Lesson: Exploring Surface Area
Context: This lesson is designed for a 7th grade class at Berkeley Elementary School. This
inclusion class contains 30 students, 5 of which have learning disabilities. The class is co-taught
by a 7th grade teacher and a special education collaborator.
Objective(s):
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The student will be able to provide a mathematically accurate definition of the terms area;
surface area, and rectangular prism.
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Using the nets of several rectangular prisms, the student will be able to construct a
rectangular prism and describe how to calculate the surface area.
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The student will be able to use the formula for surface area to determine the surface area
of a rectangular prism.
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The student will describe how changing one measured attribute of a rectangular prism
affects the calculated surface area.
SOL Strand: Geometry and Measurement – Surface Area and Volume
Related SOL:
6.10
The student will
d) describe and determine the volume and surface area of a rectangular prism.
7.5
The student will
c) describe how changing one measured attribute of a rectangular prism affects its volume and
surface area.
8.7
The student will
b) describe how changing one measured attribute of a figure affects the volume and surface
area.
Estimated class time: 40 minutes
Materials and resources
For teacher
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Elmo projector
Whiteboard
Class set of Relational GeoSolids (optional)
o A tool used to demonstrate the relationships between shape, size, and volume. Set
includes fourteen clear, hollow, plastic geometric solids [spheres, cones, cubes,
rectangles, cylinders, and pyramids]
Plastic nets that fit the Relational GeoSolids
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Chart paper for vocabulary
Markers
o Dry erase markers
Pens/Pencils for Elmo
Computer and projector
o Link: http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
3 nets for a 2in x 2in x 2in rectangular prism
3 nets for a 2in x 3in x 2in rectangular prism
Rulers
For students:
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Surface Area notes
Exploring Surface Area worksheet
Scissors
Colored pencils / crayons
Pre made colored nets for rectangular prisms (differentiation for those who need them)
1 in graph paper
1 in foam cubes (introduction to volume)
Procedures
I.
Focus and Review
1. Separate the Relational GeoSolids into two groups
a. Group 1 (Non – Prisms): Includes cylinders, cones, triangular pyramids, square
pyramids.
b. Group 2 (Prisms): Includes triangular prisms, cubes, square prisms/rectangular
prisms, hexagonal prisms
2. After laying out the prisms inform students that “these are prisms” and “these are not
prisms” (directing them to the prism and non-prism groups respectively). Ask “what do
you think makes these prisms” – Have a discussion about the properties of each group
and lead students to the following appropriate definition of a prism
Prism - A solid object, also known as a polyhedra, with parallel, congruent polygons at
each end. The sides are congruent rectangles; the shape of the ends gives the prism its
name.
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Tell students that today will begin to explore surface area and volume. Students will first
learn how to calculate the surface area and volume of rectangular prisms. Have students
come up with an appropriate definition for a rectangle.
Rectangle – A four-sided polygon (a flat shape with straight sides) where every angle is
a right angle (90°).
What other properties do we know about rectangles?
Opposite sides are parallel and of equal length.
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Thinking back upon our properties of quadrilaterals, what can we say about a
square?
A square is a special type of rectangle (all sides are equal)
So how can we use these definitions to describe/define what a rectangular prism is?
A rectangular prism is a solid (3-dimensional) object which has six faces that are rectangles
Animation: http://www.mathsisfun.com/definitions/rectangular-prism.html - Clicking explode
separates the rectangular prism into its 6 faces
a. Have a discussion to come up with a definition for the term surface area (an
example it provided below)
Surface area is composed of what two words? Surface + area
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Ask students if they recall how to find the area of a two-dimensional shape.
o Area – (simple definition) The number of square units needed to cover a
surface, area of rectangles = lw [see next two bullet points below if students
have difficulty with the definition]
o Picture the unit square (draw on board). The unit square is one unit long by one unit
wide; it can be 1 in by 1 in, 1 m x 1 m, 1 yd x 1 yd, etc.
o Area of rectangle: Let's look at a rectangle that is 4 in by 6 in (draw picture below on
whiteboard or Elmo). If we count the number of 1 in by 1 in squares that are inside
the rectangle we can easily see there are 24 of these squares. After performing the
same task with a rectangle that has different dimensions, we can see a pattern.
The total number of squares that rest within a rectangle can be found by multiplying the
length of a rectangle by its width. Therefore, 4in x 6in = 24 in2, hence the formula A =
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lw. Counting square units fits nicely with the concept of counting squares and it also
coincides with a property of algebra. In algebra, we already know (x)(x) = x2. The same
is true for mm times mm, in by in, or any unit times the same unit.
