Copyright © by The McGraw-Hill sort to put together items that have something in common Venn diagram a diagram that uses overlapping and separate circles or ellipses to organize and show
Trang 1Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber
Trang 2Copyright © 2008 by The McGraw-Hill Companies, Inc All rights reserved Except as
permitted under the United States Copyright Act, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or
retrieval system, without prior permission of the publisher.
Send all inquiries to:
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Getty Images; 33 CORBIS; 42–43 Masterfile; 48 Courtesy Miami University
Trang 3California Math Triumphs
Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting
1A Chapter 2 Place Value
1A Chapter 3 Addition and Subtraction
2A Chapter 2 Equivalence of Fractions
2B Chapter 3 Operations with Fractions
2B Chapter 4 Positive and Negative Fractions and Decimals
Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates
3A Chapter 2 Percents, Fractions, and Decimals
3B Chapter 3 Using Percents
3B Chapter 4 Rates and Proportional Reasoning
4A Chapter 1 Operations and Equality
4A Chapter 2 Math Fundamentals
4B Chapter 3 Math Expressions
4B Chapter 4 Linear Equations
4B Chapter 5 Inequalities
Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships
5A Chapter 2 Graphing
5B Chapter 3 Proportional Relationships
5B Chapter 4 The Relationship Between
Graphs and Functions
6A Chapter 1 How Measurements Are Made
6A Chapter 2 Length and Area in the Real World
6B Chapter 3 Exact Measures in Geometry
6B Chapter 4 Angles and Circles
Trang 4Authors and Consultants
iv
AUTHORS
Frances Basich Whitney
Project Director, Mathematics K–12
Santa Cruz County Offi ce of Education
Capitola, California
Kathleen M Brown
Math Curriculum Staff Developer Washington Middle School Long Beach, California
Dixie Dawson
Math Curriculum Leader Long Beach Unifi ed Long Beach, California
CONSULTANTS
Assessment
Donna M Kopenski, Ed.D.
Math Coordinator K–5
City Heights Educational Collaborative
San Diego, California
Instructional Planning and Support
Beatrice Luchin
Mathematics Consultant League City, Texas
ELL Support and Vocabulary
ReLeah Cossett Lent
Author/Educational Consultant Alford, Florida
Dinah-Might Activities, Inc.
San Antonio, Texas
Trang 5California Advisory Board
Carol Cronk
Mathematics Program Specialist
San Bernardino City Unifi ed
School District
San Bernardino, California
Audrey M Day
Classroom Teacher Rosa Parks Elementary School San Diego, California
Jill Fetters
Math Teacher Tevis Jr High School Bakersfi eld, California
Grant A Fraser, Ph.D.
Professor of Mathematics California State University, Los Angeles
Los Angeles, California
Eric Kimmel
Mathematics Department Chair
Frontier High School
Bakersfi eld, California
Donna M Kopenski, Ed.D.
Math Coordinator K–5 City Heights Educational Collaborative San Diego, California
Michael A Pease
Instructional Math Coach Aspire Public Schools Oakland, California
Chuck Podhorsky, Ph.D.
Math Director City Heights Educational Collaborative San Diego, California
Arthur K Wayman, Ph.D.
Professor Emeritus
California State University, Long
Beach
Long Beach, California
Frances Basich Whitney
Project Director, Mathematics K–12 Santa Cruz County Offi ce of Education
Capitola, CA
Mario Borrayo
Teacher Rosa Parks Elementary San Diego, California
Melissa Bray
K–8 Math Resource Teacher Modesto City Schools Modesto, California
Glencoe wishes to thank the following professionals for their invaluable
feedback during the development of the program They reviewed
the table of contents, the prototype of the Student Study Guide, the
prototype of the Teacher Wraparound Edition, and the professional
Bonnie Awes
Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California
Kathleen M Brown
Math Curriculum Staff Developer Washington Middle School Long Beach, California
Trang 6California Reviewers
vi
Each California Reviewer reviewed at least two chapters of the Student
Study Guides, providing feedback and suggestions for improving the
effectiveness of the mathematics instruction
Bobbi Anne Barnowsky
Monica S Patterson
Educator Aspire Public Schools Modesto, California
Rechelle Pearlman
4th Grade Teacher Wanda Hirsch Elementary School Tracy, California
Armida Picon
5th Grade Teacher Mineral King School Visalia, California
Anthony J Solina
Lead Educator Aspire Public Schools Stockton, California
Trang 7Volume 5A Functions and Equations
1SDAP1.1 Sort objects and data
by common attributes and describe the categories.
