volume of pyramids and cones - everyday math · • compare the properties of pyramids, prisms,...
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eToolkitePresentations Interactive Teacher’s
Lesson Guide
Algorithms Practice
EM FactsWorkshop Game™
AssessmentManagement
Family Letters
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
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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
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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.
Math Journal 2, p. 380
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
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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
Math Journal 2, p. 381
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
INDEPENDENT ACTIVITY
(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
INDEPENDENT ACTIVITY
(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.
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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.
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