1. science
2. mathematics
3. algebra

INSTRUCTOR’S TESTING
MANUAL
M ATHEMATICS FOR
E LEMENTARY T EACHERS
WITH A CTIVITIES
FOURTH EDITION
Sybilla Beckmann
University of Georgia
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2014, 2011, 2008 Pearson Education, Inc.
Publishing as Pearson, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-83672-4
ISBN-10: 0-321-83672-3
www.pearsonhighered.com
Contents
1 Numbers and the Base-Ten System
1.1 The Counting Numbers . . . . . . .
1.2 Decimals and Negative Numbers . .
1.3 Comparing Numbers in Base Ten .
1.4 Rounding Numbers . . . . . . . . .
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2 Fractions and Problem Solving
2.1 Solving Problems and Explaining Solutions
2.2 Defining and Reasoning About Fractions .
2.3 Equivalent Fractions . . . . . . . . . . . .
2.4 Comparing Fractions . . . . . . . . . . . .
2.5 Percent . . . . . . . . . . . . . . . . . . . .
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3 Addition and Subtraction
3.1 Interpretations of Addition and Subtraction
3.2 Properties of Addition and Mental Math . .
3.3 Algorithms for Addition and Subtraction . .
3.4 Adding And Subtracting Fractions . . . . .
3.5 Adding and Subtracting Negative Numbers .
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4 Multiplication
4.1 Interpretations of Multiplication . . . . . . . . . . .
4.2 Why Multiplying by 10 Is Special in Base Ten . . .
4.3 Commutative, Associative Properties, Area, Volume
4.4 The Distributive Property . . . . . . . . . . . . . .
4.5 Properties of Arithmetic, Mental Math, Basic Facts
4.6 Why Algorithms for Multiplying Work . . . . . . .
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1
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9
11
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13
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23
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29
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34
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35
36
37
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43
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48
iv
5 Multiplying Fractions, Decimals,
5.1 Multiplying Fractions . . . . . .
5.2 Multiplying Decimals . . . . . .
5.3 Multiplying Negative Numbers .
5.4 Powers and Scientific Notation .
CONTENTS
Negatives
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6 Division
6.1 Interpretations of Division . . . . . . . . . . . . . . .
6.2 Division and Fractions and Division with Remainder
6.3 Why Division Algorithms Work . . . . . . . . . . . .
6.4 Fraction Division, How Many Groups? . . . . . . . .
6.5 Fraction Division, How Many in One Group? . . . .
6.6 Dividing Decimals . . . . . . . . . . . . . . . . . . . .
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51
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59
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7 Ratio and Proportional Relationships
7.1 Motivating and Defining Ratio and Proportional Relationships
7.2 Reasoning with Multiplication and Division . . . . . . . . . .
7.3 Unit Rates and the Values of a Ratio . . . . . . . . . . . . . .
7.4 Proportional Versus Inversely Proportional . . . . . . . . . . .
7.5 Percent Revisited: Percent Increase and Decrease . . . . . . .
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75
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8 Number Theory
8.1 Factors and Multiples . . . . . . . . . . . . . . . . . . .
8.2 Even and Odd . . . . . . . . . . . . . . . . . . . . . . .
8.3 Divisibility Tests . . . . . . . . . . . . . . . . . . . . .
8.4 Prime Numbers . . . . . . . . . . . . . . . . . . . . . .
8.5 Greatest Common Factor and Least Common Multiple
8.6 Rational and Irrational Numbers . . . . . . . . . . . .
8.7 Looking Back at the Number Systems . . . . . . . . . .
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85
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93
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100
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102
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9 Algebra
9.1 Numerical Expressions . . . . . . . . .
9.2 Expressions with Variables . . . . . . .
9.3 Equations for Different Purposes . . .
9.4 Solving Equations . . . . . . . . . . . .
9.5 Solving Problems with Strip Diagrams
9.6 Sequences . . . . . . . . . . . . . . . .
9.7 Functions . . . . . . . . . . . . . . . .
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v
CONTENTS
9.8
Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
10 Geometry
10.1 Visualization . . . . . . . . . . . . . . . . . .
10.2 Angles . . . . . . . . . . . . . . . . . . . . . .
10.3 Angles and Phenomena in the World . . . . .
10.4 Triangles, Quadrilaterals, and Other Polygons
11 Measurement
11.1 Fundamentals of Measurement . . . . . . .
11.2 Length, Area, Volume, and Dimension . .
11.3 Error and Precision in Measurements . . .
11.4 Converting from One Unit of Measurement
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to Another
12 Area of Shapes
12.1 What Area Is . . . . . . . . . . . . . . . . . . . . . .
12.2 The Moving and Additivity Principles About Area .
12.3 Areas of Triangles . . . . . . . . . . . . . . . . . . . .
12.4 Areas of Parallelograms and Other Polygons . . . . .
12.5 Shearing: Changing Shapes Without Changing Area .
12.6 Areas of Circles and the Number Pi . . . . . . . . . .
12.7 Approximating Areas of Irregular Shapes . . . . . . .
12.8 Perimeter Versus Area . . . . . . . . . . . . . . . . .
12.9 The Pythagorean Theorem . . . . . . . . . . . . . . .
13 Solid Shapes
13.1 Polyhedra and Other Solid
13.2 Patterns and Surface Area
13.3 Volumes of Solid Shapes .
13.4 Volume Versus Weight . .
Shapes .
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14 Geometry of Motion and Change
14.1 Reflections, Translations, and Rotations . . . .
14.2 Congruence . . . . . . . . . . . . . . . . . . . .
14.3 Constructions With Straightedge and Compass .
14.4 Similarity . . . . . . . . . . . . . . . . . . . . .
14.5 Areas, Volumes, and Scaling . . . . . . . . . . .
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113
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126
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131
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135
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139
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157
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168
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169
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183
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vi
CONTENTS
15 Statistics
15.1 Formulating Questions and Gathering Data . . . . . . . . .
15.2 Displaying Data and Interpreting Data Displays . . . . . . .
15.3 The Center of Data: Mean, Median, and Mode . . . . . . . .
15.4 Summarizing, Describing, and Comparing Data Distributions
16 Probability
16.1 Basic Principles of Probability . . . . . . . . . . . . .
16.2 Counting the Number of Outcomes . . . . . . . . . .
16.3 Calculating Probabilities . . . . . . . . . . . . . . . .
16.4 Using Fraction Arithmetic to Calculate Probabilities
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185
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191
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195
196
197
197
198
Chapter 1
Numbers and the Base-Ten System
1
2
CHAPTER 1
1.1
The Counting Numbers
1. If a young child can correctly say the number word list to five, “one, two, three,
four, five,” will the child necessarily be able to determine how many dinosaurs
are in a collection of 4 toy dinosaurs that are lined up in a row? Discuss why
or why not.
2. If a young child can correctly say the number word list to five, “one, two, three,
four, five,” and point one by one to each bear in a collection of 4 toy dinosaurs
while saying the number words, does the child necessarily understand that there
are 4 dinosaurs in the collection? Discuss why or why not.
3. Suppose a young child can correctly say the number word list to five, “one,
two, three, four, five.” Briefly describe two key additional pieces of knowledge
the child must have to be able to determine how many hearts there are in the
collection in Figure 1.1.
Figure 1.1: Counting hearts
4. (a) Make a math drawing that shows how to organize 16 small objects in a
way that corresponds to the way we write 16 in the base-ten system.
(b) What is difficult for young students (first graders)
• about the written number 16?
• about the spoken number sixteen?
Answer briefly.
5. If a Kindergarten or first grade child counts that there are 24 sticks in a collection and you then ask the child to show you what the 2 in 24 means, the child
might show you 2 sticks. What is a way to show what the 2 in 24 means?
6. What does it mean to say that the base-ten system uses place value?
7. What problem in the history of mathematics did the development of the baseten system solve?
c 2014 Pearson Education, Inc.
Copyright SECTION 1.1
3
8. How are the values of adjacent places in the decimal representations of numbers
related? Describe how you can use collections of objects (such as toothpicks or
popsicle sticks) to show how the values of adjacent places in the decimal system
are related.
9. Make a math drawing that shows 36 beads organized in a way that corresponds
with the way we write 36 in the base-ten system.
10. According to the text, what particular difficulty do English speakers face in
learning to say some of the counting numbers?
11. (a) Describe how to organize 100 toothpicks in a way that fits with the structure of the base-ten system. Explain how your organization reflects the
structure of the base-ten system and how it fits with the way we write the
number 100.
(b) Make a math drawing showing how to organize 143 toothpicks in a way
that corresponds with the way we write 143 in the base-ten system.
12. Describe how to organize 135 toothpicks in a way that corresponds with the
way we write 135 in the base-ten system.
13. Describe how to re-organize the bundled and loose toothpicks shown in Figure 1.2 so that their new arrangement corresponds to the way we write the
number that stands for this total number of toothpicks in the base-ten system.
Figure 1.2: How Can You Rearrange These Toothpicks to fit with the Base-Ten
System?
c 2014 Pearson Education, Inc.
Copyright 4
CHAPTER 1
14. Discuss what a number line is and how to place whole numbers on a number
line.
15. Describe the difference between a “number path” (as described in the text) and
a number line.
1.2
Decimals and Negative Numbers
1. Describe how to represent 12.3 with bundled objects in a way that fits with and
shows the structure of the base-ten system.
2. Describe and draw (rough) pictures showing how to represent 1.367 as a length
in a way that fits with and shows the structure of the base-ten system.
3. Describe and show how to plot 1.367 on a number line on which 0 and 1 have
been plotted.
4. Describe three ways discussed in the text to represent a decimal such as 1.234.
For each way, show how to represent 1.234.
5. Suppose you want to show how the structure of the base-ten system remains
the same to the left and right of the decimal point. You have bundles of small
objects like the bundled toothpicks shown in Figure 1.3 and you want to use
these bundled objects to represent decimals. List at least 3 decimals that you
could use these bundles to represent and explain your answer in each case.
Figure 1.3: Which Decimals Can These Bundled Toothpicks Represent?
6. Show how to “zoom in” on smaller and smaller portions of the number line
(Figure 1.4) so that you can plot 5.7183 on each number line. Label the long
c 2014 Pearson Education, Inc.
Copyright SECTION 1.2
5
tick marks on each number line (at the ends) and plot 5.7183 on each number
line (in its approximate location).
c 2014 Pearson Education, Inc.
Copyright 6
CHAPTER 1
0
1
2
3
4
5
Long ticks:
whole numbers
Long ticks:
tenths
Long ticks:
hundredths
Long ticks:
thousandths
Figure 1.4: Zooming In on a Number Line
c 2014 Pearson Education, Inc.
Copyright 6
7
8
SECTION 1.2
7
7. Show how to “zoom in” on smaller and smaller portions of the first number line
in Figure 1.5 until you have “zoomed in” to a portion of the number line on
which the long tick marks are thousandths and so that 1.0228 can be plotted on
each number line. Plot 1.0228 on each number line. Label the long tick marks
on each number line (at both ends).
long
ticks:
whole
numbers
long
ticks:
tenths
long
ticks:
hundredths
long
ticks:
thousandths
Figure 1.5: Zooming In on a Number Line
8. Label the tick marks on the three number lines in Figure 1.6 in three different
ways. In each case, your labeling should fit with the structure of the base-ten
system and the fact that the tick marks at the ends of the number lines are
longer than the other tick marks.
9. Label the tick marks on the three number lines in Figure 1.7 in three different
ways. In each case, your labeling should fit with the structure of the base-ten
system and the fact that the tick marks at the ends of the number lines are
longer than the other tick marks.
10. Label the tick marks on the number line in Figure 1.8 so that the long tick
marks are thousandths and so that 3.8459 can be plotted on this portion of the
number line. Plot 3.8459.
c 2014 Pearson Education, Inc.
Copyright 8
CHAPTER 1
4.8
4.81
4.82
4.83
etc.
4.89
4.9
4.89
4.891
4.892
4.893
etc.
4.899
4.9
4.8991 4.8992 4.8993
etc.
4.8999 4.9
4.899
Figure 1.6: Label These Number Lines
7.1
7.1
7.1
Figure 1.7: Label These Number Lines
11. Decimals can be difficult for students to understand. Explain what a decimal
such as 3.8459 means and discuss where it is located on a number line.
12. Decimals can be difficult for students to understand. Explain what the decimal
3.8459 means and show where it is located on several number lines, including
a number line which has tick marks labeled with whole numbers and another
number line which has tick marks labeled with tenths.
13. Label the tick marks on the number line in Figure 1.8 so that the long tick
marks are hundredths and so that 3.8459 can be plotted on this portion of the
number line. Plot 3.8459.
14. Label the tick marks on the number line in Figure 1.8 so that the long tick marks
are tenths and so that 3.8459 can be plotted on this portion of the number line.
c 2014 Pearson Education, Inc.
Copyright SECTION 1.3
9
Figure 1.8: Label the Number Line and Plot the Point
Plot 3.8459.
15. Label the tick marks on the number line in Figure 1.8 so that the long tick
marks are whole numbers and so that 3.8459 can be plotted on this portion of
the number line. Plot 3.8459.
16. Draw a number line and plot 0, 1, −1, 0.83 and −0.4 on it.
17. Where are the negative numbers located on the number line? Give a few examples to illustrate.
18. Label the tick marks on the number line in Figure 1.8 so that the long tick marks
are tenths and so that −4.68 can be plotted on this portion of the number line.
Plot −4.68.
1.3
Comparing Numbers in Base Ten
1. Explain why we compare 3076 and 822 by starting at the left-most non-zero
place. What is it about the base-ten system that allows us to compare numbers
this way by starting at the left?
2. Draw (rough) pictures of objects that are bundled in a way that fits with the
structure of the base-ten system and that show that 1.1 > 0.99.
3. Draw (rough) pictures of objects that are bundled in a way that fits with the
structure of the base-ten system and that show that 1.2 > 0.45.
4. Describe and draw (rough) pictures showing how to represent 1.1 and 0.99 as
lengths in a way that fits with the structure of the base-ten system. Describe
how to use these lengths to show that 1.1 > 0.99.
5. Use a number line that has tick marks that fit with the structure of the base-ten
system to show that 1.1 > 0.99.
6. Describe two different ways to show that 1.1 > 0.99.
c 2014 Pearson Education, Inc.
Copyright 10
CHAPTER 1
7. Describe two different ways to show that 1.2 > 0.45.
8. Which of the following could the pictures in Figure 1.9 be used to illustrate?
Circle all that apply.
120 > 45
12 > 4.5
12 > .45
.12 > .45
.12 > .045
[Section 1.3]
Figure 1.9: Which Inequalities Can These Blocks Represent?
9. Which is greater, 0.0037 or 0.00086723399? Explain.
10. Jonathan says that 0.199 is greater than 0.21. Describe two different ways to
try to convince him that it’s not.
11. Is 2.37451 between 2.319 and 2.4 or not?
12. Find a number between 7.8651 and 7.8652 and plot all three numbers visibly
and distinctly on a copy of the number line in Figure 1.10. The tick marks
should fit with the structure of the base-ten system. Label all the longer tick
marks on your number line.
Figure 1.10: A Number Line
13. Find a number between 3 and 3.001 and plot all three numbers visibly and
distinctly on the number line in Figure 1.10. The tick marks should fit with
the structure of the base-ten system. Label all the longer tick marks on your
number line.
c 2014 Pearson Education, Inc.
Copyright SECTION 1.4
11
14. Find a number between 3 and 2.9999 and plot all three numbers visibly and
distinctly on the number line in Figure 1.10. The tick marks should fit with
the structure of the base-ten system. Label all the longer tick marks on your
number line.
15. Find a number between 6.55 and 6.551 if there is one. If there is no number
between them, explain why not.
16. Is there more than one number between 1.7993 and 1.7995? If not, explain why
not. If so, find two such numbers.
17. Is there are largest number? Explain your answer.
18. Is there a smallest decimal that is greater than the number 2? Explain your
answer.
19. Emily says that −5 > −2. Describe two different ways you could try to convince
Emily that this is not correct.
20. Find a number between −3 and −2.9999 and plot all three numbers visibly and
distinctly on the number line in Figure 1.10. The tick marks should fit with
the structure of the base-ten system. Label all the longer tick marks on your
number line.
1.4
Rounding Numbers
1. Round 3.546 to the nearest tenth. Explain in words why you round the number
the way you do. Use a number line to support your explanation.
2. Round 124.56 to the nearest ten (note: ten, not tenth). Explain in words why
you round the number the way you do. Use a number line to support your
explanation.
3. Describe how to round a number to the nearest hundred, using the examples
2745 and 2783 to illustrate the method you describe. Explain why the method
you describe makes sense. In other words, explain the idea underlying the
method.
4. Round 199, 999 to the nearest ten.
c 2014 Pearson Education, Inc.
Copyright 12
CHAPTER 1
5. Maya has made up her own method of rounding. Starting at the right-most
place in a number, she keeps rounding to the value of the next place to the left
until she reaches the place to which the decimal number was to be rounded.
For example, Maya would use the following steps to round 3.2716 to the nearest
tenth:
3.2716 → 3.272 → 3.27 → 3.3
Is Maya’s method a valid way to round? Explain why or why not.
6. The distance between two cities is described as 2100 miles. Should you assume
that this is the exact distance between the cities? If not, what could the distance
between the cities possibly be?
c 2014 Pearson Education, Inc.
Copyright Chapter 2
Fractions and Problem Solving
13
14
CHAPTER 2
2.1
Solving Problems and Explaining Solutions
2.2
Defining and Reasoning About Fractions
1. Anna says that the dark blocks pictured in Figure 2.1 can’t represent 41 because
there are 6 dark blocks and 6 is more than 1 but 14 is supposed to be less than
1. What should you clarify in order to interpret the dark blocks as 14 ?
Figure 2.1: Representing the Fraction
1
4
2. When Martin was asked to say what the 5 in the fraction 52 means, Martin said
that the 5 is the whole. Explain why it is not completely correct to say that “5
is the whole.” What is a better way to say what the 5 in the fraction 25 means?
3. If Harry needs 54 of a liter of dragon snot to make a full batch of potion but he
only has 53 of a liter of dragon snot, then what fraction of a batch of potion can
Harry make (assuming he has enough of the other ingredients)?
(a) Make a math drawing to help you solve the problem and explain your
solution. Use our definition of fraction in your explanation and attend to
the whole (unit amount) that each fraction is of.
(b) Describe the different wholes that occur in part (a). Discuss how one
amount can be described with two different fractions depending on what
the whole is taken to be.
4. If 43 of a cup of a food gives you your daily value of potassium, then what
fraction of your daily value of potassium is in 1 cup of the food?
Make a math drawing to help you solve the problem and explain your solution.
Use our definition of fraction in your explanation and attend to the whole (unit
amount) that each fraction is of.
c 2014 Pearson Education, Inc.
Copyright SECTION 2.2
15
5. The strips in Figure 2.2 show the relative amounts of cans that Ben and Charles
collected for the can-a-thon. (Note that each rectangle represents some fixed
number of cans, but this number may be greater than 1.)
Ben’s
amount:
Charles’s
amount:
Figure 2.2: The Amounts of Cans that Ben and Charles Collected
(a) Write a sentence in which you use a fraction to describe how the amount
of cans that Ben collected compares with the amount of can that Charles
collected. Explain briefly.
(b) Write a sentence in which you use a fraction to describe how the amount
of cans that Charles collected compares with the amount of can that Ben
collected. Explain briefly.
6. The diagram in Figure 2.3 shows a map of Mr. McGregor’s garden, which consists of two plots of different areas. Each plot is divided into 3 parts of equal area
and Mr. McGregor planted peas in the two parts that are shaded. What fraction of Mr. McGregor’s (entire) garden is planted with peas? Explain why your
answer is correct by using our definition of fraction without further subdividing
the plots.
Figure 2.3: Mr. McGregor’s Garden
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Copyright 16
CHAPTER 2
0
1
2
Figure 2.4: A Number Line
7. Plot 34 on the number line in Figure 2.4 and explain why this location fits with
our definition of fraction.
8. Plot 54 on the number line in Figure 2.4 and explain why this location fits with
our definition of fraction.
9. According to our text, what does
5
3
mean?
10. (a) Give two different fractions that you can legitimately use to describe the
shaded region in Figure 2.5. For each fraction, explain why you can use
that fraction to describe the shaded region.
Figure 2.5: What Fraction is Shaded?
(b) Write an unambiguous question about the shaded region in Figure 2.5 that
can be answered by naming a fraction. Explain why your question is not
ambiguous.
11. You showed Johnny the picture in Figure 2.6 to help Johnny understand the
meaning of the fraction 35 . But Johnny doesn’t get it. He says the picture shows
5
, not 53 .
6
Figure 2.6: A Picture for
5
3
(a) Is Johnny right that the picture can be interpreted as showing 65 ? Explain
briefly.
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Copyright SECTION 2.3
17
(b) What must you clarify in order to interpret the picture as showing 53 ?
(c) What is another way to show
12. State what
7
4
5
3
to Johnny, other than with parts of objects?
means according to our definition.
13. Discuss why it can be confusing to show an improper fraction such as
pieces of pie or the like. What is another way to show the fraction 73 ?
2.3
7
3
with
Equivalent Fractions
1. Using the example
2
2·4
=
3
3·4
and a math drawing, explain why multiplying the numerator and denominator of
a fraction by the same number results in the same number (equivalent fraction).
Give a “general” explanation, in the sense that the explanation would work the
same way if other numbers had been used.
2. Using a math drawing, explain why multiplying both the numerator and denominator of
2
3
by 4 produces the same number (an equivalent fraction). Discuss how to see
multiplication by 4 in both the numerator and denominator in terms of your
math drawing. Attend carefully to points that might be difficult for students.
3. Without using multiplication by 1, explain why multiplying the numerator and
denominator of a fraction by the same number produces an equivalent fraction.
Use the example
2
2·4
=
3
3·4
to illustrate.
4. Using the example
2
2·4
=
3
3·4
explain in two different ways why multiplying the numerator and denominator
by the same number produces an equivalent fraction.
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CHAPTER 2
5. Using the fractions 13 and 34 , describe how to give two fractions common denominators. In terms of a math drawing, what are you doing when you give
fractions common denominators?
6. Using a math drawing, explain why dividing both the numerator and denominator of
6
9
by 3 produces the same number (an equivalent fraction). Discuss how to see
division by 3 in both the numerator and denominator in terms of your math
drawing. Attend carefully to points that might be difficult for students.
7. Simplify
4
6
and use a math drawing to show the process.
8. Plot 65 , 54 , and 43 on the number line in Figure 2.7 in such a way that each
number falls on a tick mark. Lengthen the tick marks of whole numbers.
Figure 2.7: A Number Line
9. Plot 56 , 0.7, and 45 on a number line like the one in Figure 2.7 in such a way that
each number falls on a tick mark. Lengthen the tick marks of whole numbers
(if any).
10. The line segment below is 23 units long. Describe how to remove or add a portion
to this line segment to create a line segment that is 53 units long. Explain how
you know your new segment will be the correct length.
2
3
unit
11. Susie says that when you do the same thing to the top and bottom of a fraction
you get an equivalent fraction. Is Susie right, or is it possible to do the same
thing to the top and the bottom of a fraction and not get an equivalent fraction?
c 2014 Pearson Education, Inc.
Copyright SECTION 2.4
19
12. Ken ordered 45 of a ton of sand. Ken wants to receive 31 of his order now (and
2
of his order later). What fraction of a ton of sand should Ken receive now?
3
Make a math drawing to help you solve the problem. Explain how your drawing
helps you to solve the problem. In your explanation, attend carefully to the
whole (unit amount) that each fraction is of.
13. Frank is making a recipe that calls for 34 of a cup of ketchup. Frank only has 13
of a cup of ketchup. Assuming that Frank has enough of the other ingredients,
what fraction of the recipe can Frank make? Make a math drawing to help you
solve the problem. Explain how your drawing helps you to solve the problem.
In your explanation, attend carefully to the whole (unit amount) that each
fraction is of.
2.4
Comparing Fractions
1. Describe three general methods for determining which of two fractions is greater.
Illustrate the three methods with the fractions 53 and 58 .
2. Using the fractions 23 and 35 to illustrate, explain clearly and in detail why we can
determine which of two fractions is greater by giving the two fractions common
denominators. What is the rationale behind this method? In terms of pictures,
what are we doing when we give the fractions common denominators?
3. Show how to use the cross-multiplying method to determine which of
is greater.
5
8
and
7
12
4. Using the fractions 32 and 53 to illustrate, explain clearly and in detail why we
can determine which of two fractions is greater by using the cross-multiplying
method. What is the rationale behind this method? What are we really doing
when we cross-multiply in order to compare fractions?
5. Give two different methods for solving the following problem. Find two different
fractions in between 34 and 45 whose numerators and denominators are all whole
numbers.
6. Conrad says that 83 > 27 because 3 > 2 and 8 > 7. Regardless of whether or
not Conrad’s conclusion is correct, discuss whether or not Conrad’s reasoning
is valid.
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CHAPTER 2
7. Ron says that
5
8
>
3
7
because 5 > 3 and 8 > 7.
(a) Discuss whether or not Ron’s reasoning is valid (whether or not his conclusion is correct).
(b) Describe another way to reason (legitimately) to compare 58 and 73 without converting to decimals, giving the fractions common denominators, or
cross-multiplying.
8. Minju says that fractions that use bigger numbers are greater than fractions that
use smaller numbers. Make up two problems for Minju to help her reconsider
her ideas. For each problem, explain how to solve it, and explain why you chose
that problem for Minju.
9. Use reasoning other than converting to decimals, using common denominators,
19
19
or cross-multiplying to determine which of 94
and 107
is greater. Explain your
reasoning clearly and in detail.
10. Use reasoning other than converting to decimals, using common denominators,
4
5
or cross-multiplying to determine which of 19
and 17
is greater. Explain your
reasoning clearly and in detail.
