surface area - everyday math · triangular prism, square pyramid plotting and analyzing insect data...
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eToolkitePresentations Interactive Teacher’s
Lesson Guide
Algorithms Practice
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AssessmentManagement
Family Letters
CurriculumFocal Points
890 Unit 11 Volume
Advance PreparationFor 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
Key Concepts and Skills• Measure the dimensions of a cylinder
in inches and centimeters.
[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 ActivitiesStudents 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 Vocabularysurface area
MaterialsMath Journal 2, pp. 389 and 390
Study Link 11�6
slate � calculator � cardboard box �
per workstation: 2 cans, tape measure, ruler,
triangular prism, square pyramid
Plotting and Analyzing Insect DataMath 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.
Math Boxes 11�7Math Journal 2, p. 391
Students practice and maintain skills
through Math Box problems.
Study Link 11�7Math Masters, p. 341
Students practice and maintain skills
through Study Link activities.
ENRICHMENTFinding the Smallest Surface Area for a Given VolumeMath 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 VolumeMath Masters, p. 343
Student Reference Book, pp. 196 and 197
Students practice calculating area, surface
area, and volume.
Teaching the Lesson Ongoing Learning & Practice Differentiation Options
�
Surface AreaObjective To introduce finding the surface area of prisms,
cylinders, and pyramids.c
Common Core State Standards
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Surface AreaLESSON
11�7
Date Time
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.
a. Fill in the dimensions on the figure below.
b. Find the area of each side of the box. Then find the total surface area.
Area of front � in2
Area of back � in2
Area of right side � in2
Area of left side � in2
Area of top � in2
Area of bottom � in2
Total surface area � in2
2. 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?
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.
Area of top �
Area of bottom �
Area of curved side surface �
Total surface area �
38.48 cm2
878
187
187
99
99
153
153
Sample answers:
Sample answers for a canwith a 7 cm diameter and13 cm height:
First measure the diameter of the can, and divide by 2 to
find the radius. Then calculate the area (A � π º r 2).
Calculate the circumference of the base (c � π º d ), and
multiply that by the height of the can.
38.48 cm2
285.88 cm2
362.84 cm2
top
front
right
side
in.
in.
in.9
17
11
�
Math Journal 2, p. 389
Student Page
Lesson 11�7 891
Getting Started
Math MessageIf you were to wrap this box as a gift, how would you calculate the least amount of wrapping paper needed?
Study Link 11�6 Follow-UpBriefly 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.
Mental Math and Reflexes Have students write numbers from dictation and mark the indicated digits. Suggestions:
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.
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 PARTNER ACTIVITY
(Math Journal 2, p. 389)
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?
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Analyzing Fraction DataLESSON
11�7
Date Time
Kerry measured life-size photos of 15 common insects to the nearest 1
_
8 of an inch.
His measurements are given in the table below.
InsectLength
(nearest 1
_ 8 in.) Insect
Length (nearest
1
_ 8 in.)
American Cockroach 1 3
_ 8 Heelfly
3
_
8
Ant 1
_ 4 Honeybee
3
_ 4
Aphid 1
_ 8 House Centipede 1
3
_ 8
Bumblebee 1 1
_ 2 Housefly
1
_ 4
Cabbage Butterfly 1 1
_ 4 June Bug 1
Cutworm 1 1
_ 2 Ladybug
5
_ 8
Deerfly 3
_ 8 Silverfish
3
_ 8
Field Cricket 7
_ 8
1. 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.
X X X X X XX
XXX
XX
XX
X
18
14
38
12
58
34
78
0 1 181 1
41 381 1
21
Insect Lengths
Length (inches)
Num
ber o
f Ins
ects
369-392_EMCS_S_MJ2_U11_576434.indd 390A 3/3/11 9:23 AM
Math Journal 2, p. 390A
Student Page
3. Use your model of a triangular prism.
a. Find the dimensions of the triangular and rectangular faces. Then find the areas
of these faces. Measure lengths to the nearest 1 _ 4 inch.
base = in. length = in.
height = in. width = in.
Area = in2 Area = in2
b. Add the areas of the faces to find the total surface area.
Area of 2 triangular bases = in2
Area of 3 rectangular sides = in2
Total surface area = in2
4. Use your model of a square pyramid.
a. 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.
length = cm base = cm
width = cm height = cm
Area = cm2 Area = cm2
b. Add the areas of the faces to find the total surface area.
Area of square base = cm2
Area of 4 triangular sides = cm2
Total surface area = cm2
Surface Area continuedLESSON
11�7
Date Time
Formula for the Area of a Triangle
A = 1 _ 2 º b º h
where A is the area of the triangle, b is the length of its base, and h is its height.
