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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.
[Measurement and Reference Frames Goal 2]
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
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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.
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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
Math Journal 2, p. 376
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
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Math Journal 2, p. 377
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
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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
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Math Journal 2, p. 378
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
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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.
[Measurement and Reference Frames Goal 2]
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
INDEPENDENT ACTIVITY
(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
INDEPENDENT ACTIVITY
(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.
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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
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