Designing Boxes: A Lesson in Perimeter, Area and Volume
Teacher Materials (Part One)
TC-1:
Pre-assessment:
Give each student a reflection sheet. Preface the activity with the fact that their
answers are only to help you to know where they are, and that it is not an evaluative
process. Encourage them to write everything that comes to mind. Give them a few
minutes to write about the four main concepts included in the upcoming lesson.
TC-2
General Comments:
This lesson begins with the simplest instance of box design. Students will initially
design only cubes and compare the various attributes in an effort to formalize how edge
length, area and volume are related. The scenario of working as a team within a
company helps to remind students that collaborative efforts are a part of the working
world. Similarly, the need to support the scenario during the first small group activity is
important to the second small group activity when the teams will be asked higher level
questions and will have to collaborate more.
Small group activity #1:
Intent: understand how a change in one linear dimension affects surface area
and volume. (GLE 1.2.1)
A believable introduction about the reality of working in a world where
cooperation and collaboration are necessary skills to succeed will greatly increase the
success of the lesson. Discuss the need for conversation, self and group accountability
as integral parts of the experience. Depending on the comfort level and experience
your class has with collaborative work, you may want to let the students know that there
will be a formal self and peer evaluation at the end. Set the stage for the Acme Box
Company.
You may need to model the drawing of a net. If you do, emphasize the need to
draw the base first, near the center of the paper to allow room for the net to expand.
Likewise, it is helpful to give a reminder here of the importance of clear record keeping
to hopefully find shortcuts for area and volume.
Designing Boxes
Teacher Solutions Page 1 of 42
Teacher Resource
Name: ____________________________
Per: _________ Date: _______________
Take a moment to think about the following vocabulary terms. Write everything you
know about each one. Then answer the questions on the back of this sheet.
Perimeter:
Distance around the outside of a figure
Area:
Amount of “space” inside a 2-D figure
Volume:
Amount of “space” inside a 3-D figure
9
10
26
12
1. Find the missing side length.
2. If the area of a rectangle is 3. Find the missing side length.
four times larger than another,
how much longer would you
expect its side length to be if
the rectangles are similar?
Should be twice as long for 2-D
figure
c = 9 + 12
[Accept the examples the
students give you and use that
knowledge to lead them to the
goals of the lesson.]
2
c = 15
Designing Boxes
2
b = 26 2 − 10 2
b = 24
Teacher Solutions Page 2 of 42
Use as a description of the net process if needed.
The Cube
The Net
Designing Boxes
Teacher Solutions Page 3 of 42
Group A
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Group B
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Group C
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Side Lengths:
Side Lengths:
Side Lengths:
1 cm
4 cm
7cm
2 cm
5 cm
8cm
3 cm
6 cm
9cm
Group D
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Group E
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Group F
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Side Lengths:
Side Lengths:
Side Lengths:
1 cm
4 cm
7cm
2 cm
5 cm
8cm
3 cm
6 cm
9cm
Group G
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Group H
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Group I
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Design Team
Side Lengths:
Side Lengths:
Side Lengths:
1 cm
4 cm
7cm
2 cm
5 cm
8cm
3 cm
6 cm
Task Card Master [ Print on Cardstock]
Designing Boxes
Teacher Solutions Page 4 of 42
9cm
Name: Group A, B, C
Per: _______ Date: ________________
Welcome to the Acme Box Company (ABC) team. Before moving into the design
department and meeting customer demands, we are going to look at the basic
properties of all boxes. Your team will be given a number of tasks to complete in order
to understand how box sizes work. As you are given each scenario, be sure to keep an
organized list of your results so the whole team can learn from your work.
Your task will be to calculate the Total Edge Length and Total Surface Area.
Group A
Edge Length
1 cm
4 cm
7 cm
Total Edge Length
12 cm
48 cm
84 cm
Total Surface Area
6 sq cm
96 sq cm
294 sq cm
Edge Length
2 cm
5 cm
8 cm
Total Edge Length
24 cm
60 cm
96 cm
Total Surface Area
24 sq cm
150 sq cm
384 sq cm
Edge Length
3 cm
6 cm
9 cm
Total Edge Length
36 cm
72 cm
108 cm
Total Surface Area
54 sq cm
216 sq cm
486 sq cm
Group B
Group C
Explain how your team found the values you entered into the table.
