866 Unit 11 3-D Shapes, Weight, Volume, and Capacity

Teaching the Lesson materials

Key ActivitiesStudents read and discuss an essay about volume in the Student Reference Book. Studentsuse base-10 blocks and metersticks to visualize the sizes of various metric cubic units. Theyalso make open boxes and fill them with centimeter cubes to determine their volume.

Key Concepts and Skills• Use multiplication to solve volume problems. [Operations and Computation Goal 3]• Find the area of the base of a rectangular prism. [Measurement and Reference Frames Goal 2]• Count unit cubes to find the volume of a rectangular prism.

[Measurement and Reference Frames Goal 2]• Calculate the surface area of a rectangular prism.

[Measurement and Reference Frames Goal 2]• Write number models with parentheses.

[Patterns, Functions, and Algebra Goal 3]

Key Vocabulary cubic units • volume • dimensions • surface areaOngoing Assessment: Recognizing Student Achievement Use Mental Math and Reflexes.[Operations and Computation Goal 3]

Ongoing Learning & Practice materialsStudents play the Credits/Debits Game to practice addition of integers.

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students use 24 cubes tobuild as many rectangularprisms as possible, eachwith a different base.

Students create penticubesand compare their surfaceareas.

Students add volume andcubic units to their MathWord Banks.

� Teaching Aid Master (Math Masters,p. 389)

� Differentiation Handbook� centimeter cubes

ELL SUPPORTENRICHMENTREADINESS

3

� Math Journal 2, p. 297� Student Reference Book, p. 238� Study Link Master (Math Masters,

p. 330)� Game Master (Math Masters, p. 468)

2

� Student Reference Book, p. 137� Study Link 11�3� Teaching Aid Master (Math Masters,

p. 403)� base-10 blocks� centimeter cubes� tape� metersticks� slate� scissors� traffic cones and string (optional)� computer with Internet access

(optional)

See Advance Preparation

1

Additional InformationAdvance Preparation For the cubic meter demonstration in Part 1, you need 3 metersticks;for the alternative demonstration, you need 4 metersticks, 4 traffic cones, and string or 2 metersticks, 2 traffic cones, string, and tape. See the illustrations on pages 868 and 869.

Objective To review concepts and units of volume.

Technology Assessment Management System

Mental Math and Reflexes See the iTLG.

See the Web site on page 870.

� Math Message Follow-Up(Student Reference Book, p. 137)

Review the information on Student Reference Book, page 137.Once students explain the significance of the picture of thesandbox, ask them to give other examples in which it is useful toknow the volume of an object. For example:

� Buying a cooler—to decide whether it is big enough to hold thefood for a camping trip

� Renting a car—to decide if the trunk is large enough to holdthe family’s luggage

Tell students that in this lesson they will review units of volume andexplore how to use cubes to find the volume of a rectangular prism.

WHOLE-CLASS

DISCUSSION

1 Teaching the Lesson

Lesson 11�4 867

Getting Started

Study Link 11�3Follow-UpWorking in small groups, have students compare answers and pose the riddle they wrote.

Math MessageRead page 137 of theStudent Reference Book. Be prepared to explain why there is a picture of asandbox on the page.

Mental Math and Reflexes �Pose multiplication and division facts. Suggestions:

2 � 4 � 85 � 7 � 3510 � 6 � 6020 � 4 � 536 � 6 � 645 � 5 � 9

3 � 9 � 274 � 8 � 326 � 4 � 2442 � 7 � 664 � 8 � 828 � 4 � 7

9 � 9 � 818 � 7 � 566 � 9 � 5448 � 6 � 863 � 7 � 972 � 9 � 8

Volume and Capacity

VolumeThe volume of a solid object such as a brick or a ball is ameasure of how much space the object takes up. The volume of a container such as a freezer is a measure of how much thecontainer will hold.

