volume of pyramids and cones - everyday math · • compare the properties of pyramids, prisms,...

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Lesson Guide

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AssessmentManagement

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CurriculumFocal Points

872 Unit 11 Volume

Advance PreparationFor Part 1, copy Math Masters, page 334 on card stock. Cut out the templates; score the dashed lines; fold

them so that the markings are on the inside of the shapes; and tape the sides together completely to seal

the seams. If you cannot copy onto card stock, tape the master to card stock, cut along the solid lines for

each pattern, and then draw the corresponding dashed lines. Copy Math Masters, page 440. Cut out the

cone template. Curl it into position by lining up the two heavy black lines and the sets of dotted gray lines.

Seal the cone along the seams—inside and out. You will need a 15- or 16-oz food can with the top removed.

Place the cone in the can so its tip touches the base of the can. Locate the line on the inside of the cone

that touches the can’s rim. Cut along the line to remove the excess. You will need about 1 pound of dry fill:

rice, sugar, or sand.

Teacher’s Reference Manual, Grades 4–6 pp. 185, 186, 222–225

Key Concepts and Skills• Use formulas to find the volume

of geometric solids.

[Measurement and Reference Frames Goal 2]

• Compare the properties of pyramids,

prisms, cones, and cylinders.

[Geometry Goal 2]

• Describe patterns in relationships between

the volumes of prisms, pyramids, cones,

and cylinders.

[Patterns, Functions, and Algebra Goal 1]

Key ActivitiesStudents observe a demonstration in which

models of geometric solids are used to show

how to find the volumes of pyramids and

cones. Students then calculate the volumes

of a pyramid and a cone.

MaterialsMath Journal 2, p. 379

Study Link 11�3

Math Masters, pp. 334 and 440

for demonstration: 1 piece card stock, food

can, dry fill � slate � calculator

Playing Rugs and FencesMath Journal 2, p. 380

Math Masters, pp. 498–501

Class Data Pad � scissors

Students practice calculating the

perimeters and areas of polygons.

Math Boxes 11�4Math Journal 2, p. 381

Students practice and maintain skills

through Math Box problems.

Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 1. [Number and Numeration Goal 2]

Study Link 11�4Math Masters, p. 335

Students practice and maintain skills

through Study Link activities.

READINESS

Finding the Areas of Concentric CirclesMath Masters, p. 336

per partnership: crayons or colored pencils

(yellow, orange, and red) � Class Data Pad

(optional)

Students compare the areas of different

circular regions.

ENRICHMENTMeasuring RegionsMath Masters, p. 336

Students investigate the areas of concentric

circle regions in relation to their boundaries.

EXTRA PRACTICE

5-Minute Math5-Minute Math™, pp. 144, 147, and 229

Students identify the properties of

geometric solids.

ELL SUPPORT

Building a Math Word BankDifferentiation Handbook, p. 142

Students define and illustrate the term

volume of a cone.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

�

Volume of Pyramids and Cones

Objective To provide experiences with investigating the

relationships between the volumes of geometric solids.

Common Core State Standards

872_EMCS_T_TLG2_G5_U11_L04_576914.indd 872872_EMCS_T_TLG2_G5_U11_L04_576914.indd 872 3/9/11 10:59 AM3/9/11 10:59 AM

Lesson 11�4 873

Getting Started

Math MessageA rectangular prism and a cylinder each have exactly the same height and exactly the same volume. The base of the prism is an 8 cm × 5 cm rectangle. What is the area of the base of the cylinder?

Study Link 11�3 Follow-UpHave students share their volume measurements with the class.

Mental Math and Reflexes Have students complete each sentence by using the relationship between multiplication and division.

1

_ 2 ÷ 8 = 1 _ 16 because 1

_ 16 ∗ = 1 _ 2 .

3 ÷ 1 _ 2 = 6 because 6 ∗ = 3.

9 ÷ 1 _

3 = 27 because ∗ 1

_ 3 = 9.

1 _ 4 ÷ 12 = 1

_ 48 because 1 _ 48 ∗ 12 = .

5 ÷ 1 _ 5 = 25 because ∗ 1

_ 5 = 5.

7 ÷ 1 _

6 = 42 because 42 ∗ = 7.

÷ 5 = 1

_ 20 because 1 _ 20 ∗ 5 = .

