Surface Area
Objective To introduce finding the surface area of prisms,
cylinders, and pyramids.
c
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Ongoing Learning & Practice
Key Concepts and Skills
Plotting and Analyzing Insect Data
• Measure the dimensions of a cylinder
in inches and centimeters. Math Journal 2, pp. 390A and 390B
Students plot insect lengths in
fractional units on a line plot. They use
the line plot to analyze the data.
[Measurement and Reference Frames Goal 1]
• Use rectangle and triangle area formulas
to find the surface area of prisms and
cylinders. Apply a formula to calculate
the area of a circle. [Measurement and Reference Frames Goal 2]
• Identify and use the properties of prisms,
pyramids, and cylinders in calculations. [Geometry Goal 2]
Key Activities
Math Boxes 11 7
Math Journal 2, p. 391
Students practice and maintain skills
through Math Box problems.
Study Link 11 7
Math Masters, p. 341
Students practice and maintain skills
through Study Link activities.
Curriculum
Focal Points
Differentiation Options
ENRICHMENT
Finding the Smallest Surface Area
for a Given Volume
Math Masters, p. 342
per partnership: 24 cm cubes
Students find the rectangular prism with the
smallest surface area for a given volume.
EXTRA PRACTICE
Finding Area, Surface Area, and Volume
Math Masters, p. 343
Student Reference Book, pp. 196 and 197
Students practice calculating area, surface
area, and volume.
Students find the surface areas of prisms,
cylinders, and pyramids by calculating the
area of each surface of a solid and then
finding the sum.
Ongoing Assessment:
Informing Instruction See page 892.
Ongoing Assessment:
Recognizing Student Achievement
Use journal pages 389 and 390. [Measurement and Reference Frames
Goals 1 and 2]
Key Vocabulary
surface area
Materials
Math Journal 2, pp. 389 and 390
Study Link 116
slate calculator cardboard box per workstation: 2 cans, tape measure, ruler,
triangular prism, square pyramid
Advance Preparation
For Part 1, organize workstations for groups of 4 students (2 partnerships each). Each group will need 2 cans
(see Lesson 11-3) and 1 each of the triangular prism and square pyramid models (Math Masters, pages 324
and 326) constructed in Lesson 11-1. Place a cardboard box with the top taped shut near the Math Message.
Teacher’s Reference Manual, Grades 4–6 pp. 220–222
890
Unit 11
Volume
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP8
Getting Started
Content Standards
5.NF.1, 5.NF.2, 5.NF.4b, 5.NF.7c, 5.MD.2, 5.MD.5a, 5.MD.5b
Mental Math and Reflexes
Math Message
Have students write numbers from dictation and mark the
indicated digits. Suggestions:
If you were to wrap this box as a gift, how would
you calculate the least amount of wrapping paper needed?
5,024,039 Circle the digit in the thousands place, and put
an X through the digit in the hundred-thousands place.
28,794,052 Circle the digit in the ten-thousands place, and
put an X through the digit in the hundreds place.
31.005 Circle the digit in the thousandths place, and put an
X through the digit in the hundredths place.
150,247.653 Circle the digit in the thousandths place, and
put an X through the digit in the thousands place.
587.624 Circle the digit in the tens place, and put an X
through the digit in the hundredths place.
94,679,424 Circle each digit in the millions period, and
put an X through each digit in the ones period.
Study Link 11 6 Follow-Up
Briefly review the answers. Ask: How would you
explain the difference between capacity and
volume? Volume is the measure of how much space
a solid object occupies. Capacity is the measure of how
much a container can hold.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Math Journal 2, p. 389)
Ask volunteers to use geometry vocabulary to describe the box.
The box is a rectangular prism that has six rectangular faces.
Then have students explain their solution strategies. If wrapping
paper covers each face of the box exactly, the least amount of
wrapping paper needed is the sum of the area of each of the six
faces. Tell students that the sum of the areas of the faces or the
curved surface of a geometric solid is called surface area. If
wrapping paper covers each face of the box exactly, the area of
paper required is the surface area of the box. If the area of the
available wrapping paper is less than the surface area of the box,
the paper will not completely cover the box.
