Volume of Right Prisms - Everyday Math · Playing Polygon Capture ... Objective To provide experiences with using a formula for the volume of right prisms.f ... All of the cube structures

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Lesson 9�9 753

Advance PreparationFor Part 1, you will need two pieces of foam board, each at least 4 inches by 10 inches. Use the templates

on Math Masters, pages 282 and 283 to draw and cut out a triangular prism and a parallelogram prism.

With a pen or marker, draw the dashed lines (representing the heights of the bases). Draw a line segment

connecting the midpoints of two sides of the triangle.

Teacher’s Reference Manual, Grades 4–6 pp. 187–189, 222–225

Key Concepts and Skills• Use a formula to calculate the volumes

of prisms.

[Measurement and Reference Frames Goal 2]

• Define and classify prisms according

to common properties.

[Geometry Goal 2]

Key ActivitiesStudents examine a model of a prism to

verify a formula for finding the volume of right

prisms, and then use the volume formula to

calculate the volumes of right prisms.

Key Vocabularyprism � face

MaterialsMath Journal 2, pp. 324–325B

Study Link 9 �8

Math Masters, pp. 282 and 283

Class Data Pad � demonstration materials:

metric ruler, foam board, knife, tape

(optional)

Playing Polygon CaptureStudent Reference Book, p. 328

Math Masters, pp. 494 – 496 (optional);

p. 497

Polygon Capture Pieces and

Property Cards (Math Journal 1,

Activity Sheets 3 and 4)

Students practice identifying

attributes of polygons.

Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 497. [Geometry Goal 2]

Math Boxes 9�9Math Journal 2, p. 326

Student Reference Book, p. 158

Students practice and maintain skills

through Math Box problems.

Study Link 9�9Math Masters, p. 284

Students practice and maintain skills

through Study Link activities.

READINESS

Analyzing Prism Nets for Triangular PrismsMath Masters, pp. 285 and 286

scissors � tape or glue

Students build a triangular prism from

a pattern.

ENRICHMENTBuilding NetsMath Masters, pp. 287 and 288

scissors � tape or glue

Students explore the properties of prisms

by building nets for prisms.

ELL SUPPORT

Building a Math Word BankDifferentiation Handbook, p. 142

Students define and illustrate the term prism.

EXTRA PRACTICE

5-Minute Math5-Minute Math™, pp. 214 and 215

Students calculate the volumes of

rectangular prisms.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Volume ofRight Prisms

Objective To provide experiences with using a formula

for the volume of right prisms.f

��

Common Core State Standards

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754 Unit 9 Coordinates, Area, Volume, and Capacity

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

Ask volunteers to explain which of the cube structures on Study Link 9-8 are rectangular prisms and why.

� The structure in Problem 1 is a rectangular prism because all six surfaces are formed by rectangles.

� The other three structures are not rectangular prisms because their faces are not all rectangles. The bases are polygons, but not rectangles.

Remind students that geometric solids are hollow, but thinking of cube structures can help when measuring volume.

� All of the cube structures are prisms because each is a closed 3-dimensional figure made up of polygonal regions—the flat surfaces, or faces—and each has two congruent and parallel faces.

Ask students what the two congruent and parallel faces are called. Bases of the prism The other faces are formed by rectangles and connect the two bases. Ask volunteers to explain how a cube structure can be used to find the volume of a prism. Use their responses to emphasize the following points:

� You can think of each prism as being made up of layers of centimeter cubes with the same number of cubes in each layer. The number of layers is equivalent to the height of the prism.

� The number of cubes in the first layer is the same as the number of squares in the base (the area of the base).

� If you know the number of cubes in the first layer, then you can find the volume of the entire prism by multiplying the number of cubes in the first layer by the number of layers (the height of the prism).

