Module on Surface Area of Rectangular Prisms
for Grades 7-8
By
Sara Parish
Table of Contents
Math Module Outline
Pretest Form A
Lesson 1: Views and Nets of Rectangular Prisms
Lesson 2: Calculating Surface Area
Activity Sheet
Posttest Form B
Constructed Response
Module development was partially funded by the Missouri Coordinating Board for
Higher Education through the Eisenhower Professional Development Program.
Translations were partially funded by NSF ESIE SGER Project 0086580.
Math Module Outline
Strand: Geometry and Spatial Sense
Grade Level Span: 7-8
Concepts Included in Module:
UNDERSTANDING SURFACE AREA OF PRISMS: KNOWING WHAT IT IS
AND HOW TO CALCULATE IT
•
•
Views and Nets
Calculating Surface Area
Author: Sara Parish
District: Adair County R-II
Basis for Selection of Strand/Concepts:
I chose to concentrate on the NCTM strand of Geometry and Spatial Sense based on a
review of our districts Clear Access report.
Resources Used:
Korean Mathematics. (2001). Edited by Janice Grow-Maienza, translated by Sue Chung
Nugent. Kirksville, MO: Truman State University. From Ministry of Education.
Arithmetic. Seoul, Korea: National Textbooks Inc, l993.
Module Resources:
Math Text (can adapt to fit series available)
Pre-drawn Nets of Rectangular Prisms
(Other types of nets are optional)
Boxes of various shapes and sizes
(Examples: envelope boxes, cereal boxes, etc.)
Pretest Form A
1. In your own words, describe what the term area means.
2. State the formula for finding area of a rectangle. ____________________
3. State in your own words what surface area is and what it means to you.
4. Find the area of a rectangle with the following dimensions: 5 cm x 7 cm
5. Explain what the following mathematical terms mean.
NetsBase AreaSide AreaFront ViewSide ViewBack View-
6. Find the surface area of this rectangular prism. Show all work.
6
9
7. Demonstrate your understanding of views by drawing the front view, the side views,
the top view, and the bottom view of the figure above. Label them with the words front,
top, bottom, and side. Also label the dimensions of each piece. If possible draw these
connected as a net.
8. Explain the difference between a cube and a rectangular prism.
9. Based on your understanding, decide when in the real world you might use the concept
of surface area.
Scoring Guide for Items #1-5, 8, 9
2 points – student found correct answer and shows complete process
1 point – student found correct answer, but lacked complete process or
student attempted to find the correct answer , but may or may not
have made computation errors leading to an invalid answer
Scoring Guide for Items #6, #7
4 points. The student's response fully addresses the performance event.
The response demonstrates knowledge of the mathematical concepts and principles
needed to complete the event. The response communicates all process components that
lead to an appropriate and systematic solution. The response may have only minor flaws
with no effect on the reasonableness of the solution.
3 points. The student's response substantially addresses the performance event. The
response demonstrates knowledge . . . The response communicates most process
components. . . The response may have only minor flaws with minimal effect on the
reasonableness of the solution.
2 points. The student's response partially addresses the performance event. The
response demonstrates a limited knowledge. . . The response communicates some process
components . . . The response may have flaws or extraneous information that indicates
some lack of understanding or confusion.
1 point. The student's response minimally addresses the performance event. The
response demonstrates a limited knowledge . . . The response communicates few or no
process components. . . The response may have flaws or extraneous information that
indicates lack of understanding or confusion.
0 points. Other.
Lesson 1
Materials: Wrapped Present, Pre-drawn Nets, Scissors, Cereal Box
Objectives:
•
Students will gain a basic understanding of views and nets.
•
Students will construct nets of various rectangular prisms.
Standards Addressed:
•
•
•
Show-Me Goal: 1.6 Acquire the knowledge to discover patterns in ideas
3.4 Acquire the knowledge to evaluate the processes in solving problems.
MA2 Geometry and spatial sense involving area. MA4 Relationships
among geometric concepts.
Missouri Frameworks: MA- VI Geometry and Spatial Sense
2 Descriptions of two-dimensional and three-dimensional figures
3 Geometric shapes in the real world
NCTM Content Standard: Geometry: Use visualization, spatial
reasoning, and geometric modeling to solve problems.
Introduction/Demonstration:
Suppose we wanted to unwrap this present to see what
is inside. As we unwrap it we are going to unfold the
box and lay it out flat. First we are going to take the
wrapping paper off carefully trying not to tear it. Then
we are going to set it aside and talk just about the box
inside right now. (Actually perform this task in front
of the class. While doing so use this real life
application to develop the concept of nets and views.