Total Edge Length: Multiply the edge length by twelve.
Total Surface Area: Find Area of one face and multiply by six.
Find formulas that can be used to find the total edge length and the total surface area if
you know the edge length (s).
Total Edge Length = 12s
Total Surface Area: 6s2
Designing Boxes
Teacher Solutions Page 5 of 42
Name: Group D, E, F
Per: _______ Date: ________________
Welcome to the Acme Box Company (ABC) team. Before moving into the design
department and meeting customer demands, we are going to look at the basic
properties of all boxes. Your team will be given two tasks to complete in order to
understand how box sizes work. As you are given each scenario, be sure to keep an
organized list of your results so the whole team can learn from your work.
Your task will be to calculate the Single Face Area and Volume.
Group D
Edge Length
1 cm
4 cm
7 cm
Single Face Area
1 sq cm
16 sq cm
49 sq cm
Volume of Cube
1 cubic cm
64 cubic cm
343 cubic cm
Edge Length
2 cm
5 cm
8 cm
Single Face Area
4 sq cm
25 sq cm
64 sq cm
Volume of Cube
8 cubic cm
125 cubic cm
512 cubic cm
Edge Length
3 cm
6 cm
9 cm
Single Face Area
9 sq cm
36 sq cm
81 sq cm
Volume of Cube
27 cubic cm
216 cubic cm
729 cubic cm
Group E
Group F
Clearly explain below how your team found the values you entered into the table.
Single Face Area: Multiply the edge length by itself twice.
Volume of the Cube: Multiply the edge length by itself three times.
Find a formula that can be used to find the single face area and the volume of the cube
if you know the edge length.
Single Face Area = s2
Volume of the Cube = s3
Designing Boxes
Teacher Solutions Page 6 of 42
Name: Group G, H, I
Per: _______ Date: ________________
Welcome to the Acme Box Company (ABC) team. Before moving into the design
department and meeting customer demands, we are going to look at the basic
properties of all boxes. Your team will be given a number of tasks to complete in order
to understand how box sizes work. As you are given each scenario, be sure to keep an
organized list of your results so the whole team can learn from your work.
Your task will be to calculate the Length of the Diagonal of the cube. This is the
distance from one vertex, through the center of the cube to the opposite vertex.
Edge Length
1 cm
4 cm
7 cm
Length of Diagonal
1 3 ≅ 1.73
4 3 ≅ 6.93
7 3 ≅ 12.12
Edge Length
2 cm
5 cm
8 cm
Length of Diagonal
2 3 ≅ 3.46
5 3 ≅ 8.66
8 3 ≅ 13.86
Edge Length
3 cm
6 cm
9 cm
Length of Diagonal
3 3 ≅ 5.20
6 3 ≅ 10.39
9 3 ≅ 15.60
Explain below how your team found the values you entered into the table.
Length of Diagonal: Make right triangle across a face and calculate the distance. Use
that length as the leg of a right triangle along with the height as the
other leg. Use the Pythagorean Theorem to calculate the distance.
Find a formula that can be used to find the length of the diagonal if you know the edge
length.
Length of Diagonal = s 3
Designing Boxes
Teacher Solutions Page 7 of 42
TC-3
Whole Class Activity 1:
Because each team in a company has responsibilities that extend to the whole
company, you will need to tie the small group work together in a cohesive way. The
whole class activity can be handled in a number of ways depending on teacher
preference. A chart is included to help the class build a comprehensive list of the
various outcomes. The class can then discuss the relationships in the chart as a whole
Keeping the information organized will help to reinforce the following ideas:
• As the base side length is multiplied by x, the total edge length increases as a
linear function (12x).
• As the base side length edge and height are both multiplied by x, the surface
area increases as a quadratic function (6x2).
• As the base side length edge and height are both multiplied by x, the volume
increases as a cubic function (x3).
• As the base side length edge and height are both multiplied by x, the diagonal
increases as a linear function ( x 3 ).
Here the particular rule can be an extension to the activity. Students should leave
with the knowledge that changing a particular dimension has implications beyond simply
multiplying by the change. This activity can be modified to the level of the class as you
discover the limits of what they see in the chart.
Teams could also be prompted to visit other teams to find the missing parts of the
chart rather than using the whole class discussion format. This would require
conversation and discussion about the processes used. Because there are repeated
measurements, teams would have to verify their results as well.
