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

Objective To guide the development and use of a formula for

finding the volume of a rectangular prism.

Teaching the Lesson materials

Key ActivitiesStudents solve cube-stacking problems and use the results to derive a formula for the volume of a rectangular prism.

Key Concepts and Skills• Find the area of the base and the surface area of a rectangular prism.

[Measurement and Reference Frames Goal 2]• Count unit cubes and use a formula to find the volume of a rectangular prism.

[Measurement and Reference Frames Goal 2]• Solve problems involving spatial visualization. [Geometry Goal 3]• Describe a rule for a pattern and use the rule to solve problems.

[Patterns, Functions, and Algebra Goal 1]• Write number models with parentheses. [Patterns, Functions, and Algebra Goal 3]

Key Vocabulary rectangular prism • volume • formulaOngoing Assessment: Informing Instruction See page 874.

Ongoing Assessment: Recognizing Student Achievement Use journal page 302. [Measurement and Reference Frames Goal 2]

Ongoing Learning & Practice materialsStudents play Chances Are to practice using probability language to describe the likelihood of an event.

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

Differentiation Options materials

Students use interlocking cubes to build cube stacks and solve spatial visualization problems.

Students estimate the volume of a sheet ofnotebook paper.

� Teaching Master (Math Masters, p. 332)� Teaching Aid Master (Math Masters, p. 388

or 389) � interlocking cubes; sheet of notebook

paper; scissors; stick-on notes

ENRICHMENTREADINESS

3

� Math Journal 2, p. 299� Student Reference Book, pp. 236 and 237� Study Link Master (Math Masters, p. 331)� Game Master (Math Masters, p. 464)� Chances Are Event and Probability Cards

(Math Masters, pp. 462, 463, 465, and 466)

2

� Math Journal 2, pp. 298, 300–302� Study Link 11�4� centimeter cubes � centimeter ruler� slate

1

Technology Assessment Management System

Journal page 302, Problems 1, 2, and 5See the iTLG.

� Math Message Follow-Up(Math Journal 2, p. 298)

Review the answers on journal page 298. Discuss the use of variables to stand for quantities such as length and width. Twoformulas that students are likely to give are A � l � w and A � b � h. To support English language learners, write the twoformulas on the board.

Tell students that in this lesson they will apply their knowledge of area formulas to develop a formula for finding the volume of a rectangular prism.

� Solving Cube-Stacking Problems(Math Journal 2, pp. 300 and 301)

Each problem on journal pages 300 and 301 shows a picture of abox that is partially filled with cubes. Students find the number ofcubes needed to completely fill each box and record the results inthe table on journal page 300.

Fill in the column for Box 1 with the class. You might wish to usethe following prompts:● How many cubes can be placed along the longer side of

the box? 8 Along the shorter side? 4● How many cubes are needed to cover the bottom of the box? 32● How many layers of cubes are needed to fill the box? 5

How can you tell? There are 5 cubes in the stack.● How many cubes are needed to fill the box? There are 5 layers

with 32 cubes in each layer, and 5 � 32 � 160, so 160 cubesare needed to fill the box.

Have students complete the rest of the problems with partners.

PARTNER

ACTIVITY

WHOLE-CLASS

DISCUSSION

1 Teaching the Lesson

Area of a RectangleLESSON

11 5

Date Time

134

5. Find the area of the rectangle. 6. Find the height of the rectangle.

Number model: Number model:

Area cm2 height cm15.556.5

11.3 cm

5 cm

11.3 5 56.526 cm

?

403 26 15.5

Try This

7 meters

?

3. Find the height of the rectangle. 4. Find the length of the base of the rectangle.

Number model:

height m

Number model:

length of base in.7884 12 756 7 8

Area 403 cm2

1. Write a formula for the area of a rectangle. In your formula, use A for area. Use l andw for length and width, or b and h for base and height.

