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2nd Pass (B file)
Expressions
and Formulas
Algebra
FPO
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., Boon, P.; Kindt, M.; Cole, B. R.;
and Burrill, G. (2006). Expressions and formulas using algebra arrows.
In Wisconsin Center for Education Research & Freudenthal Institute (Eds.),
Mathematics in context. Chicago: Encyclopædia Britannica.
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
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regarding a license, write Encyclopædia Britannica, Inc., 331 North LaSalle Street,
Chicago, Illinois 60610.
check for
ISBN
{
ISBN 0-03-000000-0
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 Expressions and Formulas Using Algebra Arrows 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
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
Nina Boswinkel
Frans van Galen
Koeno Gravemeijer
Marja van den
Heuvel-Panhuizen
Jan Auke de Jong
Vincent Jonker
Ronald Keijzer
Martin Kindt
Jansie Niehaus
Nanda Querelle
Anton Roodhardt
Leen Streefland
Adri Treffers
Monica Wijers
Astrid de Wild
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 Expressions and Formula Using Algebra Arrows was developed by
Monica Wijers, Peter Boon, and Martin Kindt. It was adapted for use in American schools
by Gail Burrill and Teri Hedges.
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) © PhotoDisc/Getty Images; © Corbis;
© Getty Images
Illustrations
1, 6 Holly Cooper-Olds; 7 Thomas Spanos/© Encyclopædia Britannica,
Inc.; 8 Christine McCabe/© Encyclopædia Britannica, Inc.; 13 (top)
16 (bottom) Christine McCabe/© Encyclopædia Britannica, Inc.;
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
These are the credits
from EF. They will
need to be checked
for this unit.
Contents
Letter to the Student
Section A
Arrow Language
Bus Riddle
Wandering Island
Summary
Check Your Work
Section B
8
14
14
The TOC folios will be corrected on the next pass if
page breaks are OK’d.
Art for TOC will also be
picked up then.
16
18
20
22
28
28
Reverse Operations
Distances
Going Backwards
Beech Trees
Summary
Check Your Work
Section E
6
Formulas
Supermarket
Taxi Fares
Stacking Cups
Bike Sizes
Summary
Check Your Work
Section D
1
3
4
4
Smart Calculations
Making Change
Skillful Computations with
Algebra Arrows
Summary
Check Your Work
Section C
vi
32
35
36
38
38
Order of Operations
Home Repairs
Arithmetic Trees
Home Repairs Again
Flexible Computation
(Without Computer)
Return to the Supermarket
What Comes First?
Summary
Check Your Work
40
41
46
48
50
52
54
55
Additional Practice
57
Answers to Check Your Work
62
Contents v
Dear Student,
Welcome to Expressions and Formulas Using Algebra Arrows.
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
Request from Michael to add something about the applets to the letter.
vi Expressions and Formulas
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 sixth
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 sixth stop?
3. How could you determine the number of passengers on the bus
after the sixth stop?
Section A: Arrow Language 1
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.
Here is a record of people getting on and off the bus at six bus stops.
4. 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.
2 Expressions and Formulas
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
2000
2001
2002
2003
5.5
6.0
4.0
6.5
7.0
6.0
3.5
5.0
7.5
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
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
233
3/12
47
0
3/13
51
1
3/14
53
0
3/15
5
12
3/16
16
2
4 Expressions and Formulas
Total Seats Reserved
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
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?
6 Expressions and Formulas
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
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 with Algebra Arrows
Icon for tech
questions. They
haven't all been
placed. Will do
upon approval.
With the applet Algebra Arrows, you can make arrow strings and let
the computer do the computations.
This is similar to
the logo used on
the MiC Web site.
8 Expressions and Formulas
Smart Calculations B
Arrow strings made in Algebra Arrows look like the following:
In/Output
Operations
3
3
6
10
3
16
/3
1 /…
3
18
冑苳苳苳
…
7
9
23
…2
Back
Forth
5
15
Table
Graph
18
18
18
Clear
expression
value
9. Use Algebra Arrows to make the following arrow strings and
write down the result. On page XX, there is an explanation of
the applet.
83
a. 278 ⎯⎯→
b.
17
34 ⎯⎯→
15
?
⎯⎯→
?
⎯⎯→
⎯⎯
⎯⎯
69
?
⎯⎯
?
⎯⎯
Use the applet to make the following calculations:
10. a. 12.75 23.32
b. 498 29 67
Section B: Smart Calculations 9
B Smart Calculations
How To Use Algebra Arrows
1 Drag the boxes to
the workspace.
In/Output
Operations
3
3
3
3
/3
1/
In/Output
2 Click on the input
box and fill in the
starting number.
Operations
3
10
3
3
3
/3
1/
3 Click on the operation box
and fill in the number for
the operation.
In/Output
Operations
3
10
3
3
/3
1/
In/Output
4 Connect the boxes with
arrows by dragging them
to the next box.
Operations
3
3
3
/3
1/
10 Expressions and Formulas
10
6
Smart Calculations B
In problem 7, you wrote two arrow strings for the same problem. One
had many arrows, and the other had only one arrow.
11. Shorten the following arrow strings so that each string has only
one arrow. Use the applet to check your answer.
?
a.
375
50
50
?
?
375
158
1
is the same as
?
b.
?
100
is the same as
?
158
?
c.
1000
1274
?
2
is the same as
?
1274
Suggested nonessential art here
to help fill page 13.
When you use the applet, it does not matter if the computations in an
arrow string are difficult or not. The computer does the hard work for
you! When you have to do it without the computer, it can make a
difference.
Some arrow strings, such as the one in problem 11a, are easier to
calculate when they have fewer steps. Others, such as the one in
problem 11b, are easier to calculate when they are longer. You can
make some arrow strings easier to use by making them shorter or
longer.
12. For each arrow string below, make a longer string that is easier.
Then use the new arrow string to find the result. You can check
this result with the applet by making both strings.
99
a. 527 ⎯⎯→
98
b. 274 ⎯⎯→
?
⎯⎯
?
⎯⎯
13. Change each calculation into an arrow string with one arrow.
Then make a longer arrow string that is simpler to use to do the
mental calculations. Use the longer arrow strings to calculate
the answers.
a. 1,003 999
b. 423 104
c. 1,793 1,010
Section B: Smart Calculations 11
B Smart Calculations
14. a. Make the arrow string below with the applet, and write down
the result.
?
273
100
99
b. If the first number, 273, in part a is changed to 500, what is the
new result?
c. What if 273 is changed to 1,453?
d. What if 273 is changed to 76?
e. What if 273 is changed to 112?
f. Use a short string with a one-arrow calculation to show the
result for any first number.
Monica:
Add non-essential art
of arrow applet on a
computer display.
If done, show an
updated LCD computer
monitor.
15. Make the arrow string below in the applet, and find out which
numbers you could use in the operation boxes.
Make two versions: one that is difficult to calculate using mental
computation and one that is easy to calculate without the use of
a computer or calculator.
36
…
…
…
…
177
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 ⎯⎯→
12 Expressions and Formulas
⎯⎯
20
⎯⎯→
⎯⎯
1
⎯⎯→
⎯⎯
Smart Calculations B
16. a. Is Sarah’s method correct?
b. What other method using an arrow string could Sarah have
used to compute 129 521 mentally?
17. How could you rewrite 267 – 28 to make it easier to calculate
using mental computation?
Section B: Smart Calculations 13
B Smart Calculations
The small-coins-and-bills-first method is an easy way to make change.
The applet Algebra Arrows, which is a small computer program, can
be used to make arrow strings. The applet does the computations
for you.
If a computer is not available, 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 compute mentally without changing
the result.