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II.
So what does surface mean? - The outside layer of an object; a surface can also be
defined as a two-dimensional boundary that can either be flat or curved. The surfaces of
a prism are also called the faces of that prism.
So what is surface area? – The surface area is the total area of the exterior of a threedimensional geometric figure. We saw that the area of a figure is the sum of all the
unit squares that cover that figure. So the surface area is the sum of all the unit
squares that fit on the exterior of a solid.
Objectives and Instruction
Make sure each of the defined vocabulary words remain on the whiteboard or on a sheet of chart
paper in the front of the room for students to reference during the class as necessary.
3. Have students arrange themselves into groups of three; give each student a net of the 2
x 2 x 2 rectangular prism and each group a roll of tape.
Teacher: What is a net? A pattern that you can cut and fold to make a model of a solid
shape. This is a net of a cube (a special rectangular prism where all the sides are equal;
remember that ALL squares are rectangles and some rectangles are squares) – hold up
the net of the cube and show how it forms into the prism; congruent (identical) sides are
colored the same.
4. Instruct each student to fold their net into a prism so that they can get an idea of the
shape. Discuss what an edge is. Students should label the length, width, and height of
their prisms at EACH edge (“l”, “w”, or “h”) as the net remains in cube shape. Have a
discussion about whether the front HAS TO BE labeled as the length; in other words, as
long the labeling is consistent (the back is labeled the same as the front and the sides are
labeled the same as one another), does it matter whether the front is the length or the
width. Short answer, No as long as consistency in labeling remains.
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Walk around to assist students who need help. Pass out the exploring surface area
worksheet packet to each student. Place the image of the general prism from the
“Surface Area” notes worksheet on the overhead; point out how the worksheet
labels the length, width, and height.
5. Walk students through the first row of the worksheet.
a. Have students fold the prism back out into the net. Looking at one prism
alone, have students identify the length, width, and height of the prism; fill in the
worksheet as the students do the same (2in x 2in x 2in)
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b. Now as a class, identify the surface area. “We know from earlier that the
surface area is the sum of all the unit squares that fit on the exterior of a
solid. So how many total squares are there?”
i. 4 per side x 4 sides = 16, 4 on the top + 4 on the bottom = 8, so
Total = 16 + 8 = 24 squares, fill this value in last column of the first
row.
6. Now tell students that we will now double the height of our prism. How will we go about
doing this? Instruct two students per group to stack their prisms on top of one
another. Have students repeat the process for this new prism, what are our new
dimensions? Length = 2, Width = 2, Height = 4 (doubled original height). How many
cubes do we have now? 8 per side x 4 sides + 4 (top) + 4 (bottom) = 40 squares.
a. Ask students, from our previous discussions, how else can we find the surface
area without counting all the unit cubes?
i. Lead students to the conclusion that you could add up the areas
b. Place the surface area of a prism diagram on the overhead and have students
turn to the second page of their packet
i. From the diagram, how many surfaces are there to a rectangular
prism? 6
ii. Since the surface areas is the sum of all the areas of these rectangles, using
the labeling of the general prism, lets calculate the area for each surface.
a. Fill in the sheet (see key for answers)
b. Tell students they have just developed the formula for the surface area
of a prism. Use the formula to check that the surface area derived by
counting the unit cubes (after doubling) equals the surface area found
by using the formula.
7. Now tell students to stack all three of their cubes on top of each other to triple the height.
We will now use our formula to find the surface area. Write these steps on a new sheet of
paper (a blank sheet is attached in each students packet)
i. Surface area = 2 (lw) + 2(lh) + 2(wh)
ii. Surface area = 2 (2 x 2) + 2(2 x 6) + 2(2 x 6)
“length is…, width is …, etc.”
iii. Surface area = 2 (4) + 2(12) + 2(12)
iv. Surface area = 8 + 24 + 24
v. Surface area = 56
What are our units? What is the unit of area? Square units; so our
final answer is 56 in2 (notice we are counting 1 in. squares)
III.
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Technology integration:
If at all possible, each person in the group would be given a computer to use. A (one)
computer may also be given to each group for students to share. This lesson can also be
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rearranged for the one computer classroom. For purposes of this lesson, students will
work through the lesson and worksheet activities by directing your movements on the
computer, when necessary.
8. Open your browser to Surface Area and Volume activity webpage in order to
demonstrate this activity to the students.
http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
9. Show the students that you can choose between a triangular prism and a rectangular
prism; for this lesson will only focus on the rectangular prism (make sure this prism is
selected).