2. Draw a rectangle with sides measuring 3 centimeters and 9 centimeters. Find the area.

Number model: Area square centimeters27

A l w, or A b h

9 3 27

Area 56 m2

12 in

.

?

Area 84 in2

Math Journal 2, p. 298

Student Page

Lesson 11�5 873

Getting Started

Math MessageComplete journal page 298.

Study Link 11� 4 Follow-UpWorking in small groups, have students describe theitems in their home that had volumes equal to about�12� of, the same as, and 2 times the volume of the open box. Ask students to use centimeter cubes to determine the volume of thebox. 96 cm3 Ask: Suppose the box had a lid. What would be the surface area of the closed box? 136 cm2

Mental Math and ReflexesPose mental addition problems. Suggestions:

16 � 4 � 2025 � 5 � 3011 � 9 � 2037 � 3 � 4018 � 19 � 3749 � 17 � 6648 � 16 � 6432 � 18 � 50

253 � 29 � 28276 � 149 � 225126 � 259 � 385644 � 39 � 683

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

300

Cube-Stacking ProblemsLESSON

11� 5

Date Time

Each picture at the bottom of this page and on the next page shows a box that is partially filled with cubes. The cubes in each box are the same size. Each box has at least one stack of cubes that goes to the top.

Your task is to find the total number of cubes needed to completely fill each box.

Record your answers in the table below.

Table of Volumes

Placement of Cubes Box 1 Box 2 Box 3 Box 4 Box 5 Box 6

Number of cubes needed to cover the bottom

Number of cubes in the tallest stack (Be sure to count the bottom cube.)

Total number of cubes needed to fill the box

32 40 24 16 35 25

5 7 4 5 6 5

160 280 96 80 210 125

Box 1 Box 2

138

Math Journal 2, p. 300

Student Page

Ongoing Assessment: Informing InstructionWatch for students who try to find the volume by counting only the cubes shownin the picture. Remind them that the cubes that are shown will help determine theheight of the box and the number of cubes needed to cover the bottom, but thatthe volume of the box is the total number of cubes needed to fill the box.

� Deriving a Formula for the Volume of a Rectangular Prism(Math Journal 2, pp. 300 and 301)

Remind students that geometric solids, such as those pictured on journal pages 300 and 301, are called rectangular prisms.Review the properties of rectangular prisms. To support Englishlanguage learners, write students’ responses on the board. For example:

� A rectangular prism has 6 rectangular faces, 12 edges, and 8 corners.

� Pairs of opposite faces are congruent.

� Any face of a rectangular prism can be designated as the base ofthe prism. The height of the prism is the distance between thebase and the face opposite the base.

Allow students 10 to 15 minutes to complete the journal pages.Then bring the class together to discuss students’ results.

Draw a rectangular prism on the board and label the base andheight, as shown.

Ask students to look for a pattern in the table on journal page 300.To find the total number of cubes needed to fill each box, multiplythe number of cubes needed to cover the bottom of the box by thenumber of cubes in the tallest stack.

PARTNER

ACTIVITY

NOTE The properties to the right are true of right rectangular prisms. The obliquerectangular prism shown below has only tworectangular faces. In Fourth Grade EverydayMathematics, the term rectangular prism willrefer only to a right rectangular prism.

rectangle

rectangle

baseheight

Oblique rectangular prism

301

Cube-Stacking Problems continuedLESSON

11� 5

Date Time

138

base

height

Formula for the volume of a rectangular prism:

B is the area of a base.

h is the height from that base.

Volume units are cubic units.

V � B � h

Box 3 Box 4

Box 5 Box 6

Math Journal 2, p. 301

Student Page

Lesson 11�5 875

NOTE Traditionally, lowercase letters areused in formulas to represent length, anduppercase letters are used to represent areaor volume. For example, b stands for thelength of the base of a polygon, and B standsfor the area of the base of a geometric solid.

