Table of Contents
Problem-Solving Toolkit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Unit 1 Problem Solving Using Whole Numbers, Decimals,
and Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Unit Theme: Science in the World
Lesson 1
Use Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Math Focus: Add and Subtract Decimals and Fractions
Lesson 2
Look for a Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Math Focus: Greater Numbers and Scientific Notation
Lesson 3
Make an Organized List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Math Focus: Multiply and Divide Decimals and Fractions
Lesson 4
Guess, Check, and Revise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Math Focus: All Operations with Decimals and Fractions
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Team Project: Make Breakfast
Unit 2 Problem Solving Using Algebra
Unit Theme: Changes on Earth
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
S.T.E.M.
Lesson 5
Work Backward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Math Focus: Operations with Positive and Negative Numbers
Lesson 6
Make a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Math Focus: Evaluate Expressions
Lesson 7
Solve a Simpler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Math Focus: Solve 1-Step Equations
Lesson 8
Guess, Check, and Revise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Math Focus: Inequalities
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Team Project: Plan a Trip
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Unit 3 Problem Solving Using Ratio, Proportion,
and Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Unit Theme: Arts and Crafts
Lesson 9
Write an Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Math Focus: Rate, Ratio, and Proportion
Lesson 10
Draw a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Math Focus: Scale Drawings and Models
Lesson 11
Write an Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Math Focus: Percent Applications
Lesson 12
Solve a Simpler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Math Focus: Percent Change
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Team Project: Draw Your Classroom
Unit 4 Problem Solving Using Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Unit Theme: On the Move
Lesson 13
Draw a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Math Focus: Coordinate Plane
Lesson 14
Make a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Math Focus: Graphs of Linear Equations
Lesson 15
Work Backward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Math Focus: Applications of Linear Graphs and Slope
Lesson 16
Make a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Math Focus: Graphs of 2-Step Linear Equations
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Team Project: Design a Park
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Unit 5 Problem Solving Using Geometry and Measurement
. . . . . . . . 142
Unit Theme: Movies
Lesson 17
Draw a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Math Focus: Perimeter, Circumference, and Area
Lesson 18
Use Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Math Focus: Congruent and Similar Figures
Lesson 19
Draw a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Math Focus: Transformations on the Coordinate Plane
Lesson 20
Write an Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Math Focus: Surface Area and Volume
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Team Project: Design a Box
Unit 6 Problem Solving Using Data and Probability . . . . . . . . . . . . . . . . . 172
Unit Theme: Sports and Recreation
Lesson 21
Look for a Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Math Focus: Collect, Organize, and Display Data
Lesson 22
Make a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Math Focus: Measures of Central Tendency and Variation
Lesson 23
Make an Organized List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Math Focus: Permutations and Combinations
Lesson 24
Make a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Math Focus: Probability
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Team Project: Playing Fair?
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
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Strategy Focus
Lesson
1
Use Logical Reasoning
MATH FOCUS: Add and Subtract Decimals and Fractions
Learn
Read the Problem
Dana is writing a report about small adult mammals and has found
information about their average lengths:
●
A house mouse is 6.4 centimeters long.
●
A masked shrew is 1 centimeter shorter than a pygmy glider and
1.8 centimeters longer than a hognosed bat.
●
A pygmy glider is 0.8 centimeters shorter than a house mouse.
What is the length of a hognosed bat?
Reread As you read the problem, ask yourself questions.
Mark
the Text
●
What is this problem about?
●
What do I know about the lengths of the mammals?
●
What does the problem ask me to do?
Search for Information
Read the problem again. Circle the names of the animals and underline
statements that tell you about their lengths.
Record Use the details in the problem to fill in the blanks.
From the first bullet, I know that a house mouse is
long.
centimeters
From the second bullet, I know relationships between the length
of a masked shrew and the lengths of a
and
a
.
From the third bullet, I know a relationship between the length of a house
mouse and the length of a
.
You can use these facts to choose a problem-solving strategy.
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Decide What to Do
You know the length of a house mouse. You have many facts about the
relationships between the lengths of the mammals.
Ask How can I find the length of a hognosed bat?
●
The strategy Use Logical Reasoning can help me.
●
I will start with the length I know. Then I will use the given relationships
to determine an operation to use to find the lengths I do not know.
One way to
begin solving
a problem is to
ask, “What do
I know?” and
“What can I find
out from that?”
Use Your Ideas
Step 1
Start from a known length. You know that the length of
a
is
centimeters.
Step 2
You know a relationship between the length of a
house mouse and one of the other animals.
Which one?
The length of a pygmy glider is
Step 3
You know a relationship between the length of a
pygmy glider and one of the other animals.
Which one?
The length of a masked shrew is
Step 4
centimeters.
centimeters.
You know a relationship between the length of a
masked shrew and one of the other animals.
Which one?
So the length of a hognosed bat is
centimeters.
Review Your Work
Look back at the problem. Do the lengths you calculated match the
relationships in the problem?
Identify You started knowing only the length of the house mouse.
Why does it make sense to find the length of the pygmy glider next?