10. Show students how to use the scroll bars to adjust the length, width, and height of the
prisms. Highlight (place on whiteboard/Elmo) that for this computer activity the
a. Width on the site = length (in our general prism)
b. Depth on the site = width (in our general prism)
c. Height = height
11. Show students how to adjust the view of the figure using the three buttons at the top of
the applet. Talk about what each view is showing. Be sure to show the students that
clicking on the figure and rotating it will allow them to view it from any angle they
choose. (place the following on the whitboard/Elmo)
a. Width by Height = allows us to calculate the l x h area (lh)
b. Depth by Height = allows us to calculate the w x h area (wh)
c. Width by Depth = allows us to calculate the l x w area (lw)
12. Point out to students the place where the surface area and volume of the figures is
displayed.
13. Using the computer applet, show and verify that the answers obtained using the
formula for surface area after doubling and tripling the height were correct (a third
way of computing the surface area – Counting the unit squares, using a formula,
technology - computer applet)
IV.
Back to instruction (turn off the projector)
14. Now tell students, we first doubled and then tripled the height of our starting prism and
calculated the surface areas in each case. What other measured attributes of a rectangular
prism could we use to see how it affects the surface area? Answers should be length and
width (students can either choose to work with the length or the width first)
a. Length: “How would we double the length of our 2 x 2 x 2 prism?” Place two
cubes beside each other
i.
Repeat the same procedure as before, this time doubling and tripling
the length. Work with the students on completing the second row.
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Have students come up with the answers for the tripling row. Below
are the answers to the chart (for length)
Scale Factor
Starting prism
Doubling
Tripling
Length
2 in
4 in
6 in
Width
2 in
2 in
2 in
Height
2 in
2 in
2 in
Surface Area in 1 in. squares
2(4) + 2(4) + 2(4) = 24 in2
2(4 x 2 + 4 x 2 + 2 x 2) = 2(20) = 40 in2
2 (12 + 12 + 4) = 2(28) = 56 in2
b. Width: “How would we double the width of our 2 x 2 x 2 prism?” Place one cube
behind the other
ii. Repeat the same procedure as before, this time doubling and tripling
the width. Students should be able to complete each row within their
groups. Walk around to assist students who need help, ask questions to
check for understanding; “What are you doing now?”, “How is this
process similar or different than before?”, “Do you see any patterns?”
Below are the answers to the chart (for width)
Scale Factor
Starting prism
Doubling
Tripling
V.
Length
2 in
2 in
2 in
Width
2 in
4 in
6 in
Height
2 in
2 in
2 in
Surface Area in 1 in. squares
2(4) + 2(4) + 2(4) = 24 in2
2(8 + 4 + 8) = 2(20) = 40 in2
2 (12 + 4 + 12) = 2(28) = 56 in2
Brief discussion of results thus far
Allow students to share what they found and summarize the results of the lesson. It is important
that every student know the correct answers to the worksheet in order to fully understand surface
area and volume.
a. It appears that the value for the surface area in corresponding rows remain the same
regardless of which measured attribute is doubled or tripled
* S.A. after doubling height = S.A. after doubling length = S.A. after doubling width = 40 in2
* S.A. after tripling height = S.A. after tripling length = S.A. after tripling width = 56 in2
Why? Since the starting prism is a cube, all the sides are of equal length; this fact is particularly
important because it makes the following statements true: After doubling the height, if the prism
is rotated on its side, the result would be the same prism that would occur if the length was
doubled. Now if we rotate that prism (the doubled length prism) by 90 degrees counterclockwise,
you get the prism that would occur if the width were doubled. A similar argument follows for the
tripling scale factor.
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Use the computer applet to demonstrate these rotations and show students that the
value of the calculated surface area does not change.
b. In2?  We are using one inch grids, and we are using squares.
c. Does the total surface area double when you double an attribute? Triple when you
triple an attribute?
No, have students engage in a discussion as to why this may be the case.
 Looking at the values in our chart, we can see that there is an increase in total surface
area; however, 24 x 2 = 48, which does not equal 40, so we do not have a distinct
doubling of the surface area.
 The same applies for the tripling scale factor – while the surface areas increase, there is
not a direct relationship or a direct proportion of increasing given a specific scaling
factor
o Meaning: When we double or triple the original prism, we do not see a double
or tripling of the total surface area
 As you press together or combine two prisms to form a new prism, you lose two sides
that would have been used to calculate the surface area [Show students that …]
o When the height is doubled, you lose the top of one cube and the bottom of the
other;
o When the height is tripled, you lose the tops of two cubes and the bottom of the two
cubes;
o When the length is doubled, you lose two sides;
o When the length is tripled, you lose 4 sides;
o When the width is doubled, you lose two sides;
o When the width is tripled, you lose 4 sides.
