Whatcom Math Project
The Picked aPart Polyhedron
– nets and properties
Course: Geometry
Grade Level: 9 - 11th
College Readiness Standard Name(s) and Number(s):
Standard 1: Reasoning/ Problem Solving
The student uses logical reasoning and mathematical
knowledge to define and solve problems.
Standard 4: Number Sense
The student accurately describes and applies concepts and procedures related
to real and complex numbers.
Standard 5: Geometry
The student makes hypotheses, models situations, draws conclusions, and
supports claims using geometric concepts and procedures.
Student Attributes:
Demonstrates intellectual engagement
Is willing to take risks and be challenged as part of the learning process
Perseveres when faced with time-consuming or complex tasks.
Recognizes when an approach is unproductive and makes logical modifications
to that approach or switches to another approach.
Student Learning Outcome − College Readiness Standard Component and Number:
Component 1.2 Formulate a plan for solving the problem.
Component 1.3 Use logical reasoning and mathematical knowledge to obtain and justify
correct solutions.
Component 4.3 Apply estimation strategies using real numbers
Component 5.1 Make conjectures about 2-D figures and 3-D figures, or figures constructed
from these shapes.
Component 5.3 Use properties of and relationships between 2-D and 3-D figures to draw
and justify conclusions about a situation represented with such figures with
or without a coordinate system.
Learning objectives:
1.2B
Choose concepts, strategies, representations, models, and tools well suited to
solving the problem.
1.3A
Correctly execute a plan to solve the problem.
1.3B
Evaluate and revise the solution method when it appears unlikely to produce a
reasonable or suitably accurate result.
1.3D
Provide oral, written, and/or symbolic explanations of the reasoning used to obtain a
solution.
4.3A. Recognize and justify the sufficiency of estimating or approximating the result of a
calculation involving real numbers.
5.1A
Use physical, symbolic, and technological models to explore conjectures.
5.3B
Apply and justify the application of transformations, congruence, similarity, ratios,
and proportions in problem solving situations.
5.3E
Use the Pythagorean Theorem (or distance formula) in 2-D and 3-D situations when
appropriate to compute unknown distances.
5.3F
Calculate the volume and surface area of spheres, right rectangular prisms, and right
circular cylinders, right prisms ,right pyramids, and right cones
________________________________________________________________________________
Prerequisite Skills: Students should have a working knowledge and understanding of
using a protractor, a compass, and a ruler.
 Measuring in mm and cm (you could use standard measurements as well).
 Constructing and measuring specific angles with a protractor.
 Helpful to be able to construct an equilateral or isosceles triangle with a compass,
but not required.
Students should be able to find the area of rectangles, parallelograms, and triangles
(other shapes: trapezoids, circles, etc. would be helpful). Students should be able to
sketch heights for triangles and uniform quadrilaterals (parallelograms, rhombus, etc).
Lastly, the students should be able to apply the Pythagorean Theorem to solve
problems.
Materials for Students: Handout of Assignment, a compass, a ruler, a protractor, scissors,
11X17 paper, felt pens, glue sticks/tape and a piece of letter size card stock (weight not
important).
Teaching Aids: Method for getting Resource Page #1 for full class to see, props for
discussing volume of prisms, Resource Page #2 (as needed). Might be helpful:
whiteboard-size compass, meter/yard stick, large protractor.
Estimated Time For Completion: 35-50 minutes per handout. (105 -150 minutes total)
Not all work has to be done in class.
References: http://home.tiscali.nl/gibbon/exploring/ken/15.gif
http://faculty.shelton.cc.al.us/~tevans/HTML%20Powerpoints%20for%20MTH%2023
2/three_dim_figures_11_4_files/frame.htm
http://faculty.shelton.cc.al.us/~tevans/HTML%20Powerpoints%20for%20MTH%2023
2/three_dim_figures_11_4_files/slide0015_image011.gif
_______________________________________________________________________
PREASSESSMENT: Students should have been introduced to a basic net, even if it is just
during the pre-assessment questions (see the six pre-assessment questions). The preassessment also allows the teacher to find any holes in the foundation needed to
complete the lesson. If the student has a tough time measuring angles or can’t apply the
Pythagorean Theorem to find distances, these issues need to be addressed before the
lesson is started. If a large part of the class is struggling, you will need to move back
and re-teach these topics so the students can be successful with this lesson.