Use a series of questions to develop a working
vocabulary of the following terms: front view, back
view, side view, top view, bottom view, and net.) For example: When you look at this
part of the present what shape do you see? (Rectangle) What part of the present is that
shape? (Front) Proceed by drawing that size of a rectangle on the board and labeling it
“Front View.” Continue this with each face of the present until the faces are all drawn on
the board with the views labeled. For example:
Front View
Back View
Left
Side
View
Right
Side
View
Top View
Bottom View
Demonstration of Processes:
Now the present should be completely unfolded but it is still connected. The expanded
form of the box, or the unfolded box, is an example of a net. Cut the box apart into
pieces like pictured on the board. Now explain that once you have all the pieces of a box,
it’s like putting a puzzle together. Take the pieces of the box and try to assemble them by
laying them on a table or desk edge to edge. (As the pieces are put back together discuss
the reason why a top view edge and a bottom view edge cannot be placed beside each
other- because this box is a prism, a 3-D shape, there are sides that fit in between the top
and bottom faces.)
*Teacher can give each student a piece and have him/her each try to figure out where the
piece goes.
Top View
Back View
Left
Side
View
Right
Front View
Side
View
Bottom View
-Draw a 3-D rectangular prism on the board and walk them through how to draw out the
“unseen” faces.
-Draw the views and label them.
-Demonstrate how to put the pieces into a net without actually cutting them.
-Emphasize visualizing it rather than actually cutting and pasting together.
-Discuss ideas that will allow students to conclude that there is more than one correct
way to draw a net for a given prism.
Guided Practice:
Pass out copies of a pre-drawn net of a rectangular prism and have students:
1. Cut out, fold, and tape to form the prism.
2. Cut out, cut into pieces, re-form net.
Pass out copies of a pre-drawn net of a cube and have students:
1. Cut out, fold, and tape to form the prism.
2. Cut out, cut into pieces, re-form net.
See if students can conclude similarities/differences of a cube and a rectangular prism.
(Cubes- all faces are the same size, Rectangular Prisms- faces are different sizes.)
*More complex pre-drawn nets may be attempted depending on level of class and time
permitted.
Draw another 3-D shape on the board and see if students can draw the views and net
without cutting and pasting together.
Independent Practice:
Teacher holds up cereal box and asks students to draw/label the views they see. (Show
them front, back, top, bottom, left side, and right side.)
Students attempt to draw the net of this box directly underneath the views they have just
drawn. If they cannot visualize it, they may need to cut the pieces out and tape them back
together.
Give students a worksheet that has various prisms on it and have them practice drawing
the nets without the help of an actual box. For example:
*Emphasize visualizing rather than cutting and pasting.
Notes for Teachers:
Build the vocabulary in this lesson to include 2-dimensional and 3-dimensional.
This lesson could take one day or it could take two depending on level of class and
amount of practice desired.
Lesson 2
Materials: Activity Sheet, Unwrapped Present (from lesson 1)
Objectives:
•
Students will review the concepts of views, nets, and area formula.
•
Students will learn to calculate surface area.
Standards Addressed:
•
•
•
Show-Me Goal: 1.10 Apply acquired information and skills to different
contexts. 3.4 Evaluate the processes in solving problems. MA2 Geometry
and spatial sense involving area. MA4 Relationships among geometric
concepts.
Missouri Frameworks: MA- VI (Geometry and Spatial Sense)
2 Descriptions of two-dimensional and three-dimensional figures
3 Geometric shapes in the real world
NCTM Content Standard: Geometry: Use visualization, spatial
reasoning, and geometric modeling to solve problems.
Introduction/Demonstration:
Recall our present from yesterday and the net that we
formed using the pieces of the box. How many pieces
did our box have? What shape were those pieces? We
put those together to form what we call a ___? Now
lets focus on the wrapping paper that we set aside.
Further discussion:
-Review definition of area.
-Review finding area of rectangles (each piece).
-Lead into how surface area is finding area of
each piece and then adding them together.
Go back to yesterday’s lesson on nets and have students conclude why it is important to
draw out a net correctly in order to compute surface area. (*Have to be sure you add the
area of each face. Have to remember the “unseen” faces when the picture is drawn.)