11. Use reasoning other than converting to decimals, using common denominators,
13
and 14
is greater. Explain your
or cross-multiplying to determine which of 17
15
reasoning clearly and in detail.
12. Use reasoning other than converting to decimals, using common denominators,
38
or cross-multiplying to determine which of 39
and 45
is greater. Explain your
46
reasoning clearly and in detail.
13. Use reasoning other than converting to decimals, using common denominators,
21
or cross-multiplying to determine which of 43
and 41
is greater. Explain your
81
reasoning clearly and in detail.
2.5
Percent
1. If your full daily value of potassium is 3600 milligrams, then how many milligrams is 45% of your daily value of potassium? Show how to solve the problem
with the aid of a math drawing or a percent table. Explain your reasoning.
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Copyright SECTION 2.5
21
2. If the normal rainfall for August is 2.5 inches, but only 1.75 inches of rain fell in
August, then what percent of the normal rainfall fell in August? Show how to
solve the problem with the aid of a math drawing or a percent table. Explain
your reasoning.
3. If the full capacity of a tank is 25 liters and the tank is filled with only 15 liters,
then what percent full is the tank? Show how to solve the problem with the aid
of a math drawing or a percent table. Explain your reasoning.
4. If $85,000 is 40% of the budget, then what is the full budget? Show how to
solve the problem with the aid of a math drawing or a percent table. Explain
your reasoning.
5. If 34 of a cup of juice gives you 100% of your daily value of vitamin C, then what
percent of your daily value of vitamin C will you get in 1 full cup of juice? Show
how to solve the problem with the aid of a math drawing or a percent table.
Explain your reasoning.
6. If Company A sells 30% as many cars as Company B, then what are Company
B’s car sales, when they are calculated as a percentage of Company A’s sales?
Use a math drawing or percent table to help you solve the problem. Use your
drawing or table to help you explain your answer.
7. Explaining your reasoning clearly and use math drawings or tables to help you
answer the following:
a. What percent of
b. What percent of
1
5
2
5
is 25 ?
is 15 ?
8. Solve using equivalent fractions without cross-multiplying: A shirt had cost $15,
but now it is on sale at a $6 discount. What percent is the discount?
9. Solve using equivalent fractions without cross-multiplying: A company’s profits
were 12% of its revenues. If the company’s profits were $360,000, what were its
revenues?
10. Solve using equivalent fractions without cross-multiplying: 12% of a company’s
75 employees walk to work every day. How many employees walk to work every
day?
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CHAPTER 2
11. There were 70 members in a club. Of the 70 members, 60% were girls and the
rest were boys. After some more girls joined the club (and no boys joined or
left), the club was 75% girls. How many girls joined the club?
c 2014 Pearson Education, Inc.
Copyright Chapter 3
Addition and Subtraction
23
24
CHAPTER 3
3.1
Interpretations of Addition and Subtraction
1. Write an Add To problem that fits naturally with the equation 9+? = 16.
2. Write an Add To problem that fits naturally with the equation ? + 9 = 16.
3. Write an Add To, Change Unknown problem and write an equation that fits
naturally with the situation.
4. Write an Add To, Start Unknown problem and write an equation that fits
naturally with the situation.
5. Write three Add To problems: one Result Unknown, one Change Unknown, and
one Start Unknown. For each problem, write an equation that fits naturally
with the situation.
6. Write three Add To problems, one for each type. For each problem, write an
equation that fits naturally with the situation.
7. Write a Take From problem that fits naturally with the equation 16−? = 9.
8. Write a Take From problem that fits naturally with the equation ? − 9 = 7.
9. Write a Take From, Change Unknown problem and write an equation that fits
naturally with the situation.
10. Write a Take From, Start Unknown problem and write an equation that fits
naturally with the situation.
11. Write three Take From problems: one Result Unknown, one Change Unknown,
and one Start Unknown. For each problem, write an equation that fits naturally
with the situation.
12. Write three Take From problems, one for each type. For each problem, write
an equation that fits naturally with the situation.
13. Write a Put Together/Take Apart, Total Unknown problem. Write an equation
and draw a strip diagram or number bond for the problem.
14. Write a Put Together/Take Apart, Addend Unknown problem. Write an equation and draw a strip diagram or number bond for the problem.
c 2014 Pearson Education, Inc.
Copyright SECTION 3.2
25
15. Write a Put Together/Take Apart, Both Addends Unknown problem for the
number 4. Write equations that fit with the solutions.
16. (a) Write two versions of a Compare, Difference Unknown problem.
“more” in one version, and use “fewer” in the other.
Use
(b) Draw a strip diagram and write an addition equation and a subtraction
equation that fit with the problems.
(c) Identify the version of the problem you wrote in part (a) that is harder for
students and say why it is harder.
17. (a) Write two versions of a Compare, Smaller Unknown problem. Use “more”
in one version, and use “fewer” in the other.
(b) Draw a strip diagram and write an addition equation and a subtraction
equation that fit with the problems.
(c) Identify the version of the problem you wrote in part (a) that is harder for
students and say why it is harder.
18. (a) Write two versions of a Compare, Bigger Unknown problem. Use “more”
in one version, and use “fewer” in the other.
(b) Draw a strip diagram and write an addition equation and a subtraction
equation that fit with the problems.
(c) Identify the version of the problem you wrote in part (a) that is harder for
students and say why it is harder.
19. Write a word problem that a student who relies only on keywords might solve
incorrectly by adding 47 + 38 but that can be solved correctly by subtracting,
47 − 38.
20. Write a word problem that a student who relies only on keywords might solve
incorrectly by subtracting 83 − 67, but that can be solved correctly by adding,
85 + 43.
3.2
The Commutative and Associative Properties
of Addition, Mental Math, and Single-Digit
Facts
1. (a) State the commutative property of addition.
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CHAPTER 3
(b) Describe how you could use physical objects (such as snap-together cubes)
to describe and discuss the commutative property of addition.
(c) Give an example of how a young child could use the commutative property
of addition in a productive way to calculate a sum of two single-digit numbers (even if the child doesn’t know the term “commutative property”).
2. (a) State the associative property of addition.
(b) Describe how you could use physical objects (such as snap-together cubes)
to describe and discuss the associative property of addition.
(c) Give an example of how a young child could use the associative property of
addition in a productive way to calculate a sum of two single-digit numbers
(even if the child doesn’t know the term “associative property”). Write
equations to show how the associative property is used.
3. State the commutative property of addition. Use a specific example to explain
how to get the property’s equation by viewing one amount in two ways.
4. State the associative property of addition. Use a specific example to explain
how to get the property’s equation by viewing one amount in two ways.
5. (a) Describe how a young child who is still in the process of learning the
single-digit addition facts can use the “make-a-ten method” to add 7 + 5.
(b) Write equations to go along with your description in part (a).
(c) Which property of arithmetic does the “make-a-ten method” use? Show
where this property is used in your equations in part (b).
6. Figure 3.1 shows how the “make-a-10” method for adding 6 + 7 works. Write
equations that correspond to the strategy, making careful and appropriate use
of parentheses. Which property of arithmetic do your equations and the picture
of Figure 3.1 illustrate?
Figure 3.1: A Dot Picture for 6 + 7
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Copyright SECTION 3.2
27
7. Analisa has learned the following facts well:
• all the sums of whole numbers that add to 10 or less; Analisa knows these
facts “forwards and backwards,” for example, she knows not only that 4+3
is 7, but also that 7 breaks down into 4 + 3;
• 10 + 1, 10 + 2, 10 + 3, . . . , 10 + 10;
• the doubles 1 + 1, 2 + 2, 3 + 3, . . . , 10 + 10.
Describe three different ways that Analisa could use reasoning together with the
facts she knows well to calculate 7 + 6. In each case, write equations that correspond to the method you describe. Take care to use parentheses appropriately
and as needed.
8. Show how to use a property of arithmetic to make the addition problem 997+543
easy to calculate mentally. Write equations to show your use of a property of
arithmetic. State the property you use and show where you use it.
9. Give an example of an arithmetic problem that can be made easy to solve
mentally by using the associative property of addition. Write equations that
show your use of the associative property of addition. Your use of the associative
property must genuinely make the problem easier to solve.
10. In words, describe an easy way to calculate 412 − 98 mentally, without the use
of a calculator or the standard subtraction algorithm. Write equations to go
along with your description.
11. Describe a way other than using the standard subtraction method to calculate
603 − 199 without a calculator. Explain why your method makes sense.
12. To calculate 201 − 88, a student writes the following equations:
200 − 90 = 110 + 1 = 111 + 2 = 113.
Although the student has a good idea for solving the problem, his equations
are not correct. In words, describe the student’s solution strategy and discuss
why the strategy makes sense. Then write a correct sequence of equations that
correspond to this solution strategy. Write your equations in the following form:
201 − 88 = some expression
= some expression
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CHAPTER 3
.
= ..
= 113
13. To calculate 201 − 88, a student writes the following equations:
88 + 2 = 90 + 10 = 100 + 100 = 200 + 1 = 201
2 + 10 = 12 + 100 = 112 + 1 = 113
Although the student has a good idea for solving the problem, his equations are
not correct. In words, describe the student’s solution strategy and discuss why
the strategy makes sense. Then write correct equations that correspond to this
solution strategy.
14. To calculate 201 − 88, a student writes the following:
88
+2
90
+10
100
+100
200
+1
201
2
+10
12
+100
112
+1
113
201 − 88 = 113
In words, describe the student’s solution strategy and discuss why the strategy
makes sense.
15. John and Anne want to calculate $4.23−$1.97 by first calculating $4.23−$2.00 =
$2.23. John says that they must subtract $0.03 from $2.23, but Anne says that
they must add $0.03 to $2.23.
(a) Draw a number line (which need not be perfectly to scale) to help you
explain who is right and why. Do not just say which answer is numerically
correct; use the number line to help you explain why the answer must be
correct.
(b) Explain in another way who is right and why.
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Copyright SECTION 3.3
3.3
29
Why the Standard Algorithms for Adding and
Subtracting Numbers in the Base-Ten System
Work
1. Make a base-ten math drawing for 46 + 27 and show how a child could use the
drawing side by side with the standard algorithm for adding 46 + 27 to help
make sense of the algorithm.
2. Make a base-ten math drawing for 43 − 28 and show how a child could use the
drawing side by side with the standard algorithm for subtracting 43 −28 to help
make sense of the algorithm.
3. Explain clearly and concretely why we regroup the way we do when we use the
standard subtraction algorithm to subtract
102
−8
4. Using the example 305 − 7, explain the logic behind the regrouping process.
5. What specific feature or aspect of the base-ten system do we use when we
regroup in addition or subtraction? Explain.
6. Describe how to use bundled objects to explain why the standard procedure for
adding 3.5 + 0.78 makes sense. Be sure to explain the regrouping process.
7. A store owner buys small, novelty party favors in bags of one dozen and boxes
of one dozen bags (for a total of 144 favors in a box). The store owner has 7
boxes, 2 bags, and 1 individual party favors at the start of the month. At the
end of the month, the store owner has 2 boxes, 8 bags, and 6 individual party
favors left. How many favors did the store owner sell? Give the answer in terms
of boxes, bags, and individual favors.
Solve this problem by working with a sort of expanded form for these party
favors, in other words, working with
7(boxes)
2(boxes)
+
+
2(bags) +
8(bags) +
1(individual)
6(individual)
Regroup among the boxes, bags, and individual party favors, briefly indicating
your reasoning.
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CHAPTER 3
8. Bob wants to figure out how much time elapsed from 10:55 am to 11:30 am.
Bob does the following:
0 1210
1 /1: /3 /0
−10 : 55
0 : 75
and says the answer is 75 minutes. Is Bob right? If not, explain what is wrong
with his method and show how to modify his method of regrouping to make it
correct.
9. To solve 341 − 176, a student writes the following:
341
176
−5
−30
200
200
−30
170
−5
165
341 − 176 = 165
Describe the student’s solution strategy and discuss why the strategy makes
sense. Expanded forms may be helpful to your discussion.
3.4
Adding And Subtracting Fractions
1. Using the example 53 + 54 , explain why we add fractions that have the same
denominator the way we do. Use our definition of fractions in your explanation.
2. Using the example 32 + 43 , explain why we add fractions the way we do. What are
we really doing when we carry out the procedure for adding fractions? What is
the logic behind this procedure? Explain your answer clearly and in detail.
3. Using the example 13 + 41 , explain why we give fractions a common denominator
to add them. What is the logic behind that process? What are we really doing
when we give fractions a common denominator?
4. Describe how to convert a mixed number, such as 2 34 , to an improper fraction, and explain why this procedure makes sense. In other words, explain the
rationale and reasoning behind the procedure.
5. You showed Johnny the diagram in Figure 3.2 to explain why 1 35 =
8
Johnny says that it shows 10
, not 85 . What must you clarify?
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Copyright 8
5
but
SECTION 3.4
31
Figure 3.2: Showing 1 35 =
6. Frank says that
2
3
+
2
3
=
4
6
8
5
and uses the diagram in Figure 3.3 as evidence.
(a) Explain why Frank’s method is not a valid way to add fractions. Be specific
and explain where the flaw is in Frank’s reasoning.
(b) Use our definition of fractions to explain how to add
2
3
+ 32 .
add them together:
Figure 3.3:
2
3
+
2
3
=
4
6
7. Susie says that 12 + 34 = 64 and uses the diagram in Figure 3.4 as evidence.
Explain why Susie’s method is not a valid way to add fractions. Be specific.
(Do not explain how to do the problem correctly, explain where the flaw is in
Susie’s reasoning.)
Figure 3.4:
1
2
+
3
4
=
4
6
8. Which of the following problems can be solved by adding 31 + 14 ? For those
problems that can’t be solved by adding 13 + 14 , solve the problem in another
way if there is enough information to do so, or explain why the problem cannot
be solved.
(a) One third of the boys in Mrs. Scott’s class want to have a peanut butter
sandwich for lunch. One fourth of the girls in Mrs. Scott’s class want to
have a peanut butter sandwich for lunch. What fraction of the children in
Mrs. Scott’s class want to have a peanut butter sandwich for lunch?
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CHAPTER 3
(b) One third of the pizzas served at a party have pepperoni on them. One
fourth of the pizzas served at the party have mushrooms on them. What
fraction of the pizzas served at the party have either pepperoni or mushrooms on them?
(c) The pizzas served at a party all have only one topping. One third of the
pizzas served at the party have pepperoni on them. One fourth of the
pizzas served at the party have mushrooms on them. What fraction of the
pizzas served at the party have either pepperoni or mushrooms on them?
9. Is the following a valid problem for 31 + 52 ? If not, explain briefly why not and
modify the problem so that it becomes a problem for 13 + 25 .
Problem: A garden is divided into two parts. 13 of the first part is planted with
tulips and 52 of the second part is planted with daffodils. What fraction of the
garden is planted with flowers?
10. Can the following problem be solved by subtracting 13 − 41 ? If not, explain why
not, and solve the problem in another way if there is enough information to do
so.
Problem: There is 13 of a pie left over from yesterday. Julie eats
pie. Now how much pie is left?
1
4
of the leftover
11. Can the following problem be solved by subtracting 13 − 14 ? If so, use it to explain
why the procedure we use for subtracting fractions is valid. If not, explain why
not, write a different problem for 31 − 14 .
Problem: There is 13 of a pie left over from yesterday. Julie eats
pie. Now how much pie is left?
1
4
of the leftover
12. What is wrong with the following problem? Give two different ways to restate
the problem clearly and accurately. Solve your restated problems.
Problem: Tamar has
does Minghua have?
3
4
of a cup of juice. Minghua has
1
3
more. How much juice
13. Tommy says that 23 − 21 = 13 and gives the reasoning indicated in Figure 3.5 to
support his answer. Is Tommy right? If not, what is wrong with his reasoning
and how could you help him understand his mistake and fix it? Don’t explain
how to solve the problem correctly; explain where Tommy’s reasoning is flawed.
14. Write and solve a word problem for
2
3
+ 12 . (No calculator allowed.)
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First I showed 2/3. Then when you
take away half of that you have 1/3 left.
Figure 3.5: Tommy’s Idea For
15. Write and solve a word problem for
2
3
2
3
−
1
2
− 12 . (No calculator allowed.)
16. Write and solve a word problem for 2 14 + 1 12 . (No calculator allowed.)
17. Write and solve a word problem for 2 14 − 1 12 . (No calculator allowed.)
18. Show how to calculate 3 25 + 1 23 in two different ways without a calculator. In
each case, express your answer as a mixed number. Explain why why both of
your methods are legitimate.
19. Show how to calculate 3 21 − 1 32 in two different ways without a calculator. In
each case, express your answer as a mixed number. Explain why why both of
your methods are legitimate.
20. Determine the fraction of the square in Figure 3.6 that is shaded. Explain your
reasoning.
Figure 3.6: What Fraction of the Area Is Shaded?
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CHAPTER 3
21. A farmer has two plots of land. Twenty percent of the first plot is planted with
cotton and 40% of the second plot is planted with cotton. Can we calculate the
percentage of the farmer’s (total) land that is planted with cotton by adding
20% + 40%? Explain your answer.
22. A farmer has two plots of land, both of the same size. Twenty percent of the
first plot is planted with cotton and 40% of the second plot is planted with
cotton. What percentage of the farmer’s (total) land is planted with cotton?
Explain your answer.
3.5
Adding and Subtracting Negative Numbers
1. Write and solve a Compare problem for 2 − (−5) =? in which one quantity is
2, the other quantity is −5, and the difference between the two quantities is
unknown.
2. Show how to use a number line to calculate 2−(−3). Briefly explain the method.
3. Show how to use a number line to calculate −2 + (−3). Briefly explain the
method.
4. Show how to use a number line to calculate −2 − (−3). Briefly explain the
method.
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Multiplication
35
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4.1
Interpretations of Multiplication
1. Write an Array word problem that can be solved by multiplying 4 × 9. Explain
clearly why the problem can be solved by multiplying 4×9 by using the definition
of multiplication as we have described it.
2. Write an Ordered Pair word problem that can be solved by multiplying 2 × 6.
Explain clearly why the problem can be solved by multiplying 2 × 6 by using
the definition of multiplication as we have described it.
3. Write a Multiplicative Comparison word problem that can be solved by multiplying 4 × 9. Explain clearly why the problem can be solved by multiplying
4 × 9 by using the definition of multiplication as we have described it.
4. Use the definition of multiplication, as we have described it, to explain why you
can solve the following problem by multiplying.
There are 12 inches in a foot. How long in inches is a 4 foot long piece of rope?
5. Use the definition of multiplication, as we have described it, to explain why you
can solve the following problem by multiplying.
There are 5280 feet in a mile. How long in feet is a 3 mile long stretch of road?
6. Use the definition of multiplication, as we have described it, to explain why you
can solve the following problem by multiplying.
Will is driving 55 miles per hour. If he continues driving at that speed, how far
will he drive in 4 hours?
7. Use the definition of multiplication, as we have described it, to explain why you
can solve the following problem by multiplying.
How many two-letter codes can be made using only the letters A, B, C, and D?
(Double letters, such as AA are allowed. The code AB is considered different
from BA.)
8. Use the definition of multiplication, as we have described it, to explain why you
can solve the following problem by multiplying.
How many two-letter codes can be made? (Double letters, such as AA are
allowed. The code AB is considered different from BA.)
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9. How many two-entry security codes can be made using only the digits 0, 1, 2,
3, 4, 5, 6, 7, 8, 9, and the letters A, B, C, D, E, F, G, H (so for example, 3G is
one such code, so are CA, GG, B5, 57, and 00)? Explain your solution clearly
and in detail.
10. A club has 100 members. How many different pairs of members could be chosen
to be co-presidents? (So for example, if Bob, Martha and Trey are members,
then Bob and Martha could be co-presidents, or Bob and Trey could be copresidents, or Martha and Trey could be co-presidents). Explain your solution
clearly and in detail.
4.2
Why Multiplying by 10 Is Special in Base Ten
1. When we multiply a number by 10, why do the digits (in the base-ten representation) shift one place to the left?
2. Using the example 10 × 32, explain why multiplying a number in base ten by
10 shifts all the digits one place to the left.
3. (a) Describe what happens to the base ten representation of a number when
we multiply the number by 10. Give a single answer that you could use
both with students who are studying whole numbers (and not working with
decimals) as well as with students who are studying decimal multiplication.
(b) Use the definition of multiplication, as we have described it, to explain
why your description in part (a) is valid. Work with the examples 10 × 13
and 10 × 1.3, then discuss briefly what will happen in other cases.
4. Janice is confused and thinks that 10 × 1.3 is 1.30. Tell Janice a rule for
multiplying by 10 that applies to whole numbers and decimals and explain why
this rule is valid.
4.3
The Commutative and Associative Properties
of Multiplication, Areas of Rectangles, and Volumes of Boxes
1. (a) State the commutative property of multiplication.
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(b) Use a math drawing to help you explain why the commutative property of
multiplication is valid (for counting numbers). Even though your drawing
will provide only one specific example, your explanation should nevertheless be general.
2. What is the commutative property of multiplication? Use a math drawing to
help you explain why the commutative property of multiplication is valid.
3. There are 15 containers, each of which is filled with 3 tennis balls. Bob calculates
the total number of tennis balls by counting by 3s fifteen times. In other words,
Bob counts 3, 6, 9, 12, . . . 42, 45. Michael calculates the total number of tennis
balls by adding 15 + 15 + 15 = 45. Are both methods correct? Are the methods
related? Is there a relevant property of arithmetic? Explain your answers to
these questions.
4. Give an example of a problem that can be made easy to solve mentally by using
the commutative property of multiplication. Write a relevant equation.
5. A rug is 4 feet wide and 5 feet long. Use our definition of multiplication to
explain why we can calculate the area of the rug by multiplying.
6. A box is 2 feet deep, 3 feet wide, and 4 feet tall. Use our definition of multiplication to explain why we can calculate the volume of the box by multiplying.
7. A cubic foot of water weighs about 62 pounds. How much will the water in a
rectangular fish pond weigh if the fish pond is 3 feet wide, 4 feet long, and 2
feet deep? Use the definition of multiplication to explain why we can multiply
to solve this problem.
8. What is the associative property of multiplication? Explain how to use Figure 4.1 to illustrate the associative property of multiplication.
9. To calculate 4 × 60 mentally, we can just calculate 4 × 6 = 24 and then put a
zero on the end to get the answer, 240. Use the math drawing in Figure 4.2 to
help you explain why this method of calculation is valid.
10. Which property or properties of arithmetic do you use when you calculate 3×70
by first calculating 3 × 7 = 21 and then putting a zero on the end of 21 to make
210? Write equations to show which properties are used and where.
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Figure 4.1: A Picture of Dashes
Figure 4.2: Explaining a Mental Method for Calculating 4 × 60
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11. Write at least two different expressions for the total number of triangles in Figure 4.3. Each expression is only allowed to use the numbers 3, 4, and 5, the
multiplication symbol, and parentheses. In each case, use the definition of multiplication (as we have described it) to explain why your expression represents
the total number of triangles in Figure 4.3.
Figure 4.3: How Many Triangles?
12. Use the definition of multiplication, as we have described it, and some of the
math drawings in Figure 4.4 to explain clearly why
3 × (5 × 2) = (3 × 5) × 2
Be specific when you refer to the pictures.
13. Use the associative property of multiplication to make the problem 16 × 25
easy to calculate mentally. Write equations to show your use of the associative
property of multiplication.
14. Use the associative property of multiplication to make the problem 24 × 0.25
easy to calculate mentally. Write equations to show your use of the associative
property of multiplication. Explain how your solution method is related to
solving 24 × 0.25 by thinking in terms of money.
15. Give an example of how the associative property of multiplication can be used
to make a calculation easier.
16. Give an example showing how to use the associative property of multiplication to
make a calculation easy to do mentally. Write equations that go along with your
mental calculation method and show where you use the associative property.
17. (a) State the associative property of multiplication.
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A
D
B
C
E
F
G
Figure 4.4: Subdivided Boxes
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(b) Give an example to show how the associative property of multiplication
can make a calculation easier. Your example should genuinely make the
calculation easier and should not involve other properties of multiplication.
Write equations that show how the associative property is used.
18. Determine approximately how many dots are in Figure 4.5 by calculating. Explain your method. Do not count all the dots.
Figure 4.5: How Many Dots?
19. A lot of gumballs are in a glass container. The container is shaped like a box
with a square base. When you look down on the top of the container, you
see 25 gumballs at the surface, as shown in Figure 4.6. When you look at one
side of the container, you see 30 gumballs up against the glass, as shown in
Figure 4.6. You also notice that there are 6 gumballs against each vertical edge
of the container. Given this information, estimate the total number of gumballs
in the container. Explain your reasoning.
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looking down on
the top
looking at one
side
Figure 4.6: How Many Gumballs?
20. Write an estimation problem, or describe an estimation activity for elementary
school children (any grade), in which the children will use multiplication, or
ideas related to multiplication, to solve the problem or to complete the activity.
4.4
The Distributive Property
1. Describe one collection of objects whose total number is given by the expression
6×5+4
and another for the expression
6 × (5 + 4)
Make clear which is which and why each collection fits with its expression.
2. Nina divided her toy car collection into 8 groups with 4 cars in each group, but
there were 3 cars left over. Write an expression using the numbers 8, 4, and 3,
the symbols × (or ·) and +, and parentheses, if needed, for the total number
of toy cars in Nina’s collection. If you use parentheses, explain why you need
them; if you do not use parentheses, explain why you do not need them.
3. Describe a collection of objects whose total number is given by the expression
8×3+9×2
Explain briefly why your collection fits with the expression.
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4. (a) State the distributive property.
(b) Using a specific example, explain why the distributive property is valid.
Even though you use a specific example, your explanation should be general, in the sense that we should be able to see why it will hold true for
other whole numbers.
