volume of cylinders - login · pdf fileobjective to introduce the formula for the volume of...

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eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

866 Unit 11 Volume

Advance PreparationFor Part 1, collect a set of open cans or other cylindrical and watertight containers of different sizes with the

labels removed, if possible. Each partnership will need at least one container. Create a workstation in the

classroom for measuring the capacities of containers. Fill a 1-gallon container with water. Provide several

measuring cups, each marked to at least 250 mL, and several shallow pans to recover any spilled water.

Teacher’s Reference Manual, Grades 4–6 pp. 185 –192, 222–225

Key Concepts and Skills• Use tables to collect data.

[Data and Chance Goal 1]

• Apply formulas to calculate the area

of a circle and the volume of prisms

and cylinders.

[Measurement and Reference Frames Goal 2]

• Compare the volume and the capacity

of cylinders.


Key ActivitiesStudents review the formula for finding the

area of a circle. They use the formula for the

volume of a cylinder to calculate the volume

of open cans and verify the formula by

measuring the liquid capacities of the cans.

Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 414). [Measurement and Reference Frames

Goal 2]

MaterialsMath Journal 2, pp. 375 and 376

Math Masters, p. 414

Study Link 11�2

Class Data Pad � calculator � open cans or

watertight cylindrical containers � ruler �

1-gallon container of water � measuring

cups (marked in milliliters) � base-10 cube

Finding the Volumes of Rectangular PrismsMath Journal 2, p. 377

Student Reference Book,

pp. 196 and 197

Students practice finding the volumes

of rectangular prisms.

Math Boxes 11�3Math Journal 2, p. 378

Geometry Template

Students practice and maintain skills

through Math Box problems.

Study Link 11�3Math Masters, p. 333

Students practice and maintain skills

through Study Link activities.

READINESS

Comparing Volumes of Cylindersper partnership: 4 sheets of 8

1

_ 2 " by 11"

construction paper, ruler, masking tape

Students compare and relate the dimensions

of cylinders to the volume of cylinders.

ENRICHMENTCalculating Volume for Cylindersper partnership: cylindrical objects,

ruler, calculator

Students measure the dimensions and

find the volume of cylindrical objects in

the classroom.

EXTRA PRACTICE

5-Minute Math5-Minute Math™, pp. 58 and 229

Students identify geometric solids.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

� Volume of CylindersObjective To introduce the formula for the volume of cylinders.

Common Core State Standards

866_EMCS_T_TLG2_G5_U11_L03_576914.indd 866866_EMCS_T_TLG2_G5_U11_L03_576914.indd 866 4/8/11 10:02 AM4/8/11 10:02 AM

Volume of CylindersLESSON

11�3

Date Time

The base of a cylinder is circular. To find the area of the base of a cylinder,

use the formula for finding the area of a circle.

The formula for finding the volume of a cylinder is the same as the formula for

finding the volume of a prism.

Use the 2 cans you have been given.

1. Measure the height of each can on the inside. Measure the

diameter of the base of each can. Record your measurements

(to the nearest tenth of a centimeter) in the table below.

2. Calculate the radius of the base of each can.

Then use the formula to find the volume.

Record the results in the table.

3. Record the capacity of each can in the table, in milliliters.

4. Measure the liquid capacity of each can by filling the can with water. Then pour the

water into a measuring cup. Keep track of the total amount of water you pour into

the measuring cup.

Capacity of Can #1: mL Capacity of Can #2: mL

Formula for the Volume of a Cylinder

V � B º hwhere V is the volume of the cylinder, B is the area of the

base, and h is the height of the cylinder.

basediam

eter

he

igh

t

Answers vary.

Answers vary.

Height Diameter of Radius of Volume Capacity (cm) Base (cm) Base (cm) (cm3) (mL)

Can #1

Can #2

Formula for the Area of a Circle

A � � � r 2

where A is the area and r is the radius of the circle.

Math Journal 2, p. 375

Student Page

Lesson 11�3 867

Getting Started

Round to the nearest thousand.

28,152 28,000

65,680 66,000

6,580 7,000

Round to the nearest ten.

4,152 4,150

697 700

285 290

Round to the nearest tenth.

2.547 2.5

6.785 6.8

3.062 3.1

Math MessageMarble games are often played inside a circle whose diameter is 7 ft. What is the area of the playing surface? Write your solution as a number sentence.

Study Link 11�2 Follow-UpHave partners compare answers and resolve differences.