Links to the Future

Adjusting the Activity

Then call students’ attention to the following relationships:

� The number of cubes needed to cover the bottom of the box is the same as the number of squares needed to cover thebase—that is, the area of the base of the box.

� The number of cubes in the tallest stack is the same as theheight of the box.

� Therefore, you can find the volume of a rectangular prism by multiplying the area of a base by the height of the prism.

Volume of a rectangular prism � area of base � height

Written with variables, this becomes V � B � h

where V is the volume of the rectangular prism, B is the area of the base, and h is the height of the prism.

Have students record the formula at the bottom of journal page 301.

Use of a formula to calculate the volume of a prism is a Grade 5 Goal.

� Finding Volume(Math Journal 2, p. 302)

Students find the volume of stacks of centimeter cubes and calculate the volume of rectangular prisms. Ask students toexplain the strategies they used to solve the problems.For Example:

� Problem 2: A portion of the top layer is missing. Calculate the volume of a completed rectangular prism and subtract themissing blocks. Alternatively, determine the volume of the complete prism as shown and add the partial layer of cubes.

� Problem 3: There is one complete layer of cubes with threeidentical stacks on top. Add the volume of the bottom layer tothe volume of the three stacks to determine the total volume.

ELL

Have students use centimeter cubes to build the stacks and rectangularprisms on journal page 302.

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

PARTNER

ACTIVITY

Try This

302

Cube-Stacking Problems continuedLESSON

11� 5

Date Time

138

Find the volume of each stack of centimeter cubes.

1. 2.

3. 4.

5. Choose one of the problems from above. Describe the strategy that you used to find the volume of the stack of centimeter cubes.

Sample answer: For Problem 4, I found the volume of thetall part, which is 4 cm2 � 6 cm � 24 cm3, and then addedthe volume of the two extra squares to get 26 cm3.

6. 7.

Volume � cm345 Volume � cm325

Volume � cm340 Volume � cm326

2 cm

3 cm

3 cm

8 cm

2 cm

10 cm

Number model:

Volume � cm3

Number model:

Volume � cm3

(2 � 3) � 3 � 18 (8 � 10) � 2 � 160

� �

�

18 160

Math Journal 2, p. 302

Student Page

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

STUDY LINK

11� 5 Volume

Name Date Time

1. Find the volume of each stack of centimeter cubes.

a. b.

Volume � cm3 Volume � cm3

2. Calculate the volume of each rectangular prism.

a. b.

Number model: Number model:

Volume � cm3 Volume � cm3

3. What is the total number of cubes needed to completely fill each box?

a. b.

cubes cubes150150

9754(2 º 5) º 9.7 � 97(3 º 3) º 6 � 54

3039

6 cm

3 cm3 cm

2 cm

5 cm9.7 cm

4. �65 � 16 � 5. � �21 � (�19)

6. � 84 � (�55) 7. �16 � 89 � 7329�40�49

Practice

137 138

Math Masters, p. 331

Study Link Master

299

Math Boxes LESSON

11� 5

Date Time

5. Add.

a. �54 � 28 �

b. �62 � (�15) �

c. � 51 � (�139)

d. � �$23.56 � $87.45

e. $71.08 � (�$85.79) �

$63.89

6. If 4 shirts cost $76, what is the cost of

a. 2 shirts?

b. 6 shirts?

c. 1 dozen shirts?

d. 75 shirts? $1,425$228

$114$38

1. What is the total number of cubes neededto completely fill the box?

cubes125

3. When you roll a 6-sided die, about whatfraction of the time would you expect

a. a multiple of 2 to come up?

b. a factor of 20 to come up?

4. Complete.

a. 13 ft � yd ft

b. 18 ft 6 in. � yd in.

c. 972 in. � yd

d. 15,840 ft � mi

e. 24,640 yd � mi 143

2766

14

81 129

47

2. Calculate the volume.

Number model:

Volume � in330,625 138138

(25 � 35) � 35 � 30,625

25 in.