5. Explain how to use the distributive property to calculate 29 × 40 mentally by
first calculating 30 × 40. Write an equation to show your use of the distributive
property.
6. Describe a way to make 31 × 25 easy to calculate mentally. Write an equation
that corresponds to your mental method of calculation.
7. Give an example of how to use the distributive property to make a calculation easy to carry out mentally. Describe how to carry out the calculation
mentally. Write equations showing how the calculation strategy used the distributive property.
8. Give an example of a problem that can be made easier to calculate mentally by
using the distributive property. Write equations to show how the distributive
property is used. Your use of the distributive property should genuinely make
the calculation easier.
9. (a) State the distributive property.
(b) Give an example of a multiplication problem that can be made easier to
calculate by using the distributive property. Write equations to show how
the distributive property is used. Your use of the distributive property
should genuinely make the calculation easier.
10. Jim thinks that because 30 × 40 = 1200, and 1 × 1 = 1, it should follow that
31 × 41 = 1200 + 1 = 1201
Make a math drawing and use your drawing to help you explain to Jim how
30×40 and 31×41 are actually related. (A rough drawing will do, your drawing
does not have to be drawn perfectly to scale.)
11. (a) Use the distributive property several times to show why
(10 + 4) · (10 + 3) = 10 · 10 + 10 · 3 + 4 · 10 + 4 · 3
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(b) Draw an array to show why
(10 + 4) · (10 + 3) = 10 · 10 + 10 · 3 + 4 · 10 + 4 · 3
(c) Relate the steps in your equations in part (a) to your array in part (b).
4.5
Properties of Arithmetic, Mental Math, and
Single-Digit Multiplication Facts
1. Ashley knows her 1×, 2×, 3×, 4×, and 5× multiplication tables well.
Briefly describe how the three arrays in Figure ?? provide Ashley with three
different ways to determine 6 × 8 from multiplication facts that she already
knows well. In each case, write an equation that corresponds to the array and
that shows how 6 × 8 is related to other multiplication facts. For each equation,
say which property of arithmetic is used.
Figure 4.7: Different Ways to Think of 6 × 8
2. Ashley knows her 1×, 2×, 3×, 4×, and 5× multiplication tables. She also
knows the squares 1 × 1, 2 × 2, 3 × 3, . . . , 9 × 9.
Draw arrays showing three different ways that Ashley could use the multiplication facts she knows to determine 6 × 7. In each case, write an equation
that corresponds to the array and that shows how 6 × 7 is related to other
multiplication facts. For each equation, say which property of arithmetic is
used.
3. Explain how to use reasoning and the fact 25 × 25 = 625 to determine 26 × 26.
Draw a picture that helps you relate 25 × 25 and 26 × 26 (your picture need
not be drawn perfectly to scale). Then write equations that incorporate your
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CHAPTER 4
reasoning and that also show why you can calculate the answer to 26 × 26 the
way you do. Write your equations in the following form:
26 × 26 =
=
..
.
some expression
some expression
..
.
4. Keisha says that it’s easy to multiply even numbers by 5 because you just take
half of the number and put a zero on the end. Write equations that incorporate
Keisha’s method and that demonstrate why her method is valid. Use the case
5 × 8 for the sake of concreteness. Write your equations in the following form:
5×8 =
=
..
.
some expression
some expression
..
.
= 40.
5. What property of arithmetic are you using when you think of the 5× multiplication table as half of the 10× table? Explain.
6. Suppose that the sales tax where you live is 5%. Compare the total amount of
sales tax you would pay if you went to a store and bought a pair of pants and a
sweater at the same time, versus if you first bought the pair of pants and then
returned to the store later to buy the sweater.
Which property of arithmetic is relevant to this sales tax problem? Write an
equation to help you explain your answer.
7. A store is having a “30% off everything” sale. The clerk adds up the cost of all
your items and then takes 30% off this total. Do you get the same discount as
if the clerk took 30% off each individual item and then totaled? Explain your
answer, relating it to a property of arithmetic.
8. Halley calculates 45% of 280 in the following way:
Half of 280 is 140. I know 10% is 28, so 5% is half of that, which is
14. So I get 140 minus 14, which is 126.
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(a) Explain briefly why it makes sense for Halley to solve the problem the way
she does. What is the idea behind her strategy?
(b) Write a string of equations that incorporate Halley’s ideas. Which properties of arithmetic did Halley use (knowingly or not) and where? Be
thorough and be specific. Write your equations in the following format:
45% × 280 =
=
..
.
some expression
some expression
..
.
= 126.
9. Jamal calculates 7 × 48 in the following way:
7 times 5 is 35, times 10 is 350. Now I need to take away 7 times 2,
which is 14. So 350, 340, 336. So the answer is 336.
(a) Explain briefly why it makes sense for Jamal to solve the problem the way
he does. What is the idea behind his strategy?
(b) Write a string of equations that incorporate Jamal’s ideas. Which properties of arithmetic did Jamal use (knowingly or not) and where? Be
thorough and be specific. Write your equations in the following format:
7 × 48 =
=
..
.
some expression
some expression
..
.
= 336.
10. In words, describe a way to make the problem 95% × 750 easy to calculate
mentally. Then write equations that incorporate your solution strategy. Write
your equations in the following form:
95% × 750 =
=
..
.
some expression
some expression
..
.
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11. Describe two different ways to make the problem
25 × 48
easy to calculate mentally. For each of your two methods, write a corresponding
string of equations. Write your equations in the following form:
25 × 48 =
=
..
.
4.6
some expression
some expression
..
.
Why Algorithms for Multiplying Whole Numbers Work
1. (a) Use the partial products and standard algorithms to calculate 23 × 24:
24
×23
24
×23
(b) Draw an array for 23 × 24 (use graph paper). Subdivide the array in a
natural way so that the parts of the array correspond to the steps in the
partial products algorithm. Indicate the correspondence between the parts
of the array and the steps in the algorithm.
(c) Indicate which parts of the array that you drew for part (b) correspond to
the steps of the standard algorithm for 23 × 24.
(d) Solve 23 × 24 by writing equations that use expanded forms and the distributive property. Relate your equations to the steps in the partial products algorithm.
2. (a) Use the partial products algorithm to calculate
34
×27
(b) Use the meaning of multiplication and an array to explain why the partial
products algorithm gives the correct answer to the multiplication problem
in part (a). (Use graph paper for your array.)
(c) Show why the partial products algorithm calculates the correct answer
to the multiplication problem in part (a) by writing equations that use
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properties of arithmetic and that incorporate the calculations of the partial
products algorithm. (FOIL is not a property of arithmetic.) Write your
equations in the following format:
34 × 27 =
=
..
.
some expression
some expression
Identify the properties of arithmetic that you used and show where you
used them.
(d) Relate your equations for part (c) to your array for part (b).
34
, give a clear and coherent explanation for why the
×27
partial products algorithm produces correct answers to multiplication problems.
Use all of the following in your explanation: the definition of multiplication, a
math drawing (use graph paper or make a rough drawing), and the distributive
property. Your explanation should be appropriate for someone who is just
learning the partial products algorithm and doesn’t yet know the standard
multiplication algorithm.
3. Using the example
4. (a) Use the partial products and standard algorithms to calculate 82 × 437:
437
×82
437
×82
(b) Draw a rectangle to represent an array for 82 × 437 (your rectangle need
not be to scale). Subdivide the rectangle in a natural way so that the parts
of the rectangle correspond to the steps in the partial products algorithm.
Indicate the correspondence between the parts of the rectangle and the
steps in the algorithm.
(c) Indicate which parts of the rectangle that you drew for part (b) correspond
to the steps of the standard algorithm for 82 × 437.
(d) Solve 82 × 437 by writing equations that use expanded forms and the
distributive property. Relate your equations to the steps in the partial
products algorithm.
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34
using the standard multiplication algorithm, we start
×27
the second line by writing a zero:
5. When we multiply
2
34
×27
238
0
Explain why we place this zero in the second line. What is the rationale behind
the procedure of placing a zero in the second line?
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Multiplication of Fractions,
Decimals, and Negative Numbers
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5.1
Multiplying Fractions
1. Ken ordered 45 of a ton of sand. Ken wants to receive 32 of his order now (and
1
of his order later). What fraction of a ton of sand should Ken receive now?
3
Show how to use pictures to determine the answer to this problem. Explain
clearly how to interpret your pictures.
2. Write a word problem for 13 × 32 . Use the definition of multiplication and
math drawings to determine the answer to the multiplication problem. Explain clearly.
3. Write a word problem for 4 × 32 . Use the definition of multiplication (as we have
described it), and math drawings to determine the answer to the multiplication
problem. Explain clearly.
4. Write a word problem for 32 × 4. Use the definition of multiplication (as we have
described it), and math drawings to determine the answer to the multiplication
problem. Explain clearly.
5. Explain why it would be easy to interpret the math drawing in Figure 5.1
9
incorrectly as showing that 3 × 34 = 12
. Explain how to interpret the math
3
drawing as showing 3 × 4 .
Figure 5.1: A math drawing for 3 ×
3
4
6. Write a simple word problem for
3 3
×
4 5
Use your word problem and math drawings to explain clearly why it makes
sense that the answer to the fraction multiplication problem is
3×3
4×5
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In particular, use your math drawings to explain why we multiply the numerators and why we multiply the denominators.
7. Use math drawings and our definition of multiplication to explain why we multiply fractions the way we do. Use the example
2 4
2×4
× =
3 5
3×5
to explain the reasoning in detail.
8. Which of the following are word problems for 12 × 34 and which are not? Explain
briefly in each case.
(a) There is 43 of a cake left. One half of the children in Mrs. Brown’s class
want cake. How much of the cake will the children get?
(b) A brownie recipe used 34 of a cup of butter for a batch of brownies. You
ate 12 of a batch. How much butter did you consume when you ate those
brownies?
(c) Three quarters of a pan of brownies is left. Johnny eats
brownies. Now what fraction of a pan of brownies is left?
1
2
of a pan of
(d) Three quarters of a pan of brownies is left. Johnny eats
How many brownies did Johnny eat?
1
2
of what is left.
(e) Three quarters of a pan of brownies is left. Johnny eats
What fraction of a pan of brownies did Johnny eat?
1
2
of what is left.
9. Consider this word problem about baking cookies:
You are baking cookies for your class. You put pink frosting on 13 of
the cookies and you put red sugar on 14 of the cookies. How many
cookies have both pink frosting and red sugar on them?
(a) Is the cookie word problem a problem for 13 × 41 ? If so, explain briefly why
it is, if not, modify the problem so that it is a problem for 13 × 14 .
(b) Is the cookie word problem a problem for 13 + 41 ? If so, explain briefly why
it is, if not, modify the problem so that it is a problem for 13 + 41 .
10. Write a word problem for 1 13 × 1 21 . Explain how to use math drawings and the
definition of multiplication to solve your problem.
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Figure 5.2: What fraction is shaded?
11. Write an expression that uses both multiplication and addition (or subtraction)
to describe the total fraction of Figure 5.2 that is shaded. Explain briefly.
12. In order to understand fraction multiplication thoroughly, we must be able to
work simultaneously with different wholes. Using the example 23 × 52 , explain
why this is so. What are the different wholes that are associated with the
fractions in the problem 32 × 25 (including the answer to the problem)?
13. Discuss why we must we must develop an understanding of multiplication that
goes beyond seeing it as repeated addition.
14. Make the case that multiplication means the same thing whether we are multiplying fractions or whole numbers. Use word problems for 2 × 3 and 12 × 31 to
illustrate.
5.2
Multiplying Decimals
1. Johnny is solving the multiplication problem 13.8 × 1.42 by multiplying longhand. Ignoring the decimal points, Johnny gets 19596, and now he must figure
out where the decimal point goes. Explain how Johnny could reason about
the sizes of the numbers to figure out where the decimal point should go in his
answer if he doesn’t know the rule about adding the number of places behind
the decimal points in 13.8 and 1.42.
2. Write a word problem for 3.8 × 9.7.
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3. When we multiply 0.23 × 1.7 we first multiply as if the decimal points were not
there:
1.7
×0.23
51
340
391
and then we place the decimal point in the answer 391.
(a) How could a student determine where to place the decimal point if the
student has forgotten the rule about adding the number of places behind
the decimal points in 1.7 and 0.23?
(b) Explain clearly why the rule for placement of the decimal point in decimal
multiplication problems makes sense. What is the logic behind adding the
number of places behind the decimal points? Use the example 0.23 × 1.7
to discuss this logic.
4. When we multiply 2.3 × 1.7 we first multiply as if the decimal points were not
there:
1.7
×2.3
51
340
391
and then we place the decimal point in the answer 391. Explain clearly why it
makes sense to put the decimal point where we do.
5. Explain how to write 3.7 and 1.29 as improper fractions whose denominators
are products of 10s. Then use fraction multiplication to explain where to place
the decimal point in the solution to 3.7 × 1.29. Show how to use the products
of 10s in the denominators to explain why we add the number of digits behind
the decimal points in 3.7 and 1.29.
6. When we multiply decimals such as
1.36
×2.7
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we first multiply ignoring decimal points:
136
×27
952
2720
3672
Then we add the number of digits to the right of the decimal points in the two
numbers we multplied and put the decimal point that many places from the end
in the answer. In this example, we place the decimal point 1 + 2 = 3 places from
the end of 3672 to obtain the answer 3.672. Using this example, explain why
this procedure for determining where to place the decimal point makes sense.
What is the logic behind this procedure of adding the number of digits behind
the decimal points?
7. Suppose you multiply a decimal that has 2 digits to the right of its decimal
point by a decimal number that has 2 digits to the right of its decimal point.
Explain why you put the decimal point 2 + 2 places from the end of the product
calculated without the decimal points.
8. How can it be that 2.5 × 3.4 = 8.5? The decimal 8.5 has only 1 digit to the
right of its decimal point but the sum of the number of digits to the right of
the decimal points in 2.5 and 3.4 is 2. Explain the discrepancy.
9. (a) Determine the area of the rectangle in Figure 5.3 without multiplying.
(b) Use part (a) to explain why the product of 2.6 × 3.4 has a (non-zero) entry
in the hundredths place.
10. (a) Determine the area of the rectangle in Figure 5.3 without multiplying.
(b) Use the rectangle in Figure 5.3 to explain where the decimal point will be
located in the product 2.6 × 3.4.
5.3
Multiplying Negative Numbers
1. Write a word problem for 3 × (−2). Solve the word problem, thereby explaining
why 3 × (−2) is negative.
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Figure 5.3: A 2.6-unit-by-3.4-unit rectangle
2. Given that 3 × (−2) = −6, which property of arithmetic allows us to deduce
that (−2) × 3 is negative?
3. Given that 3 × (−2) = −6, use a property of arithmetic to deduce that (−3) ×
(−2) is positive.
4. Given that we know how to multiply non-negative numbers, explain why the
following make sense:
(a) 3 × −2 = −6
(b) −3 × 2 = −6
(c) −3 × −2 = 6
5. In ordinary language, the term “multiply” means “make larger,” as in:
Go forth and multiply.
In mathematics, does multiplying always make larger? Give at least three
different kinds of examples to help you explain your answer.
5.4
Powers and Scientific Notation
1. Use the definition of 10A and 10B to explain why it is always true that 10A ×
10B = 10A+B whenever A and B are counting numbers.
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2. Explain why it makes sense that we define 100 to be 1.
3. Write 425 million in ordinary decimal notation and in scientific notation.
4. Write 84 billion in ordinary decimal notation and in scientific notation.
5. Write (1.2 × 1015 ) × (4.5 × 1017 ) in scientific notation.
6. A calculator might display the answer to
4, 444, 444 × 3, 333, 333
as
1.481481 E 13
(a) What does the calculator’s display mean?
(b) What information about the solution to 4, 444, 444 × 3, 333, 333 can you
obtain from the calculator’s display? Give as thorough an answer as you
can.
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Division
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6.1
Interpretations of Division
1. Write two story problems for 28 ÷ 4, one for the “how many in each group?”
interpretation of division and one for the “how many groups?” interpretation
of division. Indicate which is which.
2. Write a multiplicative comparison problem for 28 ÷ 4.
3. (a) Explain why 12 ÷ 0 is not defined by rewriting the problem 12 ÷ 0 =? as
a multiplication problem.
(b) Explain why 12 ÷ 0 is not defined by writing a story problem for 12 ÷ 0.
4. (a) Is 0 ÷ 10 defined or not? Write a story problem for 0 ÷ 10 and use the
story problem to discuss whether or not 0 ÷ 10 is defined.
(b) Is 10 ÷ 0 defined or not? Write a story problem for 10 ÷ 0 and use the
story problem to discuss whether or not 10 ÷ 0 is defined.
5. Students often get confused about when division involving 0 is defined and when
it isn’t. Is 5 ÷ 0 defined? Is 0 ÷ 5 defined? Explain your answers clearly.
6. Is 0 ÷ 0 defined? Explain.
6.2
Division and Fractions and Division with Remainder
1. Write a simple “how many in each group?” word problem for 3 ÷ 4. Using your
word problem, explain why 3 ÷ 4 = 34 . Your explanation should be general in
the sense that you could see why the equation 34 = 3 ÷ 4 would still be true if
other counting numbers were to replace 3 and 4.
2. There are 2 identical pans of brownies to be divided equally among 3 people.
(a) What fraction of a pan of brownies will each person get? Explain your
reasoning.
(b) What fraction of the total brownies does each person get? Explain your
reasoning.
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3. Describe how to get the mixed number answer to 19÷6 from the whole-numberwith-remainder answer. By considering a simple word problem, explain why the
method you describe makes sense.
4. (a) Write a simple “how many groups?” word problem for 11 ÷ 4 for which
the answer “2, remainder 3” is appropriate. Explain what the answer “2,
remainder 3” means in the context of the word problem.
(b) Write a simple “how many in each group?” word problem for 11 ÷ 4 for
which the answer 2 34 is appropriate. Explain what the answer 2 43 means in
the context of the word problem.
5. Write a simple “how many groups?” word problem for 11 ÷ 4 for which the
answer “2, remainder 3” is appropriate. Explain what the answer “2, remainder
3” means in the context of the word problem. Then explain what the answer
2 43 would mean in that context (it might or might not make sense).
6. Write a simple “how many in each group?” word problem for 11 ÷ 4 for which
the answer 2 34 is appropriate. Explain what the answer 2 34 means in the context
of the word problem. Then explain what the answer “2, remainder 3” would
mean in that context (it might or might not make sense).
7. (a) Write a “how many in each group?” word problem for 27 ÷ 4. Solve your
problem, attending to the remainder in an appropriate way.
(b) Write a “how many groups?” word problem for 27÷4. Solve your problem,
attending to the remainder in an appropriate way.
8. Make up and solve three different word problems for 9 ÷ 4.
(a) In the first word problem, the answer should best be expressed as 2, remainder 1. Explain why this answer is best.
(b) In the second word problem, the answer should best be expressed as 2 14 .
Explain why this answer is best.
(c) In the third word problem, the answer should best be expressed as 2.25.
Explain why this answer is best.
as a mixed number
9. Describe how to use division to write the improper fraction 23
3
and explain why the procedure you describe makes sense.
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10. What day of the week will it be 100 days from today? Use mathematics to solve
this problem. Explain your solution.
11. If January 1, 2004 is on a Thursday, then what day of the week will January 1,
2005 fall on? Use mathematics to solve this problem. Explain your solution.
12. For each of the following word problems, write a division problem that solves
the problem, give an appropriate answer to the problem, and say which of the
two interpretations of division is used (the “how many groups?” or the “how
many in each group?” interpretation, with or without remainder).
(a) If a box of laundry detergent costs $5 and washes 38 loads of laundry, then
how much does the detergent for one load of laundry cost?
division problem
appropriate answer
which interpretation
of division?
(b) If a box of laundry detergent costs $5 and washes 38 loads of laundry, then
how many loads of laundry can you wash for $1?
division problem
appropriate answer
which interpretation
of division?
(c) If a box of laundry detergent costs $5 and washes 38 loads of laundry, and
if you wash 6 loads of laundry per week, then how many weeks will a box
of laundry detergent last?
division problem
appropriate answer
which interpretation
of division?
13. For each of the following word problems, write the corresponding division problem, state which interpretation of division is involved (the “how many groups?”
or the “how many in each group?”, with or without remainder), and solve the
problem.
(a) Given that 1 quart is 4 cups, how many quarts of water is 35 cups of water?
(b) If your car used 15 gallons of gasoline to drive 330 miles, then how many
miles per gallon did your car get?
(c) If you drove 250 miles at a constant speed and if it took you 4 hours, then
how fast were you going?
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(d) Given that 1 foot is 12 inches, how long in feet and inches is a board that
is 90 inches long?
6.3
Why Division Algorithms Work
1. Taylor is working on the following problem: There are 450 stickers to be put
in packages of 16. How many packages of stickers can we make, and how many
stickers will be left over? Here are Taylor’s ideas:
Ten packages will use up 160 stickers. After another 10 packages, 320
stickers will be used up. After 1 more package, 336 stickers are used.
Then there are only 14 stickers left and that’s not enough for another
package. So the answer is 21 packages of stickers with 14 stickers left
over.
Write a single equation that incorporates Taylor’s reasoning. Use your equation
and the distributive property to obtain another equation which shows that
450 ÷ 16 has whole number quotient 21, remainder 14.
2. Amanda is working on the division problem 358 ÷ 25. Amanda’s work appears
in Figure 6.1
25
×2
50
7 × 50 = 350
7 × 2 = 14
14 R 8
Figure 6.1: Amanda’s work for 358 ÷ 25
(a) Explain why Amanda’s strategy makes sense. It may help you to work
with a word problem for 358 ÷ 25.
(b) Write equations that correspond to Amanda’s work and that demonstrate
that 358 ÷ 25 has whole number quotient 14, remainder 8.
3. Arthur is working on the division problem 358 ÷ 25. Arthur writes:
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250
+50
300
+50
350
10 × 25 = 250
10 + 2 + 2 = 14 remainder 8
(a) Explain why Arthur’s strategy makes sense. It may help you to work with
a word problem for 358 ÷ 25.
(b) Write equations that correspond to Arthur’s work and that demonstrate
that 358 ÷ 25 has whole number quotient 14, remainder 8.
4. Zoe is working on the division problem 358 ÷ 25. Zoe writes:
25
×2
50
×2
100
×3
300
+50
350
8 left
2 × 2 × 3 = 12
12 + 2 = 14
14 R 8
(a) Explain why Zoe’s strategy makes sense. It may help you to work with a
word problem for 358 ÷ 25.
(b) Write equations that correspond to Zoe’s work and that demonstrate that
358 ÷ 25 has whole number quotient 14, remainder 8.
5. Assume that you don’t know any kind of longhand method of division. Explain
how you can use reasoning to calculate 495 ÷ 35. It may help you to work with
a word problem.
6. (a) Write two word problems for 1957 ÷ 6, one for the “how many in each
group?” interpretation and one for the “how many groups?” interpretation.
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(b) Assume that you don’t know any longhand method of division. Explain
how you can use reasoning to calculate 1957 ÷ 6. Use one of your word
problems from part (a) to interpret your steps.
7. (a) Write a word problem for 3458 ÷ 6 using the “how many groups?” interpretation of division.
(b) Use the scaffold method to calculate 3458 ÷ 6. Interpret each step in the
scaffold method in terms of your word problem.
8. Allie calculates 8798 ÷ 14 as follows.
3
5
20
100
500
14)8798
−7000
1798
−1400
398
−280
118
−70
48
−42
6
(a) Describe how Allie could have solved the division problem using fewer
steps.
(b) Even though Allie used more steps than necessary, is her work still mathematically valid? That is, does Allie’s work correspond to legitimate reasoning? Explain.
9. (a) Use the standard long division algorithm to calculate 471 ÷ 3.
(b) Interpret each step in your calculation in part (a) in terms of the following
problem. You have 471 toothpicks bundled into 4 bundles of one hundred,
7 bundles of ten, and 1 individual toothpick. If you divide these toothpicks
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equally among 3 groups, how many toothpicks will each group get? Be sure
to include a discussion of how to interpret the “bringing down” steps.
10. (a) Use the standard long division algorithm to calculate 2639 ÷ 3 (the whole
number with remainder answer).
(b) Interpret each step in your calculation in part (a) in terms of the following
problem. You have 2639 toothpicks bundled into 2 thousands, 6 hundreds,
3 tens, and 9 individual toothpicks. If you divide these toothpicks equally
among 3 groups, how many toothpicks will each group get and how many
toothpicks will be left over? Be sure to discuss how to interpret the “bringing down” steps.
11. Using the example 3)462 and a suitable context for that division problem, explain the logic and reasoning behind the standard division algorithm. Include
a discussion of how to interpret the “bringing down” steps. In your discussion,
do not use the phrase “goes into.”
12. Using the example 3)462 and the context of dividing up bundled toothpicks or
money, explain the logic and reasoning behind each step in the standard division
algorithm. Include a discussion of how to interpret the “bringing down” steps.
13. (a) Use the standard long division algorithm to determine the decimal answer
to 953 ÷ 6 to the hundredths place.
(b) Interpret each step in your long division calculation in part (a) in terms of
dividing $953 equally among 6 people.
14. The first step in dividing 75 by 4 using standard long division is shown below.
Finish the division to the hundredths place, explaining how to interpret each
step in terms of dividing $75 equally among 4 people. Include a discussion of
how to interpret the “bringing down” steps.
1
4) 75
−4
3
15. Use the standard long division algorithm to calculate 39 ÷ 4 (to the hundredths
place) and interpret each step in the process in terms of dividing $39 equally
among 4 people. Include a discussion of how to interpret the “bringing down”
steps.
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16. Wu has been making errors on his division problems. Here are some of Wu’s
answers:
150 ÷ 7 = 21.3
372 ÷ 8 = 46.4
154 ÷ 12 = 12.10
What is Wu likely to be confused about? Explain.
17. Use the subdivided square in Figure 6.2 to help you explain why the decimal
representation of 18 is 0.125.