Mental Math and Reflexes Play Beat the Calculator to practice rounding numbers. Have students write dictated numbers and then round the numbers to a specified place: pencil-and-paper methods versus calculators (remind students to use the fix function). Suggestions:

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

Algebraic Thinking Ask students what information is needed to solve the Math Message problem. To support English language learners, write the important ideas from the discussion on the board. The properties of a circle; the formula for finding the area of a circle (A = π ∗ r2) Ask: What is the relationship between the diameter and the radius of a circle? The radius is half of the diameter. If r represents radius and d represents diameter, what open number sentence shows their relationship? r = 1 _ 2 ∗ d, r = d _ 2 , or d = r ∗ 2 If you did not use the pi key on a calculator, what decimal number did you use in the calculations with pi? 3.14

Have volunteers write their number sentences on the board and explain their solutions. The diameter of the circle is 7 ft, so the radius is 3.5 ft; therefore, the area of the circle is π ∗ 3.5 ∗ 3.5 = 38.48 ft2 or about 38 1 _ 2 ft2.

▶ Introducing and Verifying

SMALL-GROUP ACTIVITY

the Cylinder Volume Formula(Math Journal 2, p. 375)

Algebraic Thinking Distribute the open cans, and assign groups of four students to complete Problems 1–3 on journal page 375, using at least two different cans. Remind students that 1 cubic centimeter is equal to 1 milliliter. Use a base-10 cube to show students 1 cubic centimeter. Write the following equivalencies on the Class Data Pad:

1 centimeter (cm) = 10 millimeters (mm)

1 millimeter = 0.1 ( 1 _ 10 ) centimeter

1 milliliter (mL) = 0.001 ( 1 _ 1,000 ) liter (L)

1 cubic centimeter (cm3) = 1 milliliter

ELL

NOTE See sample measurements for

containers on page 868.

867-871_EMCS_T_TLG2_G5_U11_L03_576914.indd 867867-871_EMCS_T_TLG2_G5_U11_L03_576914.indd 867 2/24/11 7:24 AM2/24/11 7:24 AM

Volume of Cylinders and PrismsLESSON

11�3

Date Time

1. Find the volume of each cylinder.

a. b.

Volume = in

3 Volume = cm3

2. Find the volume of each wastebasket. Then determine which wastebasket has the

largest capacity and which one has the smallest.

a. b.

Volume = in3 Volume = in3

c. d.

Volume = in3 Volume = in3

e. Which wastebasket has the largest capacity? Wastebasket

Which wastebasket has the smallest capacity? Wastebasket

Reminder: The same formula (V = B ∗ h) may be used to find the volume

of a prism and the volume of a cylinder.

height = 8 in.

Area of

base = 10 in2

height = 13 in.

radius = 6 in.

height = 16 in.

14 in.

12

in

.

base

9 in.9

in.

height = 14 in.

height = 4 cm

radius = 2 cm

80

1,256.6

1,470.3

1,134

1,344

c

About 50.3

height = 16 in.

radius = 5 in.

b

369-392_EMCS_S_MJ2_U11_576434.indd 376 3/4/11 7:04 PM


Student Page

1. Consider the rectangular prism shown at the right.

a. How many cubes, each 1 cm long on a side,

are needed to fill one layer of the prism? 24 cubes

b. How many cubes, each 1 cm long on a side,

are needed to fill the entire prism? 120 cubes

c. Write a number model to show how you could find the volume of the prism.

24 ∗ 5 = 120 or 8 ∗ 3 ∗ 5 = 120

2. A rectangular prism is 39 in. long, 25 in. wide, and 4 in. tall. Josh did the following

to find the volume: (39 ∗ 25) ∗ 4. Steffi turned the prism so that the base was

25 in. by 4 in. She computed 39 ∗ (25 ∗ 4) to find the volume.

a. What is the volume of the prism? 3,900 in3

(unit)

b. Do both methods give the same volume? Explain. Yes; according to the

Associative Property, (39 ∗ 25) ∗ 4 = 39 ∗ (25 ∗ 4).

Find the volume of each rectangular prism.

3. 15 m

5 m

12 m

4.

V = 1,330 ft3 (unit)

V = 900 m3

(unit)

5. a. A cube has sides 9 mm long. Circle the open number models that can be used

to find its volume.

V = 9 ∗ 9 ∗ 9 V = 81 ∗ 9 V = 18 ∗ 9 V = 93

b. What is the volume of the cube? V = 729 mm3

(unit)

6. A rectangular prism has a length of 16 units and a volume of 144 cubic units.

Give a possible set of dimensions for the prism.