⎯⎯
99
⎯⎯→
⎯⎯
becomes
⎯⎯
1
⎯⎯→
⎯⎯
100
⎯⎯→
or
⎯⎯
99
⎯⎯→
⎯⎯
becomes
100
1
⎯⎯→
⎯⎯→
⎯⎯
⎯⎯
1 a. Complete each arrow string.
1
i.
15
20 ⎯⎯→
⎯⎯
0.03
ii. 6.77 ⎯⎯⎯→
iii. This was
12.20 in the ms
?
⎯⎯→
iii. 12.28 ⎯⎯
8
⎯⎯→
⎯⎯
⎯⎯
–
2
⎯⎯
?
⎯⎯
⎯⎯→
?
⎯⎯
⎯⎯→
⎯ ⎯→
⎯⎯
⎯⎯
⎯⎯
?
⎯⎯
⎯⎯→ 20
?
⎯⎯
⎯⎯→ 20
b. Use the applet Algebra Arrows to check your answers for part
a and to make two more arrow strings, one different arrow
string for the calculation as in string ii and one different arrow
string for the same calculation as in string iii.
14 Expressions and Formulas
2. For each arrow string, either write a new string that will make the
computation easier to calculate mentally, 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. Give two examples in which a shorter string is easier to calculate
mentally. Include both the short and long strings for each
example.
4. Give 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.
You can use the applet Algebra Arrows to solve this problem.
Write an arrow string that shows how to make change for a $4.15
purchase if you handed the cashier a $20.00 bill. Show how you
would change this string if the cashier had no quarters or dimes
to use in making change.
Explain why making an arrow string shorter or longer does not make
much sense when using the applet.
Section B: Smart Calculations 15
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. Use the applet Algebra Arrows to make
an arrow string that can compute the
price of the tomatoes.
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 sticker. The price
sticker lists the fruit or vegetable, the price per pound,
the weight, and the total price.
weight
⎯⎯→
16 Expressions and Formulas
⎯⎯→
price
The scale, like an arrow string, takes the
weight as an input and gives the price
as an output.
Formulas C
Once you have built an arrow string with the applet Algebra Arrows,
you can use it as a “machine” that gives you the price when you fill in
the weight. You do not have to make the computations yourself.
price
weight
1.5
How to Make a Label
1
Click on the right mouse
button (or CRTL + click)
to open the pop-up menu
and choose Show label.
You can make labels for the input and output boxes,
to make the meaning of the numbers visible and show
what the “machine”can do.
2. Find the price for each of the following weights of
tomatoes using the arrow string in the applet.
a. 4 lb
b. 0.5 lb
36
c. 2.5 lb
Show label
Hide label
Show table
Hide table
Show Arrow string
Hide Arrow string
The prices of other fruits and vegetables are calculated
in the same way. Green beans cost $0.90 per pound.
3. a. Make a “machine” for computing the price of
green beans.
2
Click on
the label.
weight
?
price
1.5
3
Fill in
the label.
4
Press
ENTER.
36
price
b. Use it to calculate the price for the following
weights of beans:
3.5 lb, 4.5 lb, and 5.5 lb.
Section C: Formulas 17
C Formulas
The Corner Store does not have a calculating scale. The price of
tomatoes at The Corner Store is $1.20 per pound. Siu bought some
tomatoes, and her bill was $6.
4. What was the weight of Siu’s tomatoes? How did you find your
answer? Check your answer with the applet.
Taxi Fares
In some taxis, 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 fare for each ride?
a. From the stadium to the railroad station: 4 miles
b. From the suburbs to the downtown area: 7 miles
c. From the 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 “machines” will give the correct fare? Explain
your answer.
FPO
18 Expressions and Formulas
Formulas C
The Rainbow Cab Company changed its rates. The new ones are
shown:
The base price is $3.00.
The price per mile is $1.30.
7. Is a cab ride now more or less expensive than it was before?
To compare the new price and the old price, you can build the
machine below:
FPO
8. a. Build this machine with the applet and complete it.
b. Use it to compare the prices before and after the rate change
by filling in several distances.
c. Did you answer problem 7 correctly? If not, try it again.
Siluh is a taxi driver. Now that the new prices are in effect, he wants
to post a rate chart in his taxi. The rate chart will show the customers
the new fares for typical distances.
9. Use the applet to make such a rate chart for Siluh.
After the company changed its rates, George slept
through his alarm and had to take a taxi to work.
He was surprised when it cost $18.60!
10. a. Use the machine that you made in problem 8 to calculate the
distance from George’s home to work.
b. It’s also possible to find the answer to part a by making a
calculation instead of using the machine. Write an arrow
string for this calculation.
Section C: Formulas 19
C Formulas
Stacking Cups
Materials:
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.
rim
hold
base
Measure and record the following:
• the total height of a cup
• the height of the rim
• the height of the hold (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. Tell a
classmate your guess and how you arrived at it.
• Make a stack of four cups and measure it. Was
your guess correct?
11. Calculate the height of a stack of 17 cups. Describe your
calculation with an arrow string.
The cups will be stored in a space under a counter. The space is
50 centimeters high.
12. a. 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
to part a.
20 Expressions and Formulas
Formulas C
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. You can also write an arrow string for a formula.
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 a different size. The students use
this formula for finding the height of a stack of their cups:
1
number of cups ⎯⎯→
⎯⎯
3
⎯⎯→
⎯⎯
15
⎯⎯→ height of stack
You can use the applet to make this arrow string.
14. a. How tall is a stack of 10 of these cups?
b. How tall is a stack of five of these cups?
c. Sketch one of the cups. Label your drawing with the correct
height.
d. Explain what each number in the formula represents.
Now consider the following arrow string.
3
12
number of cups ⎯⎯→ ⎯⎯ ⎯⎯→ height
15. Could this arrow string be used for the same cup from problem 14?
Show your work.
FPO
To compare both arrow strings, you can also build this machine and
try several numbers.
16. Reflect Compare using this machine for solving problem 15 to
the way you solved it. What is the same? What is different?
Section C: Formulas 21
C Formulas
3
We can write the arrow string number of cups ⎯⎯→
⎯⎯
12
⎯⎯→ height
as a formula, like this: number of cups 3 12 height
17. Will the formula number of cups 12 3 height also work for
these cups? Explain your reasoning.
These cups will be stored in a space 50 cm high.
18. How many of these cups can be put in a stack? Explain how you
found your answer.
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 heights for
each customer. One number used in these formulas is the cyclist’s
inseam. 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)
19. a. Do you think you can use just any numbers at all for inseam
length? Why or why not?
b. Write an arrow string for this formula, and with Algebra Arrows,
make a machine that can calculate the saddle height for you.
FPO
Does this go with b or c?
22 Expressions and Formulas
Formulas C
c. Use the arrow machine to complete the 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?
140
Saddle Height (in cm)
120
100
80
A
60
40
20
0
20
40
60
80
100
120
140
Inseam (in cm)
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.
20. 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 with your machine, 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?
Section C: Formulas 23
C Formulas
If you plotted the points accurately, the points in the graph can be
connected by a straight line.
21. 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.
22. Write a question you can solve using this graph. Record the answer
to your own question. Exchange questions with a classmate.
Then answer the question, and discuss the questions and the
answers with your classmate.
Look at the formula for the frame height of a bicycle.
inseam (in cm) 0.66 2 frame height (in cm)
23. a. Write an arrow string for this formula, and make a machine
with the applet.
b. Use the machine to complete the table (round the frame height
to a whole number), and draw the graph for this formula on
Student Activity Sheet 2.
Since the graph that
was shown here is a
blank graph, can it be
eliminated to make the
section fit?
(Graph is placed on the
following page for
reference.)