Figure 6.2: Show
1
8
18. Describe how to use dimes and pennies to help you explain why
Explain your reasoning.
1
8
= 0.125.
19. (a) Use the standard long division algorithm to determine the decimal representation of 18 .
(b) Interpret the steps to the hundredths place in your long division in part
(a) in terms of dividing $1 equally among 8 people. Include a discussion
of how to interpret the “bringing down” steps.
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6.4
Fraction Division from the How Many Groups?
Perspective
1. Write a “how many groups?” story problem for 23 ÷ 12 and solve your problem
using either a strip diagram, a table, or a double number line. Explain your
reasoning.
2. Write a “how many groups?” word problem for 2 21 ÷ 43 and solve your problem
using either a strip diagram, a table, or a double number line. Explain your
reasoning.
3. Sally is working on the division problem 1 34 ÷ 21 by using the word problem,
“how many 12 cups of water are in 1 34 cups of water?” Sally makes a drawing
like the one in Figure 6.3 and gives the answer 3 41 . Discuss Sally’s work.
Figure 6.3: A Picture for 1 34 ÷
1
2
4. Write a simple “how many groups?” word problem for 12 ÷ 31 and use the word
problem and a math drawing to help you explain why you can solve the division
problem by first giving the fractions a common denominator and then dividing
the numerators.
5. Use the fact that we can rewrite the division problem 35 ÷ 23 =? as a multiplication
problem with an unknown factor to explain why the division problem can be
solved by multiplying 35 by 32 .
6. Seyong says that 2 ÷ 13 can’t be equal to 6 because 6 is greater than 2 but when
you divide a number, the answer must be smaller than the number you started
with. Write a story problem for 2 ÷ 13 and use simple, concrete reasoning to help
you explain why 2 ÷ 31 really is equal to 6. (Do not just “invert and multiply”
to show that 2 ÷ 13 = 6.)
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7. Discuss: is it correct to characterize division as “breaking down” and “making
smaller”?
6.5
Fraction Division from the How Many in One
Group? Perspective
1. Write a “how many in one group?” word problem for 12 ÷ 13 and use your
problem and a strip diagram, a table, or a double number line to explain why it
makes sense to calculate 21 ÷ 13 by “inverting and multiplying,” in other words,
by multiplying 21 by 13 .
2. Write a “how many in one group?” word problem for 6÷ 34 and use your problem
and a strip diagram, a table, or a double number line to explain why it makes
sense to calculate 6 ÷ 34 by “inverting and multiplying,” in other words, by
multiplying 6 by 43 .
3. Use the “how many in one group?” perspective and a drawing, a table, or
a double number line to explain why it makes sense to divide fractions by
“inverting and multiplying.” Use the example 65 ÷ 43 to illustrate the reasoning.
4. Write a word problem (any type) for 12 ÷ 23 and solve your problem by reasoning
about a strip diagram, a table, or a double number line. Explain your reasoning.
5. Consider this word problem about a potion Harry made:
In class, Harry made a liter of potion. At the end of class, 31 of a
liter was left. When Harry looked away, someone poured out 14 of the
potion that was left. Then how much potion was left?
(a) Is the potion problem a problem for 13 − 14 ? If so, explain how solve it
with the aid of a drawing or diagram. If not, explain briefly why not and
modify the problem so that it is a problem for 13 − 41 (you do not have to
solve the problem).
(b) Is the potion problem a problem for 14 · 13 ? If so, explain how solve it with
the aid of a drawing or diagram. If not, explain briefly why not and modify
the problem so that it is a problem for 41 · 13 (you do not have to solve the
problem).
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(c) Is the potion problem a problem for 14 ÷ 13 ? If so, explain how solve it
with the aid of a drawing or diagram. If not, explain briefly why not and
modify the problem so that it is a problem for 14 ÷ 31 (you do not have to
solve the problem).
6. Which of the following are word problems for the division problem 23 ÷ 21 ? For
those that are, which interpretation of division is used? For those that aren’t,
indicate briefly why not.
(a)
2
3
of a bag of jelly worms make
does it take to make one cup?
1
2
a cup. How many bags of jelly worms
(b)
2
3
1
2
a cup. How many cups of jelly worms
of a bag of jelly worms make
are in one bag?
(c) You have 23 of a container of chocolate syrup and a recipe that calls for
1
of a cup of chocolate syrup. How many batches of your recipe can you
2
make (assuming you have enough of the other ingredients)?
(d) You have 23 of a cup of chocolate syrup and a recipe that calls for 21 of a
cup of chocolate syrup. How many batches of your recipe can you make
(assuming you have enough of the other ingredients)?
(e) If 23 of a pound of nails costs 21 of a dollar, then how many pounds of nails
should you be able to buy for 1 dollar?
(f) If you have 23 of a pound of nails and you divide the nails in 12 , then how
many pounds of nails will you have in each portion?
(g) If 12 of a pound of nails costs $1, then how much should you expect to pay
for 23 of a pound of nails?
7. Discuss the difference between dividing in half and dividing by 21 . Your discussion should include:
(a) a description of ways to express these two situations using mathematical
notation,
(b) word problems for both situations; say which is which and explain how to
solve the problems by reasoning about a strip diagram, table, or double
number line.
8. Write one word problem for 43 ÷ 21 and another word problem for 34 × 12 , making
clear which problem is which. In each case, explain how to reason with the aid
of a a strip diagram, a table, or a double number line to solve your problem.
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9. For each of the following problems, write a corresponding numerical division
problem, say which interpretation of division is involved, and solve the problem
without using a calculator.
(a) Marjorie needs 34 of a pound of yarn to knit a scarf that is 6 21 feet long.
How long a scarf can Marjorie knit with 1 pound of yarn?
(b) Marjorie needs 34 of a pound of yarn to knit a scarf that is 6 21 feet long.
How many many pounds of yarn will Marjorie need for a 1 foot long section
of scarf?
(c) Marjorie needs 34 of a pound of yarn to knit a scarf that is 6 21 feet long.
How many scarfs (each of which is 6 12 feet long) can Marjorie knit with 10
pounds of yarn?
10. For each of the following problems, write a corresponding numerical division
problem, say which interpretation of division is involved, and solve the problem
without using a calculator.
(a) Liquid pours out of a hose and into a vat at a steady rate. Starting with
an empty vat, it took 2 21 hours to fill the vat 43 full. How long will it take
to fill the vat completely?
(b) Liquid pours out of a hose and into a vat at a steady rate. Starting with
an empty vat, it took 2 12 hours to fill the vat 43 full. What fraction of the
vat was filled after 1 hour?
(c) Liz needs 3 12 bags of cement mix to make a walkway but she only has 2 34
bags. What fraction of the walkway can she make with the cement mix
she has?
11. Ed says that 4÷5 doesn’t make sense because you can’t divide a smaller number
by a bigger number. Give Ed two sensible word problems for 4 ÷ 5, one for the
“how many groups?” interpretation of division and one for the “how many in
each group?” interpretation. Solve each problem.
12. Fraction division word problems involve the simultaneous use of different wholes.
Write and solve a word problem for 2 13 ÷ 32 and discuss how different wholes are
involved.
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6.6
Dividing Decimals
1. Show how to calculate 1.5 ÷ 0.0004 without a calculator.
2. Which of the division problems can be interpreted as asking:
How many groups of
are in
?
14.7 ÷ 3.8 =?
1.47 ÷ 3.8 =?
1.47 ÷ 0.38 =?
147 ÷ 38 =?
3.8 ÷ 14.7 =?
3.8 ÷ 1.47 =?
0.38 ÷ 1.47 =?
38 ÷ 147 =?
Circle all that apply and explain your reasoning.
3. (a) Calculate 2.14 ÷ 0.7 to the hundredths place without a calculator. Show
your work.
(b) Describe the standard procedure for determining where to put the decimal
point in the answer to 2.14 ÷ 0.7.
(c) Explain in two different ways why the placement of the decimal point that
you described in part (b) is valid.
4. (a) Calculate 7.3 ÷ 0.21 to the hundredths place without a calculator. Show
your work.
(b) Describe the standard procedure for determining where to put the decimal
point in the answer to 7.3 ÷ 0.21.
(c) Explain in two different ways why the placement of the decimal point that
you described in part (b) is valid.
5. Bruce must calculate 7.82 ÷ 1.6 longhand, but he can’t remember what to do
about decimal points. Instead, Bruce solves the division problem 782 ÷ 16
longhand and gets the answer 48.875. Bruce knows that he must shift the
decimal point in 48.875 somehow to get the correct answer to 7.82÷1.6. Explain
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how Bruce could reason about the sizes of the numbers to determine where to
put the decimal point.
6. Without calculating the answers, explain why the division problems
13, 000, 000, 000 ÷ 8, 000, 000, 000
and
13 ÷ 8
are equivalent.
7. If the federal budget is $2.2 trillion and this budget were paid for equally by
the 310 million residents of the U.S., then how much would each person have to
contribute to the federal budget? Assume you only have a very simple calculator
that cannot use scientific notation and that displays at most 8 digits. Describe
how to use such a calculator to solve the problem about the federal budget and
explain why your solution method is valid.
8. Will needs to cut a piece of wood 0.67 of an inch thick, or just a little less thick.
Will’s ruler shows sixteenths of an inch. How many sixteenths of an inch thick
should Will cut his piece of wood? Explain your solution method and why it
works.
9. If 2.6 liters of a liquid weigh 3.1 kilograms, then how many liters of the liquid
weigh 1 kilogram? How much does 1 liter of the liquid weigh? Explain why you
can use division to solve both of these problems. Solve the problems.
10. If cabbage costs $0.23 per pound then how many pounds of cabbage can you
buy for $3.50? Explain why you can use division to solve this problem. Solve
the problem.
11. Write and solve a “how many groups?” word problem for 3.8 ÷ 1.4. (Do not
use a calculator.) Explain how to interpret the answer.
12. Write and solve a “how many in one group?” story problem for 3.8 ÷ 1.4. (Do
not use a calculator.) Explain how to interpret the answer.
13. Write and solve a story problem (any type) for 0.85 ÷ 3.5. (Do not use a
calculator.) Explain how to interpret the answer.
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Relationships
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7.1
Motivating and Defining Ratio and Proportional Relationships
1. Use a ratio table or double number line to help you explain what it means to
say that red and yellow paint are mixed in a ratio of 7 to 3 (to make orange
paint) from the “composed unit or batch” perspective.
2. Draw a strip diagram and use it to help you explain what it means to say that
red and yellow paint are mixed in a ratio of 7 to 3 (to make orange paint) from
the “fixed numbers of parts” perspective.
3. Use a simple example (such as a paint mixture) to describe two perspectives
on what a ratio of 5 to 3 means (the “composed unit or batch” perspective
and “fixed numbers of parts” perspective). Support your descriptions with
appropriate visual aids (diagrams or tables) for each of the two ways.
4. Which of the following two mixtures will be more salty?
• 2 tablespoons of salt mixed in 7 cups of water or
• 3 tablespoons of salt mixed in 8 cups of water
Make ratio tables for the two mixtures. Explain how to use your tables in two
ways to compare the flavors without using the term “ratio” and without using
fractions.
5. There are two green paint mixtures:
Mixture A 2 liters blue paint mixed with 5 liters yellow paint;
Mixture B 7 liters blue paint mixed with 10 liters yellow paint.
A student says that the two mixtures will be the same shade of green because
in each mixture there are 3 more liters of yellow paint than blue paint. Discuss
the student’s reasoning. Is it correct?
6. There are two green paint mixtures:
Mixture A 2 liters blue paint mixed with 5 liters yellow paint;
Mixture B 7 liters blue paint mixed with 10 liters yellow paint.
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What is a common error that students make in comparing such mixtures and
what thinking might lead to the error?
7. Allie mixes 3 cups red punch with 5 cups lemon-lime soda. Benton makes the
same mixture but adds one more cup of red punch and one more cup of lemonlime soda. Discuss: what common misconception do students sometimes have
about how Allie’s and Benton’s mixtures compare and how do the mixtures
actually compare?
8. Some punch is made by mixing grape juice and bubbly water in a ratio of 5
to 2. Give three different pairs of quantities of grape juice and bubbly water
that you could use to make the punch mixture in that ratio. Be sure to include
appropriate units of measurement. Include at least one example that involves a
fraction or mixed number. Explain why all the mixtures are in that same ratio
of 5 to 2.
9. You walked 12 mile in 8 minutes. Draw a double number line and use it to show
distances and elapsed times that are in that same ratio. Explain briefly.
7.2
Solving Proportion Problems by Reasoning with
Multiplication and Division
1. A drink mixture is made by mixing cola and lemon-lime soda in a ratio of 5 to
3. How much cola and how much lemon-lime soda will you need to make 240
cups of the drink mixture?
(a) Explain how to solve this problem with the aid of a strip diagram. Include
a discussion of how we can interpret the ratio 5 to 3 as “fixed numbers of
parts” when we use a strip diagram.
(b) Explain how to solve this problem with the aid of a ratio table. Include a
discussion of how we can interpret the ratio 5 to 3 as a “composed unit or
batch” when we use a ratio table.
2. In order to get the right octane of gasoline for his car, Will mixes premium gas
and regular gas in a ratio of 3 to 2. How many gallons of regular gas should Will
add if he just added 12 gallons of premium gas? Show how to solve this problem
in two ways: with a strip diagram and with a ratio table. Briefly explain your
reasoning in each case.
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3. Susie mixed 2 cups of yellow paint with 1 cup of red paint to make an orange
paint. How many cups of yellow paint and how many cups of red paint will
Suzy need to make 12 cups of the same shade of orange paint? Explain how to
solve this problem by using logical reasoning about multiplication and division
in two ways—with the aid of a ratio table and with the aid of a strip diagram.
4. Bob mixed 3 cups of yellow paint with 1 cup of blue paint to make a green
paint. How many cups of yellow paint and how many cups of blue paint will
Bob need to make 10 cups of the same shade of green paint? Explain how to
solve this problem by using logical reasoning about multiplication and division
in two ways without setting up and solving a proportion in which you set two
fractions equal to each other and cross-multiply.
5. Carmina mixed 14 cup of strawberry frosting with 31 of a cup of lemon frosting
and thought that the mixture was just right. Carmina needs 2 cups of her
frosting mixture. How many cups of strawberry frosting and how many cups
of lemon frosting will Carmina need? Explain how to solve this problem by
using logical reasoning about multiplication and division without setting up
and solving a proportion in which you set two fractions equal to each other and
cross-multiply.
6. To make a shade of orange paint that you like, you must mix 23 of a bottle of
red paint with each 54 of a bottle of yellow paint that you use. You need 88
bottles of this orange paint. How many bottles of red paint will you need and
how many bottles of yellow paint will you need? (All bottles are the same size.)
Explain your solution.
7. A factory makes an oil mixture by mixing oils of grades A and B as follows. For
every 2.3 liters of grade A oil, 1.2 liters of grade B oil are mixed in. How many
liters of grade A oil and grade B oil will the factory need to make 1000 liters of
their oil mixture? Explain your solution.
8. Shauntay and Carla had the same number of songs on their IPODs. After
Shauntay got 23 more songs and Carla got 11 more songs, the ratio of Shauntay’s
songs to Carla’s was 5 to 3. How many songs did Shauntay and Carla have then?
Explain how to solve this problem with the aid of a strip diagram.
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79
Unit Rates and the Values of a Ratio
1. Maya mixed 2 cups of blue paint with 3 cups of yellow paint to make a green
paint. For each of the following fractions, interpret the fraction in terms of
Maya’s paint mixture in a way other than as a ratio and explain briefly why
your interpretation makes sense.
2
;
5
2
;
3
3
.
2
2. Perri’s purple paint mixture is made by mixing blue and red paint in a ratio of
2 to 3. Interpret each of the fractions below in the following two ways, using a
table and diagram as supports:
• Use the fraction to describe a unit rate in terms of the paint;
• Use the fraction to make a multiplicative comparison between amounts of
paint.
3
;
5
2
;
3
3
.
2
3. Painting at a constant rate, Will used 32 of a can of paint in 1 21 hours. Show
how to reason about double number lines to answer the following questions.
(a) At that rate, what fraction of a can of paint does Will use in an hour?
(b) At that rate, how long does it take Will to use a can of paint?
4. A drink mixture is made by mixing cola and lemon-lime soda in a ratio of 5 to
3. How much cola and how much lemon-lime soda will you need to make 240
cups of the drink mixture?
(a) Show how to solve this problem by reasoning about a ratio table. Explain
very briefly.
(b) Show how to solve this problem by reasoning a strip diagram. Explain
very briefly.
(c) Show how to solve this problem by setting up proportions in which you set
fractions equal to each other. Explain what the fractions mean in terms
of the drink mixture and explain why it makes sense to set those fractions
equal to each other.
(d) Explain the rationale for the procedure of cross-multiplying when solving
a proportion.
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5. You need 1 12 cups of grape juice for a recipe that makes 4 servings. How many
cups of grape juice will you need if you want to make 10 servings of the same
recipe?
(a) Solve this problem by setting up a proportion in which you set two fractions
equal to each other.
(b) Interpret the two fractions that you set equal to each other in part (a) in
terms of the recipe. Explain why it makes sense to set these two fractions
equal to each other.
(c) Why does it make sense to cross-multiply the two fractions in part (a)?
What is the logic behind the procedure of cross-multiplying?
6. Sam mixed 3 cups of blue paint with 4 cups of red paint to make a purple paint.
Then Sam put another 2 cups of red paint in his mixture. How many cups of
blue paint should Sam add to return his paint to its original shade of purple?
(a) Solve this problem in a way that does not involve setting up a proportion
in which you set two fractions equal to each other.
(b) Solve this problem by setting up a proportion in which you set two fractions
equal to each other.
(c) Interpret the two fractions that you set equal to each other in part (b) in
terms of the paint and explain why it makes sense to set these two fractions
equal to each other.
7.4
Proportional Relationships Versus Inversely Proportional Relationships
1. Suppose that 4 bulldozers can load 6 trucks full of gravel in 30 minutes. Assume
all bulldozers work at the same steady pace and that all trucks hold the same
amount of gravel.
(a) The relationship between the number of bulldozers and the number of
minutes it takes to load six trucks is what kind of relationship? How can
you tell?
(b) The relationship between the number of bulldozers and the number of
trucks that can be fully loaded in thirty minutes is what kind of relationship? How can you tell?
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2. Suppose that 4 bulldozers can load 6 trucks full of gravel in 30 minutes. Assume
all bulldozers work at the same steady pace and that all trucks hold the same
amount of gravel.
(a) Make a table to show the relationship between the number of bulldozers and the time it takes to fully load six trucks. Include the case of 3
bulldozers in your table and explain your reasoning.
(b) Make a table to show the relationship between the number of bulldozers
and the number of trucks they can load full of gravel in 30 minutes. Include
the case of 3 bulldozers in your table and explain your reasoning.
3. If 3 bulldozers can load 5 trucks full of earth in 30 minutes, then how long
should it take 5 bulldozers to load 7 trucks full of earth? Assume that all the
bulldozers work equally hard and all truck loads are the same size. Solve the
problem, explaining your reasoning.
7.5
Percent Revisited: Percent Increase and Decrease
1. The price of a gizmo went from $2.50 down to $1.50. Find the percent decrease
in the price of the gizmo in two ways.
2. A community goes from producing 2 12 tons of waste per month to producing 3 21
tons of waste per month.
(a) Show how to use a table or math drawing to help you calculate the percent
increase in a community’s monthly waste production.
(b) Show how to calculate the percent increase in the community’s monthly
waste production in another way.
3. The price of a computer has just been reduced by 30%. The new, reduced price
of the computer is $1400. Brittany says she can find the original price (before
the reduction) in the following way:
“First I found 10% of $1400, which is $140. Then I multiplied that
by 3, which was $420. So 30% is $420. Then I added that to $1400,
which is $1820. So the computer was $1820 at first.”
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Is Brittany’s method correct or not? If it’s correct, explain briefly why and also
explain how to solve the problem in another way. If it’s not correct, explain
briefly why not and show how to modify Brittany’s method to solve the problem
correctly.
4. The price of a ring has been reduced by 25%. The new, reduced price is $400.
What was the original price of the ring before the 25% reduction? Explain
clearly why you can solve the problem the way you do.
5. Problem: The price of a ring has been reduced by 25%. The new, reduced price
is $400. What was the original price of the ring before the 25% reduction?
Describe a common error that students make in solving this kind of problem.
Explain how to solve the problem correctly.
6. The price of an appliance has been reduced by 40%. The new price is $245.
What was the old price? Explain clearly why you can solve the problem the
way you do.
7. (a) A TV that originally cost $660 is marked down by 20%. What is its new
price? Show your work.
(b) A store raised the price of a TV by 20%. The new price is $660. What
was the price before the increase? Show your work.
(c) Discuss why we can or cannot solve problems (a) and (b) in the same way.
8. The price of a ring has been reduced by 20%. The new, reduced price is $360.
What was the price of the ring before the reduction? Show how to solve this
problem in two ways: (1) using either a percent table or a math drawing or (2)
by setting up and solving an equation.
9. Fill in the blanks so as to make the statements below correct:
(a) 600 is
% of 500.
(b) 100 is
% more than 40.
(c) 68 is
% less than 80.
10. (a) If the price of a barrel of oil goes up by 20%, and then goes back down
by 20% (of the new, raised price), will the final price of oil (after raising
and lowering) be equal to the original price of oil? If not, which will be
greater, the original price or the final price? Determine the answers to
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these questions without doing any calculations. Explain your reasoning
clearly.
(b) Now suppose that in part (a), a barrel of oil originally cost $20. Calculate
the final price of the barrel of oil after the price is raised by 20% and then
lowered by 20% (from the raised price).
11. A company produces two types of belts, style A and style B. Of the belts they
produce each day, 40% are style A and 60% are style B. The company will
continue to produce the same number of style B belts, but will increase the
number of style A belts so that 50% of the belts produced will be style A and
50% will be style B. By what percent must the company increase its production
of style A belts? Show your work and explain it briefly.
12. Explain how the commutative property of multiplication is relevant to the following question. The price of a blouse was first reduced by 25% and then
reduced by 20% (from the reduced price). What if instead, the blouse had first
been reduced by 20% and then by 25% (from the reduced price), would the final
price be the same, lower, or higher?
13. Explain how the commutative property of multiplication is relevant to the following: if the price of an item goes up by 10% and then by 15% (of the increased
price), the item costs the same as if the price of the item had first gone up by
15% and then by 10% (of the increased price).
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8.1
Factors and Multiples
1. You may answer all three parts together, or separately, as you like.
(a) What are multiples and factors?
(b) Describe an activity using physical objects that could help children learn
about multiples. Make clear how multiples are involved in the activity.
(c) Describe an activity using physical objects that could help children learn
about factors. Make clear how factors are involved in the activity.
2. (a) Write a word problem such that solving the problem will require finding
all the factors of 36.
(b) Write a word problem such that solving the problem will require finding
many or (theoretically) all the multiples of 36.
3. Show a general method for determining all the factors of a counting number in
an efficient way. Illustrate with the number 150. Explain how you know you
have found all the factors.
8.2
Even and Odd
1. If you add an odd number and an even number, what kind of number do you
get? Explain why your answer is always correct.
2. If you multiply an even number with an odd number, what kind of number do
you get? Explain why your answer is always correct.
3. One definition of the term even is the following: a counting number is even if
that number of objects can be divided into groups of 2 with none left over.
(a) What is another way to define the term even for counting numbers by
referring to objects? Why is this definition equivalent to the one above?
(b) What is a numerical way to define the term even for counting numbers?
Why is this definition equivalent to the one above?
(c) Is it possible to extend the definition of even to all integers? If so, is 0
even or not? Explain. Is −5 even or not? Explain.
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(d) Is it possible to extend the definition of even to all rational numbers (fractions)? Explain why or why not.
4. Let ABC be a three digit whole number with A hundreds, B tens, and C ones.
Use the idea of representing ABC with base-ten bundles to help you explain
why ABC is even exactly when C is either 0, 2, 4, 6 or 8.
5. Explain the even counting numbers are exactly those counting numbers that
have either a 0, 2, 4, 6, or 8 as ones digit.
6. If you add a number that has a remainder of 1 when it is divided by 3 to a
number that has a remainder of 2 when it is divided by 3, then what is the
remainder of the sum when you divide it by 3? Explain why your answer is
always correct.
8.3
Divisibility Tests
1. (a) Describe the divisibility test for 3 (the test by which we can quickly and
easily tell whether or not a counting number is evenly divisible by 3 without
actually dividing the number by 3). Give an example to show how to use
this divisibility test.
(b) Explain clearly and in detail why the divisibility test for 3 is valid. Illustrate your explanation with the three-digit counting numbers 472 and
258.
2. (a) Describe the divisibility test for 3 (the test by which we can quickly and
easily tell whether or not a counting number is evenly divisible by 3 without
actually dividing the number by 3). Give an example to show how to use
this divisibility test.
(b) Explain clearly and in detail why the divisibility test for 3 is valid for threedigit counting numbers ABC (with A hundreds, B tens, and C ones).
(c) Let ABC represent a 3-digit counting number (A hundreds, B tens, and
C ones). Relate your explanation in part (b) to the following equations:
ABC = A · 100 + B · 10 + C
= (A · 99 + B · 9) + (A + B + C)
= (A · 33 + B · 3) · 3 + (A + B + C)
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3. Beth knows the divisibility test for 3. Beth says that she can tell just by looking,
and without doing any calculations at all, that the number
444, 555, 666, 777
is divisible by 3. How can Beth do that? Explain why it’s not just a lucky guess
on Beth’s part.
4. Sam used his calculator to calculate
123, 123, 123, 123, 123 ÷ 3.