Sample answers: 16 × 3 × 3 or 16 × 9 × 1

Volume of Rectangular Prism ProblemsLESSON

11�3

Date Time

8 cm

5 c

m

3 cm

7 ft

19 ft

10 ft

369-392_EMCS_S_MJ2_G5_U11_576434.indd 377 4/7/11 3:58 PM


Student Page

Adjusting the Activity

868 Unit 11 Volume

Ask: Why do the directions say to measure the height inside the can? Some cans have recessed bottoms, so if you measure the outside of the can, it isn’t as accurate as measuring the inside. Circulate and assist.

Have students find the diameter of a can by tracing around the base

of the can, cutting out the circle, folding it in half, and measuring the length

of the fold.

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

As students finish, have them verify the capacities recorded in the table by measuring the liquid capacity of each can at the workstation. They record the results in Problem 4.

When all groups have completed the workstation activity, ask students which results they think are more accurate—the volumes obtained by the formula in Problem 3 or the volumes obtained by direct measurement in Problem 4—and why. Point out that inaccuracies might result for any of the following reasons:

� The diameter and/or the height of the can might not have been measured accurately.

� The can might not be a true cylinder. For example, the bottom of a can often has depressions, so it’s not a single flat, circular surface.

� Measuring cups are usually marked at 25-mL increments, so readings of liquid capacity are seldom exact.

The following table shows approximate dimensions for a wide range of can sizes. The formula V = B ∗ h has been used to calculate the volume and capacity of each.

Can Size* Height (cm)

Diameter(cm)

Radius(cm)

Volume(cm3)

Capacity(mL)

6.5 oz

(tuna)3.8 8.6 4.3 221 221

10.5 oz

(soup)9.9 6.6 3.3 339 339

14 oz 8.3 7.5 3.75 367 367

16 oz 10.9 7.3 3.65 456 456

20 oz 11.4 8.4 4.2 632 632

27 oz 11.4 10.2 5.1 932 932

26 oz

(salt) 13.3 8.3 4.15 720 720

Small

coffee 13.3 10.2 5.1 1,087 1,087

1.5 lb

coffee15.9 12.7 6.35 2,014 2,014

Large

coffee 17.8 15.2 7.6 3,230 3,230

*Can Size is listed for identification and not as a measure of capacity or weight.

See note on page 885 about ounces and fluid ounces.

ELL


Math Boxes LESSON

11�3

Date Time

1. Add or subtract.

a. -22 + 12 = -10

b. 18 - (-4) = 22

c. -15 - (-8) = -7

d. -4 + (-17) = -21

e. -6 - (-28) = 22

92–94 155

73 12

3. Mr. Ogindo’s students took a survey of their

favorite movie snacks. Complete the table.

Then make a circle graph of the data.

25

44%

20%

24%

4%

8%

100%

Favorite Number of Percent Snack Students of Class

Popcorn 11

Chocolate 5

Soft drink 6

Fruit chews 1

Candy with nuts 2

Total

(title)

Favorite Movie Snacks

44%popcorn

4% fruit chews

24%softdrink 20%

chocolate

8% candy

with nuts

4. Solve.

a. 3

_

8 of 40 = 15

b. 2

_

3 of 120 = 80

c. 4

_ 5 of 60 = 48

d. 7

_

9 of 54 = 42

e. 5

_

6 of 36 = 30

5. Write the prime factorization for 175.

52 ∗ 7, or 5 ∗ 5 ∗ 7

2. Which parallelogram is not congruent to

the other 3 parallelograms? Circle the

best answer.

A. B.

C. D.

47 8990 126

25

369-392_EMCS_S_MJ2_G5_U11_576434.indd 378 3/30/11 2:49 PM


Student Page

STUDY LINK

11�3 Volume of Cylinders

Name Date Time

Use these two formulas to solve the problems below.

Formula for the Volume of a Cylinder

V = B ∗ h

where V is the volume of the cylinder,

B is the area of the cylinder’s base, and

h is the height of the cylinder.

Formula for the Area of a Circle

A = π ∗ r 2

where A is the area of the circle and r is

the length of the radius of the circle.

1. Find the smallest cylinder in your home. Record its dimensions, and calculate

its volume.

radius = height =

Area of base = Volume =

2. Find the largest cylinder in your home. Record its dimensions, and calculate its volume.

radius = height =

Area of base = Volume =

3. Write a number model to estimate the volume of:

a. Your toaster

b. Your television

4. Is the volume of the largest cylinder more

or less than the volume of your toaster?

About how much more or less?