Inseam (in cm)
50
60
70
80
Frame Height
Calculated (in cm)
.....
.....
.....
.....
.....
.....
Frame Height
Rounded (in cm)
c. If you connect all points in this graph and extend your line,
does the line you drew intersect the origin (0, 0)? Why or
why not?
d. Use the graph to find the frame height for Ben, whose inseam
length is 75 cm.
24 Expressions and Formulas
140
Frame Height (in cm)
120
100
80
60
40
20
0
20
40
60
80
Inseam (in cm)
100
120
140
Formulas C
Margit did not use the applet to do problem 23, she used the formula
instead. She just used it to find the first two frame heights in the table.
She did not round the heights. Then she used the first two values to
calculate the third value.
24. a. Explain how Margit might have used the first two values to
find the third value.
b. Check to see if her method also works to find the next frame
heights.
c. How could Margit find the frame height for an inseam of 65 cm?
Formulas are often written with the result first, for example:
saddle height (in cm) inseam (in cm) 1.08
frame height (in cm) inseam (in cm) 0.66 2
81 cm
54 cm
25. Study this bike.
a. What is the frame height?
b. What is the saddle height?
c. Do both of these numbers correspond to the same inseam
length? How did you find your answer?
FPO
Section C: Formulas 25
C Formulas
A formula shows a procedure that can be applied over and over
again for different numbers in the same situation.
With the applet Algebra Arrows, you can make a machine for a
formula. This kind of machine can carry out the procedure for
different numbers you use for input.
Bike shops use formulas to fit bicycles to their riders.
inseam (in cm) 0.66 2 frame height (in cm)
The formula can also be written with the result first.
frame height (in cm) inseam (in cm) 0.66 2
Many formulas can be described with arrow strings; for example:
0.66
2
inseam ⎯⎯→ ⎯⎯ ⎯⎯→ frame height
An arrow string made in the applet Algebra Arrows is a machine
that will do the calculations.
FPO
You can use the machine to complete a table for the formula. From
the data in the table, you can draw a graph. Some problems are
easier to solve with a graph, some are easier to solve using the
formula some with an arrow string or a machine made with Algebra
Arrows, and some are easier to solve using a table. You have studied
many strategies to solve problems.
26 Expressions and Formulas
1. a. Write the following formula about taxi fares as an arrow string:
total price = number of miles $1.40 $1.90
b. Why is it useful to write a formula as an arrow string?
c. Use the applet Algebra Arrows to make a list of prices for
several distances.
The manager at The Corner Store wants customers to be able to
estimate the total cost of their purchases. She posts a table with
prices next to a regular scale.
2. a. Help the manager by copying and completing this table.
Weight
Tomatoes $1.20/lb
Green Beans $0.80/lb
Grapes $1.90/lb
0.5 lb
1.0 lb
2.0 lb
3.0 lb
b. The manager would like to have one machine to calculate the
exact prices of tomatoes, green beans, and grapes for all
weights. Build such a machine in the applet Algebra Arrows.
FPO
Section C: Formulas 27
C Formulas
This picture shows a stack of chairs.
Notice that the height of one chair is
80 centimeters, and a stack of two
chairs is 87 centimeters high.
87
cm
80
cm
Damian suggests that the following arrow string can be used to find
the height of a stack of these chairs:
1
7
80
number of chairs ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ height
3. a. Explain the meaning of each number in the arrow string.
b. Alba thinks a different arrow string could also solve the
problem.
___
___
⎯⎯→ height
number of chairs ⎯⎯→
⎯⎯
What numbers should Alba use in her arrow string? Explain
your answer.
c. Use the applet Algebra Arrows to see if your string for part b
is the same as the one Damian suggested. Show how you
did this.
28 Expressions and Formulas
This graph represents the height of stacks of chairs. The number of
chairs in the stack is on the horizontal axis, and the height of the
stack is on the vertical axis.
Height (in cm)
250
200
A
150
100
50
0
5
10
15
20
25
30
Number of Chairs
4. a. What does the point labeled A represent?
b. Does each point on the line that is drawn have a meaning?
Explain your reasoning.
c. Explain why the graph will not intersect the point (0, 0).
d. Use the graph to determine the number of chairs that can be
put in a stack that will fit in a storage space that is 116 cm high.
e. Check your answer for d using an arrow string or a machine in
the applet Algebra Arrows.
State your preferences for using a graph or an arrow string to display
the saddle height for a bicycle. Explain why you think this is the better
way to describe the data.
Section C: Formulas 29
D
Reverse Operations
Distances
Marty is going to visit Europe. He wants to prepare himself to use the
different currencies and units of measure. He knows distances in
Europe are expressed only in kilometers and never in miles.
He looks on the Internet for a way to convert between miles and
kilometers. The computer uses an estimate for the relationship
between miles and kilometers.
1. Reflect Think of a problem that Marty might do using the
converter while he is traveling in Europe.
Enter miles or kilometers and click the “Calculate” button.
Miles:
Kilometers:
Conversion
1
1.609
_____
Marty wants to convert miles to kilometers to get a better understanding
of distances in kilometers. He decides to build his own mile–kilometer
converter with the applet Algebra Arrows by making a machine.
FPO
2. a. Write down an arrow string that Marty can use to convert
miles to kilometers, and build the machine with the applet.
b. Marty lives 30 miles from his office. About how many
kilometers is that?
c. Marty’s parents live about 200 miles away. About how many
kilometers is that?
d. About how many kilometers is a distance of 10,000 miles?
30 Expressions and Formulas
Reverse Operations D
3. a. Copy this table, and complete it using the machine you have
built in the applet.
Miles
10
20
30
40
50
60
70
80
90
100
Kilometers
b. Use the table to estimate how many kilometers there are in
35 miles. Explain your reasoning.
In Europe, Marty travels 70 km from the airport to his hotel.
4. a. Use the table to estimate the distance in miles. Show
your work.
b. It is not easy to get a precise answer to part a using
the table. Check your answer with the machine. If it’s
not very accurate, try to improve it.
While traveling in Europe, Marty wants to estimate the
distances in miles. A mile–kilometer converter on the Internet
can be used for this by entering the kilometers instead of the
miles. In the machine you made with the applet, you can only
enter miles.
5. Try to make a machine in the applet Algebra Arrows that
can be used for calculating miles by entering kilometers,
and check to see if it works correctly. Write down how
you made and checked your machine.
FPO
Marty uses the fact that 80 kilometers is about 50 miles and
thinks of the following rule to estimate distances in miles:
80
number of kilometers ⎯ ⎯→
⎯⎯
50
⎯
⎯⎯→ number of miles
6. Will this rule give reasonable estimates? Explain.
Section D: Reverse Operations 31
D Reverse Operations
Marty thinks the rule is a little difficult to use with large numbers. He
wonders if he can change the numbers and use this rule instead.
8
number of kilometers ⎯⎯→
⎯⎯
5
⎯⎯→ number of miles
7. Will this new rule result in reasonable estimates? Explain.
8. a. Combine the machines for problems 5, 6, and 7, and check
to see if they all give the same answers. Write down your
conclusions, and explain why the answers are the same
or different.
FPO
b. Can you make a fourth machine that gives the same answers?
Write down the rule for this machine.
Marty uses the machine to convert the distances he will travel from
kilometers to miles.
9. Copy the table with Marty’s travel plans. Convert the distances
into miles, and record them in your table.
Amsterdam–Paris
514 kilometers
.... miles
Paris–Barcelona
839 kilometers
.... miles
Barcelona–Rome
879 kilometers
.... miles
1185 kilometers
.... miles
576 kilometers
.... miles
Rome–Berlin
Berlin–Amsterdam
32 Expressions and Formulas
Reverse Operations D
Marty’s friend Pascale from Paris will be visiting Marty in the United
States next summer. She wants to be able to convert miles into
kilometers when she is traveling in the United States.
