50

60

70

80

Saddle Height (in cm)

.....

64.8

.....

.....

d. How much does the saddle height change for every 10-cm
change in the inseam? How much for every 1-cm change?

18 Expressions and Formulas

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Formulas C
To get a quick overview of the relationship between inseam length
and saddle height, you can make a graph of the data in the table. In
this graph, the point labeled A shows an inseam length of 60 cm with
the corresponding saddle height of 64.8 cm. For plotting this point,
64.8 is rounded to 65.
140

Saddle Height (in cm)

120
100
80
A

60
40
20

0

20

40

60

80

100

120

140

Inseam (in cm)

19. a. Go to Student Activity Sheet 1. Label the point for the inseam
of 80 cm with a B. What is the corresponding saddle height in

whole centimeters?
b. Choose three more lengths for the inseam. Calculate the
saddle heights, round to whole centimeters, and plot the
points in the graph on Student Activity Sheet 1.
c. Why is it reasonable to round the values for saddle height to
whole centimeters before you plot the points?
If you complete the calculations accurately, the points in the graph
can be connected by a straight line.
20. a. Go to Student Activity Sheet 1. Connect all points in the graph
with a line.
b. If you extend your line, would it intersect the point (0, 0) in the
bottom left corner? Why or why not?
c. A line goes through an infinite number of points. Does every
point you can locate on the line you drew provide a reasonable
solution to the bike height problem? Explain your reasoning.
Section C: Formulas 19

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