8 ÷ = 16 because 16 ∗ = 8.

1 _ 10 ÷ = 1

_ 100 because 1

_ 100 ∗ =

1 _ 10

.

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

Algebraic Thinking Ask volunteers to share their solution strategies. Since the prism and cylinder each have the same volume and height, the areas of their bases must also be equal. The area of the base of the prism is 40 cm2 (8 cm ∗ 5 cm), so the area of the base of the cylinder must also be 40 cm2.

▶ Exploring the Relationship

WHOLE-CLASS ACTIVITY

between the Volumes of Prisms and Pyramids(Math Masters, p. 334)

Gather the class around a desk or table. Show them the prism and pyramid you have made. Turn the pyramid so that the apex is pointing down and show that, when the pyramid is placed inside the prism, the boundaries of their bases match and the apex of the pyramid will touch a base of the prism. The two solids you made will fit in this way because they have identical bases and heights.

Have students guess how many pyramids filled with material it would take to fill the prism. Select a pair of volunteers to follow the procedure on the next page:

PROBLEMBBBBBBBBBBOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEELEBLLBLEBLELLLBLEBLEBLEBLEBLEBLEEBLEEEMMMMMMMMMMMMMMOOOOOOOOOOOOBBBBLBBLBLBLBLLLLLLPROPROPROPROPROPROPROPROPROPROPROPROPPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROROROOPPPPPPP MMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEEELELELELEEEEEEEELLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBB ELELEELEMMMMMMMMMOOOOOOOOOBLBLBLBLBLBLBLBLBLROOOROROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLLVINVINVINVINNNVINVINVINVINNVINVINVINVINVV GGGGGGGGGGGGOLOOOOLOLOOLOO VINVINVVLLLLLLLLLLVINVINVINVINNVINVINVINVINVINVINVVINNGGGGGGGGGGGOLOLOLOLOLOLOLOLOOO VVVVVLLLLLLLLLLLVVVVVVVVVOOSOSOSOSOSOSOSOSOSOOSOSOSOOOOOOSOSOSOSOSOSOSOSOSOSOSOOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLVVVVVVVVVLLLLVVVVVVVVLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

8

1 _ 2

27

25

10 10

1 _ 4

1 _ 6

1 _ 4 1

_ 4

1 _ 2 1

_ 2

Prism and pyramid patterns from

Math Masters, page 334

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Volume of Pyramids and ConesLESSON

11�4

Date Time

1. To calculate the volume of any prism or cylinder, you multiply the area of the base

by the height. How would you calculate the volume of a pyramid or a cone?

The Pyramid of Cheops is near Cairo, Egypt. It was built about 2600 B.C. It is a square

pyramid. Each side of the square base is 756 feet long. Its height is 449.5 feet.

The pyramid contains about 2,300,000 limestone blocks.

2. Calculate the volume of this pyramid. ft3

3. What is the average volume of one limestone block?

ft3

A movie theater sells popcorn in a box for $2.75. It also sells cones of popcorn

for $2.00 each. The dimensions of the box and the cone are shown below.

4. Calculate the volume of the box. in3

5. Calculate the volume of the cone. in3

6. Which is the better buy—the box or the cone of popcorn? Explain.

I would multiply the area of the base by the height

and divide the product by 3.

756 ft

449.5 ft

7 in.

9 in

.

3 in.

$2.75

10

in.

6 in.

$2.00

85,635,144

37

189

94

The box is the better buy. 275 / 189 � 1.46 cents per cubic

inch for the box, and 200 / 94 � 2.13 cents per cubic inch

for the cone.

Try This

Math Journal 2, p. 379

Student Page

874 Unit 11 Volume

1. Fill the pyramid with dry fill so that the material is level with the top. Empty the material into the prism.

2. Fill the pyramid again and empty the material into the prism.

3. Repeat until the prism is full and level at the top. It will take about 3 pyramids of material to fill the prism.

Students need not find the actual volumes of either the prism or the pyramid. It is enough for them to discover that about 3 pyramids of material fill the matching prism. Ask students to state the relationships between the volumes of the two shapes. The volume of the prism is 3 times the volume of the pyramid. The volume of the pyramid is 1 _ 3 the volume of the prism.