I.
INTRODUCTION: There is a valuable website above. The second link in the resource
section can be used to walk students through the idea of polyhedrons and the
vocabulary that will be used. (It could be used as a starter, clicking through selected
parts (slides) and discussing with the students)
II. & III. LESSON/APPLICATION: It is assumed that students will be working in groups of
three or four. If this is not the case, the teacher needs to make the necessary
adjustments to the handouts and to her/his application of the lesson.
The lesson should start with a version of the Resource Page #1 net on the overhead as
the teacher hands a copy of it to each of the groups. During this time the teacher
should:
 Define what a net is.
 Define a polyhedron.
 Give a general overview of the three parts of the lesson.
 Inform the students of the individual product each will produce.
 Place a “tab” on the overhead net, discuss what it is and why it is placed there so
students have the proper idea.
Students should now be allowed time to work through numbers 1 to 5 from the first
handout. The teacher should walk and talk, promoting group discussions. It is up to the
teacher if she/he wants to help on the placement of the tabs. Sometimes poor
placement and later difficulties will create an environment for deeper understanding.
Having tabs in the incorrect spot does not stop the student from creating the polyhedron.
Each student is supposed to place tabs on her/his individual net later in the lesson.
Question 5 is one you want to check in on because the definition comes up again in the
lesson.
The tabs are unnecessary to complete the polyhedron. If the teacher simply wants to
use tape for sealing the polyhedrons it works. The use of tabs, however, forces the
students to visualize what the shape will look like and think about how the faces will line
up after folds are made.
Depending on students past work with volume, the teacher should plan this part of the
lesson accordingly. The teacher should bring the class together as groups reach
question six. If 80% of the students have completed five, it is time to have a discussion
about finding the volume of a right prism. Students should be asked to put down pencils
etc. A stack of paper is helpful to show how individual rectangles can stack and make a
rectangular prism. A small sticky notepad can be cut diagonally to help in discussions of
triangular prisms. The teacher might want to get out physical examples of prisms as
well. In any case, the students need to gain some understanding to be able to use the
V = Bh equation.
Question 7 should go smoothly. Question 8, finding the length of the diagonal through
the center of the polyhedron will cause some difficulties. It is up to the teacher whether
he/she wants to include the d  l 2  w2  h2 formula in the discussion or not.
Question 9 allows for the students to make choices. It is a good time to push some
students into choosing a more difficult prism. Challenge them. There are some
resources on the web mentioned in the Resource section above that the teacher can use
as examples, as well as Resource Page #2. The teacher should do a bit of reflection
on how difficult of shapes she/he wants to open the lesson up to. Pyramids?
Dodecahedrons? He/she can control this a bit by the choice of examples and checking
in with students as they move from question 9 to 10. If students choose one that is too
difficult, what do you fall back on? However, it is a student-centered learning
environment and it might be the best time to introduce the 1/3 volume idea.
A dodecahedron with regular pentagonal sides can be made by placing 12 identical right
regular pentagonal pyramids together so that each apex of the pyramids meets at the
center of the polyhedron. If you can find the volume of one of these, (a strong student
should be able to with the “1/3 volume idea” help mentioned above) you can find the
volume of the dodecahedron.
Depending on your schedule, this might be a good time to wind things down. Question
number 9 could be done at home, and possibly number 8 if the student is able to take
home one of the models.
If the teacher does not want to work with scale factors ( r; r 2 ; r 3 ) at this time in his/her
curriculum, the students could do their work directly on the card stock. However, the
teacher would need to edit the handouts a bit.
Make sure students have a sketch to work off of for the construction and have the large
paper and tools ready to go. It always seems to work best if the desks/tables are
completely cleared off. Walk and check as students construct the nets. If the nets are
sloppy or skewed the polyhedrons will be brutal. As they complete the work, hand each
a copy of Handout #2.
Questions number 1 and 2 on Handout #2 should cause some discussion. Try and stop
students from doing the actual math. Prompt them to visualize and mentally
approximate side areas and then total these. Next, estimate the area of the Base and
multiply this by the height. The rest of the questions on #2 are pretty straight forward
and should not cause too much problem. Make sure you have planned for the volume of
a pyramid dilemma.