Note: The lost spaces are not be used to calculate the total surface area!
VI.
Varying the attributes of the rectangular prism (beyond the cube)
Looking at the rows for the doubling scale factor, notice that the surface area remained the same
regardless of whether the height, length, or width attribute were doubled. The same applies after
tripling the measured attribute. We will now explore what happens with a rectangular prism that
is not a cube.
Show students the net for a 2 x 3 x 2 rectangular prism. Construct the rectangular prism
from each net. Now that students have derived the equation for surface area and explore it
in detail by this point in the lesson, use the applet to complete each chart for this prism
(first change the height, then change the length and width)
1. Fill out the data for first row of the table using the new rectangular prism attributes
(length = 2 in, width = 3 in, height = 2)
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2. Use two 2 x 3 x 2 rectangular prisms to double the height and fill out the data for the
second row – Doubling
3. Use three 2 x 3 x 2 rectangular prisms to triple the height and fill out the data for the
third row – Tripling
Scale Factor
Starting prism
Doubling
Tripling
Length
2 in
2 in
2 in
Width
3 in
3 in
3 in
Height
2 in
4 in
6 in
Surface Area in 1 in. squares
32 in2
52 in2
72 in2
Width
3 in
3 in
3 in
Height
2 in
2 in
2 in
Surface Area in 1 in. squares
32 in2
52 in2
72 in2
Changing the length:
Scale Factor
Starting prism
Doubling
Tripling
Length
2 in
4 in
6 in
So far, our values after changing the length and height attributes are the same. Ask the following
question to the class, what do you think will happen when we change the depth (the width)
of the rectangular prism? Let’s see!
Scale Factor
Starting prism
Doubling
Tripling
Length
2 in
2 in
2 in
Width
3 in
6 in
9 in
Height
2 in
2 in
2 in
Surface Area in 1 in. squares
32 in2
56 in2
80 in2
Closure - Hold a group discussion with the class
You may wish to bring the class back together for a discussion of the findings. Allow students to
share what they found and summarize the results of the lesson. It is important that every student
know the correct answers to the worksheet in order to fully understand surface area and volume.
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Were the values when we doubled and tripled the depth (width) the same in
comparison to when we doubled and tripled the length or the width? No.
Our starting rectangular prism had a length of 2, a width of 2, and a height of 2. Since
each face has a length of 2, the surface area is equal to 2(4) + 2(4) + 2(4) from the surface
area formula, or 6(2)2, which is found by first calculating the area of each square face (s2)
and then finding the sum of the areas. What is a “special” name for this special type of
rectangular prism where all the attributes are equal? A cube
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So what can we conclude from this lesson? All we can conclude is that, a. as the
measured attributes of a rectangular prism are increased, the surface area of the
rectangular prism also increases; however, there is no constant change in surface area
(i.e. there is no direct relationship between changing one measured attribute to its
changing surface area) b. a cube is a special rectangular prism in which doubling or
tripling one attribute is the same as doubling or tripling another attribute.
Extension option:
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Nets for Relational GeoSolids are a set of Cling-vinyl nets that enable students to
determine the surface area of the figures. Students can guess which net will fit each
rectangular prism and compute the area for each side or “face” of the net. Using the dry
erase markers, students can write down the area of each face on the net and then compute
the surface area. Students can then attempt to match each net with its rectangular prism,
and then state what the surface area of the solid is.
Evaluation/Assessment:
Classwork / Homework: Have students design their own rectangular prism, calculate the surface
area, and determine what happens to the surface area if one measured attribute is changed.
Differentiation and Adaptations:
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The technology is used for reinforcement, not instruction. Therefore, if the technology
fails or students are having difficulty with the resource, it will not negatively impact the
learning experience.
Modifications or accommodations can be made in the following areas:
o Setting
 Students will be paired in small groups (can be of mixed ability)
 Students may also work one-on-one with the teacher
o Materials
 Pre-cut and or colored nets / prisms
 The colors help students identify the sides that are the same, without the
reliance on labeling or numbering
 Students may also be given a copy of the teacher’s notes if necessary
o Instruction
 The difficulty of the assignment can be reduced (ex. the cube is simple
rectangular prism to begin with for students who struggle initially with 3D
designs; it is a shape they are familiar with (ex. some gift boxes)
o Student response
 Answers can be given orally
 A word processor can be used for written work
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References
Shodor Education Foundation, I. (2011). Interactivate: Surface Area and Volume. Retrieved
November/4, 2011, from
http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
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