▶ Exploring the Relationship

WHOLE-CLASS ACTIVITY

between the Volumes of Cylinders and Cones(Math Journal 2, p. 379; Math Masters, p. 440)

Algebraic Thinking Repeat the demonstration using a 15- or 16-oz food can and the cone you have made. Turn the cone upside down and show that, when the cone is placed inside the cylinder, the boundaries of their bases match and the apex of the cone will touch a base of the cylinder. The two solids fit because they have identical bases and heights.

Have students guess how many cones of material will fill the can. Expect that many students will correctly guess 3 because of the previous demonstration.

Select another pair of volunteers to follow the same procedure as before. It takes about 3 cones of material to fill the cylinder.

Ask students to complete Problem 1 on journal page 379: How would you calculate the volume of a pyramid or a cone? Then have students share their solution strategies. Emphasize the following points:

� Since the volume of a pyramid (cone) is 1 _ 3 the volume of a prism (cylinder) with an identical base and height, you can calculate the volume of a pyramid (cone) by multiplying the area of the base of the pyramid (cone) by its height and then dividing the result by 3.

� A formula for finding the volume of a pyramid or cone is V = 1 _ 3 ∗ B ∗ h.

▶ Solving Volume Problems

INDEPENDENT ACTIVITY

(Math Journal 2, p. 379)

Algebraic Thinking Assign the rest of journal page 379. Circulate and assist. When students have completed the page, ask them to share their solution strategies. Emphasize the following points:

PROBLEMBBBBBBBBBBOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEEMMMLEBLLBLEBLELLLBLEBLEBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBBBBLBBLBLBLBLLLLLLPROPROPROPROPROPROPROPROPROPROPROPRPPROPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROOOROROOOPPPPPPP MMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEELELEEEEEEEEEELLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBB ELELEELEMMMMMMMMMOOOOOOOOOBLBLBLBLBLBLBLBLBLBLROOOOROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGLLLLLLLLLLLLLVINVINVINVINVINVINNNVINVINVINVINNVINVINVINV GGGGGGGGGGGGOLOOOOLOOLOOLOO VINVINVVLLLLLLLLVINVINVINVINVINVINVINVINVINVINVIVINVINVINGGGGGGGGGGGOOLOLOLOLOLOLOLLOO VVVVLLLLLLLLLLVVVVVVVVVOOSOSOOSOSOSOSOSOSOSOOSOSOSOOOSOOOSOSOSOSOSOSOSOOSOSOSOOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLVVVVVVVVVLLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

The

shad

ed p

ortio

n ov

erla

ps o

n th

e ou

tsid

e.

Cone pattern from Math

Masters, page 440

873-877_EMCS_T_TLG2_G5_U11_L04_576914.indd 874873-877_EMCS_T_TLG2_G5_U11_L04_576914.indd 874 2/27/11 9:53 AM2/27/11 9:53 AM

Rugs and Fences: An Area and Perimeter GameLESSON

11�4

Date Time

Materials � 1 Rugs and Fences Area and Perimeter Deck (MathMasters, p. 498)

� 1 Rugs and Fences Polygon Deck (Math Masters,

pp. 499 and 500)

� 1 Rugs and Fences Record Sheet (Math Masters, p. 501)

Players 2

Object of the game To score the highest number of points by finding the area

and perimeter of polygons.

Directions

1. Shuffle the Area and Perimeter Deck, and place it facedown.

2. Shuffle the Polygon Deck, and place it facedown next to the Area and Perimeter Deck.

3. Players take turns. At each turn, a player draws one card from each deck and places

it faceup. The player finds the perimeter or area of the figure on the Polygon card

as directed by the Area and Perimeter card.

� If a Player’s Choice card is drawn, the player may choose to find either the

area or the perimeter of the figure.

� If an Opponent’s Choice card is drawn, the other player chooses whether

the area or the perimeter of the figure will be found.

4. Players record their turns on the record sheet by writing the Polygon card number, by circling A

(area) or P (perimeter), and then by writing the number model used to calculate the area or

perimeter. The solution is the player’s score for the round.

5. The player with the highest total score at the end of 8 rounds is the winner.


Student Page

Lesson 11�4 875

● Problem 2

A rectangular prism with the same base and height as the Pyramid of Cheops would have a volume of

B ∗ h = 7562 ∗ 449.5 = 256,905,432 ft3.