Ask a pair of students to measure the length, width, and height of
the box to the nearest inch. Have students record these
dimensions on the figure in Problem 1 on journal page 389, find
and record the areas of the six sides of the box, and add these to
find the total surface area. Remind students that opposite sides
(top and bottom, left and right, front and back) of the box have the
same area. This means that students need to calculate only three
different areas.
▶ Finding the Surface Area of a Can
(Math Journal 2, p. 389)
PARTNER
ACTIVITY
SOLVING
Algebraic Thinking Distribute one can to each partnership. Ask
students to imagine that the top lids of their cans have not been
removed. Ask: How would you find the surface area of your can?
Student Page
Date
Time
LESSON
11 7
Surface Area
The surface area of a box is the sum of the areas of all 6 sides (faces) of the box.
1.
Your class will find the dimensions of a cardboard box.
Fill in the dimensions on the figure below.
b.
Find the area of each side of the box. Then find the total surface area.
in2
Area of bottom =
153
153
99
99
187
187
Total surface area =
878
in2
Area of front =
Area of right side =
Area of left side =
Area of top =
2.
Sample answers:
a.
Area of back =
in2
in2
in2
top
9
in.
right
side
front
11 in.
2
in
17 in.
in2
Think: How would you find the area of the metal used to manufacture a can?
a.
How would you find the area of the top or bottom of the can?
First measure the diameter of the can, and divide by 2 to
find the radius. Then calculate the area (A = π º r 2 ).
b.
How would you find the area of the curved surface between the top and bottom of the can?
c.
Choose a can. Find the total area of the metal used to manufacture the can.
Remember to include a unit for each area.
Sample answers for a can
2 with a 7 cm diameter and
Area of top = 38.48 cm
2 13 cm height:
Area of bottom = 38.48 cm
2
Area of curved side surface = 285.88 cm
Calculate the circumference of the base (c = π º d ), and
multiply that by the height of the can.
Total surface area =
362.84 cm2
Math Journal 2, p. 389
369-392_EMCS_S_MJ2_U11_576434.indd 389
3/4/11 7:04 PM
Lesson 11 7
891
Student Page
Date
Time
LESSON
Surface Area
11 7
Remind students that in this problem they are finding the area of
the surface of their can, not the volume. Give students plenty of
time to explore solution strategies among themselves, without your
help. Add the areas of the top, bottom, and curved surface to find
the total surface area. Ask a volunteer to explain how to find the
area of the circular top and bottom. Use the formula A = π ∗ r2.
Ask another volunteer to explain how to find the area of the curved
surface between the top and bottom of the can. Calculate the
circumference of the base using the formula c = π ∗ d, and then
multiply the circumference by the height of the can.
continued
Formula for the Area of a Triangle
1
ºbºh
A=_
2
where A is the area of the triangle, b is the length of its base, and h is its height.
Use your model of a triangular prism.
3.
Find the dimensions of the triangular and rectangular faces. Then find the areas
1
of these faces. Measure lengths to the nearest _
4 inch.
1
_
2
base =
in.
length =
in.
3
_
4
in.
width =
in.
height =
3
_
4
Area =
in2
Area =
in2
a.
4
2
9
2
1
1
Add the areas of the faces to find the total surface area.
1
_
2
in2
b.
Area of 2 triangular bases =
Area of 3 rectangular sides =
1
_
2
Total surface area =
30
3
27
in2
If students are unable to suggest an approach for finding this
area, remind or show them that the label on a food can is shaped
like a rectangle if it is cut off and unrolled.
2
in
Use your model of a square pyramid.
4.
Find the dimensions of the square and triangular faces. Then find the areas
of these faces. Measure lengths to the nearest tenth of a centimeter.
a.
6.3 cm
5.5 cm
Area =17.325cm
6.3 cm
6.3 cm
Area = 39.69 cm
length =
base =
width =
height =
2
2
Add the areas of the faces to find the total surface area.
b.
39.69 cm
69.3
Total surface area =108.99cm
Area of square base =
2
Area of 4 triangular sides =
cm2
2
If you cut a can label in a straight line
perpendicular to the bottom, it will have
a rectangular shape when it is unrolled
and flattened.
Math Journal 2, p. 390
369-392_EMCS_S_MJ2_U11_576434.indd 390
3/4/11 7:04 PM
Draw a rectangle on the board:
●
How would you measure the can to find the length of the base
of the rectangle? Guide students to recognize the following
two possibilities:
1. Measure the circumference of the base of the can with a
tape measure.