Sam’s calculator displayed the answer as
4.1041041041E13
Sam says that because the calculator’s answer is not a whole number, the number 123, 123, 123, 123 is not divisible by 3. Is Sam right? Why or why not? How
do you reconcile your answer with Sam’s calculator’s display?
5. (a) Describe a divisibility test for 4, in other words describe an easy way to
check whether or not a counting number is divisible by 4 without actually
dividing the number by 4. Give an example to show how to use this
divisibility test.
(b) Explain clearly and in detail why the divisibility test for 4 is a valid way
to determine if a whole number is divisible by 4.
6. (a) Describe a divisibility test for 5, in other words describe an easy way to
check whether or not a counting number is divisible by 5 without actually
dividing the number by 5. Give an example to show how to use this
divisibility test.
(b) Explain clearly and in detail why the divisibility test for 5 is a valid way
to determine if a whole number is divisible by 5.
7. Find a divisibility test for 25 that does not require you to do any dividing in
order to determine if a number is divisible by 25. Explain why your divisibility
test is a valid way to check if a counting number is divisible by 25.
8. (a) Describe a divisibility test for 9, in other words describe an easy way to
check whether or not a counting number is divisible by 9 without actually
dividing the number by 9. Give an example to show how to use this
divisibility test.
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(b) Explain clearly and in detail why the divisibility test for 9 is valid for 3
digit counting numbers ABC (with A hundreds, B tens, and C ones).
(c) Let ABC represent a 3-digit counting number (A hundreds, B tens, and
C ones). Relate your explanation in part (b) to the following equations:
ABC = A · 100 + B · 10 + C
= (A · 99 + B · 9) + (A + B + C)
= (A · 11 + B · 1) · 9 + (A + B + C)
9. (a) Is it true that a whole number is divisible by 6 exactly when the sum of
its digits is divisible by 6? Explain briefly.
(b) If your answer in part (a) is no, then what is an easy way to determine if a
counting number is divisible by 6 other than actually dividing the number
by 6? (Hint: 6 = 2 × 3.)
(c) Check if the number
111, 222, 333, 444, 555, 666, 777, 888, 999, 000
is divisible by 6 without actually dividing the number by 6.
8.4
Prime Numbers
1. Show how to use the Sieve of Eratosthenes to find the prime numbers up to 60
and explain why the method works.
2. Explain why the Sieve of Eratosthenes produces a list of prime numbers.
3. What is the Sieve of Eratosthenes and why does it work?
4. The list of prime numbers up to 50 is as follows:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
(a) Show how to use “trial division” with the above list of prime numbers to
determine whether or not the number 2003 is prime. Determine if 2003 is
prime or not.
(b) Explain why you do not need any more prime numbers than are shown in
the above list in order to determine whether or not 2003 is prime.
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(c) Explain why you do not need to include numbers that are not prime numbers in the list above in order to determine whether or not 2003 is prime.
5. When Katie was asked to factor 26 × 77 into a product of prime numbers, she
first multiplied 26 × 77 = 2002, and then she proceeded to make a factor tree
for 2002 by first writing 2002 as 2 × 1001, and then looking for factors of 1001.
Discuss Katie’s method.
8.5
Greatest Common Factor and Least Common
Multiple
1. Describe what we mean by “the least common multiple” and “the greatest
common factor” of two counting numbers and illustrate the meaning of each
concept with well-chosen numbers. Make clear which is which.
2. (a) Using the numbers 10 and 12, describe what the least common multiple of
two counting numbers is, and describe how to determine it.
(b) Write a word problem such that solving the problem requires finding the
least common multiple of 10 and 12.
3. (a) Using the numbers 12 and 20, describe what the greatest common factor
of two counting numbers is, and describe how to determine it.
(b) Write a word problem such that solving the problem requires finding the
greatest common factor of 12 and 20.
4. You may answer both parts together, or separately, as you like.
(a) Describe what the least common multiple is and give a well-chosen example
to explain its meaning.
(b) Describe an activity using physical objects that could help children learn
about least common multiples. Make clear how least common multiples
are involved.
5. You may answer both parts together, or separately, as you like.
(a) Describe what the greatest common factor is and give a well-chosen example to explain its meaning.
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(b) Describe an activity using physical objects that could help children learn
about greatest common factors. Make clear how greatest common factors
are involved.
6. Show how to use the “slide method” to determine the GCF and LCM of 2250
and 6000.
7. Show how to use the “slide method” to determine the GCF and LCM of 240
and 360 and explain why it makes sense that this method produces the GCF
and LCM.
8. Give a good example to illustrate how to use the “slide method” to determine
the GCF and LCM of two numbers and to discuss why the method works.
9. Find the GCF and LCM of 217 ·328 ·11 and 213 ·345 ·74 and explain your reasoning.
10. Give a good example to illustrate how to find the GCF and LCM of two numbers
that are factored into products of powers of prime numbers and to explain why
the method works.
11. Susie says that she has a quick way to determine the least common multiple of
two numbers: you just multiply the numbers. Susie checked it with 3 and 5 and
with 4 and 7 and her method worked both times. Is Susie’s method a valid way
to determine the least common multiple of two counting numbers? Explain.
8.6
Rational and Irrational Numbers
1. Show how to use the standard division algorithm to calculate the decimal rep4
resentation of 37
. Describe the nature of this decimal representation.
5
2. Without actually determining the decimal representation of 31
, explain how to
reason about the process of the division algorithm to conclude that the decimal
5
representation of 31
must either terminate or repeat.
3. Explain why the decimal representation of a fraction must either terminate or
repeating. Use the fraction 71 or your own example to illustrate the explanation
(but indicate how the explanation works in general).
4. What can we say about the decimal representations of fractions whose numerator and denominator are whole numbers?
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5. Write 0.134 as a fraction whose numerator and denominator are whole numbers,
explaining your reasoning.
6. Suppose you have two decimals and that each one is either repeating or terminating. Is it possible that the product of these two decimals could be neither
repeating nor terminating? Explain your answer.
7. Tai used a calculator to solve an arithmetic problem. The calculator gave Tai
the answer 0.019607843137. Without any further information, is it possible to
tell if the answer to Tai’s arithmetic problem can be written as a fraction whose
numerator and denominator are whole numbers? Explain.
8. Give an example of an irrational number.
9. Explain why 0.9 = 1.
√
10. Prove that 5 is irrational.
8.7
Looking Back at the Number Systems
1. (a) Give an example of a numerical problem that is formulated with whole
numbers but does not have a whole number solution. Explain briefly why
your problem fulfills the request.
(b) Give an example of a numerical problem that is formulated with integers
but does not have an integer solution. Explain briefly why your problem
fulfills the request.
(c) Give an example of a numerical problem that is formulated with rational
numbers but does not have a rational solution. Explain briefly why your
problem fulfills the request.
2. Describe how the whole numbers, the integers, the rational numbers, and the
real numbers are related.
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Algebra
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CHAPTER 9
9.1
Numerical Expressions
1. Write an expression that uses addition and multiplication to describe the total
number of dots in Figure 9.1.
Figure 9.1: Write an Expression for The Number of Dots in This Picture
2. Write two different expressions for the total number of small squares in design
(a) of Figure 9.2. Each expression should use either multiplication or addition,
or both.
(a)
(b)
Figure 9.2: Write Expressions for These Designs
3. Write two different expressions for the total number of small squares in design
(b) of Figure 9.2. Each expression should use either multiplication or addition,
or both.
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4. A high-five problem: If 100 people all high-five with each other, how many
high-fives will there be?
(a) Write two different expressions for the total number of high-fives. Explain
why your expressions describe the total number of high fives.
(b) Solve the high-five problem.
5. Calculate 1 + 2 + 3 + . . . + 100 by finding and evaluating another expression
that equals this sum. Explain why the other expression has the same value.
6. Show how to calculate 1 + 2 + 3 + . . . + 200 quickly, without just adding all the
numbers. Explain briefly why your method works.
7. Show how to make the expression
19 26
500 117
·
+
·
39 19
117 600
easy to evaluate without using a calculator.
8. Show how to make the expression
(3 −
21 77
)·
22 75
easy to evaluate without using a calculator.
9. Write equations that use the rules of fraction multiplication and equivalent
fractions as well as properties of arithmetic to show why it is valid to cancel to
evaluate the expression
12 17
·
35 18
as shown below:
2
34
12
/ 17
2 · 17
·
=
=
35 18
35 · 3
105
/
3
10. Is it valid to evaluate
15 · 25
10
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CHAPTER 9
by canceling in the following way?
3
5
1
15
/ · 25
/
3·5
=7
=
2
2
10
/
2
If so, write equations to show why the canceling is valid, if not, explain briefly
why not and show a correct way to evaluate the expression.
11. Is it valid to evaluate
15 · 25
10
by canceling in the following way?
3
3 · 25
15
/ ·25
75
1
=
=
= 37
2
2
2
10
/
2
If so, write equations to show why the canceling is valid, if not, explain briefly
why not and show a correct way to evaluate the expression.
12. Is it valid to evaluate
15 + 25
10
by canceling in the following way?
3
15
/ +25
28
3 + 25
=
= 14
=
2
2
10
/
2
If so, write equations to show why the canceling is valid, if not, explain briefly
why not and show a correct way to evaluate the expression.
9.2
Expressions with Variables
1. Use mathematical terminology to describe the structure of the expression 3x2 +
4(x − 5)(x + 3) − 7.
2. In a problem about the depth of the water in a lake, a student defined the
variable W by writing “W = water.” What is wrong with this? What is a
correct way to define a variable for this context?
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3. A mail order company sells silly bands. The company charges $4 for each
package of silly bands and charges $5 for shipping, no matter how many packages
a customer buys. Define a variable and write an expression for a quantity that
is relevant to this context.
4. Initially, a farmer has 1000 pounds of wheat in bin. Then the farmer removes
25 pounds of wheat every minute. Define a variable and write an expression for
a quantity that is relevant to this context.
5. Write two different expressions for the area of the floor plan in Figure 9.3.
Explain briefly.
y meters
x meters
x meters
Figure 9.3: Write Expressions for the Area
6. Ming has pencils to distribute among goodie bags. When Ming tries to put P
pencils in each of 9 goodie bags, he is 1 pencil short. Write an expression for
the total number of pencils Ming has in terms of P .
7. There are G gallons of antifreeze in a container initially. If 13 of the antifreeze
in the container is poured out and then another 1 12 gallons are poured out, then
how many gallons of antifreeze are left in the container? Write an expression in
terms of G. Then evaluate your expression when G = 4.
8. There are G gallons of antifreeze in a container initially. If 1 12 gallons are poured
out and then 13 of the remaining antifreeze is poured out, then how many gallons
of antifreeze are left in the container? Write an expression in terms of G. Then
evaluate your expression when G = 4.
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9. There are G gallons of antifreeze in a container initially. If 1 12 gallons are poured
out and then 13 of the remaining antifreeze is poured out, then how many gallons
of antifreeze are left in the container?
(a) Show two different correct expressions that students could formulate for
the amount of antifreeze left in the container. For each, indicate how
students might reason to determine that expression (e.g., by annotating
your expressions).
(b) Show an incorrect, yet plausible, expression that students might formulate
for the amount of antifreeze left in the container. Discuss why students
might formulate the expression incorrectly this way.
10. Abby had A cards in her card collection at first. After Abby gave away 14 of
her collection she got another 12 cards. Then Abby gave away 31 of her cards.
Write an expression in terms of A for the number of cards that Abby has now.
11. Scientists determined that t seconds after launch, a projectile will be
16(3 + t)(19 − t)
feet above ground level. Describe the structure of the expression and reason
about this structure to determine when the projectile will hit the ground.
12. A widget company determined that if it sells widgets for W dollars each, then
its daily profit will be
1600 − 100(W − 5)2
dollars. Describe the structure of the expression and reason about this structure
to determine at what price the company should sell its widgets to maximize its
profit.
9.3
Equations for Different Purposes
1. We discussed four different ways that we commonly use equations. Briefly
describe two of them.
2. Give an example of an equation that is an identity.
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3. Draw, label, and shade a rectangle so that it gives rise to the identity
(x + 1) · (x + 3) = x2 + 4x + 3
Explain briefly.
4. At the University of Lower Anklescratch there are 5 times as many students as
professors. Let S stand for the number of students and P stand for the number
of professors. Write an equation relating S and P and explain briefly why your
equation is written that way.
5. To make a punch, you will mix lemon-lime soda with juice. You need 1 12 times
as much juice as soda no matter how much punch you will make. Define two
variables and write an equation that describes this relationship.
6. At a yogurt shop, you can get a yogurt cone filled with whatever amount of
frozen yogurt you want. Frozen yogurt costs $0.65 for every ounce. A cone to
hold the yogurt costs $1.05. Define two variables and write an equation to show
how the total cost of a yogurt cone is related to the number of ounces of yogurt
in the cone.
7. To get the correct octane of gas an oil company needs to mix premium gas and
low-grade gas, using 3 times as much premium gas as low-grade gas.
(a) Define two variables and write an equation to relate amounts of premium
gas and low-grade gas that the oil company can mix together.
(b) Describe two different kinds of errors that students often make in defining
variables and writing equations, as in part (a).
8. To get the correct octane of gas an oil company needs to mix premium gas and
low-grade gas, using 2 12 times as much premium gas as low-grade gas.
(a) Write an equation that relates amounts of premium gas and low-grade gas
that the oil company can mix together.
(b) Describe two different kinds of errors that students often make in writing
an equation such as the one in part (a).
9. Bob had B marbles in his marble collection. After Bob got another 15 marbles,
he gives away 15 of his marbles. Then Bob has 32 marbles. Write an equation,
involving B, that corresponds to this situation.
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Solving Equations
1. Explain how to solve the next equation by reasoning about numbers, operations, and expressions rather than by using standard algebraic equation-solving
techniques.
18 · 36 + x = 20 · 36
2. Explain how to solve the next equation by reasoning about numbers, operations, and expressions rather than by using standard algebraic equation-solving
techniques.
348 + 926 = x + 350
3. Explain how to solve the next equation by reasoning about numbers, operations, and expressions rather than by using standard algebraic equation-solving
techniques.
17(x + 48) = (50 + 984) · 17
4. Explain how to use reasoning to solve the equation x −
1
4
= 31 .
5. Explain how to use reasoning to solve the equation 34 x = 57 .
6. Solve
5x + 2 = x + 6
in two ways: with equations and with pictures of a pan balance. Relate the two
methods.
7. Solve the next two equations and explain your answers.
(a) 3x + 5 + 4x + 3 = 1 + 7x + 7
(b) 6x − 3 − 2x = 1 + 3x + x
9.5
Solving Algebra Word Problems with Strip Diagrams and with Algebra
1. All together, Meili and Kaitlyn have 160 stickers. Meili has 3 times as many
stickers as Kaitlyn. How many stickers does Kaitlyn have and how many stickers
does Meili have? Solve this problem in two ways: with the aid of a diagram and
with algebraic equations, explaining each briefly. Discuss how the two solution
methods are related.
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2. There are 450 coins divided into 3 piles. The second pile has twice as many
coins as the first. The third pile has 25 more coins than the second. How many
coins are in the first pile? Solve this problem in two ways: with the aid of a
diagram and with algebraic equations, explaining each briefly. Discuss how the
two solution methods are related.
3. All together, there are 68 frogs and toads in a tank. There are 12 more frogs
than toads. How many frogs and how many toads are in the tank? Solve this
problem in two ways: with the aid of a diagram and with algebraic equations,
explaining each briefly. Discuss how the two solution methods are related.
4. 100 children are divided into 2 groups. There are 10 more children in the first
group than in the second group. How many children are in the second group?
Solve this problem in two ways: with the aid of a diagram and with algebraic
equations. Explain your reasoning.
5. Liping has 2 stacks of books. The second stack has 3 times as many books as
the first stack. All together, Liping has 64 books. How many books are in the
first stack? Solve this problem in two ways: with the aid of a diagram and with
algebraic equations. Explain your reasoning.
6. There are 400 boxes arranged in 3 groups. The second group has twice as many
boxes as the first group. The third group has 4 more boxes than the first group.
How many boxes are in the first group? Solve this problem in two ways: with
the aid of a diagram and with algebraic equations. Explain your reasoning.
7. At a book sale, 40% of the books were sold in the first hour. In the next hour, 23
of the remaining books were sold. At that point there were 80 books left. How
many books were there at the beginning of the book sale? Solve this problem
in two ways: with the aid of a diagram and with algebraic equations. Explain
your reasoning.
8. The Pumpkin Patch has large pumpkins and regular pumpkins for sale. There
are 20% fewer large pumpkins than regular pumpkins. Altogether, there are 198
pumpkins for sale. How many large pumpkins and how many regular pumpkins
are there? Show how to solve this problem in two ways: with the aid of a strip
diagram and with equations. Explain your reasoning.
9. April spent 94 of her money. She has $20 left. How much money did she have
at first? Solve this problem in two ways: with the aid of a diagram and with
algebraic equations. Explain your reasoning.
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10. Mr. Smith gave 35 of his money to his wife and spent 41 of the remainder. If he
has $150 left, how much money did he have at first? Solve this problem in two
ways: with the aid of a diagram and with algebraic equations. Explain your
reasoning.
11. Bob had a marble collection. After Bob got another 15 marbles, he gives away
1
of his marbles. Then Bob has 32 marbles. How many marbles did Bob have
5
at first? Solve this problem and explain your reasoning.
9.6
Sequences
1. Assume that the repeating pattern of three squares followed by a circle and a
triangle in Figure 9.4 continues. What shape will be above the number 999?
Explain your answer.
...
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Figure 9.4: A Repeating Pattern of Shapes
2. Assume that the repeating pattern of three squares followed by a circle and a
triangle in Figure 9.4 continues. How many squares will there be above the
numbers 1 through 400? Explain your answer.
3. Assume that the repeating pattern of three squares followed by a circle and a
triangle in Figure 9.4 continues. How many squares will there be above the
numbers 1 through 999? Explain your answer.
4. Assume that the repeating pattern of 2 squares followed by a circle and a triangle
in Figure 9.5 continues.
...
1
2
3
4
5
6
7
8
9
10
11
Figure 9.5: A Repeating Pattern of Shapes
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13 ...
16 ...
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103
(a) Discuss and explain whether or not the following reasoning is valid:
Since there are 6 squares above the numbers 1 – 10, there will be
15 times as many squares, namely 90 squares, above the numbers
1 – 150.
(b) Find a different way to determine the number of squares above the numbers
1 – 150 and explain your reasoning.
5. What day of the week will it be 365 days from today? Explain how to figure
this out with math.
6. Five friends are sitting in a circle as shown in Figure 9.6. Antrice sings a song
that has 21 syllables, and starting with Benton, and going clockwise, points to
one person for each syllable in the song. The last person that Antrice points to
will be “it.” Who will be “it”? Explain how to predict the answer using math.
Benton
Carmina
Antrice
Doug
Ellie
Figure 9.6: Who Will Be “It”?
7. Suppose the sequence of figures in Figure 9.7 continues by adding one square
to each of the four “arms.” Explain the answers to the next questions in two
ways: (a) with algebraic equations and (b) in a way that a 4th or 5th grader
who has not yet learned to solve algebraic equations might be able to.
Is there a figure in the sequence that is made of 999 small squares? If so, which
one, if not, why not?
8. Suppose the sequence of figures in Figure 9.7 continues by adding one square
to each of the four “arms.” Explain the answers to the next questions in two
ways: (a) with algebraic equations and (b) in a way that a 4th or 5th grader
who has not yet learned to solve algebraic equations might be able to.
Is there a figure in the sequence that is made of 197 small squares? If so, which
one, if not, why not?
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9. Suppose the sequence of figures in Figure 9.7 continues by adding one square to
each of the four “arms.” For each of the following, solve the problem in a way
that a 5th grader who has not yet learned to solve algebraic equations might be
able to.
(a) Is there a figure in the sequence that is made of 102 small squares? If so,
which one, if not, why not?
(b) Is there a figure in the sequence that is made of 197 small squares? If so,
which one, if not, why not?
10. Suppose the sequence of figures in Figure 9.7 continues by adding one square
to each of the four “arms.”
1st
2nd
3rd
4th
Figure 9.7: A Sequence of Figures
(a) Find an expression for the number of small squares that the Nth figure in
the sequence is made of.
(b) Explain why your expression in part (a) is valid by relating it to the structure of the figures.
11. Draw a sequence of figures made of small circles so that the Nth figure is made
of 3N + 1 small circles. Describe how subsequent figures in your sequence would
be formed.
12. (a) Draw the first 5 entries a sequence of figures made of small circles so that
the number of small circles that the figures are made of is the following
arithmetic sequence:
6, 8, 10, 12, 14, . . .
Describe how subsequent figures in your sequence would be formed.
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(b) If the sequence were to continue forever, how many small circles would the
Nth figure be made of? Find an expression in terms of N and explain why
the expression is valid.
13. Explain how to reason to determine an expression for the Nth entry of an arithmetic sequence. Give an explanation that will work in general, but illustrate it
with the example
5, 8, 11, 14, 17, 20, . . .
14. Using the example,
1, 5, 9, 13, 17, 21, . . .
describe a general way to find an expression for the Nth entry of an arithmetic
sequence and explain why this expression must be valid.
15. Using the example,
10, 20, 40, 80, 160, 320, . . .
describe a general way to find an expression for the Nth entry of a geometric
sequence and explain why this expression must be valid.
16. Describe three different rules for determining the next entries in the following
sequence. For each rule, give the next three entries in the sequence for that
rule.
5, 7, 11, . . .
9.7
Functions
1. Some biologists are interested in studying how the number of freshwater mussels
in a lake changes over time. Use words to describe a function for this context.
(No graph or equation is expected.)
2. For each of the next two examples, determine if there could be a function
that has those inputs and associated outputs. If so, describe a rule for such a
function; if not, explain why not.
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Input:
Input:
Input:
Input:
Input:
Input:
Example 1
1 → Output:
1 → Output:
2 → Output:
2 → Output:
3 → Output:
3 → Output:
1
2
3
4
5
6
Input:
Input:
Input:
Input:
Input:
Input:
Example 2
1 → Output:
2 → Output:
3 → Output:
4 → Output:
5 → Output:
6 → Output:
1
1
1
1
1
1
3. Sketch graphs of three functions for these three different scenarios about whale
populations over a 5 year period. In each case, identify your axes and say briefly
why your graph fits with the description.
(a) In the first two years, the whale population declined rapidly. In the last
three years, the population continued to decline, but not as rapidly.
(b) Over the five-year period, the whale population dropped more and more
rapidly.
(c) The whale population declined steadily over a five-year period.
4. A tagged manatee swims up a river, away from a dock. This situation gives rise
to a distance function, for which the input is the time elapsed since the manatee
first swims away from the dock, and the output is the manatee’s distance from
the dock at that time. The graph of this function is shown in Figure 9.8. Using
the graph, describe how the manatee swims. In particular, explain clearly how
and why you can tell from the graph when the manatee swims quickly, and
when it swims slowly.
5. The paragraph below describes how Sheila ran a mile in 10 minutes.
Sheila raced away from the starting line, going faster than she’d ever
gone before. After two minutes, she slowed down just a little. Then,
just as Sheila started to pick up speed again, she tripped! Poor Sheila
fell down and stayed down for about half a minute, even though it
seemed like forever. But Sheila picked herself up and finished her mile
run, although at a slower pace than before.
Draw a graph that could be the graph of the function whose input is time
elapsed since the start of Sheila’s run, and whose output is the distance Sheila
had run at that time. Be sure to label your axes appropriately. Label the
portions of the graph that correspond to the different parts of the story.
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600
500
distance
400
from
dock
300
in feet
200
100
0
0
5
10
15
20
time in minutes
Figure 9.8: The Graph of a Manatee’s Distance Function
6. The graph in Figure 9.9 shows Kelly’s distance from school while she is walking
home. Write a story about Kelly’s walk home that fits with this graph. The
features of the graph should be reflected in your story. Discuss how your story
fits with the graph.
7. Starting at the bottom of a hill, Ruth runs up the hill, starting out fast, but
slowing down more and more as she nears the top of the hill. At the top of the
hill, Ruth stops briefly, then turns around and runs back down to the bottom
of the hill, maintaining a steady speed all the way down until she stops.
(a) Sketch a graph that could be the graph of the speed function whose input
is time elapsed since Ruth began running up the hill and whose output is
Ruth’s speed at that time. Indicate how the graph corresponds with the
story.
(b) Sketch a graph that could be the graph of the distance function whose
input is time elapsed since Ruth began running up the hill and whose
output is the total distance Ruth has run at that time. Indicate how the
graph corresponds with the story.
(c) Sketch a graph that could be the graph of the height function whose input
is the time elapsed since Ruth began running up the hill and whose output
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Kelly’s distance from school on her way home
3000
Distance
in
feet
2000
1000
5
10
15
20
25
30
Time in minutes
Figure 9.9: Kelly’s Walk Home From School
is Ruth’s height above the base of the hill at that time. Indicate how the
graph corresponds with the story.
8. The height function for rocket is given by the equation
h = 16t(25 − t)
where t is the number of seconds elapsed since launch and h is the height of the
rocket above the ground (in feet). Explain how to reason about the structure
of the equation to determine when the rocket will hit the ground.
9. A cookie company has determined that its profit function is given by the equation
y = 50 − 2(x − 5.75)2
where x is the number of dollars it sells each box of cookies for and y is the
company’s annual profit from cookies in millions of dollars. Explain how to
reason about the structure of the equation to determine how the company can
maximize its profit.
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109
Linear Functions
1. A soda mixture can be made by mixing cola and lemon-lime soda in a ratio of
3 to 2.
(a) Describe a function that is associated with this situation, write an equation
for your function, and sketch its graph.
(b) Discuss how the formula for your function in part (a), the graph of your
function, and the 3 to 2 ratio (or a different, related ratio) are related to
each other.
2. Consider the arithmetic sequence
1, 4, 7, 10, . . .
(a) Describe a function that is associated with this arithmetic sequence, write
an equation for this function, and sketch its graph.