Volume is measured in cubic units. A base-10 cube has sidesthat are 1 centimeter long; it is called a cubic centimeter.A cube with 1-inch sides is called a cubic inch.

Other cubic units are used to measure large volumes. A cubicfoot has 1-foot sides. A cubic yard has 1-yard sides and canhold 27 cubic feet. A cubic meter has 1-meter sides and canhold more than 35 cubic feet.

The volume of an object can bevery useful to know. Supposeyou wanted to buy sand to fillan empty sandbox. To estimatehow much sand to buy, you

would measure the length, width, and height of the sandbox.The length, width, and height are called the dimensions ofthe box. You would then use these dimensions to calculatehow many cubic feet (or cubic yards) of sand to order. Youcould do similar calculations to determine how much concretewould be needed to build a patio, or how much gravel to buyfor a path in the backyard.

CapacityWe often measure things that are poured into or out ofcontainers such as liquids, grains, salt, and so on. Thevolume of a container that is filled with a liquid or a solidthat can be poured is often called its capacity.

Capacity is usually measured in units such as gallons,quarts, pints, cups, fluid ounces, liters, and milliliters.These are standard units, but they are not cubic units.

The tables at the right compare different units of capacity.

Measurement

1 cubic centimeter(actual size)

1 cubic inch(actual size)

Jupiter is the largest planetin the solar system. Thevolume of Jupiter is 1,300times the volume of Earth.

U.S. Customary Units

1 gallon (gal) � 4 quarts (qt)1 gallon � 2 half-gallons1 half-gallon � 2 quarts1 quart � 2 pints (pt)1 pint � 2 cups (c)1 cup � 8 fluid ounces (fl oz)1 pint � 16 fluid ounces 1 quart � 32 fluid ounces

1 half-gallon � 64 fluidounces

1 gallon � 128 fluid ounces

Metric Units

1 liter (L) �1,000 milliliters (mL)1 milliliter � �1,0

100� liter

Student Reference Book, p. 137

Student Page

Ongoing Assessment:Recognizing Student Achievement

Use Mental Math and Reflexes to assess students’ ability to solve multiplication and division facts. Students are making adequate progress if they demonstrate automaticity with the multiplication facts and proficiencywith the division facts in the , , and problems. Some studentsmay demonstrate automaticity with the division facts.

[Operations and Computation Goal 3]

Mental Math

and

Reflexes �

868 Unit 11 3-D Shapes, Weight, Volume, and Capacity

� Visualizing Metric Cubic UnitsDiscuss the following:

� Linear measurements are usually given in standard units (suchas feet or meters), and area measurements are often given insquares of those units (such as square feet or square meters).Many volume measurements are given in cubes of standardunits, or cubic units.

� The area of a closed 2-dimensional figure is the number of unitsquares and fractions of unit squares needed to cover the interior of the figure. The volume of a 3-dimensional object isthe number of unit cubes and fractions of unit cubes needed tofill the space taken up by the object. To support English language learners, discuss the everyday and mathematicalmeanings of volume.

Use base-10 blocks and metersticks to help students visualize thesizes of various metric cubic units.

� Hold up a cm cube. Point out that each edge is 1 centimeterlong, so the volume of a cm cube is 1 cubic centimeter.

� Hold up a “big cube.” Explain that each edge is 10 centimeters,or 1 decimeter, long, so the volume of a big cube is 1 cubicdecimeter.

Ask: How many cubic centimeters are in 1 cubic decimeter? Havestudents use base-10 blocks to “prove” their answers. There are10 cubic centimeters in 1 long; there are 10 longs, or 100 cubiccentimeters, in 1 flat. You can fill a cubic decimeter container with10 flats, or 1,000 cubic centimeters. Therefore, 1 cubic decimeterequals 1,000 cubic centimeters.

You will not have enough base-10 blocks to build a 1-meter cube,but you can help students visualize such a cube using one of thesemethods:

Method 1: Place two metersticks on a flat surface at right anglesto each other. Hold up a third meterstick perpendicular to theother metersticks so that all three sticks meet in one corner.