Demonstration of Processes:
Looking back to the present and the wrapping paper, we now know that when we are
talking about surface area we are talking about how much wrapping paper is needed to
wrap our present. Looking back to our net that we drew out lets put some measurements
into the diagram. (Use a ruler and measure the actual size and label the edges of your net
on the board.) Talk about finding area of each piece and fill in the numbers below.
8
6
Top
6
6
Back View
Left
Front View
5
Right
Bottom View
Fill in the missing numbers:
Front Area:
Back Area:
Left Side Area:
Right Side Area:
Top Area:
Bottom Area:
SURFACE AREA:
__________(8x5)
__________(8x5)
__________(6x5)
__________(6x5)
__________(8x6)
__________(8x6)
__________
*Discuss other times, besides wrapping presents, that the concept of surface area might
be useful in the real world.
Guided Practice:
-Provide more examples like the one above where you show the net and have the students
calculate the area of each piece and add to find surface area. Take time to make sure
students fully comprehend how to figure out the measurement of each edge.
-Students should come to the conclusion with the rectangular prism that there are two
pieces (faces) for each set of numbers. For example: When the dimensions of the prism
measure 3x4x5 -------- 2 faces have an area of (3x4); 2 faces have an area of (3x5); and 2
faces have an area of (4x5)
Independent Practice:
Provide activity sheet like one attached.
Notes for Teachers:
Remind students who are having trouble to break it down into the net version first.
ACTIVITY SHEET
Calculating Surface Area of Rectangular Prisms
10
5
4
B
7
A
Above are two sample stands they have in a store. You want to buy one to put in your
room at home. You’ve found some design paper you wish to cover the surface with.
However, you only have 220 square feet of design paper. Do you have enough paper to
buy either box or only one? Support your decision by showing your work.
Posttest Form B
1. _________ measures the space inside a 2-dimensional figure.
2. _________ measures the space inside a 3-dimensional figure.
3. State the formula for finding area of a rectangle and tell why you think it works. You
may choose to do this in words or with pictures.
4. State in your own words what surface area is and what it means to you. Be sure to
include the mathematical concepts of base area, side area, views, and nets in your
explanation.
5. Find the area of a rectangle with the following dimensions: 7 cm x 9 cm
6. Find the surface area of this rectangular prism. Show all work.
8
5
7. Explain the difference between a cube and a rectangular prism.
8. Based on your understanding, decide when in the real world you might apply the
concept of surface area.
a. State a real world use for finding surface areab. Explain why you chose to find surface area as opposed to finding just area.
9. Pretend you are telling someone else how to figure surface area, what do you tell
them?
10. Demonstrate your understanding of views and nets by creating an example of
your own (actually construct the 3-D figure). Sketch the 3-D view of the rectangular
prism OR the net of it, choose your own dimensions for it (label them on your
sketch). Then, in an organized manner, show all the steps to calculating the surface
area.
Scoring Guides for Items on Post-Test
Item #3
2 points. States the formula for finding the area of a rectangle and tells why they think it
works.
1 point. States the formula for finding the area of a rectangle or tells why they think it
works.
Item #4
2 points. Student explains what surface area is, including 4 of the required terms and
what it means to them.
1 point. Student explains what surface area is including less than 4 of the required terms,
or what surface area means to them.
Item #6
2 points. Student found correct answer and shows complete process
1 point. Student found correct answer, but lacked complete process or student attempted
to find the correct answer , but may or may not have made computation errors leading to
an invalid answer
Item #8a and 8b
2 points. Student’s explanation is complete.
1 point. Student’s explanation is partial.
Scoring Guide for Student Performance Item
Item #9
4 points. The student's response fully addresses the performance event.
The response demonstrates knowledge of the mathematical concepts and principles
needed to complete the event. The response communicates all process components that
lead to an appropriate and systematic solution. The response may have only minor flaws
with no effect on the reasonableness of the solution.
3 points. The student's response substantially addresses the performance event. The
response demonstrates knowledge . . . The response communicates most process
components. . . The response may have only minor flaws with minimal effect on the
reasonableness of the solution.
2 points. The student's response partially addresses the performance event. The
response demonstrates a limited knowledge. . . The response communicates some process
components . . . The response may have flaws or extraneous information that indicates
some lack of understanding or confusion.
1 point. The student's response minimally addresses the performance event. The
response demonstrates a limited knowledge . . . The response communicates few or no
process components. . . The response may have flaws or extraneous information that
indicates lack of understanding or confusion.
0 points. Other.