1SDAP2.1 Describe, extend, and explain ways to get to a next element in simple repeating patterns (e.g., rhythmic, numeric, color, and shape).
2SDAP2.1 Recognize, describe, and extend patterns and determine a next term
in linear patterns (e.g., 4, 8, 12 , the number of ears on one horse, two horses, four horses).
3AF2.1 Solve simple problems involving a functional relationship between two quantities (e.g., fi nd the total cost of multiple items given the cost per unit).
3AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may
be calculated by counting by 4s or by multiplying the number of horses by 4).
4AF1.5 Understand that an
Dana Meadows near Yosemite National Park
in This Chapter
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Chapter
2-1 Bar Graphs and Picture Graphs .44
1SDAP1.2, 2SDAP1.1, 2SDAP1.2
2SDAP1.1 Record numerical data in systematic ways, keeping track of what has been counted.
2SDAP1.2 Represent the same data set in more than one way (e.g., bar graphs and charts with tallies).
3SDAP1.3 Summarize and display the results of probability experiments in a clear and organized way (e.g., use a bar graph or a line plot).
4MG2.0 Students use dimensional coordinate grids to represent points and graph lines and simple fi gures.
two-4MG2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the
graph of the equation y = 3x and connect
them by using a straight line).
5SDAP1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.
5SDAP1.5 Know how to write
ordered pairs correctly; for example, (x, y).
Golden Gate Bridge, San Francisco
Trang 9Wind turbines, Altamont
Chapters 1 and 2 are contained in Volume 5A Chapters 3 and 4 are contained in Volume 5B.
3AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may
be calculated by counting by 4s or by multiplying the number of horses by 4).
6NS1.3 Use proportions to solve problems (e.g., determine the value of N
Trang 107AF3.0 Students graph and interpret linear and some nonlinear functions.
7AF3.1 Graph functions of the y = nx2
and y = nx3 and use in solving problems.
7AF3.3 Graph linear functions, noting that the vertical change (change
in y-value) per unit of horizontal change (change in x-value) is always the same
and know that the ratio (“rise over run”) is called the slope of a graph.
7AF3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle) Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities.
Redwood National Park
Trang 11Let’s Get Started
Use the Scavenger Hunt below to learn where things are
located in each chapter
p 59?
you where you can take the Online Readiness Quiz
Trang 122 Chapter 1 Patterns and Relationships
Have you ever compared prices?
Two DVDs cost $30 If you buy an
individual DVD, it costs $16.99 Is it
a better deal to buy 2 DVDs or to
buy an individual DVD?
Patterns and Relationships Chapter
1
Trang 13Copyright © by The McGraw-Hill Companies, Inc.
Quiz at ca.mathtriumphs.com to find out
them with what you’ll learn in this chapter
You know how to identify items that are alike
Example:
23, 25, 31, 26, 34, 35, 36, 28, 39
The numbers above can be sorted into
twenties and thirties
A pattern is a sequence of numbers
or figures that repeats You can predict what will happen next
The pattern shows multiples of sixes
In order to find the next number, you will add six Based on the pattern, the next number will be 30
You know how to add the same numbers several times
Example: If 4 apples cost $3, how much does it cost to buy 16 apples?
Trang 14Copyright © by The McGraw-Hill
sort
to put together items that have something in common
Venn diagram
a diagram that uses overlapping and separate circles or ellipses to organize and show data
KEY Concept
Sort and Classify
All objects have specific attributes , or characteristics Objects
can be sorted by their attributes A Venn diagram is one way
to show how objects are sorted
Follow these steps to sort and classify objects
1 Place objects with the same attribute into a group
2 Use a Venn diagram to show how the objects are sorted
These objects are sorted by color and by shape
Red Square
These objects are red and they are squares.
These objects are not red
and they are
not squares.