(b) Discuss how the components of your formula in part (a) is related to the
arithmetic sequence and to the graph in part (a).
3. Match the following descriptions of functions to the appropriate graphs in Figure 9.10. Explain your reasoning.
(a) Whenever the input increases by 1 the output increases by 2.
(b) Whenever the input increases by 1 the output decreases by 2.
(1)
(2)
(3)
Figure 9.10: Fit Descriptions With These Graphs
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4. Consider a function that has the following properties:
• When the input is 3, the output is 5.
• Whenever the input increases by 2, the output increases by 3.
(a) Sketch the graph of a function that has the bulleted properties and explain
how the bulleted items are reflected in the graph.
(b) Find an equation for a function that has the bulleted properties.
5. For each of the tables in Table 9.1, determine if it could be the table for a linear
function. If so, find an equation for the linear function. If not, explain why not.
f
input output
0
4
1
3
2
2
3
1
4
0
g
input output
2
3
4
6
6
9
8
12
10
15
input
1
2
3
4
5
h
output
1
4
9
16
25
Table 9.1: Which of These Are Tables of Linear Functions?
6. For each of the tables below, determine if it could be the table for a linear
function. If so, find an equation for the linear function (no explanation needed).
If not, explain why the function is not linear.
f
input
2
4
6
8
10
output
17
14
11
8
5
input
1
2
4
8
16
g
output
3
6
12
24
48
7. Assuming that the tables in Table 9.2 are ones for linear functions, fill in the
blank outputs and find equations for these functions.
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input
0
1
2
3
4
5
6
111
output
7
13
input output
0
1
2
4
3
4
5
9
6
input output
-3
9
-2
-1
0
1
7
2
3
Table 9.2: Find the Other Entries in the Tables for Linear Functions
8. For each of the following, either define variables x and y for quantities that vary
together in a realistic situation or say why there can’t be such an example.
(a) As x increases, y decreases and the relationship between x and y is not
linear.
(b) As x increases, y decreases and the relationship between x and y is linear.
9. Describe a real-world scenario in which two quantities vary together in a linear
relationship. Define an independent and dependent variable for the quantities,
write an equation to show how they are related, and explain why the equation
is valid.
10. A mail-order bead company sells 2 pounds of beads for $10. Each additional
pound of beads costs $3. Some students are trying to write equations to describe
the relationship between the cost and the number of pounds of beads.
Discuss each of the following students’ work. Explain whether what has been
written so far could make sense and why or why not. Indicate also if anything
still needs to be added, clarified, or modified.
(a) Phillip writes the equation C = 3P + 10.
(b) Shauntay writes the equation C = 3(P − 2) + 10.
11. Compare and contrast inversely proportional relationships with linear relationships that have negative slopes. Illustrate with an example of each. Include a
table and a (real-world) context as part of each example.
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12. For each of the following tables, determine what kind of relationship the table
exhibits and explain how you can tell. You do not need to find equations for
the relationships.
Table A
x
y
1
2
2
7
3 14
4 23
5 34
Table B
x
y
1
7
2 14
3 28
4 56
5 112
Table C
x
y
1
7
3 12
7 22
13 37
21 57
13. For each of the following tables, determine what kind of relationship the table
exhibits and explain how you can tell. You do not need to find equations for
the relationships.
Table A
x
y
1 −2
2 −2
3 −2
4 −2
5 −2
Table B
x
y
0
2
2
6
8 18
26 54
80 162
Table C
x
y
0
1
1
3
2
7
3 13
4 21
14. Give an example of a table that represents a linear relationship but that at first
does not appear to be a linear relationship (i.e., students might misidentify it
if they are not thinking carefully enough about what linear relationships are).
Explain why the relationship really is linear and say briefly what might be tricky
for students.
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113
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10.1
Visualization
1. The diagram in Figure 10.1 shows the earth as seen from outer space, looking
down on the North Pole (labeled N) at one point in time.
(a) What time of day is it at points A and B? Explain how you can tell based
on where people at those locations see the sun in the sky and from the fact
that the sun rises in the east and sets in the west.
(b) Based on part (a), which way does the earth rotate (looking down on the
North Pole)?
sun rays
A
N
B
Figure 10.1: The earth from outer space
2. The diagram in Figure 10.2 shows the earth and moon as seen from outer space,
looking down on the North Pole (not to scale!) at one point in time.
(a) What does the moon look like to people on earth who can see it? Explain
how you can tell from the diagram.
(b) Is the moon waxing (getting bigger) or waning (getting smaller)? Explain
how you can tell from the diagram.
3. The diagram in Figure 10.3 shows the earth and moon to scale, as might be
seen from outer space. Approximately what fraction of the moon’s surface will
people on earth be able to see when the sun’s rays are hitting the earth and
moon as indicated? Explain your answer in detail. (Consider only those people
on that part of the earth where it is possible to see the moon at the time.)
4. The diagram in Figure 10.4 shows the earth and moon as seen from outer space
above the North Pole (not to scale!).
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115
moon
earth
sun rays
Figure 10.2: The earth and moon from outer space
Sun rays
Moon
Earth
Figure 10.3: The Earth and Moon Shown to Scale
(a) Approximately what time of day is it at point P? Explain how you can tell
from the diagram and the fact that the sun rises in the east and sets in
the west.
(b) How does the moon appear to a person at point P? Explain your answer.
5. The diagram in Figure 10.5 shows the earth and moon as seen from outer space,
looking down on the north pole (not to scale!).
(a) For a person located at point P, approximately what time is it? Explain
how you can tell from the diagram and the fact that the sun rises in the
east and sets in the west.
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sun rays
N
P
Moon
Figure 10.4: The Earth and Moon
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(b) Is the moon waxing or waning in Figure 10.5? Explain how you can tell
from the diagram.
Moon
P
sun rays
N
Figure 10.5: The Earth and Moon
6. The diagram in Figure 10.6 shows the earth as seen from outer space, looking
down on the north pole (labeled N).
(a) What time of day is it at point P? Explain how you can tell from the
diagram and the fact that the sun rises in the east and sets in the west.
(b) If a person at point P in the picture above can see the moon, and if the
moon is neither new nor full, then is the moon waxing or is it waning?
Explain how you can tell from the diagram.
7. The diagram in Figure 10.7 shows the earth as seen from outer space, looking
down on the North Pole (labeled N).
(a) What time of day is it at point P? Explain how you can tell from the
diagram and the fact that the sun rises in the east and sets in the west.
(b) If a person at point P in the diagram can see the moon, and if the moon
is neither new nor full, then is the moon waxing or is it waning? Explain
how you can tell from the diagram.
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moon’s orbit
sun rays
N
P
Figure 10.6: The Earth
moon’s orbit
N
sun rays
P
Figure 10.7: The Earth
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8. Use the fact that the sun rises in the east to explain clearly why the time of day
on the west coast of the U.S. is earlier than the time of day on the east coast
of the U.S. Draw diagrams or simple pictures to aid your explanation.
10.2
Angles
1. What are the two ways of defining (or thinking about) the concept of angle?
Briefly describe how these are related. Then give an example of a situation
where both ways of thinking about angles is needed.
2. Discuss the two different ways of defining the concept of angle. Describe how
to use an “angle explorer,” shown in Figure 10.8, to relate the two definitions.
Figure 10.8: An “Angle Explorer”
3. Explain clearly why Keisha’s “angle explorer” in Figure 10.9 does not show a
bigger angle than Aaron’s.
Aaron’s
Keisha’s
Figure 10.9: Two “Angle Explorers”
4. Suppose that two lines in a plane meet at a point, as in Figure 10.10. Use the
fact that the angle formed by a straight line is 180◦ to explain why a = c and
b = d.
5. Brad got in his car at point A and drove to point B along the route indicated
in Figure 10.11.
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b
c
a
d
Figure 10.10: Lines Meeting at a Point
(a) Show all of Brad’s angles of turning along his route.
(b) What is the total amount of turning that Brad did along his route? Describe how you can determine this without measuring individual angles and
adding them up.
A
N
E
W
S
B
Figure 10.11: Brad’s Route
6. Given that the indicated lines in Figure 10.12 are parallel, determine angle a
without actually measuring it. Explain your reasoning briefly.
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121
55°
a
parallel
125˚
Figure 10.12: Determine angle a
7. Given that the indicated lines in Figure 10.13 are parallel, determine angle a
without actually measuring it. Explain your reasoning briefly.
8. (a) Describe an informal hands-on way to help students see why it is plausible
that the sum of the angles in every triangle is 180◦.
(b) Give at least two reasons why the method of part (a) is not a proof that
the sum of the angles in every triangle is always 180◦ .
(c) Prove that the sum of the angles in every triangle is always 180◦ .
9. Explain why the sum of the angles in every triangle must always be 180◦ .
10. Use a “walking and turning” method to explain why the sum of the angles in
every triangle is always 180◦ .
11. Use the Parallel Postulate to prove that the sum of the angles in every triangle
is always 180◦ .
12. Determine angle x in Figure 10.14 without measuring it. Explain your reasoning.
13. Given that the lines in Figure 10.15 are parallel, determine the sum a + b + c
without measuring the angles. Explain your reasoning.
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95˚
parallel
a
35°
Figure 10.13: Determine angle a
155˚
parallel
x
110˚
Figure 10.14: Determine angle x
14. Determine the sum of the labeled angles in Figure 10.16 (without measuring).
Explain your reasoning.
10.3
Angles and Phenomena in the World
1. The drawing in Figure 10.17 shows Ashley in a room with two wall mirrors as
seen from the point of view of a fly looking down from the ceiling. What will
Ashley see when she looks at point P in the mirror? Explain.
2. The drawing in Figure 10.18 is of a person, a full length wall mirror, and a
chair, shown from the point of view of a fly on the ceiling. Can the person see
the chair in the mirror? If so, show a place on the mirror where the person can
see the chair. Either way, explain briefly why or why not.
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a
parallel
b
c
Figure 10.15: What is the sum of the angles?
a
b
c
f
e
d
Figure 10.16: Determine the sum of the labeled angles
1. (a) Informally, we might describe a circle as a perfectly round shape. What is
the mathematical definition of a circle?
(b) Solve the following problem and explain how the mathematical definition
of circle is involved in solving the problem:
The towns of Kneebend and Anklescratch are 10 miles apart, as shown on
the map in Figure 10.19. The Big Savings store is 6 miles from Kneebend
and 8 miles from Anklescratch. Where is the Big Savings store?
2. Which of the following provide a correct definition of the term circle. Check all
that apply.
(a) A collection of points in a plane that are all one fixed distance away from
a point.
(b) All the points in a plane that are one fixed distance away from each other.
(c) All the points that are one fixed distance away from a point.
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P
mirror 2
mirror 1
Ashley
Figure 10.17: Ashley in a Room with Two Wall Mirrors
mirror
chair
person
Figure 10.18: A Person, a Chair, and a Mirror
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(d) All the points in a plane that are one fixed distance away from a point.
3. A new Giant Superstore is being planned somewhere in the vicinity of Kneebend
and Anklescratch, towns which are 10 miles apart (as shown on the map in
Figure 10.19). The developers will only say that all the locations they are
considering are less than 7 miles from Kneebend and more than 5 miles from
Anklescratch. Indicate all the places where the Giant Superstore could be located. Explain your answer.
Anklescratch
Kneebend
Figure 10.19: Where Could the Giant Superstore be Located?
4. John says that his house is more than 5 miles from Walmart and more than 3
miles from KMart. Indicate all possible locations for John’s house on the map
in Figure 10.20. Explain your answer.
5 miles
Walmart
KMart
Figure 10.20: Where Could John’s House be Located?
5. A GPS unit receives information from two satellites. The GPS unit learns that
it is 10,000 miles from one satellite and 15,000 from the other satellite. Without
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any further information, describe the nature of all possible locations of the GPS
unit. Explain your answer.
10.4
Triangles, Quadrilaterals, and Other Polygons
1. Students sometimes get confused about the relationship between rectangles and
squares. Describe and explain this relationship as clearly, thoroughly, and precisely as you can by referring to our (short) definitions of rectangles and squares.
2. (a) Give the (short) definitions of square, rectangle, and parallelogram.
(b) Either describe in words or draw a clear and specific diagram to show how
the sets of squares, rectangles, and parallelograms are related. Explain
how you know these sets of shapes are related the way they are.
3. Use words and a clear diagram to describe how the sets of squares, rectangles,
and rhombuses are related. Then use the (short) definitions of these shapes to
explain how you know the sets of shapes are related the way they are.
4. Draw a detailed Venn diagram (or other clear diagram) showing how the set of
rectangles and the set of rhombuses are related. Explain clearly why these two
sets are related the way they are.
5. Draw a Venn diagram or other clear diagram showing the relationships between
the sets of squares, rectangles, and parallelograms.
6. Use a compass and straightedge to construct an equilateral triangle in a careful
and precise fashion. Explain why your construction must necessarily produce
an equilateral triangle. (Do not just show that your triangle is equilateral by
measuring.)
7. Use a compass and ruler to help you draw an isosceles triangle that has two
sides of length 3 inches and one side of length 2 inches.
8. (a) Use a compass and ruler to help you draw a triangle that has one side
of length 4 inches, one side of length 2 inches, and one side of length 3.5
inches.
(b) Explain why your method of construction must produce a triangle with
the required side lengths.
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9. (a) Describe how 4 children could use 4 pieces of string to show a variety of
different rhombuses.
(b) Describe how you could show a variety of different rhombuses by threading
straws onto string or by fastening sturdy strips of cardboard with brass
fasteners.
(c) Describe how to fold and cut an ordinary piece of paper so that when you
unfold the paper, the resulting shape is necessarily a rhombus. Explain
why the resulting shape must be a rhombus.
10. The line segments AB and AC in Figure 10.21 have been constructed so that
they could be two sides of a rhombus.
(a) Use a compass and straightedge to finish constructing a rhombus that has
AB and AC as two of its sides.
C
A
B
Figure 10.21: The Beginning of a Construction of a Rhombus
(b) By referring to the definition of rhombus, explain why your construction
in part (a) must produce a rhombus.
11. Give an example to show how the definition of circle is used in constructing
shapes (other than circles). Explain how the definition of circle is used in your
example.
12. Give instructions to Robot Rob for how to move and turn so that his path
makes an equilateral triangle (with your choice of side lengths). Explain briefly
why the instructions will result in the desired path.
13. Give instructions to Automaton Audrey for how to move and turn so that her
path makes a parallelogram that has an angle of 50◦ (with your choice of side
lengths). Explain briefly why the instructions will result in the desired path.
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14. Suppose you use Geometer’s Sketchpad (GSP) to construct two circles with
centers A and B, in such a way that the circles will always have the same radius,
no matter how you move them. Suppose that the two circles meet at points C
and D, and suppose that you construct line segments to make a quadrilateral
ACBD (by connecting connecting A to C, C to B, B to D, and D to A), as
shown in Figure 10.22. What kind of special quadrilateral must ACBD be, no
matter how you move the points in your construction (as long as the circles
still meet at two points)? Explain your answer clearly and in detail, as if you
were explaining to someone who was just learning about the geometric concepts
involved.
C
A
B
D
Figure 10.22: A Construction
15. Assume that the line segments AB and AC in Figure 10.23 have been constructed using Geometer’s Sketchpad so that they could be two sides of a rhombus. Explain how to use Geometer’s Sketchpad in order to finish the sketch
in Figure 10.23 so that it will always be a rhombus, no matter how points are
moved around. In doing so, use only the definition of rhombus (not other properties that rhombuses have). Refer to the following list of Geometer’s Sketchpad
commands from the construct menu:
construct segment,
construct perpendicular line,
construct parallel line,
construct circle by center and point,
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construct circle by center and radius.
C
A
B
Figure 10.23: Construct a Rhombus
16. Emily wants to use Geometer’s Sketchpad to construct a square in such a way
that no matter how she moves the points in her construction, it will always
remain a square. So far, Emily has constructed a line segment AB and two
lines that are perpendicular to AB and pass through A and B, as shown in
Figure 10.24. Which of the GPS commands listed below should Emily use
next to finish her construction? Explain how and why she should use those
commands.
construct segment
construct perpendicular line
construct parallel line
construct circle by center and point
construct circle by center and radius
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A
B
Figure 10.24: Emily’s Construction
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11.1
Fundamentals of Measurement
1. Jenny wants to know what it means when we say that a tank is 284 cubic feet.
What can you tell Jenny?
2. Which of the following describe the same volume, or mean the same as, or are
the correct way to read 2 in3 ?
• a 2 inch by 2 inch by 2 inch cube
• 2 inches cubed
• 2 cubic inches
• 2 in × 2 in × 2 in
3. When Joe was asked to draw a shape that has an area of 3 square centimeters,
he drew a 3 cm by 3 cm square. Is Joe right or not? Explain.
4. You may discuss parts (a), (b), (c) together or separately, as you like.
(a) What error (that we discussed) do students commonly make in interpreting
an area measurement such as “4 square inches”?
(b) Explain clearly what “4 square inches” means.
(c) What is another way to write “4 square inches,” how is it said correctly,
and what is a common error in the way it is said?
(d) Students sometimes say, “area is length times width.” Explain why this
statement is not fully accurate.
5. State the meaning of each of the following prefixes, which are used in the metric
system: kilo, deci, hecto, milli, centi, deka.
6. Give examples of two units in the metric system that use the prefix milli. State
the attributes that the units are used to measure. For each unit, give an example
of some actual thing whose size could be appropriately described using that unit.
7. What is special about the way units in the metric system are named? Give
several examples to illustrate.
8. How is one milliliter related to one gram and to one centimeter?
9. What is the difference between an ounce and a fluid ounce?
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Length, Area, Volume, and Dimension
1. Explain why either of the two rectangles in Figure 11.1 can be considered the
larger of the two.
Figure 11.1: Two Rectangles
2. Describe how it could happen that three different buildings could each be
claimed—rightfully—to be the largest of the three. Choose your examples to
show a range of geometric ways of thinking about the size of an object. Give
specific details about each building and explain why each can be considered
largest. Discuss the implications of this kind of situation for teaching students
about measurement.
3. Describe one-dimensional, two-dimensional, and three-dimensional parts or aspects of a bottle. In each case, name an appropriate U.S. customary unit and
an appropriate metric unit for measuring or describing the size of that part or
aspect of the bottle. What are practical reasons for wanting to know the sizes
of these parts or aspects of the bottle?
4. Suppose there are two rectangular pools: one is 30 feet wide, 40 feet long, and
3 feet deep throughout, the other is 20 feet wide, 40 feet long, and 5 feet deep
throughout. Compare the sizes of the pools in two meaningful ways other than
by comparing one-dimensional aspects.
5. Explain why each one of the two blocks in Figure 11.2 can be considered the
“biggest” of the two by first comparing the blocks’ sizes with respect to a 2dimensional aspect of the blocks, and then comparing the blocks’ sizes with
respect to a 3-dimensional aspect of the blocks.
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5 in
1 in
3 in
5 in
3 in
3 in
Figure 11.2: Two Blocks
6. Explain why each one of the two blocks pictured in Figure 11.2 can be considered
the “biggest” of the two by comparing the blocks with respect to two different
measurable attributes other than a one-dimensional attribute.
11.3
Error and Precision in Measurements
1. What is the difference between reporting that something weighs 2 pounds and
that it weighs 2.0 pounds? Give a detailed, thorough answer.
2. Does a food label that says “0 grams trans fat in one serving” mean that the
food contains no trans fat? If not, how much trans fat could be in one serving
of the food? (Assume that food labels follow the usual scientific conventions on
reporting measurements.) Answer specifically and explain your answer.
3. If the distance between two cities is reported as 3460 miles, does that mean that
the distance is exactly 3460 miles? If not, what can you say about the exact
distance?
4. Roger is calculating the distance from town A to town C. Roger is given that
the distance from town A to town B is 240 miles, the distance from town B to
town C is 350 miles, that town B is due east of town A, and that town C is due
north of town B. Roger does some calculations and concludes that the distance
from town A to town C is 430.116 miles. Should Roger leave his answer like
that? Why or why not? If not, what answer should Roger give? Explain. (You
may assume that Roger has done his calculations correctly.)
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135
Converting from One Unit of Measurement
to Another
1. Sam is confused about why we multiply by 3 to convert 6 yards to feet. Sam
thinks we should divide by 3 because feet are smaller than yards. Explain in at
least two different ways why it makes sense to multiply by 3 to convert 6 yards
to feet.
2. Sam is confused about why we multiply by 3 to convert 6 yards to feet. Sam
thinks we should divide by 3 because feet are smaller than yards. Address
Sam’s misconception and explain in a clear, simple, non-technical way why we
multiply by 3 to convert 6 yards to feet.
(a) Explain how to convert 24 feet to inches without using dimensional analysis, explaining in a clear and simple way why you calculate the way you
do.
(b) Explain how to convert 24 feet to yards without using dimensional analysis,
explaining in a clear and simple way why you calculate the way you do.
3. Explain how to convert 52 inches to feet without using dimensional analysis,
explaining why you calculate the way you do. Describe at least two different
correct ways to write the exact answer to this conversion problem. Explain
briefly why these different ways of writing the answer mean the same thing.
4. Explain how to convert 11 quarts to gallons without using dimensional analysis,
explaining why you calculate the way you do. Describe at least two different
correct ways to write the exact answer to this conversion problem. Explain
briefly why these different ways of writing the answer mean the same thing.
5. The distance between two cities is described as 370 kilometers. What is this
distance in miles? Calculate your answer using the basic fact that 1 inch =
2.54 cm.
6. What is 1 billion seconds in terms of years, days, hours, and minutes? Explain
why you can solve this problem the way you do. (Even though it’s not quite
accurate, take each year to be 365 days long.)
7. One meter is 100 cm. Does it therefore follow that one square meter is 100
square centimeters? Explain.
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8. One yard is 3 feet. Does it therefore follow that one cubic yard is 3 cubic feet?
Explain.
9. Using the example of converting 1 square foot to square inches, describe and
discuss a common error that students make when converting area measurements.
10. How many cubic inches are in 2 cubic feet? Explain your answer and discuss
errors that students may make in solving this problem.
11. (a) Convert 2 square meters to square centimeters and explain why your
method makes sense.
(b) Describe a misconception about area measurements such as 2 square inches
that some students sometimes have.
(c) Describe a misconception about converting 1 square foot to square inches
that some students sometimes have, even if they know how feet and inches
are related.
(d) Describe a misconception about converting 6 yards to feet that some students sometimes have, even if they know how yards and feet are related.
12. Use the basic fact 1 inch = 2.54 cm in order to determine what 1 cubic yard is
in terms of cubic meters.
13. Use the basic fact 1 inch = 2.54 cm in order to determine what 100 square
meters is in terms of square feet.
14. The floor area of a house is 2500 square feet. Find the floor area of the house
in square meters, using the fact that 1 inch = 2.54 cm. Show your work, taking
care to use correct notation.
15. The area of land is often measured in acres. One acre is 43,560 square feet.
(a) What is a square mile in acres? Explain.
(b) What is a square kilometer in acres? Explain.
16. Analyze the following calculations which intend to convert 900 square meters
to square feet. Which ones use legitimate methods and are correct, and which
are not? Explain.
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(a)
900 m2 = 900 m ×
1 in
1 ft
100 cm
×
×
= 2953 ft2 .
1m
2.54 cm
12 in
(b)
100 × 100 cm2
1 in2
1 ft2
2
900 m = 900 m ×
×
×
2 = 9688 ft .
2
2
1m
2.54 × 2.54 cm 12 × 12 in
2
2
(c)
900 m = 900 m ×
100 cm
1 in
1 ft
×
×
= 2953 ft
1m
2.54 cm 12 in
Therefore,
900 m2 = 29532 ft2 = 8, 718, 767 ft2
(d) 900 square meters is the area of a square that is 30 meters wide and 30
meters long.
30 m = 30 m ×
100 cm
1 in
1 ft
×
×
= 98.43 ft
1m
2.54 cm
12 in
Therefore,
30 m2 = 98.43 × 98.43 ft2 = 9688 ft2 .
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Area of Shapes
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12.1
What Area Is and Areas of Rectangles Revisited
1. What does it mean to say that a shape has an area of 30 square inches?
2. You have a 3 foot by 4 foot rectangular rug in your classroom. You also have a
bunch of square foot tiles and some tape measures.
(a) What is the most primitive way for your students to determine the area of
the rug?
(b) What is a less primitive way for your students to determine the area of the
rug and why does this method work?
3. Figure 12.1 shows a 3-cm-by-8-cm rectangle decomposed into 1-cm-by-1-cm
squares. Discuss the difference in the units we attach to the 3 and the 8 in these
two situations: (a) when applying the rectangle area formula to determine the
area of the rectangle, and (b) when viewing the rectangle as decomposed into
equal groups in order to apply our definition of multiplication to determine the
area of the rectangle.
Figure 12.1: What units should we attach to the 3 and the 8 in using the area formula
and in describing equal groups?
4. (a) Explain how to see the the large rectangle in Figure 12.2 as decomposed
into 2 12 groups with 5 21 squares in each group, so as to describe the area of
the rectangle as 2 12 × 5 21 cm2 .
(b) Calculate 2 12 × 5 12 without a calculator, showing your calculations. Then
verify that this calculation has the same answer as when you determine
the area of the rectangle in Figure 12.2 by counting squares.
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1 cm2
Figure 12.2: Explain why the area is 2 12 × 5 12 cm2
12.2
The Moving and Additivity Principles About
Area
1. Use only the formula for areas of rectangles and the moving and additivity
principles about area to determine the area of the shaded shape in Figure 12.3.
Explain your method.
3 cm
4 cm
2 cm
2 cm
9 cm
Figure 12.3: A shape
12.3
Areas of Triangles
1. One way to use the moving and additivity principles to determine the area of
the shaded triangle in Figure 12.4 is to subdivide the triangle into small pieces,
as shown on the right, and move and recombine these pieces so as to make 3
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squares. Explain how to use the moving and additivity principles to determine
the area of the triangle in a significantly different way.