Mental Math and Reflexes Have students show thumbs up if the statement is true and thumbs down if the statement is false. Suggestions:

Prisms are 3-dimensional shapes. Thumbs up; true

A rectangular prism is formed by eight flat surfaces. Thumbs down; false

The amount of space enclosed by a 3-dimensional shape is measured in cubic units. Thumbs up; true

The volume of a prism is the area of its base. Thumbs down; false

Math Message Are all of the cube structures shown on Study Link 9 �8 rectangular prisms? Be prepared to explain your answer.

Study Link 9�8 Follow-Up Have partners compare answers and resolve any differences.

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LESSON

9�9

Name Date Time

Triangular Prism Base Template

This template is a pattern for the base of a triangular prism. Place the

template on a sheet of foam board. Using a pencil or ballpoint pen,

mark the position of the six dots by piercing the template. Remove

the template and connect the points with solid or dashed

lines, the same as on the template. Cut out the prism

along the outer solid line, using a serrated knife or

saw, making sure that your cuts are made

perpendicular to the base.

Math Masters, p. 282

Teaching Master

LESSON

9�9

Name Date Time

Parallelogram Prism Base Template

py

gg

p

This template is a pattern for the base of a parallelogram prism.

Place the template on a sheet of foam board. Using a pencil or

ballpoint pen, mark the position of the five dots by piercing the

template. Remove the template and connect the points with

solid or dashed lines, the same as on the template.

Cut out the prism along the solid line, using a

serrated knife or saw, making sure that

your cuts are made perpendicular

to the base.


Teaching Master

Lesson 9�9 755

Refer to Study Link 9-8, Problem 1, and pose the following question: What would the volume of the prism be if it were 5 centimeters high? 75 cm3 2 1 _ 2 centimeters high? 37.5 cm3 Have volunteers demonstrate their solution strategies on the board.

NOTE Slanted prisms are a second type of rectangular prism. They are

distinguished from right prisms because all the lateral faces are not all rectangles

and the lateral edges are not perpendicular to the bases. Slanted prisms are

defined by transformations of a polygonal base.

Right prisms

Slanted prisms

▶ Verifying the Volume

WHOLE-CLASS ACTIVITY

Formula for Prisms(Math Masters, pp. 282 and 283)

Algebraic Thinking Show students the two prisms that you cut out of foam board. Ask volunteers to describe the shapes. Expect that students might respond that the shapes are 3-dimensional figures, geometric solids, and/or prisms. Explain that prisms are named by the shapes of the bases. Ask: What are the names of these two prisms? Triangular prism and parallelogram prism Ask students how we know that the triangular faces on the triangular prism are the bases. Because the bases of a prism must be parallel and congruent

Write the formula V = B ∗ h on the Class Data Pad. Tell students that this formula can be used to calculate the volume of any prism regardless of its base shape. The volume of any prism is the product of the area of its base times its height. Write this definition under the formula.

Have volunteers use the following activity to verify the volume formula:

1. Use a metric ruler to measure the base and height of one of the parallelogram prism faces. This face will be one of the bases. Use these measures to calculate the area of the base (B).


Volume of Prisms continuedLESSON

9 �9

Date Time

7. 8.

Volume � cm3 Volume � cm3

9. 10.

Volume � in3 Volume � cm3

11. 12.

Volume � in3 Volume � m3

Area ofbase = 35 cm2

9 cm 8 cm

6.5 cm

3 cm

5.5 in.

11 in.

3 in.

4 cm

6 cm

5 cm

5 cm

3 cm

3 in.Area of base = 8.3 in2

5 mArea of base = 14 m2

315 156

190

70

90.75

24.9

Math Journal 2, p. 325

Student Page

Volume of PrismsLESSON

9 �9

Date Time

The volume V of any prism can be found with the formula V � B � h, where B is the

area of the base of the prism, and h is the height of the prism from that base.

Find the volume of each prism.

1. 2.

Volume � cm3 Volume � cm3

3. 4.

Volume � cm3 Volume � in3

5. 6.

Volume � ft3 Volume � cm3

height � 8 cm

Area ofbase � 10 cm2

height � 4 cm

Area ofbase � 20 cm2

10 cm

12 cm

6 cm

9 in.

