California math triumphs measurement, volume 6a

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Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber

Jupiter Images

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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:

Cover, i Jupiter Images; iv (tl)File Photo, (tc tr)The McGraw-Hill Companies,

viii Mitchell Funk/Getty Images; ix S Alden/PhotoLink/Getty Images; x Peter

82 (t)Getty Images, (b)Lawrence Manning/CORBIS; 83 GK & Vikki Hart/Getty

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California Math Triumphs

iii

Graphs and Functions

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Authors 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

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California Advisory Board

v

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

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California 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

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Volume 6A Measurement

3AF1.4 Express simple unit conversions

in symbolic form (e.g., _ inches = _ feet × 12).

3MG1.4 Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes).

6AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).

7MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters)

7MG1.3 Use measures expressed

as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.

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2MG1.3 Measure the length

of an object to the nearest inch and/or centimeter.

3MG1.2 Estimate or determine the area and volume of solid fi gures by covering them with squares or by counting the number of cubes that would fi ll them.

3MG1.3 Find the perimeter of a polygon with integer sides.

4MG2.2 Understand that the length of a horizontal line segment equals

the difference of the x-coordinates.

4MG2.3 Understand that the length of a vertical line segment equals the

difference of the y-coordinates.

Alamo Square, San Francisco

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4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm 2 ), square meter (m 2 ), square kilometer (km 2 ), square inch (in 2 ), square yard (yd 2 ), or square mile (mi 2 ).

5MG1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle (i.e., two

of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle

on the parallelogram).

5MG1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to complete the surface area for these objects.

5MG1.3 Understand the concept

of volume and use the appropriate units

in common measuring systems (i.e., cubic centimeter [cm 3 ], cubic meter [m 3 ], cubic inch [in 3 ], cubic yard [yd 3 ]) to compute the volume of rectangular solids.

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x

Chapter

4 Angles and Circles

4-1 Lines 5MG2.1 54

4-2 Angles 5MG2.1 63

Progress Check 1 72

4-3 Triangles and Quadrilaterals 5MG2.1 73

4-4 Add Angles 5MG2.1, 5MG2.2, 6MG2.2 81

Progress Check 2 90

4-5 Congruent Figures 7MG3.4 91

4-6 Pythagorean Theorem 5MG2.1, 7MG3.3 99

Progress Check 3 108

4-7 Circles 6MG1.2 109

4-8 Volume of Triangular Prisms and Cylinders 117

6MG1.3 Progress Check 4 127

Assessment Study Guide 128

Chapter Test 134

Standards Practice 136

in This Chapter

5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straight edge, ruler, compass, protractor, drawing software).

5MG2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.

6MG1.2 Know common estimates

of π (3.14, _22

7 ) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.

6MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid

6MG2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle

to solve problems involving an unknown angle.

7MG3.3 Know and understand the Pythagorean theorem and its

converse and use it to fi nd the length

of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.

7MG3.4 Demonstrate an understanding of conditions that indicate two geometrical fi gures are congruent and what congruence means about the relationship between the sides and angles

of the two fi gures.

Mono Lake Tufa State Reserve

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1

Let’s Get Started

Use the Scavenger Hunt below to learn where things are

located in each chapter

benchmark for one inch?

on p 52?

you can take the Online Readiness Quiz

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How tall are you?

How much does your dog weigh? How far do you travel

to school? These questions ask for measurements of weight and length Other measurements include capacity, time, and temperature.

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3

What You Know What You Will Learn

You know how to multiply and divide

The metric system is a measurement

system in which units differ from the base unit by a power of ten

1l 1,000 ml

1 liter of juice = 1,000 milliliters of juice

So, 4 liters of juice = 4 × 1,000, or 4,000 milliliters of juice

You know how to multiply and divide

The customary system of

measurement uses units such as foot and quart You multiply or divide to change units

Quiz at ca.mathtriumphs.com to find out

them with what you’ll learn in this chapter

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a measurement system that includes units such

as meter, kilogram, and liter

Prefixes used for units of metric measurement always have

the same meaning The meter is the basic unit of length in

the metric system Each prefix shows the size of a unit

compared to a meter

Prefix Meaning Metric

Unit Symbol

Real-World Benchmark

deci one-tenth decimeter dm length of

Use a ruler to help you understand how the units of

Sometimes it is necessary to convert from one unit of measurement to

another Prefixes can help you understand the relationship between the

two units A metric place-value chart can also be useful

To convert a larger unit to a smaller unit, you

should multiply

To convert a smaller unit to a larger unit, you

should divide thousands hundreds tens ones tenths hundredths thousandths

1000 100 10 1 0.1 0.01 0.001

3AF1.4 Express simple unit conversions in symbolic form 3MG1.4 Carry out simple unit conversions within a system of measurement.