Venn diagrams can show common objects among at least three
categories This diagram shows the factors of 18, 24, and 32
1 and 2 are common factors of 18, 24, and 32
To sort and classify, think about how the objects are alike Put the items
that are alike together in a group
KAF1.1 Identify, sort, and classify objects by attribute and identify objects that do not belong to a particular group.
1SDAP1.1 Sort objects and data by common attributes and describe the categories.
Trang 15Copyright © by The McGraw-Hill Companies, Inc.
Example 1
Use a Venn diagram to sort
the cubes Sort them as
2-cube stacks or red-cube
These stacks are red and have 2 cubes.
2 Cube Stacks
This stack is not red and does not have 2 cubes.
are sorted Identify the objects in each group,
in both groups, and in neither group
YOUR TURN!
Use a Venn diagram to sort the figures Sort them as
rectangles or striped figures.
1 Sort and classify the objects
2 Use a Venn diagram to show how the objects are sorted Identify
the objects in each group, in both groups, and in neither group
GO ON
The cube stacks listed
in both categories have to be in the overlapping part of the circles.
Trang 16Copyright © by The McGraw-Hill
1 Sort and classify the numbers
Even: Multiples of 5: Neither:
10, 12, 14, 16, 18, 20 10, 15, 20 11, 13, 17, 19
2 Use a Venn diagram to show how the numbers are sorted Identify
the numbers in each group, in both groups, and in neither group
Even Multiples of 5
These numbers are even and are multiples of 5.
These numbers are not even
and are not multiples of 5.
2 Use a Venn diagram to show how the numbers are sorted
Identify the numbers in each group, in both groups, and in
neither group
Trang 17Copyright © by The McGraw-Hill Companies, Inc.
2, 4, 6, and 8 are even numbers.
Circle correct answer(s) Cross out incorrect answer(s)
Trang 18Copyright © by The McGraw-Hill
5 Create a Venn diagram to sort the numbers Classify them as
multiples of 10 or 2-digit numbers
50, 67, 80, 93, 106, 110, 125, 200
Step 1 Sort and classify the numbers
Multiples of 10: 2-Digit Numbers: Neither:
Step 2 Use a Venn diagram to show how the numbers are
sorted Identify the numbers in each group, in both groups, and in neither group
Multiples of 10 2-Digit Numbers
Step by Step Practice
6 Create a Venn diagram to sort the numbers Classify them as
Trang 19Copyright © by The McGraw-Hill Companies, Inc.
7 Create a Venn diagram to sort the numbers Classify them as
multiples of 9 or odd 17, 18, 23, 27, 36, 37, 40, 45, 50
Solve.
8 FOOD Amado classified fruit into two categories: fruit
you can peel and fruit that does not have to be peeled
He sorted apples, grapes, pears, oranges, bananas,
strawberries, pineapples, and raspberries Which two
fruits did Amado put in both categories?
Understand Read the problem Write what you know You need to find the
two fruits that can be peeled or do not have to be peeled
Plan Pick a strategy One strategy is to draw a diagram
Create a Venn diagram to classify types of fruit
Solve
These fruits can be peeled or do not have to be peeled.
Fruit You Can Peel
Fruit That Does Not Have to Be Peeled
and are fruits that can be peeled
or do not have to be peeled
Check Is your answer reasonable? If Amado had fruit in his lunch
box, which fruit would he have to peel?
Step by Step Problem-Solving Practice Problem-Solving Strategies
✓ Draw a diagram.
Look for a pattern.
Guess and check.
Act it out.
Solve a simpler problem.
GO ON
Trang 20Copyright © by The McGraw-Hill
9 NUMBER SENSE Curt is learning how to write numbers He
noticed that some were made of straight lines and some had
curved lines
0 1 2 3 4 5 6 7 8 9
Create a Venn diagram to show how these numbers are sorted into
numbers made with straight lines and numbers made with curves
categories
10 List three ways to classify the numbers 5, 10, 15, 20, 25
and 30
Skills, Concepts, and Problem Solving
List the objects in each category.
Trang 21Copyright © by The McGraw-Hill Companies, Inc.
15 Create a Venn diagram to sort the numbers Classify them as
17 FOOD During lunch, Betsy wrote down how many students chose
green beans, potatoes, or both Betsy made this Venn diagram
HJSMT Green Beans Both Potatoes
Out of 49 students, how many chose both vegetables? GO ON
Trang 22Copyright © by The McGraw-Hill
Vocabulary Check Write the vocabulary word that completes each
sentence.