25
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Try
Solve the problem.
1
Mark
the Text
Bart learned that the first stage of a type of rocket had a burn time
that was 28.5 seconds longer than the burn time during the second
stage. The total burn time for both stages was 150.3 seconds. What
was the burn time for the first stage?
Read the Problem and Search for Information
Locate key mathematical details that describe relationships. Restate the
relationships in your own words.
Decide What to Do and Use Your Ideas
You can use logical reasoning to connect the facts in the problem.
Step 1
Draw a bar diagram to show how the burn times for the first
stage and the second stage are related.
The total burn time was
seconds.
Step 2
Step 3
Ask
Yourself
If I know the time
for the second stage,
how can I find the
time for the first
stage?
150.3
First Stage
Use logical reasoning to
extend the bar diagram.
The burn time for the first
stage was
Second Stage
seconds longer than the
burn time for the second stage.
150.3
28.5
Use the combined diagram to find the missing times. The time
for two second stages is 150.3 seconds minus 28.5 seconds.
150.3 − 28.5 =
The burn time for the second stage of the rocket was
seconds.
So the burn time for the first stage of the rocket was
seconds.
Review Your Work
Check that the burn times for the two stages have the correct sum.
Describe Explain how you used logical reasoning to solve the problem.
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Apply
2
Some hospital nurses recorded the birth weights of babies. Baby A
weighed 1__34 pounds less than Baby B. Baby C weighed 1__12 pounds
less than Baby D. Baby B weighed 2__18 pounds more than Baby C.
Baby A weighed 7__14 pounds. Which baby weighed the least?
Baby A’s birth weight was
pounds. This was
less than Baby
. So Baby
weighed
at birth.
▲
Solve the problems.
pounds
pounds
Then, I can find Baby
’s birth weight was
pounds.
Then, I can find Baby
’s birth weight was
pounds.
Answer
Hint You know
Baby A’s birth
weight. Look for
another baby whose
birth weight has a
relationship with
Baby A’s.
Ask
Yourself
Which operation
should I choose when
I use one weight to
find another?
Was it necessary to find all of the birth weights in order to
answer the question? Explain.
Examine
3
Kyle learns that concrete starts as a paste made of cement and
water. The paste is mixed with gravel and sand. The mass of cement
paste and gravel is about __23 of the total mass. The mass of cement
paste and sand is about __12 of the total mass. What fraction of the
mass of the mixture is cement paste?
Gravel
Paste
Sand
1
2
Cement and sand is
Sand by itself is
When I add each set
of fractions in the
diagram, what should
I get?
▲
2
3
Ask
Yourself
Hint Use the bar
diagram to find the
missing fractions.
of the mass of the mixture.
of the mass of the mixture.
Answer
Classify
What kinds of problems can be solved using bar diagrams?
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▼
Hint Look for
4
sentences that relate
the numbers of signs
that two apes know.
Ask
Yourself
Scientists have been able to teach some apes to use sign language
to communicate. Lucy knows 25 fewer signs than Michael. Michael
knows 100 more signs than Nim, but 50 fewer signs than Koko. Lucy
knows 850 signs. Which ape knows the greatest number of signs
and how many signs does it know?
Lucy knows
What can I figure out
next?
signs.
So Michael knows
signs.
Answer
How could you use logical reasoning to find which ape knows
the most sign language without finding the number of words any of
them know?
▼
Analyze
Hint Use the
relationships shown
in the bar diagram
to help you solve the
problem.
5
Mr. Bell has researched energy-saving appliances. If he buys a new
refrigerator, washer, and dryer, he could save $7.80 per week. If he
only replaced the refrigerator and washer, he could still save $5.10
per week. If he only replaced the washer and dryer, he would save
$6.10 per week. How much would Mr. Bell save if he only replaced
his washer?
Ask
Yourself
Refrigerator
Washer
Dryer
Mr. Bell saves $5.10
per week by replacing
his refrigerator and
washer. Where in the
bar diagram should
I write $5.10?
Answer
Sequence What steps did you take to solve this problem?
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Practice
Solve the problems. Show your work.
6
Ella is studying the top speeds of mammals. She has recorded this information.
●
A zebra can run 8 miles per hour faster than a giraffe can.
●
A lion can run at a speed of 50 miles per hour, which is 20 miles per hour
slower than a cheetah’s top speed.
●
A zebra’s top speed is 10 miles per hour slower than a lion’s.
How much less is a giraffe’s top speed than a cheetah’s?
Answer
Differentiate
7
Which statement is the best to start with in this problem? Why?
Manuela is interviewing scientists about their jobs. __23 of the scientists she has
talked to are either chemists or physicists. The rest are biologists. She figures
out that __34 of the scientists are either biologists or physicists. What fraction of the
scientists that Manuela has interviewed are physicists?
Answer
What is a question that you could answer about this problem after reading
only the first three sentences?
Relate
Look back at Problem 4. Write a new problem by changing two
numbers in the problem. Solve the problem you wrote.
Create
Lesson 1 Strategy Focus: Use Logical Reasoning
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