--------- The first section of the lesson is complete -------------------
If you haven’t covered the material yet in your curriculum, you will need to have a
discussion and some practice with scale factors of 2 and 3 dimensional shapes. Some
texts call this the r; r 2 ; r 3 theorem. The scale reduction (r value) is 11/17 for this lesson.
Students will need to square it to find the new surface area and cube it to find the new
volume. You will want to check your copy machine and make sure you can do 65% or
better yet if it has a reduction to “letter” size option.
Between the second and the third handouts, the teacher needs to go to the copy
machine and reduce all constructions to letter size using card stock (if it works in your
copier). You probably want to choose a construction that is of high quality and make
some extra versions to account for absent students and any other mishaps. The
corresponding values from Handout #2 questions 7 and 8 needs to be made available to
these students.
The students can pick up the reduced versions and Handout #3 at the same time.
Questions 1 and 2 are a big part of the lesson. Does the r; r 2 ; r 3 theorem really work?
Students need to have access to their work on the second handout for questions 7 and
8.
Question number 4 is trying to stop the students from simply using the ruler to measure
from a vertex to one that is two or more edges away. At the same time, depending on
the net, the students might not have the tools (law of sines, cosines, etc) to find all
the answers and the teacher needs to be ready to address questions about this!
How far is it between the farthest two points in an octahedron? The teacher might have
to allow students to make their best estimation.
Questions 5 and 6 allow the students to compare the two similar polyhedrons and see
how the total surface area changes at a different rate than the volume.
Question 8 might need some support depending on the polyhedron.
Once again, not all questions need to be done in class. The students have their own
model and could take it home and use it.
Follow-up questions that might be asked:
 We did a reduction of the polyhedron, would the scale factor method of finding
surface area and volume work if we increased the polyhedron’s size?
 Does this method only work for polyhedrons or is it transferable to models of
buildings, statues, or blueprints of the city?
 What if we only shrunk the net in one direction? Meaning that the net kept the same
width but vertically it was reduced to 11/17ths of the original size.
 Could we still make the polyhedron?
 Would our scale factors need to change since we only reduced in one
direction? If so, how?
 Are there any other problems that might arise from doing this?
IV.
EVALUATION:
During Class –
Walking and working with students
Correct completion of the assignment
In-between Handouts Checking assignments as needed
Later –
Students work on all three handouts
Assessment questions verbal and written
V.
EXTENSIONS:
Can the students transfer the new knowledge/understanding to other class work?
Written word problems regarding nets and ratios between changes in edge lengths of
three dimensional models and the effect it has on the surface area and volume.
Example: Michael has made a model of a statue he is planning on making for his
senior project. The model is 16 inches tall, took one gallon of paint to cover, and
weighs 3.5 pounds. If he makes the actual statue 6 feet tall, how much paint will he
need and what will it weigh?
If you have access to a computer lab and a CAD (computer added design) software
package, the students could “construct” their polyhedron in this environment.
Resource Page #1
Resource Page #2
Handout # 1
The Picked aPart Polyhedron – nets and properties
Name
Please look at the net (resource page #1). The lines with the numbers 1 through 5 on them are the
ones you will fold along. The angles labeled A through F will be referred to in some of the following
questions. The heavy lines mark the net for the 3-dimensional shape we will be investigating but
please study all the lines and marks. They were used to help in the construction.
Visualize
1. If you simply cut out the net and fold it there will be no way to keep it together. Your first job is to
decide as a group where to put “tabs” so that a little glue can be used to get nice edges on your final
product. It is best if the tabs look like isosceles trapezoids with the bigger base being the edge of the
net.
2. Once you agree on the placement of the tabs, sketch them in and cut out your net. Don’t fold it
yet.
3. Please make some predictions about the net. Estimate the answers to the following questions.
Answer to the nearest half cm. After it is folded into a prism . . .
a) how far will vertex A be from vertex B?
b) how far will vertex A be from vertex C?
c) how far will vertex A be from vertex D?
d) how far will vertex A be from vertex E?
e) how far will vertex A be from vertex F?