2. Measure the diameter of the base of the can. Calculate the
circumference of the base, using the formula c = π ∗ d.
●
Student Page
Date
Have partners measure the dimensions of their cans and complete
Part 2 on journal page 389. Circulate and assist.
Time
LESSON
Analyzing Fraction Data
11 7
How would you find the other dimension (the width) of the
rectangle? Measure the height of the can.
1
Kerry measured life-size photos of 15 common insects to the nearest _
8 of an inch.
His measurements are given in the table below.
Length
1
(nearest _8 in.)
Insect
1 _8
3
American Cockroach
_1
Aphid
_3
Honeybee
4
4
3
Housefly
_1
Cabbage Butterfly
1 _4
June Bug
1
Cutworm
1 _2
Ladybug
_5
Deerfly
_3
Silverfish
_3
Field Cricket
_7
1
1.
1
1
8
Watch for students who seem to have difficulty with visualizing the cutting and
unrolling of the can label. Have them use the following procedure to cut a sheet
of paper to use as a label for the side of the can:
1 _8
House Centipede
8
1 _2
Bumblebee
Ongoing Assessment: Informing Instruction
3
_
8
Heelfly
_1
Ant
Length
1
(nearest _8 in.)
Insect
4
1. Mark the top and bottom rims of the can at the can’s seam.
8
8
2. Lay the can on the sheet of paper with the marks touching the paper. Mark
these points on the paper.
8
Make a line plot of the insect lengths on the number line below. Be sure to label the
axes and provide a title for the graph.
3. Roll the can one complete revolution, until the marks touch the paper again.
Mark these points on the paper.
Number of Insects
Insect Lengths
4. Connect the four marks on the paper to form a rectangle. Cut out
the rectangle.
0
X
X
X
X
X
X
1
8
1
4
3
8
1
2
X
X
X
X
5
8
3
4
7
8
1
1
X
X
X
X
X
1
3
1
18 14 18 12
Length (inches)
Math Journal 2, p. 390A
369-392_EMCS_S_MJ2_U11_576434.indd 390A
892
Unit 11
3/3/11 9:23 AM
Volume
Student Page
▶ Finding the Surface Area
PARTNER
ACTIVITY
of a Prism and a Pyramid
Date
Time
LESSON
continued
1 _2 in.
1
(Math Journal 2, p. 390)
Algebraic Thinking Have partners measure each solid constructed
in Lesson 11-1 from Math Masters, pages 324 and 326 and record
the dimensions on journal page 390. Then have them calculate and
record the face areas and total surface area for each solid.
Circulate and assist.
2.
What is the maximum length of the insects measured?
3.
What is the minimum length?
4.
What is the range of insect lengths?
5.
What is (are) the mode length(s) of the insects?
6.
What is the median length of the insects?
7.
How much longer is the American cockroach
that Kerry measured than the field cricket?
8.
Suppose 5 ladybugs were placed end to end. How long would they be?
9.
How many times as long is the bumblebee than the housefly?
10. a.
Ongoing Assessment:
Recognizing Student Achievement
Analyzing Fraction Data
11 7
Use the data and the line plot you made on journal page 390A to solve these
problems. Give all answers in simplest form.
Journal pages
389 and 390
in.
1 _8 in.
3
_3 in.
8
_3 in.
4
_1 in.
2
1
3_
8 in.
6 times as long
3
Suppose all of the insects less than _4 inch long were placed end-to-end.
How long would they be?
2 _8 in.
3
Use journal pages 389 and 390 to assess students’ ability to measure to the
1
nearest _
4 inch and 0.1 cm and to find the area of circles, triangles, and
rectangles. Students are making adequate progress if they correctly answer
Problems 2 and 3 and use rectangle, circle, and triangle area formulas to find the
surface area of prisms and cylinders.
_1
8
5
Suppose all of the insects greater than _8 inch long were placed end-to-end.
How long would they be?
b.
9 _8 in.
5
12 in.
What is the total length of all 15 insects placed end to end?
c.
11.
What is the mean length of the 15 insects that were measured?
_4 in.