10. a. How can Marty’s last rule be changed to convert miles into
kilometers? Write the new rule as an arrow string, and make
it into a machine using the applet Algebra Arrows.
b. Use the applet to check whether the machine you built in
part a gives the same results as the machine you built in
problem 2a. Write down how you checked this, and give
your conclusions.
Pascale wants to tour the United States. She wants to visit some
interesting places, such as national parks, theme parks, and
major cities.
11. Reflect Make a list of five interesting places Pascale might like to
visit. Find the distances in miles between them (use the Internet
or an atlas), and convert them to kilometers using one of the
machines you built in the applet Algebra Arrows.
Going Backwards
Pat and Kris are playing a game. One player writes down an arrow
string and the output (answer) but not the input (starting number).
The other player has to determine the input.
Here are Pat’s arrow string and output.
+4
x 10
–2
÷2
_?_ ____> __ ____> __ ____> __ ____> 29
12. a. What should Kris give as the input? Explain how you found
this number.
b. One student found an answer for Kris by using a reverse
arrow string. What number should go above each one of
the reversed arrows?
?
⎯⎯
←⎯⎯
⎯⎯
←⎯⎯
⎯⎯
←⎯⎯
←⎯⎯ 29
⎯⎯
Section D: Reverse Operations 33
D Reverse Operations
When it was her turn, Kris wrote this.
?
⎯⎯
3
⎯⎯→
⎯⎯
6
⎯⎯→
⎯⎯
5
⎯⎯→
⎯⎯
2
⎯⎯→ 6
13. a. What will Pat give as the input? Explain how you found this
number.
b. Write the reverse arrow string that can be used to find the input.
To make a reverse arrow string with Algebra Arrows, it is possible to
choose the opposite direction for the arrows.
FPO
14. a. Use the applet to make the arrow string in the picture above,
and complete and write down the reverse arrow string.
b. Use the reverse arrow string to find the unknown input. Use
the first arrow string to check it.
Beech Trees
The park near Jessica’s house is full of beech trees. Some botanists
have observed that a beech tree grows pretty evenly when it is
between 20 and 80 years old. They have developed two formulas
that describe the growth of beech trees if the age is known. They are
written as arrow strings.
0.4
age ⎯⎯→
⎯⎯
2.5
⎯⎯→ thickness
0.4
age ⎯⎯→
⎯⎯
1
⎯⎯→ height
In these arrow strings, age is in years, and height is in meters;
thickness (diameter) is in centimeters and is measured at 1 meter (m)
above the ground.
To answer the next questions, you can use the applet Algebra Arrows
to make machines fitting the arrow strings and if necessary, the
reverse arrow strings.
34 Expressions and Formulas
Reverse Operations D
15. Find the heights and thicknesses of trees that are 20, 30, and 40
years old.
16. Jessica wants to know the age of a tree. How can she find it?
Jessica estimates the height of a beech tree as about 20 meters.
17. Use this estimate of the height to find the age of the tree.
Jessica uses some straight sticks to
help her measure the thickness of
another tree. She finds that that the
tree is 25.5 centimeters thick.
18. About how old is the tree?
Show your work.
Jessica realizes that she can make
a new formula. Her new formula
gives the height of a tree if the
thickness is known.
19. Write Jessica’s formula as an
arrow string. Make the machine
with Algebra Arrows, and then
use it to fill in the table below.
FPO
Thickness (in cm)
Height (in cm)
30
34
45
52
20. a. Find a (beech) tree in your area and measure its thickness.
b. Use the formula, or the machine, to estimate its height.
c. How well do you think the formula predicted the height?
Give reasons for your thinking.
Section D: Reverse Operations 35
D Reverse Operations
Every arrow has a reverse arrow. A reverse arrow represents the
opposite operation.
4
4
For example, the reverse of ⎯⎯→ is ←⎯⎯.
Reverse arrows can be used to make reverse arrow strings.
For example,
⎯⎯
⎯⎯
4
←⎯⎯
3
⎯⎯
⎯⎯
⎯⎯→
4
⎯⎯→
3
←⎯⎯
⎯⎯
4
⎯⎯
⎯⎯
⎯⎯→
3
⎯⎯→
⎯⎯
reverses to
, which is the same as
.
⎯⎯
In the applet Algebra Arrows, you can also reverse the arrows.
From Amsterdam in the Netherlands to Chicago, Illinois, the distance
is 4,090 miles. Marty wants to convert this distance to kilometers. The
Internet converter, shown in the beginning of this section, uses 1.609
as the number of kilometers per mile. Marty wonders if he could use
the rounded number 1.6 too.
1. a. Use the applet Algebra Arrows to find out how many kilometers
Marty will find for the distance if he uses the Internet conversion
number.
b. How many kilometers will Marty find if he applies the rounded
number 1.6?
c. What is the difference between the answers for parts a and b?
Why is the difference not really important?
36 Expressions and Formulas
Carmen and Andy are at the store buying ham and cheese for
sandwiches. Carmen sees some Swiss cheese that costs $4.40 per
pound. She decides to buy 0.75 lb, but she wants to calculate the
cost before she orders it.
2. Write an arrow string to show the cost.
Carmen wrote this arrow string.
4
3
$4.40 ⎯⎯→ $1.10 ⎯⎯→ $3.30
3. Is Carmen’s arrow string correct? Why or why not?
4. Write the reverse arrow string for each of these strings.
2
a. input ⎯⎯→
2
b. input ⎯⎯→
⎯⎯
⎯⎯
3
⎯⎯→
5
⎯⎯→
⎯⎯
⎯⎯
4
⎯⎯→ output
7
⎯⎯→ output
5. Try sample numbers to test your reverse arrow strings. You might
want to use the applet Algebra Arrows. Show your work.
Explain when it may be important to have exact calculations and
when a reasonable estimate is acceptable. You can use examples in
your explanation.
Section D: Reverse Operations 37
E
Order of Operations
Home Repairs
Jim is a contractor specializing in small household repairs that require
less than a day to complete. For most jobs, he uses a team of three
people. For each one of the three people, Jim charges the customer
$25 in travel expenses and $37 per hour. Jim usually uses a calculator
to calculate the bills. He uses a standard form for each bill.
1. Use the forms on Student Activity Sheet 3 to show
the charge for each plumbing repair job.
a. Replacing pipes for Mr. Ashton: 3 hours
b. Cleaning out the pipes at Rodriguez and Partners:
212– hours
c. Replacing faucets at the Vander house: 34– hour
38 Expressions and Formulas
Order of Operations E
People often call Jim to ask for a price estimate for a particular job.
Because Jim is experienced, he can estimate how long a job will take.
He then uses the table to estimate the cost of the job.
Labor Cost
per Worker
Travel Cost
per Worker
Cost
per Worker
Total for
Three Workers
(in dollars)
(in dollars)
(in dollars)
(in dollars)
1
37
25
62
186
2
74
25
99
297
3
111
25
136
408
4
148
25
173
519
Hours
5
6
7
2. a. What do the entries in the first row of the table represent?
b. Copy the table, and add the next row for five hours to
the table.
3. a. Reflect Explain the regularity in the column for the labor
cost per worker.
b. Study the table. Make a list of all of the regularities you can
find. Explain the regularities.
4. a. Draw an arrow string that Jim could use to make more rows
for the table.
b. Use your arrow string to make two more rows (for 6 and
7 hours) on the table.
Section E: Order of Operations 39
E
Order of Operations
Arithmetic Trees
While working on the home repair cost problems, Enrique writes this
arrow string to find the cost of having three workers for two hours
of repairs.