2
1 3 _
4
1 3 _
4
4 1 _
2
3 1 _
2
27
2
9
108.99
30 1 _
2
6.36.3
39.69
39.6969.3
6.35.5
17.325
�
369-392_EMCS_S_MJ2_U11_576434.indd 390 3/4/11 7:04 PM
Math Journal 2, p. 390
Student Page
892 Unit 11 Volume
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.
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.
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.
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.
● How would you find the other dimension (the width) of the rectangle? Measure the height of the can.
Have partners measure the dimensions of their cans and complete Part 2 on journal page 389. Circulate and assist.
Ongoing Assessment: Informing Instruction
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. Mark the top and bottom rims of the can at the can’s seam.
2. Lay the can on the sheet of paper with the marks touching the paper. Mark
these points on the paper.
3. Roll the can one complete revolution, until the marks touch the paper again.
Mark these points on the paper.
4. Connect the four marks on the paper to form a rectangle. Cut out
the rectangle.
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Analyzing Fraction Data continuedLESSON
11�7
Date Time
Use the data and the line plot you made on journal page 390A to solve these problems. Give all answers in simplest form.
2. What is the maximum length of the insects measured? 1 1 _ 2 in.
3. What is the minimum length? 1 _ 8 in.
4. What is the range of insect lengths? 1 3 _ 8 in.
5. What is (are) the mode length(s) of the insects? 3 _ 8 in.
6. What is the median length of the insects? 3 _ 4 in.
7. How much longer is the American cockroach that Kerry measured than the field cricket?
1 _ 2 in.
8. Suppose 5 ladybugs were placed end to end. How long would they be? 3 1 _ 8 in.
9. How many times as long is the bumblebee than the housefly? 6 times as long 10. a. Suppose all of the insects less than 3 _ 4 inch long were placed end-to-end.
How long would they be?
2 3 _ 8 in.
b. Suppose all of the insects greater than 5 _ 8 inch long were placed end-to-end. How long would they be?
9 5 _ 8 in.
c. What is the total length of all 15 insects placed end to end? 12 in.
11. What is the mean length of the 15 insects that were measured? 4 _ 5 in.
369-392_EMCS_S_MJ2_G5_U11_576434.indd 390B 4/7/11 3:58 PM
Math Journal 2, p. 390B
Student Page
3. Find the volume of the rectangular prism.
Volume of rectangular prism
V = B * h
V = 12 cm3
Math Boxes LESSON
11�7
Date Time
5. Solve the pan-balance problems below.
a.
One weighs as much as 3 Xs. One weighs as much as 6 Xs.
b.
One weighs as much as 5 One weighs as much as 5
paper clips. paper clips.
1. Write each number in simplest form.
a. 80
_ 5 = 16
b. 53
_ 6 =
c. 8 24
_ 48 =
d. 11 38
_ 54 =
4. Find the area and perimeter of
the rectangle.
Area = 24.5 cm2
(unit)
Perimeter = 21 cm
(unit)
2. How are cylinders and cones alike?
Both have a circularbase and onecurved surface.
8 5 _
6
8 1 _
2
11 19
_
27
4 cm3
cm
1 cm
X XX X X X X
X X X X X X
7 cm
1
23 cm
62 63 147 148
197 186 189
228 229
369-392_EMCS_S_MJ2_U11_576434.indd 391 3/4/11 7:04 PM
Math Journal 2, p. 391
Student Page
Lesson 11�7 893
▶ Finding the Surface Area PARTNER ACTIVITY
of a Prism and a Pyramid(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.
Ongoing Assessment: Journal pages
389 and 390 �Recognizing Student Achievement
Use journal pages 389 and 390 to assess students’ ability to measure to the nearest 1 _ 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. [Measurement and Reference Frames Goals 1 and 2]
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.
▶ Math Boxes 11�7
INDEPENDENT ACTIVITY
(Math Journal 2, p. 391)
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.
▶ Study Link 11�7
INDEPENDENT ACTIVITY
(Math Masters, p. 341)
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.
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LESSON
11�7
Name Date Time
A Surface-Area Investigation
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. Dimensions (cm) Surface Area (cm2) Volume (cm3)
2 × 6 × 1 40 12
3 × 4 × 1 38 12
3 × 2 × 2 32� 12
12 × 1 × 1 50 12
Dimensions (cm) Surface Area (cm2) Volume (cm3)
1 × 2 × 12 76 24
1 × 3 × 8 70 24
1 × 6 × 4 68 24
2 × 3 × 4 52� 24
2 × 2 × 6 56 24
24 × 1 × 1 98 24
2.