Designing Boxes
Teacher Solutions Page 8 of 42
Acme Box Company Cube Compilation Sheet
Name: Teacher Resource
Look carefully at the table and look for patterns in the dimensions. Your work will
benefit from seeing how the length of the edge affects the various measurements. Use
the table to answer the questions below.
Edge
Length
1cm
Total Edge 12
Length
cm
2 cm
3 cm
4 cm
5 cm
6 cm
7 cm
8 cm
9 cm
s
24
cm
36
cm
48
cm
60
cm
72
cm
84
cm
96
cm
108
cm
12s
Face Area
1
cm2
4
cm2
9
cm2
16
cm2
25
cm2
36
cm2
49
cm2
64
cm2
81
cm2
s2
Total
Surface
Area
6
cm2
24
cm2
54
cm2
96
cm2
150
cm2
216
cm2
294
cm2
384
cm2
486
cm2
6 s2
Volume
1
cm3
8
cm3
27
cm3
64
cm3
125
cm3
216
cm3
343
cm3
512
cm3
729
cm3
s3
Diagonal
1 3
cm
2 3
cm
3 3
cm
4 3
cm
5 3
cm
6 3
cm
7 3
8 3
cm
9 3
cm
s 3
cm
1. Describe the change in the Face Area as the Length of an Edge doubles.
The Face Area increases by a factor of 4 or 22
2. Describe the change in the Face Area as the Length of an Edge triples.
The Face Area increases by a factor of 9 or 32
3. Describe the change in the Face Area if the Length of an Edge were to be made
15 times longer?
The Face Area increases by a factor of 225 or 152
4. Describe the change in the Volume as the Length of an Edge doubles.
The Volume increases by a factor of 8 or 23
5. Describe the change in the Volume as the Length of an Edge triples.
The Volume increases by a factor of 27 or 33
6. Describe the change in the Volume if the Length of an Edge were to be made 15
times larger?
The Volume increases by a factor of 3375 or 153
Designing Boxes
Teacher Solutions Page 9 of 42
TC-4
This is the opportunity to split the lesson into two parts. After using the chart as a whole
class discussion activity, you can conclude the lesson for the day. The next day, can be
started with the students revisiting the chart and copying the information onto their own
table. They can then answer the accompanying questions which will help to reinforce
how changing a linear measurement affects surface area and volume.
TC-5
Small group activity #2:
Start by explaining the work order to the class. This is a standard component of
all business operations. Remind them that they have to be clear in their calculations
and results. They can be faced with changes and must roll with the changes.
Communication among team members is critical.
This activity asks teams to meet customer constraints and work backwards to
meet the customer’s needs. Let them puzzle these out. The teacher has significant
control over the difficulty and frustration level of the teams. Use the excel spreadsheet
calculator to fill out some customer request sheets prior to the activity. You can remove
various measurements to vary the level of the question. Be ready to become a fickle
customer and drop off a change in the order just as teams finish calculations. This can
be a very real experience in the frustration of meeting customer demands.
This activity could be adapted to be used as an assessment as well.
Designing Boxes
Teacher Solutions Page 10 of 42
This is an example of how the excel calculator works. To produce an unlimited number
of solutions to the small group activity #2, simply change the values of A, B, and C to
generate outputs. The teacher can control the difficulty of the experience by assigning
various combinations to force groups to solve for different attributes.
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol. scalar
1
2
3
0.5
3
4
12
6
9
1.5
8
12
2
24
36
6
13
26
39
6.5
396
1512
3348
108
1
4
9
0.25
144
1152
3888
18
1
8
27
0.125
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
1
2
3
0.5
6
12
18
3
b
c
d
S.A.
S.A. scalar
Vol
Vol. scalar
8
9
16
24
4
18
27
4.5
13.45362
26.90725
40.36087
6.726812
972
3744
8316
261
1
4
9
0.25
432
3456
11664
54
1
8
27
0.125
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol. scalar
1
2
3
0.5
3
9
10
6
9
1.5
18
27
4.5
20
30
5
13.78405
27.5681
41.35215
6.892024
540
2088
4644
144
1
4
9
0.25
270
2160
7290
33.75
1
8
27
0.125
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol. scalar
1
2
3
0.5
6
2
12
12
18
3
4
6
1
24
36
6
13.56466
27.12932
40.69398
6.78233
648
2448
5400
180
1
4
9
0.25
144
1152
3888
18
1
8
27
0.125
Designing Boxes
Teacher Solutions Page 11 of 42
T-1:
Pre-assessment:
Give each student a reflection sheet. Preface the activity with the fact that their
answers are only to help you to know where they are, and that it is not an evaluative
process. Encourage them to write everything that comes to mind. Give them a few
minutes to write about the four main concepts included in the upcoming lesson.