18 A(n) is a characteristic of an object such as
color, size, shape, or thickness
19 A(n) is a diagram that uses overlapping and
separate circles or ellipses to organize and show data
20 Writing in Math Classify the numbers 3, 6, 7, 9, 12, 14, 21, and 28
Explain how to sort them
21 Create a Venn diagram to sort the numbers Classify them as even
or multiples of 5
100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110
Solve
22 NUMBER SENSE Donna listed all the multiples of 4 up to 50 Lia
listed all the multiples of 6 up to 50
The girls made a Venn diagram of the lists Which multiples were
in both categories? Explain your answer
Trang 23Copyright © by The McGraw-Hill Companies, Inc.
terms
each of the quantities that forms a series or pattern
KEY Concept
Patterns
A pattern is a sequence of numbers or figures that repeats
according to a rule Because patterns repeat, you can figure
out how to continue a pattern To identify and continue a
pattern, follow these steps
1 Say each number or figure out loud
2 Listen for the terms of the pattern that repeat
3 Use the terms of the pattern that repeat to find the
⎭ ⎫
The next term in the pattern is 1
To complete this pattern of shapes, the next figure must
complete the repeating part of the pattern and then the
pattern starts over again
The next three figures in the pattern are
Patterns can be related to many attributes, such as shape, color, and
1 Read the pattern out loud
2 The repeating terms are A, A, B This is
the pattern
3 Use the terms of the pattern that repeat to
find the next term
The next term is A
YOUR TURN!
What is the pattern? Write the next term
in the pattern.
5, 10, 10, 5, 10, 10, 5, 10, 10
1 Read the pattern out loud
2 The repeating terms are , ,
3 Use the terms of the pattern that repeat
to find the next term
The next term is GO ON
1SDAP2.1 Describe, extend, and explain ways to get to a next element in simple patterns 2SDAP2.1 Recognize, describe, and extend patterns and determine a next
term in linear patterns.
Trang 24Copyright © by The McGraw-Hill
What is the pattern? Write the next three
terms in the pattern
1 Read the pattern out loud
2 The repeating terms are “square, circle,
circle.” This is the pattern
3 Use the terms of the pattern that repeat to
find the next term
A circle is needed to complete the repeating
terms Then begin the repeating terms again
The next term is , followed by ,
YOUR TURN!
What is the pattern? Write the next three terms in the pattern
1 Read the pattern out loud
2 The repeating terms are ,
3 Use the terms of the pattern that repeat
to find the next term
Another circle is next, followed
by two circles
Trang 25Copyright © by The McGraw-Hill Companies, Inc.
4 What is the pattern? Write the next term in the pattern.
21, 22, 22, 21, 22, 22, 21, 22
Step 1 Read the pattern out loud
Step 2 The repeating terms are , ,
Step 3 Use the terms of the pattern that repeat to find the next
term The number is needed to complete the repeating terms
The next term is
Step by Step Practice
What is the pattern? Write the next term in each pattern.
5 A, a, B, b, A, a, B, b, A, a
The repeating terms are , , ,
The letters , are needed to complete the repeating
terms
The next term is
6
The repeating terms are , , ,
The next term is
What is the pattern? Write the next three terms in each pattern.
7 r, S, t, T, t, r, S, t, T, t, r
The repeating terms are , , , ,
The next three terms are , ,
8
The repeating terms are , ,
The next three terms are , , GO ON
Trang 26Copyright © by The McGraw-Hill
Solve.
9 SCHOOL Tim’s English grades for the first four weeks of
school are listed below If this pattern continues, what will
be the twenty-third Tim will receive in English?
B, B, C, A, B, B, C, A, B, B, C
Understand Read the problem Write what you know
Tim’s English grades were , , ,
,
Plan Pick a strategy One strategy is to look for a pattern
Solve The repeating pattern is , , ,
According to the pattern, the twenty-third grade Tim will receive is a(n)
Check Look at the pattern Does your answer follow the
pattern?
Step by Step Problem-Solving Practice
10 LANDSCAPING While shopping for flowers, Simona noticed that
the florist had displays arranged by color In this pattern of flower
colors, which color is missing?
?