The pyramid’s volume is 1 _ 3 as much, or 85,635,144 ft3.

● Problem 3

To find the average volume of a block, divide the total volume by the number of blocks: 85,635,144

_ 2,300,00 = slightly more than 37 cubic feet per block.

● Problem 4

The popcorn box is a rectangular prism whose volume equals 189 in3.

● Problem 5

A cylinder with the same base and height as the popcorn cone would have a volume of B ∗ h = (π ∗ 32) ∗ 10 = 283 in3. The cone’s volume is 1 _ 3 as much, or 94 in3.

● Problem 6

The box is the better buy. Ask students to calculate a cost-volume ratio for each container: 200 _ 94 = 2.13 cents per cubic inch for the cone, and 275 _ 189 = 1.46 cents per cubic inch for the box.

2 Ongoing Learning & Practice

▶ Playing Rugs and Fences PARTNER ACTIVITY

(Math Journal 2, p. 380; Math Masters, pp. 498–501)

Algebraic Thinking Students practice calculating the perimeter and area of polygons by playing Rugs and Fences. Write “P = perimeter,” “A = area,” “b = length of base,” and “h = height” on the Class Data Pad. Ask students to define perimeter. The distance around a closed, 2-dimensional shape Then have volunteers write the formulas for the area of a rectangle, A = b ∗ h, or A = bh a parallelogram, A = b ∗ h, or A = bh and a triangle A = 1 _ 2 ∗ (b ∗ h), or A = 1 _ 2 bh on the board or Class Data Pad. Read the game directions on journal page 380 as a class. Partners cut out the cards on the Math Masters pages and play eight rounds, recording their score on Math Masters, page 501. See the margin for the area and perimeter of each figure.

Polygon Deck B Polygon Deck C

Card A P Card A P

1 35 24 17 48 28

2 36 26 18 22 20

3 14 18 19 48 36

4 60 32 20 17 20

5 64 32 21 28 28

6 8 18 22 40 36

7 36 24 23 28 32

8 54 30 24 24 24

9 48 32 25 23 26

10 6 12 26 28 32

11 54 36 27 86 54

12 192 64 28 48 32

13 32 26 29 22 30

14 64 36 30 48 52

15 20 25 31 60 32

16 216 66 32 160 70

The area (A) and perimeter (P) of the polygons in

Rugs and Fences


Math Boxes LESSON

11�4

Date Time

1. Solve.

a. �1

3� of 36 �

b. �2

5� of 75 �

c. �3

8� of 88 �

d. �5

6� of 30 �

e. �2

7� of 28 � 8

25

33

30

12

3. Lilly earns $18.75 each day at her job.

How much does she earn in 5 days?

Open sentence:

Solution: d � $93.75

18.75 � 5 � d

4. Solve.

a. 2 c � fl oz

b. 1 pt � fl oz

c. 1 qt � fl oz

d. 1 half-gal � fl oz

e. 1 gal � fl oz128

64

32

16

16

6. Jamar buys juice for the family.

He buys eight 6-packs of juice boxes. His

grandmother buys three more 6-packs.

Which expression correctly represents how

many juice boxes they bought?

Circle the best answer.

A. (8 � 3) � 6

B. 6 � (8 � 3)

C. 6 � (8 � 3)

2. Find the volume of the solid.

5. Make a factor tree to find the prime

factorization of 32.

Volume � B � h where B is the area

of the base and h is the height.

area of base

30 units2

3 units

Volume � 90 units3

16

8

4

2

32

º2

ºº 22

ºº 22º2

ºº 22º2º2

�

73 197

38–40243 397

12 219


Student Page

STUDY LINK

11� 4 Comparing Volumes

Name Date Time

Use �, �, or � to compare the volumes of the two figures in each problem below.

1.

2.

3.

4. Explain how you got your answer for Problem 3.

Because both pyramids have the same height,compare the areas of the bases. The base area of the square pyramid is 5 º 5 � 25 m2. The base area of thetriangular pyramid is �1

2� º 5 º 5, or 12.5 m2.