1 cm
Figure 12.4: One way to determine the area
2. One way to use the moving and additivity principles to determine the area of
the shaded triangle in Figure 12.5 is to subdivide the triangle into small pieces,
as shown on the right, and move and recombine these pieces so as to make 4
squares. Explain how to use the moving and additivity principles to determine
the area of the triangle in a significantly different way.
1 cm
Figure 12.5: One way to determine the area
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3. Show two ways to use the moving and additivity principles to determine the area
of the triangle in Figure 12.6: one primitive way (that relies directly on what
area means) and one more advanced way (that can be generalized to explain
the triangle area formula). Explain each method.
Figure 12.6: Determine the area in two ways
4. Use the area formula for rectangles and principles about area that we have
studied to find and explain two different formulas for the area of the right
triangle in Figure 12.7 in terms of the side lengths b and h. (Your two formulas
should look different even though they will be equivalent. In each case, your
explanation should fit with the way you write the formula.)
h
b
Figure 12.7: A right triangle
5. Using the area formula for rectangles and principles about area that we have
studied, give a clear and thorough explanation for why the area of the triangle
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in Figure 12.8 is
h.
1
2
· b · h square units for the given choices of base b and height
h
b
Figure 12.8: A triangle
6. Using the area formula for rectangles and principles about area that we have
studied, give a clear and thorough explanation for why the area of the triangle
in Figure 12.9 is 12 · b · h) square units for the given choices of base b and height
h.
h
b
Figure 12.9: A triangle
7. Using the area formula for rectangles and principles about area that we have
studied, give a clear and thorough explanation for why the area of the triangle
in Figure 12.9 is 12 · b · h) square units for the given choices of base b and height
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h. Your explanation should be general, in the sense that we could see why it
would work for other triangles whose height is outside the triangle.
8. Determine the area of the shaded triangle in Figure 12.10. Indicate your reasoning.
1 cm
Figure 12.10: A triangle
9. The shaded triangle in Figure 12.11 is enclosed in a 10 unit by 10 unit square.
Determine the area of the shaded triangle in two different ways. Explain your
reasoning in each case.
10 units
4 units
10 units
Figure 12.11: Determine the area
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10. Determine the area of the triangle in Figure 12.12 in two different ways. Explain
your reasoning in each case. Adjacent dots on the grid are 1 cm apart.
1 cm
Figure 12.12: A triangle
12.4
Areas of Parallelograms and Other Polygons
1. Determine the area of the quadrilateral in Figure 12.13 using only the principles
we have studied about area and the area formula for rectangles (do not use any
other area formulas). Explain your reasoning. Adjacent dots on the grid are
1 cm apart.
2. Determine the area of the parallelogram in Figure 12.14 using only the principles
we have studied about area and the area formula for rectangles (do not use any
other area formulas). Explain your reasoning. Adjacent dots on the grid are
1 cm apart.
3. Determine the area of the shaded shape in Figure 12.15, explaining your reasoning.
4. Determine the area of the shaded shape in Figure 12.16, explaining your reasoning.
5. Determine the area of the shaded shape in Figure 12.17. Explain your reasoning
briefly.
6. Explain clearly why there can be no formula for areas of parallelograms that is
only in terms of the lengths of the sides of the parallelogram.
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1 cm
Figure 12.13: A Quadrilateral
1 cm
Figure 12.14: A parallelogram
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10 units
5 units
20 units
Figure 12.15: Determine the Area
1 cm
Figure 12.16: Determine the area
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1 cm
Figure 12.17: Determine the area
7. Use the moving and additivity principles to explain why the area of the shaded
parallelogram in Figure 12.18 is b × h for the given choice of base b and height
h without subdividing the parallelogram.
h
b
Figure 12.18: Explain the area formula for a parallelogram
8. Explain why the area of the parallelogram in Figure 12.18 is b × h for the given
choice of base b and height h. You may use the moving and additivity principles
and the area formulas for rectangles and triangles in your explanation (use any
or all of these).
9. Find a formula in terms of a, b, and h for the area of the trapezoid in Figure 12.19. Explain why your formula is valid by using some or all of the following: the area formula for rectangles, the area formula for triangles, and the
moving and additivity principles.
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a
h
b
Figure 12.19: A trapezoid.
12.5
Shearing: Changing Shapes Without Changing Area
12.6
Areas of Circles and the Number Pi
1. Explain how to see why the formula for the area of a circle of radius r makes
sense if one already knows the formula for the circumference of a circle of radius
r.
2. Given that the circumference of a circle of radius r units is 2πr units, explain
how to subdivide and rearrange a circle of radius r units in order to show why
the area of this circle is πr 2 square units.
3. Using only the area formula for rectangles and the moving and additivity principles, find a formula for the area of a regular pentagon (see Figure 12.20) in
terms of b and h and explain why your formula is valid. Here b is the length of
each side and h is the distance from the center of the pentagon to the midpoint
of a side.
4. In the gym there is a round metal pole that is 8 inches in diameter. The gym
teacher wants to wrap the pole tightly with 1 inch thick rope from the ground
up to a height of 6 feet. (The rope will be wound around the pole over and over
until it reaches a height of 6 feet.) About how much rope will the gym teacher
need? Explain your reasoning.
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b
h
b
b
b
b
Figure 12.20: A Regular Pentagon
5. A garden path surrounds a circular garden, as shown in Figure 12.21. The
garden path is 5 feet wide all the way around and the distance around the
outside of the garden path is 80 feet.
Figure 12.21: A Garden Path
(a) What is the area of the garden path? Explain.
(b) What is the area of the garden (inside the path)? Explain.
6. A concrete patio will be made in the shape of a 12 foot by 12 foot square with
half-circles attached at two opposite ends, as pictured in Figure 13.8. What is
the area of this patio?
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Figure 12.22: A Patio
12.7
Approximating Areas of Irregular Shapes
1. Suppose you have a map with a scale of 1 inch ↔ 25 miles. You trace a county
on the map onto 41 inch graph paper. (The grid lines are spaced 14 inch apart.)
You figure that on the map, the county takes up about 65 squares of graph
paper. Approximately what is the area of the actual county? Explain.
2. Some students have a map with a scale of 1 inch ↔ 30 miles. A county on the
map has been traced onto 51 inch graph paper. (The grid lines are spaced 15 inch
apart.) The students agree that the county takes up about 55 squares of graph
paper. The students have several initial ideas and unfinished calculations for
determining the area of the county. For each of these initial calculations, either
use them to finish determining the area of the county or explain why they will
probably lead to an error.
(a) 30 ÷ 5 = 6
(b) 55 ÷ 5 = 11
(c) 55 ÷ 25 = 2 51
(d) 55 = 5 · 11
6 · 6 = 36
11 · 30 =
2 15 · 30 =
5 ↔ 30
11 ÷ 5 = 2 51
3. Suppose you have a map with a scale of 1 cm ↔ 25 km. You trace a country on
the map onto a sheet of card stock, cut out the tracing, and weigh it. The tracing
of the country weighs 7 grams. You find that a 30 cm by 40 cm piece of the
same card stock weighs 20 grams. Based on these measurements, approximately
what is the area of the country? Explain.
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153
Contrasting and Relating the Perimeter and
Area of Shapes
1. Explain the difference between perimeter and area in a way that could help
someone learn.
2. A student in your class wants to know why we multiply only two of the lengths
of the sides of a rectangle in order to determine the rectangle’s area. When we
calculate the perimeter of a rectangle we add the lengths of the four sides of the
rectangle, so why don’t we multiply the lengths of the four sides in order to find
the area? Explain to the student what perimeter and area mean and explain
why we carry out the perimeter and area calculations for a rectangle the way
we do.
3. Describe a common error that students make when finding the perimeter of
a shape such as the shaded shape in Figure 12.23 and describe the misunderstanding that is at the root of this error. Discuss what you will need to draw
students’ attention to in order to help them avoid this misunderstanding.
Figure 12.23: What error do students often make concerning perimeter?
4. (a) On the 14 inch graph paper in Figure 12.24 (meaning that adjacent grid
lines are 41 inch apart), draw two different rectangles that have perimeter
5 12 inches.
(b) Without using a calculator, determine the areas of your rectangles. Show
your calculations, or explain briefly how you determine the areas.
5. (a) Describe a concrete way to demonstrate that many different shapes can
have the same area.
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Figure 12.24: Quarter-Inch Graph Paper
(b) Describe a concrete way to demonstrate that many different shapes can
have the same perimeter.
6. The distance around a piece of property is 5.3 miles. With only this information
about the property, can you determine the area of the property? If so, explain
how, if not, explain why not.
7. A piece of property has a perimeter of 5.3 miles.
(a) If you don’t know anything about the shape of the property, then what is
the largest area that the property could have? Explain.
(b) If you know that the property is in the shape of a rectangle, then what is
the largest area that the property could have? Explain.
(c) If you know that the property is in the shape of a rectangle, then what is
the range of possible areas that the property could have? Explain.
(d) Is it possible that the area of the property is 1 square mile? Explain.
8. A piece of property has a perimeter of 5.3 miles.
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(a) If you know that the property is rectangular (but you don’t know its dimensions) then what can you say about its area? (Give as much information
as possible.)
(b) If you don’t know anything about the shape of the property, then what
can you say about its area? (Give as much information as possible.)
9. Sam wants to find the area of an irregular shape. Sam cuts a piece of string
to the length of the perimeter of the shape. Sam measures that the string is
about 40 cm long. Provide an informative discussion about what can and can’t
be deduced about the area of Sam’s shape.
10. Sam wants to find the area of an irregular shape. Sam cuts a piece of string to
the length of the perimeter of the shape. Sam measures that the string is about
40 cm long. Sam then forms his string into a square on top of centimeter graph
paper. Using the graph paper, Sam determines that the area of his string square
is about 100 cm2 . Sam says that therefore the area of the irregular shape is also
100 cm2 . Is Sam’s method for determining the area of the irregular shape valid
or not? Explain. If the method is not valid, what can you determine about the
area of the irregular shape from the information that Sam has? Explain.
12.9
Using the Moving and Additivity Principles
to Prove the Pythagorean Theorem
1. Use Figure 12.25 to explain why a2 +b2 = c2 , where a and b are the lengths of the
short sides of a right triangle, and c is the length of the triangle’s hypotenuse.
(You may assume that all shapes that look like squares really are squares, and
that in each drawing, all four triangles with side lengths a, b, c are identical right
triangles.)
2. (a) State the Pythagorean theorem.
(b) Prove the Pythagorean theorem.
3. What is the longest pole that can fit in a box that is 4 feet wide, 3 feet deep,
and 5 feet tall? Explain.
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a
b
a
a
c
a
b
a
a
b
c
c
a
b
c
b
c
b
a
b
b
b
c
a
a
Figure 12.25: Shapes Forming Identical Large Squares
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Solid Shapes and their Volume and
Surface Area
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13.1
Polyhedra and Other Solid Shapes
1. What is a prism?
2. What is a pyramid?
3. What is a cylinder?
4. What is a cone?
5. How are prisms, cylinders, pyramids, and cones related?
6. What is the difference between a rectangle and a rectangular prism? Discuss a
relationship between those shapes.
7. What is the difference between a square and a cube? Discuss a relationship
between those shapes.
8. Answer the following without using a model.
(a) How many faces does a prism with a pentagon base have? What shapes
are the faces? Explain briefly.
(b) How many edges does a prism with a pentagon base have? Explain.
(c) How many corners does a prism with a pentagon base have? Explain.
9. Answer the following without using a model.
(a) How many faces does a pyramid with a pentagon base have? What shapes
are the faces? Explain briefly.
(b) How many edges does a pyramid with a pentagon base have? Explain.
(c) How many corners does a pyramid with a pentagon base have? Explain.
10. A cube is one of the Platonic solids. Name one other Platonic solid and describe
its characteristics. How many faces does it have? What shape are the faces?
How many faces come together at a vertex?
11. A dodecahedron has 3 regular pentagon faces coming together at each corner,
but there is no Platonic solid that has 4 or more regular pentagon faces coming
together at each corner. Why not? (See Figure 13.1.)
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108◦
Figure 13.1: A Pentagon
13.2
Patterns and Surface Area
1. For each of the patterns in Figure 13.2, name the shape it would make if it were
cut out, folded, and taped to make a closed shape. (Do this without cutting
and folding!) In each case, label the base(s) (if any). Determine whether the
first shape is oblique or right.
2. What shapes would the patterns in Figure 13.3 make if they were cut out along
the heavy lines, folded along the dotted lines, and taped together to make closed
shapes? Name each shape as precisely as you can.
3. Which of the patterns in Figure 13.4 could be cut out and folded up without
overlaps to make a closed cube? Which cannot? (Circle does or does not.) Solve
this problem by visualizing (do not cut out patterns!).
4. Draw two (fairly) precise patterns: one for a triangular prism and one for a
pyramid with a triangle base. Indicate which is which. Indicate which sides
would be joined together if you were to fold up the patterns to form the shapes.
5. Suppose you take a rectangular piece of paper, roll it up, and tape two ends
together, without overlapping them, to make a cylinder (without bases). If the
cylinder is 8 21 inches long and has a diameter of 3 12 inches, then what were the
length and width of the piece of paper? Explain your reasoning.
6. Draw a a pattern for a triangular prism that has the triangle in Figure 13.5 as
a base. On your pattern, label the sides that have length a, the sides that have
length b, and the sides that have length c. (You may copy the triangle to other
locations on the page.)
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precise name of shape:
label the base(s)
oblique or not?
precise name of shape:
label the base(s)
precise name of shape:
label the base(s)
Figure 13.2: Patterns
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precise name of shape:
precise name of shape:
precise name of shape:
Figure 13.3: Patterns
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does
does not
does
does not
Figure 13.4: Which Would Make a Cube?
b
a
c
Figure 13.5: A Triangular Base
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7. Use a ruler and compass to draw a pattern that could be cut out and taped
together without overlaps to make a cone without a base. Indicate which sides
should be taped together.
8. Draw a pattern for a triangle-base pyramid. You may leave the base off your
pattern. Indicate which sides will be joined to form the pyramid.
9. Draw small sketches of patterns for the shapes listed in parts (a), (b), and (c).
Each pattern should be such that if it were cut out, and if appropriate sides were
attached without overlapping, the result would be the desired closed shape. For
each pattern, provide enough details, such as the lengths of various parts of
the pattern, so that someone who had the proper tools (compass, ruler, graph
paper) could make a precise pattern of the correct size.
(a) A rectangular prism that is 2 inches by 3 inches by 4 inches.
(b) A cylinder that has bases of radius 1 inch and is 3 inches high.
(c) A pyramid that has a square base.
10. Describe a pattern for a box that is 20 cm wide, 30 cm long, and 40 cm tall and
use the pattern to explain how to determine the surface area of the box.
11. Describe a pattern for a 2 inch tall cylindrical tin can that has diameter 3 inches
and use the pattern to explain how to determine the total surface area of the
can (including the top and the bottom). Explain your reasoning.
12. Determine the total surface area of a cone that has a circular base of radius
3 cm if the lateral portion of the cone (the side that does not include the base)
is made from a half-circle. Explain your reasoning.
13.3
Volumes of Solid Shapes
1. What does it mean to say that a solid shape has a volume of 20 cubic centimeters?
2. Describe a concrete way to demonstrate that many different solid shapes can
have the same volume.
3. Students often get confused between the surface area and the volume of a 3dimensional object.
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(a) Explain the difference between the surface area of a water tower and the
volume of a water tower in a way that could help someone learn.
(b) Determine the total surface area as well as the volume of a box that is 3
feet wide, 2 feet deep, and 4 feet tall. Explain why you calculate as you
do.
4. Students often get confused between the surface area and the volume of a solid
shape. Explain the difference between the surface area and volume of a 3 foot
by 2 foot by 4 foot box in a way that could help someone understand why we
calculate these the way we do.
5. Students often get confused between the surface area and the volume of a solid
shape. Describe what surface area and volume are and discuss how they are
different.
6. Your students have an open-top box that has a 3 inch by 4 inch rectangular
base and is 2 inches high. They also have a bunch of cubic inch blocks and some
rulers.
(a) What is the most primitive way for your students to determine the volume
of the box?
(b) What is a less primitive way for your students to determine the volume of
the box and why does this method work?
7. Explain why the Volume = (height) · (area of the base) formula is valid for right
prisms using the example of a prism that is 4 cm high and that has a base of
area 6 cm2 .
8. Use the idea of decomposing a prism into layers to explain why the volume of a
right prism is (height) · (area of base). Use the example of a prism that is 4 cm
high and has a base of area 6 cm2 to illustrate the reasoning.
9. Use the idea of breaking a prism into layers to explain why the volume of a
right prism is (height) · (area of base). Illustrate with a prism that is 4 cm high
and has a base of area 6 cm2 .
10. A student in your class wants to know why we multiply only three of the lengths
of the edges of a box in order to calculate the volume of the box. Why don’t
we have to multiply all the lengths of the edges? Explain to this student why
it makes sense to calculate the volume of a box as we do.
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11. The front (and back) of a greenhouse have the shape and dimensions shown in
Figure 13.6. The greenhouse is 40 feet long and the angle at the top of the roof
is 90◦ . Determine the volume of the greenhouse in cubic feet. Explain your
solution.
15 ft
9 ft
12 ft
Figure 13.6: A Greenhouse.
12. A volume problem:
The front and back of a storage shed are shaped like isosceles triangles
with dimensions shown on the left in Figure 13.7. The storage shed
is 15 feet long. Determine the volume of the shed, explaining your
reasoning.
Tommy solves the volume problem as follows: The base of the shed is a 12 foot
by 15 foot rectangle, which has area 12 · 15 = 180 square feet. The shed is 8
feet high, so the volume of the shed is 8 · 180 = 1440 cubic feet.
(a) Is Tommy’s solution to the volume problem correct? Explain why or why
not.
(b) Solve the volume problem using a different method than Tommy did, explaining your reasoning.
13. A concrete patio will be made in the shape of a 12-foot-by-12-foot square with
half-circles attached at two opposite ends, as pictured in Figure 13.8. If the
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8 feet
15 feet
12 feet
side view of shed
front and back of shed
Figure 13.7: A Storage Shed.
concrete will be 4 inches thick, then how many cubic yards of concrete will be
needed for the patio? Explain your solution.
Figure 13.8: A Patio.
14. Caulking is often used to seal around bathtubs and showers in order to make
them waterproof. When you squeeze caulking out of a tube, it comes out in the
shape of a (very long) cylinder whose diameter is the diameter of the hole in
the tube where the caulking comes out. Suppose that a tube of caulking has a
hole of diameter 18 of an inch and suppose that you can use the tube of caulking
to seal 50 feet worth of edges around bathtubs and showers. How many cubic
inches of caulking must have been in the tube when it was full? Explain your
answer.
15. The front and back of a storage shed are shaped like half-circles of diameter 16
feet, as shown in Figure 13.9. The shed is 25 feet long.
(a) Determine the volume of the storage shed, indicating your reasoning.
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front view
167
side view
Figure 13.9: A Storage Shed.
(b) The top of the storage shed is to be covered with plastic sheeting (not
including the front and back). What size piece of plastic is needed to cover
the shed? Explain your reasoning.
16. Why is there a 13 in the volume formula for a pyramid? Give some idea for
where the 31 comes from.
17. A sand and gravel company has a cone-shaped pile of sand. The company
measures that the distance around the pile of sand at the base is 85 feet and
the “slanted” distance from the edge of the pile at ground level to the top of the
pile is 25 feet. Determine the volume of sand in the cone-shaped pile, explaining
your solution.
18. A cone without a base is made from a half-circle of paper by joining two radii.
The base of the cone is a circle of radius 3 cm. What is the volume of the cone?
Explain your reasoning.
19. The lateral portion of a cone (the curved side that does not include the base)
is made from a half-circle of paper by joining two radii. The base of the cone is
a circle of radius 3 cm.
(a) Determine the total surface area of the cone, explaining the reasoning.
(b) Determine the volume of the cone, explaining the reasoning.
20. A right pyramid has a height of 50 meters and has a square base with sides 50
meters long.
(a) Determine the volume of the pyramid.
(b) Determine the surface area of the pyramid, not including the base.
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21. A right pyramid has a square base with sides 100 meters long. The distance
from a corner of the base to the top of the pyramid is 125 meters.
(a) Determine the volume of the pyramid.
(b) Determine the surface area of the pyramid (not including the base).
22. Draw a small sketch of a pattern for a cone that has a base of radius 3 cm and
a volume of 12π cm3 . Your pattern should be such that if it were cut out, and
if appropriate sides were attached without overlapping, the result would be the
desired cone. Provide enough details, such as the lengths of various parts of the
pattern, so that someone who had the proper tools (compass, ruler, protractor)
could make a precise pattern of the correct size. Show relevant calculations.
13.4
Volume of Submersed Objects Versus Weight
of Floating Objects
1. Suppose you have a paper cup floating in a measuring cup that contains water.
When the paper cup is empty, the water level in the measuring cup is at 250 ml.
When you put 9 quarters in the paper cup (which is still floating), the water
level in the measuring cup is at 300 ml. What information about a quarter can
you deduce from this experiment? Explain.
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Geometry of Motion and Change
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14.1
Reflections, Translations, and Rotations
1. For each figure in Figure 14.1, describe a single transformation that will take
shape A to shape B. Describe each transformation in detail (you may draw on
the pictures to help you).
A
A
A
B
B
B
Figure 14.1: Three Pictures
2. For each figure in Figure 14.2, describe a single transformation that will take
shape A to shape B. Describe each transformation as precisely as you can.
A
A
B
A
B
B
Figure 14.2: Three Pictures
3. For each of the three pictures in Figure 14.3, describe a single transformation
that takes shape A to shape B. Draw marks on the pictures to help you describe
the transformations as precisely as possible.
4. Draw the result of reflecting the shaded shape in Figure 14.2 across the heavy
line.
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A
A
A
B
B
B
Figure 14.3: Three Pictures
Figure 14.4: Draw the Result of Reflecting Across the Heavy Line
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5. Draw the result of translating the shaded shape in Figure 14.5 in the direction
and by the distance indicated by the arrow.
Figure 14.5: Draw the Result of Translating
6. Draw the result of rotating the shaded shape in Figure 14.6 90◦ counterclockwise
around the origin.
1. What symmetries does the design in Figure 14.7 have? Describe each symmetry
as precisely as you can. (Consider the design as a whole.)
2. What kinds of symmetry does the design in Figure 14.8 have? Describe each
kind of symmetry as precisely as you can.
3. Say what it means for a shape or design to have 4-fold rotation symmetry. On
graph paper, draw a shape or design that has 4-fold rotation symmetry.
4. (a) Say what it means for a shape or design to have 4-fold rotation symmetry.
(b) Say what it means for a shape or design to have reflection symmetry.
(c) Say what it means for a shape or design to have translation symmetry.
5. Draw a design that is made with copies of the curlicue in Figure 14.9 (and its
reflection) so that the design as a whole has both 3-fold rotation symmetry and
reflection symmetry. Neither artistry nor an explanation is needed.
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y-axis
x-axis
Figure 14.6: Draw the Result of Rotating 90◦ Counterclockwise
Figure 14.7: A Design
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Figure 14.8: A Design
Figure 14.9: A Curlicue
6. Draw a design that is made out of (approximate) copies of the curlicue shown
in Figure 14.9 (and its reflection) and that has 4-fold rotation symmetry as well
as reflection symmetry. (Artistry is not required—a rough sketch will do as
long as it shows the desired features clearly.)
7. Draw a design that is made out of copies of the shaded shape in Figure 14.10
and has 4-fold rotation symmetry but no reflection symmetry.
Figure 14.10: Create a Symmetrical Design
8. Draw a design that is made out of copies of the shaded shape in Figure 14.10
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and has both 4-fold rotation symmetry and reflection symmetry.
9. In the grid in Figure 14.11, draw a design that is made out of copies of the
shaded shape to the left, that has both 2-fold rotation symmetry and reflection
symmetry, but that does not have 4-fold rotation symmetry. No explanation is
needed.
Figure 14.11: Create a Symmetrical Design
10. On the graph paper in Figure 14.12, draw a single design that has the following
3 types of symmetry: translation symmetry, reflection symmetry, and rotation
symmetry.
11. Explain clearly the distinction between the concept of translation and the concept of translation symmetry. How are these two concepts related?
12. Explain clearly the distinction between the concept of rotation and the concept
of rotation symmetry. How are these two concepts related?
13. Explain clearly the distinction between the (mathematical) concept of reflection
and the concept of reflection symmetry. How are these two concepts related?
14.2
Congruence
1. Ada, Bada and Cada are three cities. Bada is 15 miles from Ada, Cada is 20
miles from Bada, and Ada is 25 miles from Cada. There are straight line roads
between Ada and Bada, Ada and Cada, and Bada and Cada.
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Figure 14.12: Graph Paper
(a) Draw a careful and precise map showing Ada, Bada, and Cada and the
roads between them, using a scale of 10 miles = 1 inch. Describe how to
use a compass to make a precise drawing.
(b) If you were to draw another map, or if you were to compare your map to
a classmate’s, how would they compare? In what ways might the maps
differ, in what ways would they be the same? Which criterion for triangle
congruence is most relevant to these questions?
2. Suppose you make a triangle by threading three pieces of straw onto a string
and tying the ends of the string together to make a loop. Similarly, suppose
you make a quadrilateral by threading four pieces of straw onto a string and
tying a loop. Describe the structural difference between the triangle and the
quadrilateral (other than the fact that the triangle is made out of three pieces of
straw and the quadrilateral is made out of four), and explain how this structural
difference is related to the concept of congruence.