11 WEATHER Wilma recorded the pattern shown below for the last
eight days of weather The weather reporter predicted rain on
Sunday Do you think that Wilma will agree with the reporter’s
✓ Look for a pattern.
Guess and check.
Act it out.
Work backward.
Trang 27Copyright © by The McGraw-Hill Companies, Inc.
12 Is the next number in the pattern 5? Explain
5, 55, 505, 5, 5, 55, 505, 5
Skills, Concepts, and Problem Solving
Write the repeating terms of each pattern.
The repeating terms are , , ,
The next term is
18
The repeating terms are , , ,
The next term is
What is the pattern? Write the next three terms in each pattern.
Trang 28Copyright © by The McGraw-Hill
Solve.
21 TRAVEL To travel from Phil’s Pharmacy to Betty’s Bagels, there
are 10 turns Ivan noticed a pattern
If the pattern continues, what is the 10th turn to
Betty’s Bagels?
22 NOTEBOOKS Katrina was doodling in her notebook She made
the following pattern of marks What is the next mark that Katrina
will make in her notebook if she continues the pattern?
x — x —
23 MUSIC In a musical piece for the school marching band, the
trumpets play these notes:
E F G E E F G E E
If the pattern continues, what are the next three notes?
24 CAFETERIA The school lunch menu has a pattern
Monday: chicken fingers
Tuesday: hamburgers
Wednesday: pizza
Thursday: spaghetti
Friday: fish sticks
If this pattern continues next week, what will be served
next Wednesday?
Vocabulary Check Write the vocabulary word that completes each
sentence.
25 A(n) is a sequence of numbers, figures, or symbols that
follows a rule or design
26 are the quantities that form a sequence or pattern
Trang 29Copyright © by The McGraw-Hill Companies, Inc.
27 Writing in Math Explain how identifing the patterns in problems
12 and 14 were similar
31 WEATHER Kimi recorded the high temperature each day in June
at 1:00 p.m The temperatures are listed in the table
76 77 77 74 79 80 80 81 79 83 84 84 81 83 84
87 88 90 91 92 88 84 85 82 83 79 78 81 83 84
Use a Venn diagram to sort the temperatures into the categories
Temperatures Between 70°F and 85°F and Temperatures Between
80°F and 95°F
Temperatures Between 70˚F and 85˚F
Temperatures Between 80˚F and 95˚F
Trang 30Copyright © by The McGraw-Hill
The repeating terms are , ,
The next term is
8
The repeating terms are , , , ,
The next term is
Solve
on the board: dog, lion, eagle, cat, elephant,
horse He asked his class to sort these animals
as pets or wild animals Create a Venn diagram
to sort these animals Some animals can be in
both categories
the front doors on the houses were painted using a color pattern
The door colors were blue, brown, red, red, blue, brown, red If the
pattern continues, what color should Darron see on the next door?
Progress Check 1 (Lessons 1-1 and 1-2)
Trang 31Copyright © by The McGraw-Hill Companies, Inc.
Number patterns and visual patterns follow a rule
You can use a rule to answer questions about the pattern and
to predict what comes next in the pattern
12, 24, 36, 48, 60, 72, 84, …
The rule for the pattern above is “Add 12.”
To continue the pattern, add 12 to the last term
84 + 12 = 96
96 + 12 = 108
108 + 12 = 120The next three terms in the pattern are 96, 108, and 120
Rules define relationships between numbers For example,
there are 12 inches in 1 foot So, there are 24 inches in 2 feet
(12 × 2), 36 inches in 3 feet (12 × 3), 48 inches in 4 feet
1 Each car has 4 wheels
2 One rule is “add 4 for each car.”
3 Another rule is
× 8 =
4 There are legs on
2SDAP2.1 Recognize, describe, and extend patterns and determine a next term in linear patterns.
3AF2.2 Extend and recognize a linear pattern by its rules.
Trang 32Copyright © by The McGraw-Hill
Example 2
Write the next three terms in the pattern.
1, 3, 7, 15
1 Find the rule
The rule is multiply by 2, and then add 1
2 Continue the pattern
The next three terms are and
Trang 33Copyright © by The McGraw-Hill Companies, Inc.