5
[Measurement and Reference Frames Goals 1 and 2]
Math Journal 2, p. 390B
369-392_EMCS_S_MJ2_G5_U11_576434.indd 390B
4/7/11 3:58 PM
2 Ongoing Learning & Practice
▶ Plotting and Analyzing
INDEPENDENT
ACTIVITY
Insect Data
(Math Journal 2, pp. 390A and 390B)
Students plot insect lengths in fractional units on a line plot. They
use the line plot to analyze data and solve problems involving
addition, subtraction, multiplication, and division with fractions.
As needed, review how to make a line plot based on fractional
units and how to perform operations with fractions.
Student Page
Date
Time
LESSON
Math Boxes
11 7
▶ Math Boxes 11 7
1.
INDEPENDENT
ACTIVITY
Write each number in simplest form.
a.
b.
(Math Journal 2, p. 391)
c.
d.
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 11-5. The skill in Problem 5
previews Unit 12 content.
2.
How are cylinders and cones alike?
4.
Find the area and perimeter of
the rectangle.
16
8 _56
_=
8 _12
8_ =
19
11 _
27
11 _ =
80
_
=
5
53
6
Both have a circular
base and one
curved surface.
24
48
38
54
147 148
62 63
3.
Find the volume of the rectangular prism.
7 cm
Volume of rectangular prism
1
V=B*h
1 cm
4 cm
V=
(Math Masters, p. 341)
5.
Home Connection Students practice using formulas to
calculate volume and surface area. If students use
calculators, remind them to enter 3.14 for π if necessary.
Area =
cm
INDEPENDENT
ACTIVITY
12
3
▶ Study Link 11 7
3 2 cm
24.5 cm2(unit)
Perimeter =
cm3
21 cm
(unit)
186 189
197
Solve the pan-balance problems below.
XXXXX
XXXXXX
XX
a.
One
weighs as much as
3
Xs.
One
weighs as much as
6
Xs.
b.
weighs as much as
One
paper clips.
5
weighs as much as
One
paper clips.
5
228 229
Math Journal 2, p. 391
369-392_EMCS_S_MJ2_U11_576434.indd 391
3/4/11 7:04 PM
Lesson 11 7
893
Study Link Master
Name
Date
STUDY LINK
Time
3 Differentiation Options
Volume and Surface Area
11 7
Circumference of circle:
c=πºd
Area of rectangle:
A=lºw
197 198
200 201
Area of circle:
A = π º r2
Volume of rectangular prism:
V=lºwºh
Kesia wants to give her best friend a box of chocolates.
Figure out the least number of square inches of wrapping
paper Kesia needs to wrap the box. (To simplify the
problem, assume that she will cover the box completely
with no overlaps.)
1.
Amount of paper needed:
2 in.
6 in.
4 in.
▶ Finding the Smallest Surface
(Math Masters, p. 342)
Sample answer: I found the area of each of
the 6 sides and then added them together.
Algebraic Thinking To apply students’ understanding of volume
and surface area, have them use given volumes to find the
dimensions of rectangular prisms. Partners use the patterns in
the dimensions to identify the dimensions of a prism that would
have the smallest surface area.
Could Kesia use the same amount of wrapping paper to
cover a box with a larger volume than the box in Problem 1?
Yes Explain.
1
_
A 4 in. × 4 in. × 3 2 in. box has a volume of
56 in3 and a surface area of 88 in2.
Find the volume and the surface area of the two figures in Problems 3 and 4.
Volume:
3.
4.
502.4 cm3
2
351.7 cm
Volume:
216 in3
m
8c
When students have finished, discuss any difficulties or curiosities
they encountered.
6 in.
Surface area:
10 cm
Surface area:
15–30 Min
Area for a Given Volume
88 in2
Explain how you found the answer.
2.
PARTNER
ACTIVITY
ENRICHMENT
Volume of cylinder:
V = π º r2 º h
216 in2
cube
Math Masters, p. 341
INDEPENDENT
ACTIVITY
EXTRA PRACTICE
323-347_EMCS_B_MM_G5_U11_576973.indd 341
3/9/11 8:44 AM
▶ Finding Area, Surface Area,
5–15 Min
and Volume
(Math Masters, p. 343; Student Reference Book, pp. 196 and 197)
Students practice calculating area, surface area, and volume of
various geometric figures.