6AF2.1 Convert one unit of measurement to another.

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Lesson 1-1 Unit Conversions: Metric Length 5

Example 2

Convert 4.5 kilometers to meters

1 Use a chart Place 4 in the km column

and 5 in the next column to the right

2 Place zeros in the columns between 5 and

the decimal point

3 Read the number from the chart for the

2 Place a zero in the column

3 Read the number from the chart for the conversion

8.2 dm = m GO ON

Example 1

Convert 6 centimeters to meters

1 Use a chart Place 6 in the cm column

2 Place zeros in the m and dm columns

3 Read the number from the chart for

the conversion 6 cm = 0.06 m

YOUR TURN!

Convert 3 millimeters to meters

1 Use a chart Place in the

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6 Chapter 1 How Measurements Are Made

Example 3

Complete the conversions using the Metric Equivalents table below.

kilometers (km) meters (m) decimeters (dm) centimeters (cm) millimeters (mm)

1 km = 1,000 m = 10,000 dm = 100,000 cm = 1,000,000 mm

0.001 km = 1 m = 10 dm = 100 cm = 1,000 mm

0.00001 km = 0.01 m = 0.1 dm = 1 cm = 10 mm

0.000001 km = 0.001 m = 0.01 dm = 0.1 cm = 1 mm

Convert from 8 meters to kilometers using division

You are converting from a smaller unit to a larger unit, so you divide

8 m = km

YOUR TURN!

Convert from 8 meters to centimeters using multiplication

You are converting from a unit to a unit, so you

Silo7.3 × 1,000 = 7,300 mm

Circle correct answer(s) Cross out incorrect answer(s)

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Solve.

14 SPORTS A soccer field is 120 meters long How many

decimeters long is a soccer field?

Understand Read the question Write what you know

A soccer field is meters long

Plan Pick a strategy One strategy is to look for a pattern

decimeters is equal to 1 meter Find a rule One rule is to add

Solve The pattern begins with the numbers 10, 20, and

30 Continue the pattern until the final term is 120

10, 20, 30, The number 120 is the term

The soccer field is decimeters long

Check Think: Decimeters are a smaller unit of measure

than meters, so the number of decimeters of a soccer field is greater than the number of meters

The answer makes sense

Step by Step Problem-Solving Practice

15 SEWING Frances bought 1,850 millimeters of ribbon to

make a pillow The pillow required 170 centimeters of ribbon

In centimeters, how much extra ribbon is left?

Check off each step

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16 SHOES The sales clerk measured Wayne’s foot to be

2.4 decimeters long How many millimeters long is Wayne’s foot?

17 Is 600 millimeters equal to 6 meters? Explain

Skills, Concepts, and Problem Solving

Convert using a place-value chart

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34 TRAVEL It is 49 kilometers from Jesse’s house to his

grandmother’s house How many meters is it to Jesse’s

grandmother’s house?

35 AIRPLANES Hernando’s paper airplane traveled

3,400 centimeters How many meters did it travel?

36 PETS Ginny’s cat was found wandering around a park that was

2,200 meters from her home How many kilometers away was

Ginny’s cat?

37 TRAVEL Ataro passed a sign that said “Albany 192 km.” How

many meters did he have left to drive?

Vocabulary Check Write the vocabulary word that completes each

sentence.

38 The system is a measurement system that includes

units such as meter, gram, and liter

39 A is the standard unit of measurement for length in the

metric system

40 Writing in Math Explain how to convert 5.2 meters to centimeters

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Lesson 1-2 Unit Conversions: Customary Length 11

Lesson

1-2

VOCABULARY customary system

as foot, pound, and quart

benchmark

an object or number used

as a guide to estimate or reference

convert

to switch or exchange for something equal in value (Lesson 1-1, p 4)

inch in small paper clip

foot ft 1 ft = 12 in. standard ruler

Sometimes it is necessary to convert from one unit of measure to

another Knowing customary conversions can help you understand

the relationship between two units

Use the last column of the table to help you understand the

relative size of a unit by comparing it to everyday objects

Use a ruler to see how the units of length compare

1 There are 12 inches in 1 foot

2 Fill in the table

1 There are 3 feet in 1 yard Enter the number of feet in the chart by using multiples of three

2 Fill in the table

feet is equal to 5 yards

GO ON

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Example 2

Convert 7 yards to feet.

1 You are converting from yards to feet,

which is a larger unit to a smaller unit

You should multiply

2 1 yard is equal to 3 feet

So, 7 yards is equal to 3 × 7, or 21 feet

YOUR TURN!

Convert 156 inches to feet.