37
25
3
2 ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯
Karlene is working with Enrique, and she writes this expression.
2 37 25 3
Karlene finds an answer of 149. Enrique is very surprised.
5. a. How does Karlene find 149 as her answer?
b. Why is Enrique surprised?
Karlene and Enrique decide that the number sentence 2 37 25 3
is not necessarily the same as the arrow string:
37
2 ⎯⎯→
⎯⎯
25
⎯⎯→
⎯⎯
3
⎯⎯→
⎯⎯
There is more than one way to interpret the number sentence.
The calculations can be completed in different orders.
6. Solve the problems below. Compare your answers with your
classmates’ answers.
a. 1 11 11
b. 10 10 1
10
c. 10 ⎯⎯→
⎯⎯
2
⎯⎯→
⎯⎯
d. How can you be sure that everyone will get the same answer?
Sometimes the context of a problem helps you understand how to
calculate it. For example, in the home repair problem, Karlene and
Enrique know that the 3 represents the number of workers. So it
makes sense to first calculate the subtotal of 2 37 25 and then
multiply the result by 3. Sometimes people are not careful how they
write the calculations for a problem.
2 37 74 25 99 3 297
7. Why is this a not a good way to write the calculations?
40 Expressions and Formulas
Order of Operations E
So that everyone gets the same answer to a string of calculations with
different operations, mathematicians have agreed that multiplication
and division should be completed before addition and subtraction in
an expression.
8. Use the mathematicians’ rule to find the value for each expression:
a. 32 5 20
b. 18 3 2 5
c. 47 11 6 8
Calculators and computers nearly always follow the mathematicians’
rule. Some old or very simple calculators, however, do not use
the rule.
9. a. Use the mathematicians’ rule to find 5 5 6 6 and
6 6 5 5.
b. Does your calculator use the mathematicians’ rule? How did
you decide?
c. Reflect Why do you think calculators have built-in rules?
To make sure that everyone agrees on the value of an expression, it is
important to have a way to write expressions so that it is clear which
calculation to do first, which next, and so on.
6
4
A
B
C
D
This is a very simple map, not drawn to scale. Suppose the distance
from A to D is 15 miles. In the drawing, you can see that the distance
from A to B is 6 miles, and the distance from B to C is 4 miles.
10. What is the distance from C to D? Write down your calculations.
Section E: Order of Operations 41
E
Order of Operations
Telly found the distance from C to D by adding 6 and 4. Then she
subtracted the result from 15. She could have used an arithmetic tree
to record this calculation.
•
To make an arithmetic tree, begin by writing down all of the
numbers.
In/Output
15
6
4
Operations
……
……
……
…/…
…2
冑苳苳苳
…
…
•
Then pick two numbers. Telly picked 6 and 4. (Sometimes there is
no choice in picking the numbers, and sometimes there is. This
depends on the problem and the calculations to be performed.)
In/Output
15
6
4
Operations
……
……
……
…/…
…2
冑苳苳苳
…
…
…
•
Telly added the numbers and found the sum of 10.
In/Output
15
6
4
Operations
……
……
……
…/…
……
10
…
2
…
…
42 Expressions and Formulas
冑苳苳苳
…
Order of Operations E
•
Telly selected the 15 and the new 10.
15
In/Output
6
4
Operations
……
……
……
…/…
……
10
…
冑苳苳苳
…
2
…
…
Graph
Clear
•
She subtracted to find a difference of 5.
15
In/Output
6
4
Operations
……
……
……
…/…
……
10
…
冑苳苳苳
…
2
…
……
…
5
Graph
Clear
11. Complete Student Activity Sheet 4.
0.8
10
0.6
16
1.2
4
5
7
3
0.6
In/Output
……
Operations
……
……
……
…/…
…2
冑苳苳苳
…
……
……
……
……
……
……
…
Section E: Order of Operations 43
E
Order of Operations
You can make arithmetic trees using the applet Algebra Trees. This
applet works in almost the same way as the applet Algebra Arrows,
which you used before. When you build a tree for a calculation and fill
in the input boxes, the computer does the computation for you.
In what picture? Do you
mean the image below?
In the picture, you see a screenshot of the applet Algebra Trees with
the tree for the calculation of problem 10.
FPO
12. a. Start the applet Algebra Trees, and build the tree shown in the
screenshot.
b. In the bottom-left corner, you can choose expression or value.
Try these and describe what they do.
c. Change the tree in such a way that it becomes the tree for the
calculation: 15 6 4. By activating expression, you can check
to see if the tree is correct.
In/Output
25
2
37
3
Operations
……
……
……
…/…
…2
冑苳苳苳
…
……
…
…
……
Graph
Clear
Should this be highlighted
to point it out?
expression
value
44 Expressions and Formulas
……
……
Order of Operations E
13. a. Make a tree in the applet Algebra Trees that shows the proper
calculation of 1 2 3 4. Remember you should multiply
first, and then add the products.
FPO
Until this art is redrawn,
ignore the arrow string
beneath the tree.
This is how the manuscript
appears.
FYI
Because this arrow string
only appears beneath the
other piece of art, I can’t
really read the numbers.
This may not be correct.
b. Make the tree that shows the same calculation as this
arrow string.
2
1 ⎯⎯→
⎯⎯
3
⎯⎯→
⎯⎯
4
⎯⎯→
⎯⎯
c. With the same four input numbers and the same three
operations, you can make other trees. Make at least two
other trees with different results from the trees you made
in parts a and b. Write down the corresponding expressions
and their results.
Home Repairs Again
Alex feels that the calculation for home repair bills that Jim the
contractor makes (see page xx) should begin with the travel costs.
Travel costs are part of the base rate; the customer always has to
pay them. Alex wants to write the calculation as an arrow string.
Flo thinks that it is impossible to begin an arrow string for the
home repair bills with the travel costs.
14. Is it impossible, as Flo thinks, to begin an arrow string for the
home repair bills with the travel costs? Explain your reasoning.
Section E: Order of Operations 45
E
Order of Operations
Alex decides to make an arithmetic tree with the applet Algebra Trees
for calculating the bill for a two-hour home repair job expressing the
travel costs first.
15. Does this arithmetic tree give
the costs (as shown in the table
for problem 2 on page 41)?
Explain why or why not.
FPO
It is possible to build an arithmetic tree using
words instead of numbers. The picture shows
how this can be done on paper.
hours
wage per hour
labor costs
In the applet Algebra Trees, you can do this by using labels (as you
did in the applet Algebra Arrows). In the table, you can see how such
a tree is built step by step.
FPO
46 Expressions and Formulas
Order of Operations E
By making some of the “branches” longer, the tree looks like the
one from problem 15, with all of the input boxes on the top.
FPO
Once you have built this tree, you have created a machine that can
do the computations for all kinds of bills. You only have to change
the input numbers.
16. a. Make this tree for the home repair bills in the applet
Algebra Trees.
b. Use the tree to investigate what would happen to the total
costs of a job when the travel costs double. Would the total
costs double too, or does something else happen? Explain
how you found out.
c. Think of something else you can investigate using the tree.
Investigate it, and write a few lines about your investigation
and the results.
Section E: Order of Operations 47
E
Order of Operations
Flexible Computation (Without Computer)
Arithmetic trees are another strategy to make it easier to mentally
calculate some addition and subtraction problems.
18
23
7
18
____
7
23
____
____
____
17. a. Compare the two trees.
b. Design a tree that makes adding the three numbers easier
to do.
Addition problems with more numbers have many possible
arithmetic trees. Here are two trees for 12 14 43 32 .
18. a. Copy the trees, and find the sum.