Teaching Master
Teaching Master
Name
LESSON
11 7
Date
Name
Time
LESSON
A Surface-Area Investigation
11 7
In each problem below, the volume of a rectangular prism is given. Your task is to find
the dimensions of the rectangular prism (with the given volume) that has the smallest
surface area. To help you, use centimeter cubes to build as many different prisms as
possible having the given volume.
Record the dimensions and surface area of each prism you build in the table. Do not
record different prisms with the same surface area. Put a star next to the prism with
the smallest surface area.
1.
Date
Area, Surface Area, and Volume
Area of rectangle: A = l ∗ w
Circumference of circle: c = π ∗ d
Volume of rectangular prism: V = B ∗ h
or V = l ∗ w ∗ h
Area of circle: A = π ∗ r 2
Record the dimensions and find the area.
Volume of cylinder: V = π ∗ r 2 ∗ h
2.
Record the dimensions and find the volume.
2 8 ft
Dimensions (cm)
Surface Area (cm2)
Volume (cm3)
2×6×1
40
12
1 × 2 × 12
1×3×8
1×6×4
2×3×4
2×2×6
24 × 1 × 1
3.
76
70
68
52
56
98
12
Width =
8
7
_
Area = 14 16
12
12
Volume (cm3)
24
20 cm2
2 cm
Number of
1-centimeter unit cubes
needed to fill the box:
Volume =
40
40 cm3
Record the dimensions, and find the volume and surface area for each
figure below. Round results to the nearest hundredth.
24
24
3.
24
Rectangular prism
4.
Cylinder
7 cm
8 cm
8 cm
5 cm
Width of base =
7 cm
Height of prism =
3
Volume = 280 cm
2
Surface area = 262 cm
Length of base =
Find the area of each face. Then
find the sum of these areas.
7 ft
7 ft
9 ft
3
Volume = 346.36 ft
2
Surface area = 274.89 ft
Diameter =
Height =
Math Masters, p. 343
Math Masters, p. 342
323-347_EMCS_B_MM_G5_U11_576973.indd 343
3/9/11 8:44 AM
Volume
5 cm
4 cm
Height of prism =
24
Describe a general rule for finding the surface area of a rectangular prism in words or
Unit 11
Area of base =
24
with a number sentence.
894
ft2
If the volume of a prism is 36 cm3, predict the dimensions that will result in the
smallest surface area. Explain.
323-347_EMCS_B_MM_G5_U11_576973.indd 342
Length of base =
Width of base =
How many square tiles, each 1 ft long
on a side, would be needed to fill the
rectangle?
7
14 _
16
3 º 3 º 4; I used the smallest 3 dimensions
that would result in the given volume.
4.
5 _12 ft
2 _5 ft
9 ft
Surface Area (cm2)
Dimensions (cm)
1
5 2 ft
Length =
cm
2.
38
32
50
5 cm
cm
5
3×4×1
3×2×2
12 × 1 × 1
4
2 cm
5
1.
Time
4/7/11 9:43 AM
Name
LESSON
11 7
1.
Date
Time
Area, Surface Area, and Volume
Area of rectangle: A = l ∗ w
Circumference of circle: c = π ∗ d
Volume of rectangular prism: V = B ∗ h
or V = l ∗ w ∗ h
Area of circle: A = π ∗ r 2
Record the dimensions and find the area.
Volume of cylinder: V = π ∗ r 2 ∗ h
2.
Record the dimensions and find the volume.
4
5 cm
cm
2 cm
5
2 8 ft
1
5 2 ft
Length =
Length of base =
Width =
Width of base =
Area =
Area of base =
How many square tiles, each 1 ft long
on a side, would be needed to fill the
rectangle?
Height of prism =
Number of
1-centimeter unit cubes
needed to fill the box:
Record the dimensions, and find the volume and surface area for each
figure below. Round results to the nearest hundredth.
Rectangular prism
4.
Cylinder
9 ft
7 cm
7 ft
8 cm
cm
3.
5
Copyright © Wright Group/McGraw-Hill
Volume =
Length of base =
Diameter =
Width of base =
Height =
Height of prism =
Volume =
Volume =
Surface area =
Surface area =
343