With the help of this partial frame, students can imagine a cube whose edges are the length of a meterstick. The volume of this cube is 1 cubic meter.

Method 2: Place four hollow traffic cones on the floor at thecorners of a square with 1-meter sides. Put a meterstick throughthe top of each cone so that each stick stands straight up. Connectthe tops of the metersticks with string to form a square. Thestring should be as close to the top of the metersticks as possible.(See margin.)

WHOLE-CLASS

ACTIVITY

1 m

eter

1 meter 1 meter

Lesson 11�4 869

2 cm

3 cm

4 cm

P attern for open box

Adjusting the Activity

Method 3: A variation of the above method uses two cones andtwo metersticks. Place the cones 1 meter from a wall and 1 meterapart. Connect the tops of the metersticks with string. Run astring from the top of each meterstick to the wall at a height of 1 meter, and tape the string to the wall.

Ask: How many cubic decimeters are in 1 cubic meter? 1,000 Howmany cubic centimeters are in 1 cubic meter? 100 � 100 � 100 � 1,000,000

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

Remind students of alternative ways of writing square units: square m, sq m, or m2; square cm, sq cm, or cm2; and square in., sq in., or in2. Similarly, cubic units may be written as m3, cm3, orin3. These are read as “cubic meter,” “cubic centimeter,” and “cubic inch.”

� Using Cubes to Find the Volume of a Rectangular Prism(Math Masters, p. 403)

Model the following activity for students before they work in partnerships to make their own boxes:

1. On a sheet of centimeter grid paper (Math Masters, page 403),draw a pattern for an open box. For example, the bottom of thebox might be a rectangle 4 centimeters long and 3 centimeterswide, and the box might be 2 centimeters high.

2. Cut out the pattern. Fold up the sides, and tape them together.

3. Fill the box with centimeter cubes. The number of cubes needed to fill the box is the volume of the box.

Partners may make boxes with any dimensions they choose, but the height must be at least 2 centimeters.

PARTNER

ACTIVITY

1 m

eter

1 meter 1 meter

870 Unit 11 3-D Shapes, Weight, Volume, and Capacity

297

Math Boxes LESSON

11� 4

Date Time

5. Round each number to the nearest tenth.

a. 3.46

b. 0.71

c. 4.35

d. 9.60

e. 22.89 22.99.64.40.73.5

6. Jake can ride his bike 5 miles in 40 minutes. At this rate, how long does it take him to ride 1 mile? Circle the best answer.

A. 200 minutes

B. 40 minutes

C. 20 minutes

D. 8 minutes

1. The object below has the shape of ageometric solid. Name the solid.

cylinder

3. Write a number model to estimate the answer. Then correctly place thedecimal point.

a. 7.56 � 4 � 3 0.2 4

Number model:

b. 563.2 ÷ 4 � 1 4 0.8

Number model:

4. Insert �, �, or � to make a true numbersentence.

a. �14 �6

b. �123 �241

c. �8.9 �5.7

d. ��14� ��

25�

e. ��39� ��

13��

�

�

�

�

106 107

6 60

182 183

2. Draw the figure after it is rotatedcounterclockwise �

14�-turn.

101

8 � 4 � 32

600 � 4 � 150

Math Journal 2, p. 297

Student Page

STUDY LINK

11� 4 Volume

137 138

Name Date Time

TAB

TAB

TAB

TAB

Cut out the pattern below and tape it together to form an open box.

1. Find and record two items in your home that have volumes equal to about �

12� of the volume of the open box.

2. Find and record two items in your home that have about the same volume as the open box.

3. Find and record two items in your home that have volumes equal to about 2 times the volume of the open box.

Answers vary.

Answers vary.

Answers vary.