1
2
1
4
3
4
3
2
____
____
1
2
1
4
3
4
3
2
____
____
____
____
b. Design two other arithmetic trees for the same problem,
and find the answers.
c. Which arithmetic tree makes 12 14 43 32 easiest to
calculate? Why?
19. Design an arithmetic tree that makes each problem here easy
to calculate.
a. 17 3 22 8
b. 4.5 8.9 5.5 1.1
4
1
1
3
2 10
4
c. 10
20. How are different arithmetic trees for the same problem the same?
How could they differ?
48 Expressions and Formulas
Order of Operations E
You may have noticed that if a problem has only addition, the answer
is the same no matter how you draw the arithmetic tree. You might
wonder if this is true for subtraction.
21. Do the following trees give the same result? What can you
conclude about different arithmetic trees for subtraction?
18
7
4
18
7
____
4
____
____
____
Return to the Supermarket
The automatic calculating scale at Veggies-R-Us is out of order.
Ms. Prince buys 0.5 lb of grapes and 2 lb of tomatoes.
22. a. What is the total cost for the grapes and tomatoes?
b. Can you write an arrow string to show how to calculate
Ms. Prince’s bill? Why or why not?
c. Can Ms. Prince’s bill be calculated with an arithmetic tree?
If so, make the tree. If not, explain why.
Section E: Order of Operations 49
E
Order of Operations
Dr. Keppler buys 2 lb of tomatoes,
0.5 lb of grapes, and 12 lb of green
beans.
23. Make an arithmetic tree for the
total bill for the tomatoes, grapes,
and green beans.
The store manager provides calculators for the cashiers. The
calculators use the rule that multiplication is calculated before
addition. Then the store manager wrote these directions.
amount of tomatoes x 1.50 + amount of grapes x 1.70 + amount of green beans x 0.90 =
(in pounds)
(in pounds)
(in pounds)
FPO
24. a. If the cashiers punch in a calculation using these directions,
will they find the correct total for the bill?
b. Make a tree with the applet Algebra Trees that calculates the
total if you enter the weights.
50 Expressions and Formulas
Order of Operations E
What Comes First?
Arithmetic trees are useful because they resolve any question about
the order of the calculation. The problem is that they take up a lot of
room on your paper.
15
6
4
____
____
25. a. Copy the first tree.
b. Since the 6 4 is simplified first, circle it on your copy.
15
6
4
____
____
The tree can then be simplified.
15
10
____
Instead of the second arithmetic tree, you could write: 15 6 4
c. What does the circle represent?
The whole circle is not necessary. People often write 15 (6 4).
This does not require as much space, but the parentheses show
how the numbers are grouped together.
Section E: Order of Operations 51
E
Order of Operations
If you make a tree with the applet Algebra Trees, you can see the
expression with parentheses when you click on expression in the
bottom-left corner.
FPO
26. a. Rewrite the tree as an expression using parentheses to
indicate which numbers are grouped together.
b. Check your answer with the applet.
52 Expressions and Formulas
Order of Operations E
c. Use the applet to make a tree for 6 4 2.
d. Use the applet to make a tree for 30 5 (84 79).
e. Choose a set of numbers as input in the same tree that you
made in part d that will give you a negative answer. Now
choose a set of numbers that will give you an answer between
0 and 1.
27. Use parentheses in the expression 2 37 25 3 to find the
correct total for Karlene’s problem in the Home Repair section.
Section E: Order of Operations 53
E Order of Operations
The beginning of this unit introduced arrow language and the applet
Algebra Arrows to represent formulas. This section introduced
arithmetic trees and the applet Algebra Trees to represent formulas
that cannot be made using arrow language. There are several
ways to write these types of formulas.
You can express formulas with words.
cost tomatoes $1.50 grapes $1.70 green beans $0.90
(in lb)
(in lb)
(in lb)
You can express formulas with arithmetic trees, either on paper or in
the applet Algebra Trees.
If you make a formula in the applet, you can easily investigate what
happens if you change the numbers. The applet does the calculations
for you.
Arithmetic trees show the order of calculation. If a problem is not in
an arithmetic tree and does not have parentheses, there is a rule for
the order of operations: Complete multiplication and division before
addition and subtraction.
5 4 3 2 1 is represented in this tree.
5
4
3
____
2
____
____
____
54 Expressions and Formulas
1
Check for first
instance of these
terms.
You can use parentheses to convert an arithmetic tree into an
expression that shows which operations to do first.
(5 4) (3 2) 1 20.5
If you make this tree in the applet Algebra Trees and click on
expression, the expression with parentheses is shown in the
output box.
1. Design an arithmetic tree on paper that makes each problem
easier to solve using mental calculation.
a. 17 6 3 7 4
3
1
1
12 10
43 10
c. 10
b. 4.5 8.9 5.5 1.1
2. a. Use the mathematicians’ rule to simplify this expression.
24 3 5 8 10
You can use an arithmetic tree and the applet Algebra Trees if
you wish.
b. Write 24 3 5 8 10 ____ using parentheses to show
in which order the operations should be performed.
3. a. Use the applet Algebra Trees to make a tree that can calculate
the height of a stack of cups.
FPO
b. Use this tree or machine to calculate the height of a stack of
16 cups with a rim of 3.5 cm and a hold of 9.5 cm.
Section E: Order of Operations 55
E Order of Operations
Adult men can use the following rule to estimate their ideal weight.
Can "kilograms"
be abbreviated?
weight (in kilograms) height (in cm) 100 (4 circumference of wrist in cm)
2
For women, the rule is slightly different: change 100 to 110.
4. a. Use the applet Algebra Trees to make an arithmetic tree to
represent the general rule for men.
b. Matthew is 175 cm tall. The circumference of his wrist is 17 cm.
Use the tree to estimate Matthews ideal weight.
c. Andrew is 162 cm tall. The circumference of his wrist is 16 cm.
His weight is 54 kilograms (kg). Does Andrew weigh too much
or too little, according to the general rule? Show your work.
d. Investigate using your tree to find how much the difference
in “ideal” weight is if the circumference of the wrist changes
by 1 cm.
Some formulas can be written using an arrow string, and some
cannot. What is the difference between these two types of formulas?
Find some examples of each type to illustrate this difference.
56 Expressions and Formulas
Additional Practice
Section A
Arrow Language
1. Here is a record for Mr. Kamarov’s bank account.
Date
Deposit
Withdrawal
Total
10/15
$210.24
10/22
$523.65
$140.00
10/29
$75.00
$40.00
a. Find the totals for October 22 and October 29.
b. Write arrow strings to show how you found the totals.
c. When does Mr. Kamarov first have a minimum of $600 in his
account?
2. Find the results for these arrow strings.
3
a. 15 ⎯⎯→
1.9
b. 3.7 ⎯⎯→
⎯⎯
⎯⎯
1,520
c. 3,000 ⎯ ⎯⎯→
Section B
11
⎯⎯→
8.8
⎯⎯→
⎯⎯
⎯⎯
⎯⎯
600
1.6
⎯⎯→
⎯⎯⎯→
⎯⎯
⎯⎯
5,200
⎯ ⎯⎯→
⎯⎯
Smart Calculations
1. For each shopping problem, write an arrow string to show the
change the cashier owes to the customer. Be sure to use the
small-coins-and-bills-first method. Then write another arrow
string that has only one arrow to show the total change.
a. A customer gives $20.00 for a $9.59 purchase.
b. A customer gives $5.00 for a $2.26 purchase.
c. A customer gives $16.00 for a $15.64 purchase.