1 You are converting from inches to feet, which is a smaller unit to a larger unit

To convert a larger unit to a smaller unit, multiply

To convert a smaller to a larger unit, divide

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Convert.

3 9 yd = ft

unit, so you should

Step by Step Practice

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Solve.

18 HOMES The bedroom in Teri’s apartment is 144 inches

long How many yards long is the room?

A bedroom is inches long

Plan Pick a strategy One strategy is to work backward

You know the total number of inches

Subtract repeatedly until the answer is 0

Count the number of times you subtracted 36

The room is yards long

Check Think: An inch is a smaller unit of measure than a

yard So the number of inches should be greater than the number of yards The answer makes sense

19 SCHOOL Ina’s desk is 42 inches wide How many feet wide is

her desk? Check off each step

Understand

Plan

Solve

Check

20 SPORTS During Saturday’s football game, James set the school

record by running 96 yards to score a touchdown How many feet

did James run for the touchdown?

21 Is 108 inches equal to 9 feet? Explain

Problem-Solving Strategies

Draw a diagram.

Look for a pattern.

Guess and check.

Solve a simpler problem.

✓ Work backward.

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Convert using a table.

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Solve.

35 RACES Carla measured a bicycle course in her neighborhood

It was 7,040 yards How many miles was the bicycle course?

36 HISTORY One of the largest balls

of string is in Branson, Missouri

How many inches is the circumference

41.5 ft

of the ball of string?

37 DECORATING Olivia is redecorating her

bedroom She measured the length as

138 inches She measured the width as

114 inches What are the dimensions of

Olivia’s room in feet?

38 SCHOOL At Wakefield Junior High School during a fire drill,

students have to go to the football field and stand single-file in

lines One line was 12 feet long Another line was 15 feet long

A third line was 24 feet long How many yards were the lines

formed by the students?

sentence.

39 The system is a measurement system that

includes units such as foot, pound, and quart

40 To means to switch or exchange for

something equal in value

41 A(n) is an object or number used as a guide

to estimate or reference

42 Writing in Math Explain how to convert 288 inches to yards

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58 ADVERTISING Keeley placed an ad in the newspaper The ad

could be no longer than 75 millimeters long How many

centimeters long could the ad be?

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ten thousands ones tenths hundredths thousandths

meters (m) deci (dm) centi (cm) milli (mm)

kilometers long is the road?

How many inches long is the pool?

Progress Check 1 (Lessons 1-1 and 1-2)

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Lesson 1-3 Unit Conversions: Metric Capacity and Mass 19

Lesson

1-3

VOCABULARY metric system

as meter, gram, and liter (Lesson 1-1, p 4)

capacity

the amount of dry or liquid material a container can hold

Metric Capacity and Mass

Prefixes used for standard units of measurement in the metric

system always have the same meaning

The base unit of capacity in the metric system is the liter

Metric Units for Capacity

Unit for

Capacity Abbreviation Equivalents

milliliter mL 1 mL = 0.001 L drop of water

water bottle

kiloliter kL 1 kL = 1,000 L bathtub filled

with water

The base unit of mass is the gram

Metric Units for Mass

Unit for

milligram mg 1 mg = 0.001 g grain of salt

kilogram kg 1 kg = 1,000 g watermelon

Sometimes it is necessary to convert from one unit of measure to

another Prefixes can help you understand the relationship between

two units A metric place-value chart can also be useful

GO ON

7MG1.1 Compare weights, capacities, geometric measures, times, and

temperatures within and between

measurement systems.

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2 You are converting from a larger to a

smaller unit You need to multiply

2 You are converting from a

to a unit You need to

Convert 5,500 milliliters to liters

1 Use a chart Place 5,500 in the chart so that the

zero that is farthest right is in the mL column

1 Use a chart Place in the chart so

that the zero that is farthest right is in the mL

column

conversion

270 mL = L

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unit You need to

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14 NUTRITION Hershel bought a giant turkey sandwich for a

party The giant sandwich has 200 grams of protein How

many milligrams of protein are in the giant turkey sandwich?

Understand Read the problem Write what you know The

giant sandwich has grams of protein

Plan Pick a strategy One strategy is to solve a simpler

problem Work with 100, and then multiply your answer by 2 to find the total milligrams in the sandwich

milligrams is equal to 1 gram

Solve You are converting from to

Check A milligram is a smaller unit of measure than a

gram, so the number of milligrams of protein should be greater than the number of grams

Step by Step Problem-Solving Practice Problem-Solving Strategies

Draw a diagram.

Look for a pattern.

Guess and check.

✓ Solve a simpler problem.

Work backward.

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15 HEALTH Wendy weighs 45 kilograms How many grams does

she weigh?

Check off each step

Understand Plan

Solve Check

16 NUTRITION Elijah drank all of the water in the bottle shown How

many milliliters of water did he drink?