35 in.

35 in.

�36�, or �

12�

�46�, or �

23�

�$14.71

�77�88

�26

Math Journal 2, p. 299

Student Page

Journal page 302

Problems

1, 2, and 5 �Ongoing Assessment:Recognizing Student Achievement

Use journal page 302, Problems 1, 2, and 5 to assess students’ ability to findthe volume of stacks of centimeter cubes. Students are making adequateprogress if they are able to build the stacks with actual centimeter cubes or lookat the pictures to find the volume and then describe the strategy used. Some students may be able to solve Problems 3 and 4, which involve more difficultarrangements of cubes, and Problems 6 and 7, which involve representations of rectangular prisms that do not show individual cubes.

[Measurement and Reference Frames Goal 2]

� Playing Chances Are(Student Reference Book, pp. 236 and 237; Math Masters, pp. 462–466)

Students play Chances Are to practice using probability language to describe the likelihood of an event. See Lesson 7-11 for additional information.

� Math Boxes 11� 5(Math Journal 2, p. 299)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 11-7. The skill in Problem 6previews Unit 12 content.

Writing/Reasoning Have students write a response to thefollowing: Use probability terms to describe the likelihoodof each of the events in Problem 3. Explain your choice of

language. Sample answer: There is a 50-50 chance of rolling amultiple of 2 because 2, 4, and 6 are multiples of 2. There are 3favorable outcomes out of 6 possible outcomes. It is likely that afactor of 20 will come up. 1, 2, 4, and 5 are factors of 20. Thereare 4 favorable outcomes out of 6 possible outcomes.

� Study Link 11� 5(Math Masters, p. 331)

Home Connection Students find the volume of stacks of centimeter cubes, calculate the volume of rectangularprisms, and determine the number of cubes that are needed to fill boxes.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

PARTNER

ACTIVITY

2 Ongoing Learning & Practice

� Solving Spatial-Visualization Puzzles(Math Masters, p. 332)

To explore the representation of 3-dimensional figures with 2-dimensional drawings, have students use interlocking cubes to build cube stacks and solve spatial-visualization problems.

� Estimating the Volume of a Sheet of Paper(Math Masters, p. 388 or 389)

To apply students’ understanding of volume, have themestimate the volume of a sheet of notebook paper. In aMath Log or on an Exit Slip, have students write a briefreport describing their strategy. One possible strategy isgiven below.

1. Cut the sheet of paper into 1-inch squares.

2. Stack the squares into a neat pile. Measure the height of thepile of squares. About �

14� -inch high The area of the base of the

pile is 1 square inch.

3. Use the formula: V � B � h

V � 1 � �14�

V � �14�

4. The volume of a sheet of notebook paper is about 0.25 cubic inches.

5–15 Min

INDEPENDENT

ACTIVITYENRICHMENT

PARTNER

ACTIVITYREADINESS

3 Differentiation Options LESSON

11� 5

Name Date Time

Hidden Cubes

1. The stacks of cubes shown below are called soma cubes and were first designed in 1936 by Piet Hein, a Danish poet and scientist.

Use interlocking cubes to build the stacks shown below. Use a small stick-on note to label each stack with the appropriate letter. Then record the number of cubes needed to build each stack.

A cubes B cubes C cubes D cubes

E cubes F cubes G cubes

Use the cube stacks that you made above to build each of the figures below. The figures do not have any hidden holes. Record the number of cubes needed to build each figure and the cube stacks that you used.

2. cubes

I used the following cube stacks to build the figure:

3. cubes

I used the following cube stacks to build the figure:

4. cubes

I used the following cube stacks to build the figure: All of them, A–G27

C, G, E or C, G, F12

E and F8

444

4443

Try This

Math Masters, p. 332

Teaching Master

Lesson 11�5 877

1-inch squares cut froma single sheet of paper

1-inch cube