Additional Practice 57
Additional Practice
2. Rewrite these arrow strings so that each one has only one arrow:
35
a. 750 ⎯⎯→
3
b. 63 ⎯⎯→
1
c. 439 ⎯⎯→
⎯⎯
⎯⎯
⎯⎯
40
⎯⎯→
50
⎯⎯→
20
⎯⎯→
⎯⎯
⎯⎯
⎯⎯
3. Use the applet Algebra Arrows to make two machines for each
arrow string: one machine that has the one arrow that is shown
and another machine that will make the computation easier to
calculate mentally. Explain why your new string makes the
computation easier or why it is not possible to simplify the string.
66
a. 74 ⎯⎯→ ⎯⎯
Section C
58
b. 231 ⎯⎯→
⎯⎯
27
c. 459 ⎯⎯→
⎯⎯
Formulas
Clarinda has a personal computer at home, and she subscribes to
Tech Net for Internet access. Tech Net charges $15 per month for
access plus $2 per hour of usage. So if Clarinda is connected to
the Internet for a total of 3 hours one month, for example, she pays
$15 plus 3 times $2, or $21, for the month.
1. Which string shows the cost for Internet service through Tech
Net? Explain your answer.
$2
number of hours
→ total cost
a. $15 ⎯⎯→ ⎯⎯ ⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯
$15
$2
b. number of hours ⎯⎯→ ⎯⎯ ⎯⎯→ total cost
$15
$2
c. number of hours ⎯⎯→ ⎯⎯ ⎯⎯→ total cost
2. In the applet Algebra Arrows, make a machine for Clarinda’s Tech
Net costs, and use it to calculate the costs for these monthly
usage amounts.
a. 5 hours
b. 20 hours
c. 6 12 hours
58 Expressions and Formulas
Additional Practice
Another Internet access company, Online Time, charges only $10 per
month, but $3 per hour.
3. a. In the applet Algebra Arrows, make another machine for the
cost of Internet access through Online Time.
b. Use the machines to find out for what numbers of hours
of Internet use per month Clarinda should use which
company—Tech Net or Online Time—if she wants to pay the
least in monthly charges? Show your work.
Carlos works at a plant nursery that sells flower pots.
One type of flower pot has a rim height of 4 cm and
a hold height of 16 cm.
4 cm
16 cm
4. a. How tall is a stack of two pots? Three pots?
b. Write a formula using arrow language that
can be used to find the height of any stack
if you know the number of pots.
c. Carlos has to stack these pots on a shelf
that is 45 cm high. How many pots can
he place in a stack that high? Explain your
answer.
5. Compare the following pairs of arrow strings, and determine
whether they provide the same results. You may use the
applet Algebra Arrows if you wish.
8
2
2
8
5
3
2
1
a. input ⎯⎯→ ⎯⎯ ⎯⎯→ output
input ⎯⎯→ ⎯⎯ ⎯⎯→ output
b. input ⎯⎯→ ⎯⎯ ⎯⎯→ output
3
5
input ⎯⎯→ ⎯⎯ ⎯⎯→ output
6
c. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output
2
6
1
input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output
Additional Practice 59
Additional Practice
Section D
Reverse Operations
Ravi lives in Bellingham, Washington. He travels to Vancouver,
Canada, frequently. When Ravi is in Canada, he uses this rule to
estimate prices in U.S. dollars.
4
3
number of Canadian dollars ⎯⎯→ ⎯⎯ ⎯⎯→ number of U.S. dollars
1. With the applet Algebra Arrows, make a machine for Ravi’s
formula, and use it to estimate U.S. prices for these Canadian
prices.
a. A hamburger for $2 Canadian
b. A T-shirt for $18 Canadian
c. A movie for $8 Canadian
d. A pair of shoes for $45 Canadian
2. a. Write a formula and make the machine for it that Ravi can use
to convert U.S. dollars to Canadian dollars.
b. Using the applet, how can you check to see if the machine you
made in part a is correct?
3. Write the reverse string for each one of these strings. You may
use the applet to check your strings.
1
2.5
4
a. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output
6
2
5
b. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output
4. Find the input for each string, and show your work.
10
2
3
a. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ 9
4
5
3
1
b. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ 10
60 Expressions and Formulas
Additional Practice
Section E
Order of Operations
1. In your notebook, copy and complete the arithmetic trees.
a.
12
b.
3
2
____
____
24
4
c.
1.5
3.5
3
7
____
____
____
____
8
2
____
____
2. In the applet Algebra Trees, make trees to find the answers to the
following:
a. 10 1.5 6
b. (10 1.5) 6
c. 15 (2 2 1)
3. a. Suzanne took her cat to the veterinarian for dental surgery.
(Her cat had never brushed his teeth!) Before the surgery, the
veterinarian gave Suzanne an estimate for the cost: $55 for
anesthesia, $30 total for teeth cleaning, $18 per tooth pulled,
$75 per hour of surgery, and the cost of medicine. Use the
applet Algebra Trees to make an arithmetic tree to represent
the total cost of Suzanne’s bill from the veterinarian. Use
labels in your arithmetic tree.
b. Use your tree to make a list of the costs Suzanne has to pay for
different possible treatments.
c. Write an expression using parentheses to calculate the costs
for pulling 3 teeth and for a total time in surgery of half an
hour. The cost of the medicine is $13.
Additional Practice 61
Section A
Arrow Language
1.
Date
Seats
Requested
Cancellations
Total
Seats Reserved
3/11
233
3/12
47
0
280
3/13
51
1
330
3/14
53
0
383
3/15
5
12
376
3/16
16
2
390
2. Arrow strings will vary. Sample response:
47
0
51
1
53
0
5
12
16
2
3/12
233 ⎯
⎯⎯→ 280 ⎯
⎯⎯→ 280
3/13
280 ⎯
⎯⎯→ 331 ⎯
⎯⎯→ 330
3/14
330 ⎯
⎯⎯→ 383 ⎯
⎯⎯→ 383
3/15
383 ⎯
⎯⎯→ 388 ⎯
⎯⎯→ 376
3/16
376 ⎯⎯⎯→ 392 ⎯
⎯⎯→ 390
3. The airline needs to begin a waiting list on March 14.
4. Answers will vary. Sample response:
One advantage is that it quickly tells you how many people are
booked for the flight on the 16th. One disadvantage is that you
do not know on what day the waiting list was started.
1.40
0.62
5.83
1.40
5. a. 12.30 ⎯⎯⎯→ 13.70 ⎯ ⎯⎯→ 13.08 ⎯ ⎯⎯→ 18.91 ⎯⎯⎯→ 17.51
b. Discuss your answer with a classmate. Sample response:
Vic had $12.30 in his pocket. His mom gave him $1.40 for bus
fare. On the way to the bus stop, he bought a pen for $0.62.
Then he sold his lunch to Joy for $5.83. He paid the bus driver
$1.40. How much did Vic have left?
62 Expressions and Formulas
Answers to Check Your Work
6. Discuss your answer with a classmate. Sample response:
Fourteen people got on the empty bus at the first stop. At the
second stop, two got off and eight got on. How many were still
on the bus? [20 people, or 21 people if you count the driver]
2
8
14 ⎯⎯→ 12 ⎯ ⎯→ 20
7. Sample response:
Arrow language shows all the steps in order so that you can find
answers that are in the middle of a series of calculations.
Section B
Smart Calculations
1. a.
15
8
1
–
2
i. 20 ⎯⎯⎯→ 35 ⎯ ⎯→ 27 ⎯ ⎯→ 27.5
ii. Your arrow string may be different from the one shown here.
Check to see if you get $20 as the output; also see part b.
0.03
0.20
13
6.77 ⎯⎯
⎯⎯→ 6.80 ⎯⎯
⎯⎯→ 7 ⎯⎯
⎯→ 20
iii. Your arrow string may be different from the one shown here.
Check to see if you get $20 as the output; also see part b.
