goal of this lesson is provide students hands on experience building
nets and creating rectangular prisms. To help build this understanding,
students will use manipulatives to explore the relationships between
nets, 3-dimensional figures, volume, and surface area. Polydrons are
plastics pieces with hinged edges, which connect to each other to form
polyhedra, and should be used if available. Alternately, use Klikkos, a
similar type of manipulative.
Cube Net Built out of Polydrons
Cube Built out of Polydrons
If Polydrons and Klikkos are unavailable, square pieces of paper
and tape can be used. While it is possible to build the shapes out of
paper, they will not be as easy to alter. Each group will need many
squares, so cut these out ahead of time. Keep the squares relatively
small with a side length of about 2 inches. Also, interlocking cubes,
if available, can be used for the surface area analysis.
Hand out the Building Blocks
activity sheet. Divide students into groups of 3 and give each student
in the group 6 Polydrons. Ask students to connect their Polydrons to
form the shape shown in Question 1. Draw this net on the board so
students can use this as a comparison when creating their additional 10
nets in the questions that follow.
Have students fold the net up into a cube and have a discussion
about nets and how they are used to create 3-dimensional objects. Do
not mention surface area at this point. Challenge students to create 10
additional nets that create a cube. When a student suggests that they
have new net, have them first ensure it folds up into a cube and then
bring their net to the board rotating it to make sure it is not a
duplicate. Then, have them draw their net on the board. Students should
also sketch these nets on the graph in Question 2. After students
believe they have the 11 nets, they can use the Cube Nets tool to check their solutions if computers are available.
Allow time for groups to complete the activity sheet. Walk among
the groups, and observe how students answered Question 5. Did they use
the formula and subtract the top or did they add up the areas of the
sides? Ask students how they can use their Polydrons to answer the
question. Students may also have other ideas. Use this example for a
class discussion on how the surface area formula will not calculate the
correct answer unless students understand its meaning. Ask students how
they calculated surface area and use the opportunity to discuss how
students can use different methods to solve the same problem.
If groups finish early, ask them to predict the ratio of two
surface areas and the ratios of the two volumes for a cube in which
they assume each side has length 3 and go through the same steps they
used for the 2×2×2 cube analysis.
Have a classroom discussion about Question 12. Many students
intuitively think the surface areas and volumes double. If students
have not discovered it on their own, discuss that the ratio of the
volumes is the cube of the ratio of the sides, and the ratio of the
surface areas is the square of the ratio of the sides. This will be
more apparent to groups that have time to consider the 3×3×3 situation.
Distribute the Fishing for the Best Prism
activity sheet. As a class, students will suggest 3 different
rectangular prisms that have a volume of 8 cubic units. Based on the
total number of groups in the class, divide up the 3 configurations.
Multiple groups can work on the same configuration if there are more
than 3 groups. Each group should complete pages 1 and 2.
Students may need help creating the net in Question 2 because
they may start immediately to build the prism. Have models of both a
net and an isometric drawing of a rectangular prism that students can
reference as they work on page 1. Review students’ answers for the
volume, surface area, and surface area of a fish tank. If necessary,
show student how to rotate their rectangular prism to find the various
configurations for the fish tank.
As a class, review the answers to the questions on pages 1
and 2. The discussion should build on their discoveries from the
Building Blocks activity. Students investigated a real-life open ended
problem in Questions 5–9. A tropical fish company hired the students to
build the most appealing fish tank. In Question 9, students may not
choose the cheapest fish tank because it may not be the most appealing.
It is an important question. Students apply math to a practical
problem, which should help them realize that a mathematical solution is
an option but there are other factors that influence decisions.It is
important for this activity that you allow students to be creative.
To conclude the activity, write the questions on page 3 on the
board, allowing ample space under each for answer. Assign each group to
begin with a different question, and then have groups walk from
question to question and write their answers underneath. If for a
particular question their answer is the same as another group, they
should be encouraged to create a picture or alternative explanation
that helps illustrate the answer. Once all groups have contributed
their answers to all questions, review the answers as a class.