Designing Boxes
Teacher Solutions Page 12 of 42
Teacher Resource
Name: ____________________________
Per: _________ Date: _______________
Take a moment to think about the following vocabulary terms. Write everything you
know about each one. Then answer the questions on the back of this sheet.
Perimeter:
Distance around the outside of a figure
Area:
Amount of “space” inside a 2-D figure
Volume:
1. Find the missing side length.
2. If the area of a rectangle is
3. Find the missing side length.
four times larger than another,
how much longer would you
expect its side length to be if
9
10
26
the rectangles are similar?
12
Should be twice as long for 2-
c = 9 + 12
c = 15
2
2
D figure
b = 26 2 − 10 2
b = 24
[This will be a difficult
question. Let them just throw
something out there so you
have something to work with
at the end of the lesson.]
Designing Boxes
Teacher Solutions Page 13 of 42
Amount of “space” inside a 3-D figure
Designing Boxes
Teacher Solutions Page 14 of 42
T-2
General Comments:
This lesson begins with the simplest instance of box design. Students will initially
design only cubes and compare the various attributes in an effort to formalize how edge
length, area and volume are related. The scenario of working as a team within a
company helps to remind students that collaborative efforts are a part of the working
world. Similarly, the need to support the scenario during the first small group activity is
important to the second small group activity when the teams will be asked higher level
questions and will have to collaborate more.
Small group activity #1:
Intent: understand how a change in one linear dimension affects surface
area and volume. [GLE 1.2.1]
A believable introduction about the reality of working in a world where
cooperation and collaboration are necessary skills to succeed will greatly increase the
success of the lesson. Discuss the need for conversation, self and group accountability
as integral parts of the experience. Depending on the comfort level and experience
your class has with collaborative work, you may want to let the students know that there
will be a formal self and peer evaluation at the end. Set the stage for the Acme Box
Company.
You may need to model the drawing of a net. If you do, emphasize the need to
draw the base first, near the center of the paper to allow room for the net to expand.
Designing Boxes
Teacher Solutions Page 15 of 42
Likewise, it is helpful to give a reminder here of the importance of clear record keeping
to hopefully find shortcuts for area and volume.
Designing Boxes
Teacher Solutions Page 16 of 42
The Cube
The Net
Designing Boxes
Teacher Solutions Page 17 of 42
Designing Boxes
Teacher Solutions Page 18 of 42
Group A
Investigate the following
edge lengths and be
prepared to report your
findings to the whole
Group B
Group C
Investigate the following
Investigate the following
edge lengths and be
edge lengths and be
prepared to report your
prepared to report your
findings to the whole
findings to the whole
Design Team
Design Team
Side Lengths:
Side Lengths:
Design Team
Side Lengths:
1 cm
4 cm
7cm
2 cm
5 cm
8cm
3 cm
6 cm
9cm
Group D
Group E
Group F
Investigate the following
Investigate the following
Investigate the following
edge lengths and be
edge lengths and be
edge lengths and be
prepared to report your
prepared to report your
prepared to report your
findings to the whole
findings to the whole
findings to the whole
Design Team
Design Team
Design Team
Side Lengths:
Side Lengths:
Side Lengths:
1 cm
4 cm
7cm
2 cm
5 cm
8cm
3 cm
6 cm
9cm
Group G
Group H
Group I
Investigate the following
Investigate the following
Investigate the following
edge lengths and be
edge lengths and be
edge lengths and be
prepared to report your
prepared to report your
prepared to report your
findings to the whole
findings to the whole
findings to the whole
Design Team
Design Team
Design Team
Side Lengths:
Side Lengths:
Side Lengths:
1 cm
4 cm
Designing Boxes
7cm
2 cm
5 cm
8cm
3 cm
6 cm
Teacher Solutions Page 19 of 42
9cm
Task Card Master [ Print on Cardstock]
Designing Boxes
Teacher Solutions Page 20 of 42
Name: Group A, B, C
Per: _______ Date: ________________
Welcome to the Acme Box Company (ABC) team. Before moving into the design
department and meeting customer demands, we are going to look at the basic
properties of all boxes. Your team will be given a number of tasks to complete in order
to understand how box sizes work. As you are given each scenario, be sure to keep an
organized list of your results so the whole team can learn from your work.