7 in.

3 in.

3 ftArea of base = 15 ft2

4 cm4 cm

6 cm

5.7 cm

80

720 189

45 48

80


Student Page


2. Measure the height (h) of the parallelogram prism. Record the area of the base (B) and the height (h) of the prism on the board.

3. Calculate the volume (V) of the prism, and record it on the board.

Cut the prism into two parts along the dashed height line. Move the triangular piece to the other end to form a rectangular prism. (See Figure 1.) Tape the two pieces together if you wish. Ask students if the volume of the prism has changed. No. Cutting and rearranging the prism has not changed its volume.

Figure 1

4. Use a metric ruler to measure the base and height of one of the rectangular prism faces. This face will be one of the bases. Use these measures to calculate the area of the base (B).

5. Measure the height (h) of the rectangular prism. Record the area of the base (B) and the height (h) of the rectangular prism on the board.

6. Calculate the volume (V) of the prism, and record it on the board.

Explain that when the parallelogram prism was cut and rearranged, the volume did not change. Because the formula V = B ∗ h applies to any prism, using this formula should result in the same volume for the parallelogram prism and the rectangular prism. Ask: When applied to the parallelogram prism and the rectangular prism, did the formula V = B ∗ h result in the same volume? Yes

Repeat this procedure with the triangular prism. (See Figure 2.) This time cut the prism into three sections. First cut through the line connecting the midpoints of two sides. This creates a trapezoidal prism and a smaller triangular prism. Then cut the resulting triangular prism through the dashed height line, creating two smaller triangular prisms. Rotate and reattach these small triangular prisms at the ends of the trapezoidal prism to form a rectangular prism. Tape the pieces together if you wish.

Figure 2


Adding Volumes of SolidsLESSON

9 � 9

Date Time

Find the volume of each solid figure. First find the volumes of the two rectangular prisms that make up each solid. Use either of these two formulas: V = B ∗ h or V = l ∗ w ∗ h. Then add the two volumes to find the volume of the entire solid. In Problems 1– 4, you can count the cubes to check your work.

1. Volume of a 2 × 2 × 2 prism: cubic units

Volume of a 1 × 2 × 1 prism: cubic units

Volume of entire solid: cubic units








Volume of a 3 1 _ 2 × 2 × 2 prism: cubic units


8

24

36

60

2

24

5

14

10

48

41

74

325A-325B_EMCS_S_MJ2_G5_U09_576434.indd 325A 3/22/11 12:42 PM

Math Journal 2, p. 325A

Student Page

Adding Volumes of Solids continuedLESSON

9 � 9

Date Time

5. Volume of a 12 cm × 3 cm × 2 cm prism: cm3

Volume of a 8 cm × 3 cm × 5 cm prism: cm3

Volume of the entire solid: cm3

6. Volume of one ream of paper: in3

Volume of two reams of paper: in3

7. Volume of 100 ft × 60 ft × 220 ft prism: ft3

Volume of 210 ft × 60 ft × 180 ft prism: ft3

Volume of entire solid: ft3

8.

Estimate the volume of the water in this pool by doing the following:

a. Underestimate the volume by using the dimensions from the shallow end.

50 m by 25 m by 1.5 m: m3

b. Overestimate the volume by using the dimensions from the deep end.

50 m by 25 m by 2 m: m3

c. Find the average of the two volumes to estimate the volume of the pool.

m3

12 cm 3 cm

8 cm

5 cm

2 cm

60 ft

40 ft

100 ft 210 ft

180 ft

Each ream of paper measures

8 1 _ 2 in. by 11 in. by 2 in.

50 m

25 m

2 m

1.5 m

Shallow End Deep End

Try This

1,320,000

2,268,000

192

187

374

3,588,000

1,875

2,500

2,187.5

72

120

325A-325B_EMCS_S_MJ2_G5_U09_576434.indd 325B 4/6/11 11:02 AM

Math Journal 2, p. 325B

Student Page

Lesson 9�9 757

▶ Finding the Volumes of Prisms PARTNER ACTIVITY

(Math Journal 2, pp. 324 and 325)

Algebraic Thinking Have students complete journal pages 324 and 325.