17 Are 65 liters equal to 0.065 kiloliters? Explain

Convert using a place-value chart.

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Solve.

28 COOKING Norma needed 1,500 milliliters of vegetable oil to cook

a chicken for the family reunion She bought a 2-liter bottle of oil

How many liters of oil did Norma have left over?

29 TRAVEL At the airport, you can only have 32 kilograms of mass

per bag How many grams are you able to carry in each bag?

sentence.

30 is the amount of matter in an object

31 is the amount of dry or liquid material a container

can hold

32 A(n) is a metric unit for measuring volume or capacity

33 A(n) is a metric unit for measuring mass

34 Writing in Math Explain how to convert 6.07 grams to kilograms

37 TRAVEL It is 2.5 miles from Kiki’s house to Laurie’s house How

many feet is this?

Convert (Lesson 1-1, p 4)

40 7.01 m = dm 41 546 m = km

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Lesson 1-4 Unit Conversions: Customary Capacity and Weight 25

VOCABULARY customary system

as foot, pound, and quart (Lesson 1-2, p 11)

capacity

the amount of dry or liquid material a container can hold (Lesson 1-3, p 19)

weight

a measurement that tells how heavy or light an object is

benchmark

an object or number used

as a guide to estimate or reference

KEY Concept

Lesson

1-4 Unit Conversions: Customary

Capacity and Weight

The customary system of measurement is not based on

powers of ten It is based on numbers like 12 and 16, which

have many factors

Customary Units for Capacity

Unit for

fluid ounce fl oz eye dropper

Customary Units for Weight

Customary Units for Weight Unit for

pound lb 1 lb = 16 oz bunch of grapes

Sometimes it is necessary to convert from one unit of measure to

another Knowing customary conversions can help you understand

the relationship between two units

GO ON

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1 8 pints is equal to 1 gallon

2 Fill in the table

1 pints is equal to 1 quart

2 Fill in the table

pints is equal to 3 quarts

Example 2

Convert 9 tons to pounds.

1 You are converting from tons to pounds,

You need to multiply

2 1 ton is equal to 2,000 pounds

So, 9 tons are 9 × 2,000 pounds, or

18,000 pounds

YOUR TURN!

Convert 48 ounces to pounds.

1 You are converting from to

, which is a unit

to a unit

You need to

2 1 pound is equal to ounces

So, 48 ounces are

48 ÷ pounds,

or pounds

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Convert 2 tons to ounces.

1 You are converting from tons to ounces,

You need to multiply

2 1 ton is equal to 2,000 pounds

1 pound is equal to 16 ounces

So, 1 ton is equal to 32,000 ounces

2 × 32,000 = 64,000 ounces

So, 2 tons equals 64,000 ounces

YOUR TURN!

Convert 3.2 tons to ounces.

Convert 56 fluid ounces to pints.

1 You are converting from fluid ounces to

pints, which is a smaller unit to a larger

unit You need to divide

2 1 cup is equal to 8 fluid ounces 1 pint is

equal to 2 cups So, 1 pint is equal to

16 fluid ounces

56 ÷ 16 = 3.5

So, 56 fluid ounces equals 3.5 pints

YOUR TURN!

Convert 22 pints to gallons.

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unit You need to

Step by Step Practice

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Solve.

16 MEASUREMENT A bathtub for a baby can hold 7 gallons

of water How many quarts of water can the bathtub hold?

A baby bathtub holds gallons of water

Plan Pick a strategy One strategy is to look for a

pattern

How many quarts are in 1 gallon?

quarts = gallon Find a rule One rule is to add

Solve The pattern begins with the numbers 4, 8, and 12

Continue the pattern until you find the seventh term

4, 8, 12, , , , The seventh term is

quarts = galThe baby bathtub can hold quarts of water

Check Think: A quart is a smaller unit of measurement

than a gallon So the number of quarts of water is greater than the number of gallons of water The answer makes sense

17 ZOO ANIMALS An animal at the city zoo weighs 7,000 pounds

How many tons does the animal weigh? Check off each step

Understand Plan

Solve Check

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18 COOKING For the baking contest this year, each baker will be

given 48 ounces of flour Diedra needs more flour than that for her

recipes She is bringing 32 ounces of flour How many pounds of

flour will Diedra have altogether?

19 Are there 64 cups in 2 gallons? Explain

Convert using a table.

30 ART Claus mixed the paint

shown to make a shade of gray

How many gallons of gray paint

did Claus make?

31 PETS Vincent feeds his dog

one cup of dog food in the morning

and one cup of dog food in the 16 pints 8 pints

evening How many ounces of

food will Vincent’s dog eat in

14 days?

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