6 cm

9 c

m

9 c

m

6 cm6

cm

24

ft

3 yd

height of base � 2 yd

5 m5

m

height � 6 m

base is

a square

8 y

d

6 ft

5 m

height of

base � 5 m

he

igh

t �

6 m

�

�

�

5. 4�1

3� � 2�

4

9� � 6. 2�

6

7� � 1�

1

3� �

7. 6 º 105 � 8. 584 � 23 � 25.39600,000

1�1211�6�

79

�

Practice

196–199

Math Masters, p. 335

Study Link Master

876 Unit 11 Volume

▶ Math Boxes 11�4


(Math Journal 2, p. 381)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 11-2 and 11-6. The skill in Problem 5 previews Unit 12 content.

Ongoing Assessment: Math Boxes

Problem 1 �Recognizing Student Achievement

Use Math Boxes, Problem 1 to assess students’ ability to calculate a fraction of

a whole. Students are making adequate progress if they correctly identify each of

the five values.

[Number and Numeration Goal 2]

▶ Study Link 11�4


(Math Masters, p. 335)

Home Connection Students compare the volumes of geometric solids.

3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Finding the Areas of 5–15 Min

Concentric Circles(Math Masters, p. 336)

Algebraic Thinking To explore the relationship between radius and the area of circles, have students compare the areas of different circular regions. Read the introduction to Math Masters, page 336. Write the formula for finding the area of a circle: A = π ∗ r ∗ r or A = π ∗ r2 on the board or the Class Data Pad, and then make the table shown below. Ask students to use the formula to find the area for each circle.

Radius Area

1 in. 3.14 in2

2 in. 12.56 in2

3 in. 28.26 in2

4 in. 50.24 in2

5 in. 78.50 in2

Have partners solve Problem 1. Suggest that they discuss their strategy before they begin. Circulate and assist.


LESSON

11� 4

Name Date Time

Finding the Area of Concentric Circles

Concentric circles are circles that have the same center,

but the radius of each circle has a different length.

The smallest of the 5 concentric circles below has a

radius of 1 in. The next largest circle has a radius of 2 in.

The next has a radius of 3 in. The next has a radius of 4 in.,

and the largest circle has a radius of 5 in. The distance from

the edge of one circle to the next larger circle is 1 in.

1. Use colored pencils or crayons to

shade the region of the smallest 3

circles red. Shade the region that you

can see of the next circle yellow, and the region that you can see of the

largest circle orange.

Which region has the greater area, the red region or the orange region?

2. a. How can you change the distance between the circles to make the area of the

yellow region equal to the area of the red region? Explain your answer on the back

of this page.

b. How can you change the distance between the circles to make the area of

the yellow region equal to the area of the orange region? Explain your

answer on the back of this page.

They are equal.

1 in. 1 in. 1 in. 1 in.1 in.

orange

yellow

red

Math Masters, p. 336

Teaching Master

Lesson 11�4 877

When students have finished, have volunteers explain their solution strategies. Sample answers: In Problem 1, we know that the area for the red region is the same as the area of the third circle. The area for the orange region is the same as the area for the fifth circle minus the area for the fourth circle. The area for the red region is the same as the area for the orange region.

ENRICHMENT PARTNER ACTIVITY

▶ Measuring Regions 15–30 Min

(Math Masters, p. 336)

To apply students’ understanding of area, have them modify the distance between concentric circles to enlarge or shrink regions. Have partners complete Problem 2 on Math Masters, page 336.

When students have finished, have them share and discuss their solution strategies.

For Problem 2a, I would make a table to record region areas and the distance between the circles. Then I would use guess-and-check to increase the radius of the circle for the yellow region until it is about twice the area of the red region.

For Problem 2b, I would decrease the red region to make the areas of the yellow and orange regions equal.

EXTRA PRACTICE

SMALL-GROUP ACTIVITY

▶ 5-Minute Math 5–15 Min

To offer students more experience with identifying the properties of geometric solids, see 5-Minute Math, pages 144, 147, and 229.

ELL SUPPORT PARTNER ACTIVITY

▶ Building a Math Word Bank 15–30 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 phrase volume of a cone and write words, numbers, symbols, or draw pictures that are related to the term. See the Differentiation Handbook for more information.

Planning Ahead

Be sure to collect the materials listed at the end of Lesson 11-3 before the start of the next lesson.


volume of pyramids and cones - everyday math · • compare the properties of pyramids, prisms,...

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