▶ Adding Volumes of Solids

WHOLE-CLASS ACTIVITY

(Math Journal 2, pp. 325A and 325B)

To solve problems on journal pages 325A and 325B, students find the volumes of rectangular prisms and then use those results to find the volumes of solid figures composed of two non-overlapping rectangular prisms. Write 3 × 2 × 4 on the board. Ask:

● What does it mean if a rectangular prism has the dimensions 3 × 2 × 4? Sample answer: The length is 3 units, the width is 2 units, and the height is 4 units.

● Does the notation suggest how you can find the volume? Yes Explain your answer. Sample answer: The notation suggests that you multiply the three dimensions together to find the volume.

Have students look at the prism in Problem 1, Math Journal 2, page 325A. Explain that this figure can be viewed as two rectangular prisms put together. Ask: How could you find the volume of this solid using the formula for finding the volume of a rectangular prism? Sample answer: Use the formula to find the volumes of the two smaller prisms, and then add the two volumes together.

Have students complete journal pages 325A and 325B. The solids on page 325A show stacks of cubes so students can count cubes to verify their results. The solids on page 325B provide the dimensions of the prisms but do not show the cubes.

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STUDY LINK

9�9 Volumes of Prisms

197

Name Date Time

The volume V of any prism can be found with the formula V = B ∗ h, where B is the area of the base of the prism, and h is the height of the prism from that base.

1. 2.

Volume = cm3 Volume = cm3

3. 4.

Volume = in3 Volume = cm3

5. 6.

Volume = in3 Volume = m314045

16270

14472

3.5 in.

8 in.

5 in.

5 cm

7.2 cm

4 cm6 cm

6 cm

4 cm

4 cm

6 cm

5 cm

3 cm

3 cm

7 mArea of base = 20 m2

3 in. Area of base = 15 in2

Practice

Solve each equation.

7. 36 ∗ r = 144 8. 3,577 - t = 3,822

9. 3,577 - m = 3,417 10. d ∗ 68 = 340 5160-2454

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Study Link Master

Math Boxes

Date Time

1. a. Plot the following points on the grid:

( – 4, – 1); ( – 3,1); (1,3); (2,1); ( – 2, – 1)

b. Connect the points with line segments in the order given above. Then connect ( – 4, – 1) and ( – 2, – 1).

What shape have you drawn?

pentagon

2. Find the diameter of the circle. Choose the best answer.

48 units

66 units

44 units

11 units

0 1 2 3 4 5

1

-1-1-2-3-4-5

-2-3-4-5

2345

22 units

3. What transformation does the figure show? Circle the best answer.

A translation

B reflection

C rotation

D dilation

preimage image

4. a. What is the perimeter of the rectangle?

30 units

9 units

6 un

its

b. What is the area?

54 square units

5. Two cups of flour are needed to make 24 oatmeal cookies. How many cups of flour are needed to make...

a. 4 dozen cookies? 4 cups

b. 6 dozen cookies? 6 cups

c. 120 cookies? 10 cups

LESSON

9 � 9

208

157 158153

186 108 109

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Student Page


2 Ongoing Learning & Practice

▶ Playing Polygon Capture PARTNER ACTIVITY

(Student Reference Book, p. 328; Math Masters, pp. 494–497)

Students practice identifying properties of polygons by playing Polygon Capture. Students first played this game in Unit 3 and stored the sets of Polygon Capture pieces and cards. If additional sets are needed, copy Math Masters, pages 494–496. Have students cut out the pieces and cards.

