Summary
This activity will teach the basics of surface area and volume.
Materials
Invitation to Learn
The Great Cover Up
- X/Y-axis Dry Erase
Mats
- Key Graphing Cling
- Pre-made rectangular
prisms
- Rectangular prism net
patterns
- Markers
- Cubes
- The Great Cover Up (pdf)
- Graph paper
Cut it Out!
Additional Resources
Books
Math Dictionary-The Easy, Simple, Fun Guide to Help Math Phobics Become Math Lovers, by
Eula Ewing Monroe; ISBN 978-1-59078-413-6
Background for Teachers
In order for students to visualize and determine surface area and
volume of three-dimensional shapes, it is important to manipulate
actual objects. They must also have experience with the concepts of
area, nets and rectangular prisms prior to the introduction of this
standard and objective (Standard IV Objective 2). When talking about
area, we are referring the measurement of a two-dimensional shape.
When talking about surface area, we are referring to the measurement
of a three-dimensional shape.
The surface area of a prism is the sum of the areas of all the faces,
including the bases. The surface area is measured in square units.
Although the students may not be familiar with this concept, they
have actually experienced it as they've worked with nets. Surface area
takes nets one step further by determining actual measurements. The
purpose of this lesson is to help the students make this connection.
The mathematical formula for surface area is:
SA= 2(l · w) +2(l · h) + 2(w · h)
The actual formula for surface area involves using length, width
and height, at this level. Having the students find the area of each face
and then adding them together as square units to find the total will be
less confusing for fifth graders.
The volume of a prism tells how many cubic units it takes to fill the
prism. Volume is measured in cubic units. The mathematical formula
for volume is:
V= l · w· h
Instructional Procedures
Invitation to Learn
- Put students into small groups of two to four, and give each
group a container with at least 48 cubes.
- Challenge each group to build as many different regular
rectangular prisms that have a volume of 12 cubic units.
- Once a prism has been built, ask them to set it aside to keep
as an example so that the same prisms are not repeated. (With
12 cubes, they can build four different rectangular prisms:
1x2x6, 1x3x4, 1x1x12, and 2x2x3.)
- Demonstrate the four different prisms that twelve cubes can
make by having a few volunteers stand and describe their
prisms -- length, height, and width. Because of previous lessons
on volume, the students should be able to describe their prisms
using length, width and height. A discussion on whether a
different orientation makes a difference in the dimensions may
be needed.
- Review the definition of volume with your students.
- As a class, find the volume for the four different prisms using
length, width, and height.
Length |
Width |
Height |
Volume |
|
|
|
|
- On chart paper, record the students findings . Highlight the fact
that because each prism is made of twelve cubes, each has a
volume of 12 cubic units.
Instructional Procedures
The Great Cover Up
Use the concept of area with 2-D measurement, using a 3-D
prism (1 x 2 x 3), to introduce the concept of surface area or 3-D
measurement. Ask the students to predict if the prisms have the
same volume, do they have the same surface area. Come back to this
question at the end of the Cut it Out! activity.
- Show the class the pre-made prism or 3-D object and ask
the students how they can use area to determine how much
wrapping paper would be needed to cover the entire prism
without any overlapping.
-- This measurement is called surface area and can only be found
for 3-D objects.
- Discuss their ideas for ways to measure the surface area.
- If they struggle coming up with a solution, bring out nets that
were used in previous lessons, to visualize the connection
between nets and surface area.
- Review area and its formula by finding the area of each
face. This discussion is critical in helping students make the
connection between area and surface area.
- With the help of the students, use the Key Graphing Cling to
model how to draw the net of the pre-made prism.
- Use the concept of a room to help them visualize each surface
as it is being drawn: floor, ceiling, four walls: two front/back
walls, two side walls.
- Use these same terms to label each part of the net.
- Discuss how to find the area of each face, leading to the idea
that adding all areas would give the total area or surface area.
- Discuss how using just letters rather than whole words as
labels can be simpler.
- Have students come up with a formula for surface area using
only letters:
SA = a + a + b + b + c + c
(f=area of floor and ceiling; b=area of front and back wall; c=area
of side walls)
- Plug in actual area for each face underneath formula. Give
students The Great Cover Up to record data with you.
- Repeat this process of creating nets together using different
rectangular prisms.
- Next, pass out individual 9x11 Double sided X/Y-Axis Dry Erase
Mats and dry erase markers so students can draw more nets
with you. Continue to use worksheet to determine and record
the surface areas.
- Now using graph paper and the cubes.
- Have students work in small groups to reconstruct the original
four 12-cube prisms and have them draw nets for each one.
- Fill out The Great Cover Up for each prism.
- Save their nets for the next activity.
Cut it Out!
- Use chart made in Invitation to Learn and referring to the
first prism listed, have all students find that net and cut it
out.
- Fill out #1 on Cut it Out together.
- Students will repeat the process for the remaining three
prisms.
- Review their predictions of the connection between surface
area and volume.
- In their math journals, have students write what they learned
about surface area and volume. Have them analyze the pattern
they learned from their Cut it Out. Have the students explain
how to use a formula to find volume, and how they would find
the surface area of a prism.
Extensions
- Prism Race
(pdf)
- Challenge students to find out how many different rectangular
prisms they can make with a volume of 36 cubes. Encourage
them to think of ways to make sure they have made all possible
prisms.
- Challenge students to make as many prisms as possible with the
same surface area. Have them record the volumes of their solids
and note any patterns.
- Art: Draw three-dimensional rectangular prisms on isometric
dot paper.
- Design a net that can be folded into a rectangular prism that can
hold 24 Multilink cubes.
Family Connections
- Have students find at least five rectangular prisms from
household items. Have them measure the length, height, and
width of each item. Record the measurements in their math
journals, and then find the volume and surface area of each of
the items.
Assessment Plan
- Informal assessment includes class discussion, math journals
and observation of group/partner work.
- The Great Cover Up
- Cut it Out
- Birthday Boxes
Bibliography
Sowel, E.J. (1989). Effects of manipulative materials in mathematics instruction. Journal for
research in mathematics education, 20 (4), 498-505.
This review of research sums up the result of sixty studies
addressing the effectiveness of manipulatives on student learning and
attitudes in mathematics teaching. Sowell concludes that the more
concrete the manipulatives, and the longer the time spent using them,
the better instructional outcomes.
Hinzman, K.P. (1997). Use of manipulatives in mathematics at the middle school level
and their effects on students' grades and attitudes. ERIC Source (ERIC # ED411150).
Retrieved December 10, 2006, from http://www.eric.ed.gov
This paper reports on a study that examines mathematical scores
when hands on manipulatives and group activities were used in the
classroom. Results indicate that student performance was enhanced
by the use of manipulative materials; and students' attitudes toward
mathematics were significantly more positive than those in previous
years when manipulatives were not used.