3. Annie will give Benton instructions for drawing the triangle in Figure 14.13.
Annie’s instructions will involve 3 pieces of information about the triangle, each
of which is either a length or an angle. Then Benton will draw a triangle using
Annie’s instructions. Give three different sets of instructions that Annie could
give Benton so that when Benton constructs that triangle, it will necessarily be
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congruent to the triangle in Figure 14.13.
1st set of instructions involving 3 pieces of information:
2nd set of instructions involving 3 pieces of information:
3rd set of instructions involving 3 pieces of information:
2.8 cm
110°
4.5 cm
25°
45°
6 cm
Figure 14.13: A Triangle
4. Give an example to show why there is not a side-side-angle congruence criterion.
5. Is there a side-side-side-side congruence criterion for quadrilaterals? Explain
why or why not.
6. (Refer to Figure 14.14.) Given that sides AB and AC have the same length
and that angles BAD and CAD are the same size, apply a triangle congruence
criterion to prove that the angles at B and C are the same size. Explain your
reasoning clearly and in detail.
A
B
D
C
Figure 14.14: Triangle ABC and an angle bisector at A.
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7. (Refer to Figure 14.14.) Given that sides AB and AC have the same length,
that angles BAD and CAD are the same size, and that D lies on the line
segment between B and C, apply a triangle congruence criterion to prove that
the angles at D, namely angles BDA and CDA, are both right angles. Explain
your reasoning clearly and in detail.
8. (Refer to Figure 14.15.) Given that the sides with the same markings are the
same length, apply a triangle congruence criterion to prove that angles BAC
and DAC are the same size (i.e., prove that the diagonal AC bisects the angle
at A).
B
C
A
D
Figure 14.15: A quadrilateral and a diagonal.
9. Given that the four angles of the quadrilateral ABCD in Figure ?? are right
angles, apply a triangle congruence criterion and a fact about angles in triangles
to prove that opposite sides of the quadrilateral are the same length (i.e., prove
that sides AB and CD are the same length and that sides BC and DA are the
same length). Note: you may add line segments.
A
B
D
C
Figure 14.16: Proving that opposite sides of a rectangle are the same length.
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Constructions With Straightedge and Compass
1. (a) Use a compass and straightedge (ruler) to construct a line that is perpendicular to the line segment AB in Figure 14.17 and that divides the line
segment AB in half.
A
B
Figure 14.17: A Line Segment
(b) Show a rhombus that has A and B as vertices and that arises naturally
from your construction in part (a). Use the definition of rhombus, and the
way you carried out your construction, to explain why your shape really is
a rhombus.
(c) Which property of rhombuses explains why your construction in part (a)
produces a line that divides the line segment AB in half and is perpendicular to AB? Explain.
2. (a) Use a compass and straightedge (ruler) to divide the angle at A in Figure 14.18 in half.
A
Figure 14.18: Bisect Angle BAC
(b) Show a rhombus that arises naturally from your construction in part (a).
Use the definition of rhombus, and the way you carried out your construction, to explain why your shape really is a rhombus.
(c) Which property of rhombuses explains why your construction in part (a)
produces a line that divides the angle at A in half? Explain.
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3. Using a straightedge (ruler) and compass, construct a line that is perpendicular
to the line segment AB in Figure 14.19 and passes through the point A. Leave
the marks showing your construction.
A
B
Figure 14.19: A Line Segment
4. Use a compass and straightedge (ruler) to construct a 45 degree angle in a
careful and precise fashion.
5. Use a compass and straightedge (ruler) to construct a hexagon for which all six
sides have the same length and all six angles are equal. Leave your construction
marks to show how you accomplished your construction (don’t just show the
finished hexagon).
14.4
Similarity
1. A design on a rectangular piece of paper that is A cm wide and B cm tall will
be scaled up to fit on a similar rectangular poster that is C cm wide. How tall
should this poster be?
(a) Choose numbers for A, B, and C so that the problem is especially easy to
solve using the scale factor method. Show how to solve the problem with
the scale factor method and explain clearly the logic and reasoning of the
method.
(b) Choose numbers for A, B, and C so that the problem is especially easy
to solve using the internal factor method. Show how to solve the problem
with the internal factor method and explain clearly the logic and reasoning
of the method.
2. Annemarie designed her own flag in the shape of a rectangle that is 4 inches
by 7 inches. Now Annemarie wants to make a scaled up version of her flag in
such a way that the 4 inch side becomes 12 inches long, and so that the larger
flag is similar to the original flag. Annemarie figures that because the 4 inch
side will become 8 inches longer in the enlarged flag (because 4 + 8 = 12), she
should also make the 7 inch side 8 inches longer in the enlarged flag. Therefore
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Annemarie figures that the enlarged flag should be 12 inches by 15 inches. Is
Annemarie’s reasoning valid? Explain.
3. A flag problem: The children in your class have drawn flag designs on 4 inch
by 6 inch cards. If the children will scale up their flags so that the 4 inch side
will become 18 inches long, then how long should the 6 inch side become in the
larger flag?
Solve the flag problem in two ways, (1) with the scale factor method, and (2)
with the internal factor method. In each case:
• explain clearly the idea and the reasoning of that method (i.e., why that
method makes sense) as if you were explaining it to students who know
about multiplication and division, but not about setting up proportions;
• link the method to a proportion that more advanced students might set
up (in which they set two fractions equal to each other).
4. A poster problem: A painting that is 100 inches by 150 inches will be reproduced
on a poster. On the poster, the 100 inch side will become 20 inches long.
Determine how long the 150 inch side will become on the poster.
Solve the poster problem in two ways, (1) with the scale factor method, and (2)
with the internal factor method. In each case:
• explain clearly the idea and the reasoning of that method (i.e., why that
method makes sense) as if you were explaining it to students who know
about multiplication and division, but not about setting up proportions;
• link the method to a proportion that more advanced students might set
up (in which they set two fractions equal to each other).
5. A scaling problem: Audrey has designed a banner that is 4 inches wide and 20
inches long. Now Audrey wants to make a scaled up version of the banner. The
scaled up banner will be 12 inches wide. How long should the scaled up banner
be?
Determine the answer to the scaling problem in two ways other than by setting
up a proportion in which you set two fractions equal to each other. In each case,
explain why the method makes sense as if you were explaining it to a student
who understands about multiplication and division (but who does not know
about setting up proportions). Also, in each case show how the method relates
to a proportion that you can set up to solve the problem.
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6. A scaling problem: Jessica wants to make a scaled up version of a picture that
is on a postcard. The postcard is 4 inches wide and 6 inches long. If Jessica
wants to make the scaled up version 10 inches wide, then how long should she
make it?
Determine the answer to the scaling problem in two ways other than by setting
up a proportion in which you set two fractions equal to each other. In each case,
explain why the method makes sense as if you were explaining it to a student
who understands about multiplication and division (but who does not know
about setting up proportions).
7. Briefly describe two different ways that you could use the theory of similar
triangles to determine the height of a flagpole. For each way:
• draw a sketch that shows the similar triangles and explain why the triangles
must be similar;
• indicate on your sketch which measurements you would make and show
how to use those measurements to determine the height of the flagpole
(use any method).
8. Suppose you are looking down a road and you see a person ahead of you. You
hold out your arm and “sight” the person with your thumb, finding that the
person appears to be as tall as your thumb is long. Let’s say that your thumb is
2 inches long, and that the distance from your sighting eye to your thumb is 22
inches. If the person is 6 feet tall, then how far away are you from the person?
Solve this problem using either the scale factor method or the internal factor
method, and say clearly what the idea behind the method is. In other words,
explain why it makes sense to solve the problem the way you do.
9. A vertical pole that is 2 meters tall casts a shadow that is 3 meters long. Nearby,
at the same time, another vertical pole casts a shadow that is 9 meters long.
How tall is this pole?
(a) Make a drawing to show the similar triangles that are relevant to solving
this problem and explain why the triangles are similar.
(b) Explain how to solve the problem in two ways, using the scale factor
method and the internal factor method.
10. A vertical pole that is 1 yard long casts a shadow that is 2 feet 6 inches long. At
the same time, Heather’s shadow is 3 feet 9 inches long. How tall is Heather?
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(a) Make a drawing to show the similar triangles that are relevant to solving
this problem and explain why the triangles are similar.
(b) Determine Heather’s height in an elementary way without setting up a
proportion in which you set two fractions equal to each other. Explain
your reasoning in detail, as if you were teaching someone the method you
use.
11. Ms. Nice’s 5th grade class wants to figure out how tall the school flagpole is.
On a sunny day, the class goes outside and measures that the shadow of the
flagpole is about 21 feet long. At the same time, Juan’s shadow is 3 feet, 4
inches long. Juan is 4 feet, 3 inches tall.
(a) Determine the (approximate) height of the flagpole using a method that
the children in Ms. Nice’s class could find plausible. The children know
about multiplication and division, but they don’t know any more advanced
mathematics, such as setting up proportions. The class is allowed to use
calculators when multiplying and dividing decimals or fractions.
(b) Now add details to your explanation in part (a) in order to make your
explanation more thorough for someone who has a more advanced knowledge of mathematics. Draw a picture showing that the flagpole problem
involves similar triangles. Explain why the triangles must be similar.
12. A new store is being planned near a neighborhood. The store will have a round
sign, 10 feet in diameter, and will be up on a tall pole to make it widely visible.
The sign will be 1000 feet away from some of the residents in the neighborhood
(line of sight distance). Since the sign will be illuminated at night, the residents
want to know which will appear bigger: the sign or the moon? The residents
know you are an excellent math teacher, so they would like you to answer their
question and explain the solution so that they can learn about the math that’s
involved. Please help out the residents! The moon is 384,000 km away and its
diameter is 3,500 km.
14.5
Areas, Volumes, and Scaling
1. Kelsey made a scale model of one of the Egyptian pyramids. Kelsey’s model
is 10 cm tall. The actual Egyptian pyramid is 1000 times as tall as Kelsey’s
model.
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(a) Originally, the Egyptian pyramid had been covered with a thin layer of
alabaster, which has since washed away. Kelsey covered her pyramid model
with shiny paper in order to simulate the alabaster. Kelsey determined that
she used approximately 200 square centimeters of shiny paper to cover her
pyramid. Approximately how many square meters of alabaster had been
needed to cover the actual Egyptian pyramid? Explain.
(b) Kelsey filled her pyramid with rice and determined that her pyramid holds
about 300 ml. What is the volume of the original Egyptian pyramid in
cubic meters? Explain.
2. A large iguana can be 7 feet long and weigh 16 pounds. Suppose that from
excavated bones, a dinosaur was found to have been 25 feet long and proportioned like a large iguana. Give a good estimate for the weight of the dinosaur.
Explain your reasoning.
3. A typical adult male gorilla is about 5 21 feet tall and weighs about 400 pounds.
King Kong was supposed to have been about 20 feet tall. Assuming that King
Kong was proportioned like a typical adult male gorilla, approximately how
much should King Kong have weighed? Explain your reasoning.
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15.1
Formulating Statistical Questions, Gathering
Data, and Using Samples
1. What characterizes statistical questions? Use an example and a non-example
to illustrate.
2. (a) Why are samples used in statistical studies?
(b) In statistical studies, why are random samples preferred over other kinds
of samples?
3. At a factory that produces light bulbs, a batch of 1200 light bulbs has just been
produced. To check the quality of the light bulbs, a random sample of 50 light
bulbs are selected to test for defects. Out of these 50 light bulbs, 2 were found
to be defective. Based on these results, what is the best estimate you can give
for the number of defective light bulbs in the batch of 1200? Explain how to
solve this problem in two different ways, including at least one way that does
not involve cross-multiplying.
4. There is a large bin filled with ping pong balls, but we don’t know how many.
There are 30 orange ping pong balls in the bin; the rest are white. Yoon-He
reaches into the bin and randomly picks out 20 ping pong balls. Out of the 20
Yoon-He picked, 3 are orange. Based on Yoon-He’s sample, what is the best
estimate we can give for the number of ping pong balls in the bin? Explain how
to solve this problem in two different ways, including at least one way that does
not involve cross-multiplying.
5. A researcher wants to estimate the number of fish in a small pond. She throws
a net in the water, and when she pulls it out, she finds 30 fish in the net. The
researcher marks these 30 fish and throws them back in the pond, unharmed.
The next day, the researcher uses her net to catch some fish again. This time
she catches 40 fish and finds that 2 of the fish are marked (so these two fish are
two of the fish she had caught on the previous day). Assuming that the fish
mix freely in the pond, give the best estimate you can for the number of fish
in the pond based on the researcher’s experiment. Explain how to solve this
problem in two different ways, including at least one way that does not involve
cross-multiplying.
6. Your 5th grade class is having a “guess how many candies are in the jar” contest.
Initially, there are only red candies in the jar. Then you show the children that
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you put 20 green candies in the jar. (The green candies are the same size as the
red candies and are thoroughly mixed in with the red ones.) Tyler is blindfolded
and allowed to pick 20 candies out of the jar. Of the candies Tyler picked, 4 are
green; the other 16 are red. Based on this experiment, what is the best estimate
we can give for the total number of candies in the jar? Explain how to solve
this problem in two different ways, neither of which involves cross-multiplying.
15.2
Displaying Data and Interpreting Data Displays
1. Briefly discuss the use of line graphs: when is it appropriate to use a line graph
and when is it not appropriate to use a line graph? Give two examples to
illustrate.
2. The table below gives information about some of the activities that the children
at Franklin Elementary say they enjoy.
Activity
Jumping rope
Playing in sandbox
Climbing on the play structure
Playing catch
Percent of children
saying they enjoy
the activity
20%
15%
35%
30%
(a) Would it be appropriate to use a single pie graph to display this information? Explain your answer.
(b) Roughly sketch a way to display the data above in a graph other than a
pie graph.
3. The students in Mrs. Brown’s class rolled a pair of dice 30 times. They added
the number of dots on both dice and made a dot plot of the data, as shown in
Figure 15.1.
(a) Write two “read the data” questions for the dot plot. Answer your questions.
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X
2 3
X
X
X
4
X
X
X
X
5
X
X
X
X
X
6
X
X
X
X
X
X
7
X
X
X
X
8
X
X
X
X X X X
9 10 11 12
Figure 15.1: A Dot Plot of the Number Rolled with 2 Dice
(b) Write two “read between the data” questions for the dot plot. Answer
your questions.
(c) Write two “read beyond the data” questions for the dot plot. Answer your
questions (if possible).
4. Ms. Smith has a bag containing 200 paper squares. The squares are identical
except that some squares are red and the rest are white. The students in
Ms. Smith’s class each picked 10 paper squares out of the bag (replacing the
squares after picking them). Each child wrote the number of red squares he or
she picked on a sticky note. The class then made a dot plot on the chalkboard
that looked like the one in Figure 15.3 (each X represents a sticky note).
0
1
X
2
3
X
X
X X
X X X X
4 5 6 7
X
X
X
X
X
X
8
X
X
X
X
X
9
X
X
X
X
10
Figure 15.2: A Dot Plot of Number of Red Squares Picked out of 10 Squares
(a) Write two “read the data” questions for the dot plot. Answer your questions.
(b) Write two “read between the data” questions for the dot plot. Answer
your questions.
(c) Write two “read beyond the data” questions for the dot plot. Answer your
questions (if possible).
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5. Ms. Smith has a bag containing 300 square tiles. The tiles are identical except
that some are red and the rest are white. The students in Ms. Smith’s class
each picked 10 tiles out of the bag (replacing the tiles after picking them). Each
student wrote the number of red tiles he or she picked on a sticky note. The class
then made a dot plot on the chalkboard that looked like the one in Figure 15.3
(each X represents a sticky note).
0
1
X
2
3
X
X
X X
X X X X
4 5 6 7
X
X
X
X
X
X
8
X
X
X
X
X
9
X
X
X
X
10
Figure 15.3: A dot plot of number of red tiles out of 10 tiles
Write 4 questions you could ask students about the dot plot, including at least
one at each of the three levels of data reading we discussed. Label each of your
questions with its approximate level. Answer your questions (to the extent
possible).
15.3
The Center of Data: Mean, Median, and Mode
1. (a) Why do we have the mathematical concept of the mean (average)? Why
is it a useful concept?
(b) Describe how we calculate a mean.
(c) Describe how to view a mean concretely as “leveling out” or “making
even.”
(d) We can calculate a mean numerically or we can determine a mean concretely by “leveling out” or “making even.” Explain why we must get the
same answer either way.
2. (a) Explain what the mathematical term average or mean means.
(b) Describe a way that you could use concrete objects to help 4th or 5th
graders learn about the mathematical concept of the mean.
3. Describe how to use physical objects to determine the mean of a list of numbers
by “leveling out.” Then explain why this “leveling out” way of determining the
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mean must give the same result as calculating the mean numerically. Use the
example
1, 1, 4, 6
to illustrate.
4. Explain how to use physical objects to describe the mean of a list of numbers
in terms of “leveling out.” Explain why this way of thinking about the mean
agrees with the way we calculate the mean. Use the example
1, 1, 4, 6
to illustrate.
5. Juanita read an average (mean) of 2 books a day for 4 days. How many books
will Juanita need to read on the 5th day so that she will have read an average
of 3 books a day over 5 days? Solve this problem in two different ways and
explain both of your solutions.
6. Explain how you can quickly calculate the mean of the following test scores
without adding the numbers.
76, 73, 78, 75
7. Frank’s average score on his math tests in the first quarter is 70. Frank’s average
score on his math tests in the second quarter is 90. Frank’s semester score in
math is the average of all his test scores from the first and second quarters. Can
Frank necessarily calculate his semester score by finding the average of 70 and
90? If so, explain why; if not, explain why not, say what other information you
would need to calculate Frank’s semester score, and show how to calculate this
score.
8. Briefly discuss how to view the mean of a set of numerical data as a “balance
point.” Illustrate with a simple example.
9. (a) When numerical data are displayed in a dot plot, what is an especially
appropriate way to think about the mean?
(b) What is a common error that students make when they compute the mean
of a set of numerical data that are displayed in a dot plot? Illustrate with
an example.
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10. Describe a common error that students make with the median. Illustrate with
an example.
11. For each of the following scenarios, either create a dot plot for the scenario or
explain why the scenario cannot occur.
(a) A 10 point test was taken by 11 children. The median score was 8 but
more children scored a 10 than an 8.
(b) A 10 point test was taken by 11 children. The median score was 8 but
more children scored a 9 or a 10 than scored 8 or lower.
12. Suppose that all 4th graders in a state take a writing competency test that is
scored on a 5 point scale. Is it possible for 80% of the 4th graders to score below
average? If so, show how that could occur; if not, explain why not.
13. Ten children take a 10 point test. The mean score is 7.
(a) Is it possible for the median score to be 10, given that the mean score is
7? If so, give an example of how this could occur; if not, explain why not.
(b) Is it possible for the median score to be 6 or less, given that the mean score
is 7? If so, give an example of how this could occur; if not, explain why
not.
(c) Is it possible for the median score to be 3 or less, given that the mean score
is 7? If so, give an example of how this could occur; if not, explain why
not.
15.4
Summarizing, Describing, and Comparing Data
Distributions
1. For each of the following situations, draw a rough picture to indicate the shape
that you would expect the data distribution to take. Say briefly why you would
expect the data to be distributed that way.
(a) Family arrival times for honors night at school.
(b) Scores on an exam that students described as “you either got it or you
didn’t.”
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(c) The number of push-ups that the 5th graders at a school could do in a
minute.
2. At a math center in a class, there is a bag filled with 30 red blocks and 20 white
blocks. Each student in the class of 25 will do the following activity at the math
center: randomly pick 10 blocks out of the bag without looking and write the
number of red blocks picked on a sticky note.
Make a hypothetical dot plot that could arise from this situation. Make your
dot plot so that it has characteristics you would expect to see in a dot plot
arising from actual data. Briefly describe these characteristics of the dot plot.
3. A bag was filled with 60 black chips and 40 white chips. A sample of 10 chips was
picked randomly from the bag, the number of black chips picked was recorded
on a sticky note, the chips were returned to the bag, and this process of picking
a sample of 10 chips was repeated many times. The sticky notes were used to
make a histogram.
Draw roughly what you would expect this histogram to look like (you may draw
a rough shape instead of an actual histogram). Briefly describe the characteristics that you would expect the histogram to have.
4. (a) Determine the 25th, 50th, and 75th percentiles for the hypothetical test
scores shown in the dot plots of Figure 15.4.
0
X
X X
X X X
X X X X
1 2 3 4
X
X
X X
X X X X
X X X X X X
5 6 7 8 9 10
0
X
X X
X X X
X X X
1 2 3
4
5
6
X
X X
X X X
X X X
7 8 9
Figure 15.4: Hypothetical Test Results
(b) Suppose you only have the percentiles from part (a) and you don’t have
the dot plots in Figure 15.4. Discuss what you can tell about the data sets.
Can you tell which data set is more tightly clustered and which is more
dispersed?
5. What is the difference between getting 90% correct on a test and being at the
90th percentile on a test? Give some specific examples to illustrate.
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6. What is the purpose of reporting a student’s percentile on a state or national
standardized test? How is this purpose different from reporting the student’s
percent correct on a test?
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Probability
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16.1
CHAPTER 16
Basic Principles of Probability
1. What is the probability of spinning a red on the spinner in Figure 16.1? Explain.
red
yellow
blue
red
red
green
Figure 16.1: A Spinner
2. What is the probability of spinning either a red or a green on the spinner in
Figure 16.1? Explain.
3. A family math night at school features the following game. There are two
opaque bags, each containing red blocks and yellow blocks. Bag 1 contains 3
red blocks and 4 yellow blocks. Bag 2 contains 5 red blocks and 8 yellow blocks.
To play the game, you pick a bag and then you pick a block out of the bag
without looking. You win a prize if you pick a red block. Jake thinks he should
pick from bag 2 because it has more red blocks than yellow blocks. Is Jake more
likely to pick a red block if he picks from bag 2 than from bag 1? Why or why
not?
4. At a math center in a class, there is a bag filled with 40 red blocks and 10 blue
blocks. Each child in the class of 25 will do the following activity at the math
center: pick a random block out of the bag without looking, record the block’s
color, and put the block back into the bag. Each child will do this 10 times in
a row. Then the child will write the number of blue blocks picked on a sticky
note. Describe a good way to display the whole class’s data. Show roughly
what you expect the display you suggest to look like, and say why.
5. There is a bag with 200 cubes in it. Some of the cubes are red and some are
black. You reach into the bag 20 times, each time picking a random cube,
recording its color, and putting the cube back in the bag. You picked 7 black
cubes and 13 red cubes. With this information, what is the best estimate you
can make for how many red cubes and how many black cubes are in the bag?
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Describe how to calculate this best estimate and explain why your method of
calculation makes sense.
16.2
Counting the Number of Outcomes
1. A certain type of key is made with 3 notches on one side of the key. Each notch
can be any one of 5 depths (shallow, medium shallow, medium, medium deep,
deep). How many keys can be made this way? Explain why you can solve the
problem the way you do.
2. Allie and Betty want to know how many three-letter codes, such as BMW, or
DDT are possible (letters are allowed to repeat, as in DDT or BOB). Allie
thinks there can be 26 + 26 + 26 three-letter combinations whereas Betty thinks
the number is 26 × 26 × 26. Which girl, if either, is right and why? Explain
your answer.
3. How many different three-letter codes are there that do not contain any repeated
letters? (So for example, neither DDT nor BUB are allowed.) Explain your
answer.
4. How many 3-digit counting numbers can be made using only the odd digits 1,
3, 5, 7, and 9? (A digit may be used more than once within a number.)
5. Karen buys 4 skirts, 5 blouses, and 6 sweaters, and 7 pairs of shoes, all of which
are coordinated to go together. How many different outfits consisting of a skirt,
a blouse, a sweater, and a pair of shoes can Karen make? Explain why you can
solve the problem the way you do.
16.3
Calculating Probabilities in Multi-Stage Experiments
1. A child’s game has a spinner that is equally likely to land on any one of 4
regions. One region is red, one region is yellow, one region is blue, and one
region is green. To win a game, Nate must spin either green followed by red or
a red followed by a green in the next two spins. What is the probability that
Nate will win the game? Explain.
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2. You have two dice, each numbered from 1 to 6. What is the probability of
rolling a total of 6? Explain.
3. Suppose you have 5 marbles in a bag: 3 red and 2 black. If you reach into the
bag without looking and randomly pick out two marbles, what is the probability
that both marbles will be red? Explain.
4. John flips a fair coin 10 times in a row and it comes up heads all 10 times. John
says that he’s more likely to get a tail on his next flip because a tail is “due.”
Is John more likely to get a tail than heads on his next flip? Why or why not?
5. Kelsey made up the following game for a fundraiser. She put 20 plastic fish in
a bag. The fish are identical except that 19 are blue and 1 is red. A contestant
plays the game by “going fishing” twice: the contestant reaches in the bag and,
without looking, picks out a fish. Then the contestant reaches in again and
picks out another fish (the first fish is not returned, it is left out of the bag
before picking the second fish). The contestant pays $1 to play the game and
wins a prize worth $5 if they pull out a blue fish followed by a red fish.
(a) What is the probability of winning the game? Explain.
(b) If 100 people play the game, about how many people would you expect to
win? Why?
(c) Based on your answer to (b), about how much money should Kelsey expect
to earn for the fundraiser if 100 people play her game?
16.4
Using Fraction Arithmetic to Calculate Probabilities
1. A children’s game has a spinner that is equally likely to land on any one of four
colors: red, blue, yellow, or green. What is the probability of spinning a blue
followed by a green in 2 spins? Explain how to solve this problem with fraction
multiplication and explain why this method makes sense.
2. There are two bags. The first bag contains 2 green tiles, 3 yellow tiles. The
second bag contains 4 green tiles and 1 yellow tile. A game consists of picking
1 tile from each bag. To win the game, you must pick the same color tile from
each bag. Use fraction arithmetic to determine the probability of winning the
game. Explain why you can use that arithmetic to solve this problem.
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