Ongoing Assessment:Recognizing Student Achievement

Math Masters

Page 497 �

Use the Record Sheet from Polygon Capture (Math Masters, page 497) to assess students’ ability to match polygons with their properties. On their first two draws of Polygon Capture, have students write the properties and list or draw all of the shapes that fit the properties. Students are making adequate progress if they have correctly matched properties and shapes. [Geometry Goal 2]

▶ Math Boxes 9�9

INDEPENDENT ACTIVITY

(Math Journal 2, p. 326; Student Reference Book, p. 158)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 9-6. The skill in Problem 5 previews Unit 10 content.

Writing/Reasoning Have students do the following: Draw and label a preimage of a simple figure and then draw the image rotated 180° clockwise about a point. Have students refer to Student Reference Book, page 158. Sample answer:

Preimage Image

▶ Study Link 9�9

INDEPENDENT ACTIVITY

(Math Masters, p. 284)

Home Connection Students calculate the volume of various types of prisms.


LESSON

9�9

Name Date Time

Unfolding Geometric Solids

If you could unfold a prism so that its faces are laid out as a set attached at their

edges, you would have a flat diagram for the shape. Imagine unfolding a triangular

prism. There are different ways that you could make diagrams, depending on how you

unfold the triangular prism.

Which of the following are diagrams that could be folded to make a triangular prism?

Write yes or no in the blank under each diagram.

1. 2.

3. 4.

Yes No

Yes

No


Teaching Master

LESSON

9�9

Name Date Time

Using Faces and Bases

The flat diagram formed from unfolding a prism so that its faces are laid out flat and

attached at their edges is called a geometric net. For a given prism, there are different

nets, depending on how you think about unfolding the prism.

1. Cut out the figures on Math Masters, page 287. You and your partner will use

the figures to build nets for the prisms below.

2. Take turns to select, draw, and place figures to form a net for a prism.

3. The partner who places the figure that completes the net states the number of

faces and the number of bases. For example, if the net for a cube were

completed, the partner would say, “4 faces, 2 bases.” This ends the round.

4. A partner can also block the completion of a net. In this case, the partner

would put down a figure that would prevent completing the net in the following

placement and say “block.” The blocked partner then has the opportunity to

complete the net by placing two figures and stating the number of faces and

bases. Again, this would end the round.

Example:

Student 1 Student 2 Student 1

Draw 1: Draw 2: Draw 3:

Student 2 Student 1 Student 1 states,

Draw 4: Draw 5: “3 faces, 2 bases.”

This ends the round.


Teaching Master

Lesson 9�9 759

3 Differentiation Options

READINESS

SMALL-GROUP ACTIVITY

▶ Analyzing Prism Nets for 15–30 Min

Triangular Prisms(Math Masters, pp. 285 and 286)

To explore the properties of triangular prisms, have students build a prism from a net and identify other nets that will form a triangular prism. This activity mirrors activities in Lesson 9-8, except that students are now working with triangular prisms instead of cubes. Read and discuss the introduction on Math Masters, page 286 as a group.

Discuss students’ solutions, and then have them identify the prism surfaces as either faces or bases.

ENRICHMENT PARTNER ACTIVITY

▶ Building Nets 30+ Min

(Math Masters, pp. 287 and 288)

To extend students’ ability to visualize, name, and describe geometric solids, have students build nets for prisms. Partners cut out the individual shapes on Math Masters, page 287 and follow the instructions on Math Masters, page 288 to form nets for rectangular, triangular, pentagonal, hexagonal, or octagonal prisms.

Ask students to discuss any discoveries or curiosities they encountered during the activity. Consider having partners cut out and fold their completed nets to check the configurations.

ELL SUPPORT


▶ Building A Math Word Bank 5–15 Min

(Differentiation Handbook, p. 142)

To provide language support for volume, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the term prism, draw pictures related to the term, and write other related words. See the Differentiation Handbook for more information.

EXTRA PRACTICE


▶ 5-Minute Math 5–15 Min

To offer students more experience calculating the volumes of rectangular prisms, see 5-Minute Math, pages 214 and 215.


Documents

Volume of Right Prisms - Everyday Math · Playing Polygon Capture ... Objective To provide experiences with using a formula for the volume of right prisms.f ... All of the cube structures