5 There are 36 inches to one yard Write the next three terms in
the pattern
Step 1 Find a rule Multiply by
Step 2 Multiply the number of yards by to continue the
pattern
2 × = 3 × = 4 × = The next three terms are , , and
Step by Step Practice
In each sequence, find a rule Then, write the next three terms.
Trang 34Copyright © by The McGraw-Hill
12 BOOKS Fidel bought 6 books The first book cost $5 Each
additional book cost $1 more than the previous book How
much did Fidel spend on books in all?
Understand Read the problem Write what you know
The first book costs Each additional book costs more than the previous book
Fidel bought books
Plan Pick a strategy One strategy is to make a table
Label the rows Book and Cost
Solve One book costs The cost increases by
for each additional book
Check Does your answer make sense?
Step by Step Problem-Solving Practice Problem-Solving Strategies
Trang 35Copyright © by The McGraw-Hill Companies, Inc.
13 FISH A swordfish grows at a regular rate for the first year of life
Suppose it weighs 14 pounds at the age of one month, 28 pounds
at the age of 2 months, and 42 pounds at the age of 3 months
What is the weight of a swordfish at the age of 6 months?
Understand Solve Plan Check
14 FITNESS Martha runs for 30 minutes each day except for
Saturday and Sunday After 2 weeks, how much time will Martha
have spent running?
15 Explain a rule for the pattern 10, 20, 40, 80
Skills, Concepts, and Problem Solving
Find a rule for each pattern.
Next terms: , , Next terms: , ,
24 Write the next three conversions in the pattern
Number of Quarts 4
GO ON
Trang 36Copyright © by The McGraw-Hill
26 BOOKS Dimitri placed boxes of books on 5 shelves in the library
He put 1 box on the top shelf, 3 boxes on the second shelf, and
5 boxes on the third shelf If he continues this pattern, how many
boxes will he put on the fifth shelf?
27 MUSIC For a band concert, chairs were set up for the musicians
There were 4 chairs in every row How many chairs are there in
28 A(n) tells how numbers are related to each other
29 A(n) is a sequence of numbers, figures, or symbols that
follows a rule or design
30 Writing in Math Explain how to find the next three terms in the
sequence 2, 4, 8, 16 using 2 different rules
Trang 37Copyright © by The McGraw-Hill Companies, Inc.
Equations show number relationships To solve an equation,
substitute a value for a variable.
Solve the equation y = x + 1 when x = 6, 7, and 8.
The equation shows the relationship between x and y.
Substitute 6 for x Substitute 7 for x.
Substitute 8 for x Each solution is an ordered
pair of numbers Making
a table may help you understand the solutions
The number that you substitute is the first number in the ordered
pair, and the solution you get after you substitute is the second
number For example, the solutions of the three equations above
are written: (6, 7), (7, 8), and (8, 9)
GO ON
3AF2.1 Solve simple problems involving a relationship between two quantities 4AF1.5 Understand that an equation
such as y = 3x + 5 is a prescription for
determining a second number when a first number is given.
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There are 5,280 feet in 1 mile How many
feet are in 3 miles? Use the equation
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Step 1 How many pints equal one quart?
Write an equation using that information
number of number of number of pints in all = pints in 1 quart · quarts
Dan has pints of lemonade
Step by Step Practice
Write an equation for each conversion using x and y Solve for the
number of units given.
6 number of tons = number of pounds ÷ 2,000; 7,000 pounds
7 number of meters = number of kilometers · 1,000; 8.2 kilometers
8 number of fluid ounces = number of cups · 8; 40 cups
9 number of miles = number of yards ÷ 1,760; 7,040 yards
GO ON
Trang 40Copyright © by The McGraw-Hill
14 AGES Gretchen is 3 times as old as Vito Gretchen is
24 years old How old is Vito?
Understand Read the problem Write what you know
Gretchen is years old
Gretchen is times as old as Vito
Plan Pick a strategy One strategy is to use logical
reasoning to write an equation
Solve Let x represent Vito’s age and y represent
Gretchen’s Write the equation
Substitute for y
= 3x = x
Vito is years old
Check Multiply Vito’s age by 3
× 3 =
Step by Step Problem-Solving Practice Problem-Solving Strategies
✓ Use logical reasoning.
Look for a pattern.
Guess and check.
Solve a simpler problem.
Work backward.