5. Is the volume of the largest cylinder more or

less than the volume of your television set?

About how much more or less?

80 ∗ 70 ∗ 45 = 252,000

30 ∗ 30 ∗ 18 = 16,200

more

more

202,683 cm2

1,400 cm 254 cm

105.9 cm3

4.3 cm 2.8 cm

1.315

Sample answers:

Sample answers:

283,756,200 cm3

283,740,000 cm3

283,500,000 cm3

194197 198

6. 6 1 _

3 ∗

2 _

5 =

7. 10

6 _

8 ∗

1 _

2 =

8. 4 - 2.685 =

Practice

5 3

_ 8

24.6 cm2

2 8

_ 15

323-347_EMCS_B_MM_G5_U11_576973.indd 333 3/9/11 12:48 PM

Math Masters, p. 333

Study Link Master

Lesson 11�3 869

▶ Finding the Volumes of Prisms PARTNER ACTIVITY

and Cylinders(Math Journal 2, p. 376)

Algebraic Thinking Remind students that in Lessons 9-8 and 9-9 they found the volume of a prism by multiplying the area of its base by its height. Discuss whether students think it is reasonable that the same formula is used to find the volumes of cylinders (the only difference being the way the area of the base is calculated).

Assign the problems on journal page 376. Circulate and assist.

Ongoing Assessment: Exit Slip �Recognizing Student Achievement

Use an Exit Slip (Math Masters, page 414) to asess students’ ability to

explain what is similar and what is different between finding the volume

of cylinders and finding the volume of prisms. Students are making adequate

progress if their explanations refer to using the same formula to find volumes but

different formulas to calculate the area of the bases.


2 Ongoing Learning & Practice

▶ Finding the Volumes of

INDEPENDENT ACTIVITY

Rectangular Prisms(Math Journal 2, p. 377; Student Reference Book, pp. 196 and 197)

Students practice finding the volumes of rectangular prisms by packing them with unit cubes, using formulas, and solving real-world problems.

▶ Math Boxes 11�3


(Math Journal 2, p. 378)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 11-1. The skill in Problem 5 previews Unit 12 content.

▶ Study Link 11�3


(Math Masters, p. 333)

Home Connection Students find the volume of two cylindrical objects in their homes. They compare the volume of the larger object to the volume of their toaster and their TV set.

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870 Unit 11 Volume

3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Comparing Volumes 15–30 Min

of CylindersTo explore comparing the volume of cylinders, have students construct and compare paper cylinders. Give partners 4 sheets of construction paper. Explain that, to make a cylinder from the paper, they will first draw a line 1 inch from an edge. Roll the rectangle into a cylinder, and tape the paper along the line. Then they will tape the cylinder to a second piece of paper. Reinforce the concept that the volume of a container is a measure of how much the container will hold.

Ask partners to construct two cylinders from the paper. One cylinder should have the largest volume they can make. The other cylinder should have the smallest volume they can make.

When students have finished, have them display their cylinders and explain their solution strategies. Emphasize the relationship between the height and the area of the bases in determining volume. The taller cylinder does not necessarily have a larger volume than a shorter cylinder.

ENRICHMENT PARTNER ACTIVITY

▶ Calculating Volume 5–15 Min

for CylindersTo apply students’ understanding of the formula used to calculate the volume of cylinders, have them work with a partner to find the volume of cylindrical objects in the classroom. They should measure the height and diameter, record the measurements in a table, and calculate the volume.

When students have finished, have partners estimate the volume of the objects from other partnerships and check their estimates against the calculated volumes.


Lesson 11�3 871

EXTRA PRACTICE

SMALL-GROUP ACTIVITY

▶ 5-Minute Math 5–15 Min

To offer students more experience with identifying geometric solids, see 5-Minute Math, pages 58 and 229.

Planning Ahead

For Lesson 11-5, you will need the following materials:

Containers

� About 7 two-liter soft-drink bottles, made out of clear or light-colored plastic

� About 7 large-mouthed containers that hold up to 2 liters of water each, for example, large coffee cans; gallon milk containers that are cut to provide a large opening, while retaining the handle; other large soft-drink bottles with the tops cut off about 9 in. from the bottom

Solid Objects

� About 30 to 40 rocks (enough to fill a gallon container), each about half the size of a fist, for example—landscape rocks

� A few unopened cans of nondiet soft drink

� Other objects for displacement activities, such as baseballs, golf balls, apples, or oranges

� A supply of paper towels


volume of cylinders - login · pdf fileobjective to introduce the formula for the volume of...

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