0.20
0.70
7
12.10 ⎯⎯
⎯⎯→ 12.30 ⎯⎯
⎯⎯→ 13 ⎯⎯→ 20
b. To check if your strings are correct, you can make them into
“machines” in the applet Algebra Arrows. They will look
like this:
FPO
Note that some numbers look different; for example, 0.20 has been
changed to 0.2, and no dollar signs are used.
Answers to Check Your Work 63
Answers to Check Your Work
You can make a number of different strings for ii and iii. Here you see
an example for each.
FPO
2. a. Sample response:
7
30
200
423 ⎯⎯→ 430 ⎯⎯⎯→ 460 ⎯⎯⎯→ 660
This string is easier because you can add the numbers in the
ones place, then add the numbers in the tens place, and finally
add the numbers in the hundreds place.
b. Sample response:
This string is already easy because you can easily subtract 24
from 44 to get 20, so the answer is 520.
c. Sample response:
25
54
29 ⎯ ⎯→ 4 ⎯⎯⎯→ 58
This string is easier because when you subtract 25 first, it
leaves an easy number to work with.
d. Sample response:
2
32
998 ⎯⎯→ 1,000 ⎯⎯⎯→ 1,032
This string is easier because when you add 2 to 998, you get
an easy number with a lot of zeros that are easy to work with.
It’s easy to add numbers to 1,000.
3. Check your answer with a classmate. Sample answer:
(long)
31
19
232 ⎯⎯⎯→ 263 ⎯⎯⎯→ 282
50
(short) 232 ⎯⎯⎯→ 282
Shorter strings are easier when the total of the numbers above
the arrows is a multiple of 10 or a number between 1 and 10.
64 Expressions and Formulas
Answers to Check Your Work
4. Check your answer with a classmate. Sample answer:
98
(short) 232 ⎯⎯⎯→ 330
(long)
100
2
232 ⎯ ⎯⎯→ 332 ⎯⎯⎯→ 330
Longer strings are easier when the number above the one arrow
is not a multiple of 10 or a number between 1 and 10.
5. Sample explanation:
The shortened arrow string shows the total amount of change.
Section C
Formulas
1.40
1.90
1. a. number of miles ⎯⎯⎯⎯→ ______ ⎯⎯⎯⎯→ total price
If you use the applet Algebra Arrows, your arrow string (or
machine) might look like this.
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b. Using an arrow string makes calculations easier.
c. Your lists may be different. Compare your list with a classmate.
You can make the list by filling in a different number of miles
each time. In the applet, instead of filling in a number of miles,
if you click on table (in the menu on the left), you will see lists
(or a table) with miles and prices.
FPO
Answers to Check Your Work 65
Answers to Check Your Work
2. a.
Weight
Tomatoes $1.20/lb
Green Beans $0.80/lb
Grapes $1.90/lb
0.5 lb
$0.60
$0.40
$0.95
1.0 lb
$1.20
$0.80
$1.90
2.0 lb
$2.40
$1.60
$3.80
3.0 lb
$3.60
$2.40
$5.70
b.
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3. a. The – 1 means that one chair is subtracted from the total
number of chairs in the stack; the 7 means that for every
chair that is added, the height of the stack will grow 7 cm.
The 80 represents the height of the first chair in the stack.
b. Alba should use 7 and 73 above the arrows. She wrote
7 because every chair adds 7 cm to the height of the stack.
Next, 73 is added for the height of the first chair minus the
7 cm that was already added in the first step.
c. To check if the arrow strings are the same, you can make this
machine in the applet algebra arrows. If you check for some
different numbers of chairs, both strings should give the
same answers. (You need to check several numbers; one
is not enough.)
FPO
66 Expressions and Formulas
Answers to Check Your Work
Note that in the arrow strings in this machine, the intermediate
solutions are not shown. You can leave out or add extra input/
output boxes as you wish.
You can also use the table option in the menu on the left to check
several numbers at once.
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4. a. The point on the graph labeled A represents a stack of
15 chairs with a total height of about 175 cm.
b. Not every point on the line has a meaning. For example, you
cannot add half a chair, and the total height of the stack cannot
be 100 cm.
c. The graph will not intersect (0, 0) because a stack of zero chairs
make no sense. Also if 0 is used for the number of chairs in
Damian’s arrow string, the result is 73 cm, which is impossible
of course. So the line should start at one chair.
d. About six chairs
e. Checking with the arrow string in the applet will show the
heights when entering the number of chairs (see the
illustration for problem 3c.): for five chairs, the required
space is 108 cm, and they will fit; for six, it is 115 cm, and
these will probably fit too; and for seven, it is 122 cm,
but these will clearly not fit.
Answers to Check Your Work 67
Answers to Check Your Work
Section D
Reverse Operations
1. a.
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Your answer should not contain decimals because the original
measurement is rounded to the nearest mile, so your answer
has to be rounded to 6,581 km.
b.
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c. 6581 6544 37 km
You may have given several reasons why this difference is not
important. Discuss your answer with a classmate. Sample
answers:
•
You do not know how 4,090 miles was measured. Was it as
an airplane flies? Making computations using a model of the
earth ? Using and converting sea miles? They will all result
in different outcomes, and so 4,090 will most likely be a
rounded number itself.
•
37 km compared to 6,500 (or 6,581 or 6,544) is less than 1%.
That is not a important difference.
•
If you are traveling that far, 37 km will not make as important
difference.
0.75
2. $4.40 ⎯⎯⎯⎯→ $3.30
3
3. Carmen’s string is correct because 0.75 is the same as 4.
By dividing by 4, she found one-fourth of the price. Next
she multiplied by three, which gave her three-fourths of
the price.
4
3
2
7
5
2
⎯⎯→ ______ ⎯ ⎯→ input
4. a. output ⎯⎯→ ______ ⎯
b. output ⎯⎯→ ______ ⎯⎯⎯→ ______ ⎯⎯→ input
68 Expressions and Formulas
Answers to Check Your Work
5. You can check your arrow strings in several ways using the
applet Algebra Arrows. Two different ways are shown here.
Check your answers with a classmate. You should have used
several numbers to see if your arrow string works. Note that
the intermediate results are not shown in these strings.
FPO
FPO
Section E
Order of Operations
1. a.
17
3
6
20
4
7
10
30
37
b.
4.5
5.5
8.9
10
1.1
10
20
Answers to Check Your Work 69
Answers to Check Your Work
c.
3
10
1
10
1
10
1
2
3
4
4
10
5
10
1
13
4
2. a. You can simplify this expression to 8 40 10 by calculating
the division and multiplication first. The result will then be
48 10 38. Using the applet Algebra Trees, you may have
made this tree.
FPO
70 Expressions and Formulas
Answers to Check Your Work
b. (24 3) (5 8) 10 _____
You can find this expression using the applet by clicking
on expression in the menu on the left once you have made
the tree. In the output box, the expression will be presented
this way.
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3. a. Your tree may look like this.
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b. The height of this stack is 65.5 cm.
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Answers to Check Your Work 71
Answers to Check Your Work
4. a.
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b. If you enter the data for Matthew, you find out he should
weigh 71.5 kg.
c. If you enter 192 cm for Andrew’s height and 16 cm for his wrist
circumference, his ideal weight will come out as 63 kg.
According to the rule, he does not weigh enough.
d. You can choose one weight and fill in different numbers for
wrist circumference, with a difference of 1 cm between them,
to see what happens. If you do this with a height 180 cm and
wrist circumferences of 14, 15, 16, 17 cm, this is the result:
Wrist Circumference (cm)
Ideal Weight (kg)
14
15
16
17
68
70
72
74
So if the wrist circumference is 1 cm larger/smaller, the ideal
weight is 2 kg more/less.
72 Expressions and Formulas