4. Now it is time to fold up and glue your net. Fold it up so that the letters (A, B, C,) are on the
outside. It works best if you “pre-fold” each edge and tab.
5. You should now have a “Right Rectangular Prism.” Using your book and your teammates, come
up with a definition for that term and write it below.
6. Please pay attention as the teacher leads a discussion on the equation V = Bh. Now use this
equation to find the volume of the Right Rectangular Prism you just folded from the net.
Use this rectangle as a model of your Base, label it with the measurements from your prism, and find
its area. Then use this answer to find the Volume. Show all work and circle your answer. Make sure
you have the correct units.
7. Find the surface area of the Right Rectangular Prism. For all problems involving surface area,
please use the method indicated below. Show all work. (units?)
2 sides
2x
=
2 sides
2x
=
2 sides
2x
=
Grand total of all sides =
8. Calculate the answers to the following questions (no rulers). Show work when needed.
a) How far is vertex A from vertex B?
b) How far is vertex A from vertex C?
c) How far is vertex A from vertex D?
d) How far is vertex A from vertex E? (Things to think about: Is triangle ACE a right triangle?
Triangle ADE? Triangle ABE?)
e) How far is vertex A from vertex F?
f) Verify answers with a ruler.
9. Now it is time for each of you to make your own net. It needs to be a polyhedron and it can’t be a
Right Rectangular Prism. Your teacher has some examples for you. Please decide on one that
interests you personally and get out a piece of paper and make a sketch of its net.
10. Have members in your group decide if your net will work or not. Once you find a net that will
work, get a piece of the large paper and construct your net using a protractor, compass, and a ruler.
Handout # 2
The Picked aPart Polyhedron – nets and properties
Name
Please leave all marks used in the construction of your net on the paper. When you are happy with
your construction, place tabs where they are needed. Ask the teacher for a felt pen and, using a
straight edge, trace your net’s outline. Do not trace the tabs.
1. Estimate the total surface area of your polyhedron in square cm.
2. Estimate the volume of your polyhedron in cubed cm.
3. Place an ‘A’ and a ‘B’ on vertices of your net in such a way that after the net is folded and taped
they will be as far away from each other as possible.
4. Place a ‘C’ on a vertex of your net so that after it is folded it will be at the same vertex A.
5. Place a ‘D’ on a vertex of your net so that after it is folded it will be one edge length away from A.
6. Place an ‘E’ on a vertex of your net so that after it is folded it will be two edge lengths away from
A.
7. Compute the total surface area of your polyhedron. Please use the method discussed and used
for the Right Rectangular Prism.
8. Compute the volume of your polyhedron. (This may be tough for some of you, especially if you
made a type of pyramid. We will discuss it.)
Your net is going to be reproduced on the copy machine at 11/17 (about 65%) of its present size.
You are then going to cut it out and fold it and find its surface area and volume.
Handout # 3
The Picked aPart Polyhedron – nets and properties
Name
Referring back to your answers to numbers 7 and 8 on Handout #2 and the fact that the net was
reduced to 11/17 (65%) of the original, answer the following questions.
1. What should be the total surface area of your reduced polyhedron? Please use the appropriate
scale factor (r, r squared, r cubed) talked about in class. Show all work.
2. What should be the total volume of your reduced polyhedron? Please use the appropriate scale
factor talked about in class. Show all work.
You have the reduced version of the net you constructed. Cut it out, fold it, and glue or tape it
together.
3. Does the polyhedron look as you visualized it when you were making the net? If ‘No,’ why not?
4. Find the answers to the following questions in cm. You can only measure edge lengths; you must
compute any diagonals needed. Show all work. You might not be able to find them all.
a) How far is vertex A from vertex C?
b) How far is vertex A from vertex D?
c) How far is vertex A from vertex E?
d) How far is vertex A from vertex B? (Try to visualize and sketch any right triangles that
might help in the calculation.)
5. Please compute the surface area of the polyhedron using the method we have discussed in class.
6. Now compute the volume of the polyhedron using the V = Bh method.
7. Check your answers against your computations to questions #1 and #2, are they close? Explain
any discrepancies.
8. Try and name your polyhedron with the correct geometric term. Think of the one you did in
groups.
Congratulations you have completed the full assignment.