3. If the volume of a prism is 36 cm3, predict the dimensions that will result in the
smallest surface area. Explain.
4. Describe a general rule for finding the surface area of a rectangular prism in words or
with a number sentence.
find the sum of these areas.Find the area of each face. Then
that would result in the given volume.3 º 3 º 4; I used the smallest 3 dimensions
323-347_EMCS_B_MM_G5_U11_576973.indd 342 3/9/11 8:44 AM
Math Masters, p. 342
Teaching MasterLESSON
11�7
Name Date Time
Area, Surface Area, and Volume
Record the dimensions, and find the volume and surface area for each figure below. Round results to the nearest hundredth.
Area of rectangle: A = l ∗ w
Volume of rectangular prism: V = B ∗ hor V = l ∗ w ∗ h
Circumference of circle: c = π ∗ d
Area of circle: A = π ∗ r 2
Volume of cylinder: V = π ∗ r 2 ∗ h
1. Record the dimensions and find the area.
Length = 5 1 _ 2 ft Width = 2 5 _ 8 ft Area = 14 7
_ 16 ft2
How many square tiles, each 1 ft long on a side, would be needed to fill the rectangle?
14 7
_ 16
2. Record the dimensions and find the volume.
Length of base = 5 cm
Width of base = 4 cm
Area of base = 20 cm2
Height of prism = 2 cm
Number of 1-centimeter unit cubes needed to fill the box: 40
Volume = 40 cm3
3. Rectangular prism
Length of base = 8 cm Width of base = 5 cm Height of prism = 7 cm Volume = 280 cm3
Surface area = 262 cm2
4. Cylinder
Diameter = 7 ft Height = 9 ft Volume = 346.36 ft3
Surface area = 274.89 ft2
125 ft
582 ft
5 cm 2 cm4 cm
8 cm 5 cm
7 cm
7 ft
9 ft
323-347_EMCS_B_MM_G5_U11_576973.indd 343 4/7/11 9:43 AM
Math Masters, p. 343
Teaching Master
STUDY LINK
11�7 Volume and Surface Area
Name Date Time
1. 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.)
Amount of paper needed:
Explain how you found the answer.
Sample answer: I found the area of each of the 6 sides and then added them together.
2. Could Kesia use the same amount of wrapping paper to
cover a box with a larger volume than the box in Problem 1? Explain.
A 4 in. × 4 in. × 3 1
_ 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.
3. Volume: 4. Volume:
Surface area: Surface area:
Yes
216 in2
216 in3
351.7 cm2
502.4 cm3
88 in2
6 in.
2 in.4 in.
10 c
m
8 cm
6 in.
cube
197 198200 201
Area of rectangle: A = l º w
Volume of rectangular prism: V = l º w º h
Circumference of circle:c = π º d
Area of circle: A = π º r 2
Volume of cylinder:V = π º r 2 º h
323-347_EMCS_B_MM_G5_U11_576973.indd 341 3/9/11 8:44 AM
Math Masters, p. 341
Study Link Master
894 Unit 11 Volume
3 Differentiation Options
ENRICHMENT PARTNER ACTIVITY
▶ Finding the Smallest Surface 15–30 Min
Area for a Given Volume(Math Masters, p. 342)
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.
When students have finished, discuss any difficulties or curiosities they encountered.
EXTRA PRACTICE
INDEPENDENT ACTIVITY
▶ 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.
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LESSON
11�7
Name Date Time
Area, Surface Area, and Volume
343
Cop
yrig
ht ©
Wrig
ht G
roup
/McG
raw
-Hill Record the dimensions, and find the volume and surface area for each
figure below. Round results to the nearest hundredth.
Area of rectangle: A = l ∗ w
Volume of rectangular prism: V = B ∗ hor V = l ∗ w ∗ h
Circumference of circle: c = π ∗ d
Area of circle: A = π ∗ r 2
Volume of cylinder: V = π ∗ r 2 ∗ h
1. Record the dimensions and find the area.
Length =
Width =
Area =
How many square tiles, each 1 ft long on a side, would be needed to fill the rectangle?
2. Record the dimensions and find the volume.
Length of base =
Width of base =
Area of base =
Height of prism =
Number of 1-centimeter unit cubes needed to fill the box:
Volume =
3. Rectangular prism
Length of base =
Width of base =
Height of prism =
Volume =
Surface area =
4. Cylinder
Diameter =
Height =
Volume =
Surface area =
125 ft
582 ft
5 cm 2 cm4 cm
8 cm 5 cm
7 cm
7 ft
9 ft
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