Practice

4. 96 ÷ 4 � 5. 86 / 5 �

6. �2382

� � 7. 4��3�5�8� �2924 17 R1, or 17�

15�

89 R2, or 89 �24�, or 89 �

12�

Math Masters, p. 330

Study Link Master

Adjusting the Activity

Technology Link Alternatively, have students visit theWeb site at http://www.illuminations.nctm.org/tools/tool_

detail.aspx?id�6 to create boxes of varying dimensions and manipulate and count unit cubes, rows of unit cubes, or layers of unit cubes.

Have students imagine that each of the boxes has a lid. Have them calculate the surface area of the closed boxes by determining the sum of theareas of the faces. The surface area of the sample box on page 869 is 52 square centimeters: (2 ∗ (2 ∗ 3)) � (2 ∗ (2 ∗ 4))� (2 ∗ (3 ∗ 4)) � 52.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

� Playing the Credits/Debits Game(Student Reference Book, p. 238; Math Masters, p. 468)

Students play the Credits/Debits Game to practice adding positiveand negative numbers. See Lesson 10-6 for additional information.

� Math Boxes 11�4(Math Journal 2, p. 297)

Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 11-2 and 11-6. The skills in Problems 5 and 6 preview Unit 12 content.

Writing/Reasoning Have students write a response to thefollowing: How did you round each number to the nearesttenth in Problem 5? Sample answer: I found the digit in

the tenths place. Then I looked at the digit to the right. If it wasless than 5, I kept the digit in the tenths place the same. If thenumber was 5 or greater, I rounded up the number in the tenths place.

� Study Link 11�4(Math Masters, p. 330)

Home Connection Students cut out and assemble anopen box. They search for items at home that have volumes equal to about �

12� of, the same as, and 2 times

the volume of the open box.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

PARTNER

ACTIVITY

2 Ongoing Learning & Practice

� Finding Rectangular Prisms (Math Masters, p. 389)

To explore the concept of volume, have students use 24 centimetercubes to build as many rectangular prisms as possible, each with a different base. The area of the base must be greater than 1 cm2,and the height of the prism must be greater than 1 cm.

Have students create a table on an Exit Slip (Math Masters,page 389) to organize their work. The table should include thearea of the base, the height, and the volume of each prism theymake. Remind students to include the units. (See margin.)

� Exploring Penticubes(Math Masters, p. 389)

To investigate volume and surface area, have students build penticubes, which are 3-dimensionalfigures with a volume of 5 cubic units. They are

constructed from 5 cubes connected by at least one face. There are 29 possible penticubes.

Have students compare the surface areas of the penticubes anddescribe on an Exit Slip (Math Masters, page 389) anything theynotice about the figures with similar surface areas. Expectresponses such as the following:

� All but two of the penticubes have a surface area of 22 squareunits. The penticubes that are circled to the right have a surface area of 20 square units.

� If a cube touches only 1 other cube face, then the surface area ofthat individual cube is 5 square units. If a cube touches two faces,then the surface area of that individual cube is 4 square units.

� The more faces the cubes touch, the smaller the surface area is. For example, in the circled penticubes, 4 of the cubes touch2 faces each, thereby creating a smaller surface area.

� Building a Math Word Bank(Differentiation Handbook)

To provide language support for volume, have students use theWord Bank Template found in the Differentiation Handbook. Askstudents to write the terms volume and cubic units, draw picturesrelating to each term, and write other related words. See theDifferentiation Handbook for more information.

5–15 Min

SMALL-GROUP

ACTIVITYELL SUPPORT

15–30 Min

PARTNER

ACTIVITYENRICHMENT

15–30 Min

PARTNER

ACTIVITYREADINESS

3 Differentiation Options

Lesson 11�4 871

Area of Base Height Volume (sq cm) (cm) (cu cm)

2 12 24

12 2 24

3 8 24

8 3 24

4 6 24

6 4 24

Example:

There are 29 possible penticubes.