Your task will be to calculate the Total Edge Length and Total Surface Area.
Group A
Edge Length
1 cm
4 cm
7 cm
Total Edge Length
12 cm
48 cm
84 cm
Total Surface Area
6 sq cm
96 sq cm
294 sq cm
Edge Length
2 cm
5 cm
8 cm
Total Edge Length
24 cm
60 cm
96 cm
Total Surface Area
24 sq cm
150 sq cm
384 sq cm
Edge Length
3 cm
6 cm
9 cm
Total Edge Length
36 cm
72 cm
108 cm
Total Surface Area
54 sq cm
216 sq cm
486 sq cm
Group B
Group C
Explain how your team found the values you entered into the table.
Total Edge Length: Multiply the edge length by twelve.
Total Surface Area: Find Area of one face and multiply by six.
Find formulas that can be used to find the total edge length and the total surface area if
you know the edge length (s).
Total Edge Length = 12s
Designing Boxes
Teacher Solutions Page 21 of 42
Total Surface Area: 6s2
Designing Boxes
Teacher Solutions Page 22 of 42
Name: Group D,E,F
Per: _______ Date: ________________
Welcome to the Acme Box Company (ABC) team. Before moving into the design
department and meeting customer demands, we are going to look at the basic
properties of all boxes. Your team will be given two tasks to complete in order to
understand how box sizes work. As you are given each scenario, be sure to keep an
organized list of your results so the whole team can learn from your work.
Your task will be to calculate the Single Face Area and Volume.
Group D
Edge Length
1 cm
4 cm
7 cm
Single Face Area
1 sq cm
16 sq cm
49 sq cm
Volume of Cube
1 cubic cm
64 cubic cm
343 cubic cm
2 cm
5 cm
8 cm
Single Face Area
4 sq cm
25 sq cm
64 sq cm
Volume of Cube
8 cubic cm
125 cubic cm
512 cubic cm
3 cm
6 cm
9 cm
Single Face Area
9 sq cm
36 sq cm
81 sq cm
Volume of Cube
27 cubic cm
216 cubic cm
729 cubic cm
Group E
Edge Length
Group F
Edge Length
Clearly explain below how your team found the values you entered into the table.
Single Face Area: Multiply the edge length by itself twice.
Volume of the Cube: Multiply the edge length by itself three times.
Find a formula that can be used to find the single face area and the volume of the cube
if you know the edge length.
Designing Boxes
Teacher Solutions Page 23 of 42
Single Face Area = s2
Volume of the Cube = s3
Designing Boxes
Teacher Solutions Page 24 of 42
Name: Group G,H,I
Per: _______ Date: ________________
Welcome to the Acme Box Company (ABC) team. Before moving into the design
department and meeting customer demands, we are going to look at the basic
properties of all boxes. Your team will be given a number of tasks to complete in order
to understand how box sizes work. As you are given each scenario, be sure to keep an
organized list of your results so the whole team can learn from your work.
Your task will be to calculate the Length of the Diagonal of the cube. This is the
distance from one vertex, through the center of the cube to the opposite vertex.
Edge Length
Length of Diagonal
Edge Length
Length of Diagonal
Edge Length
Length of Diagonal
1 cm
4 cm
7 cm
1 3 ≅ 1.73
4 3 ≅ 6.93
7 3 ≅ 12.12
2 cm
5 cm
8 cm
2 3 ≅ 3.46
5 3 ≅ 8.66
8 3 ≅ 13.86
3 cm
6 cm
9 cm
3 3 ≅ 5.20
6 3 ≅ 10.39
9 3 ≅ 15.60
Explain below how your team found the values you entered into the table.
Length of Diagonal: Make right triangle across a face and calculate the distance. Use
that length as the leg of a right triangle along with the height as the
other leg. Use the Pythagorean Theorem to calculate the distance.
Find a formula that can be used to find the length of the diagonal if you know the edge
length.
Length of Diagonal = s 3
Designing Boxes
Teacher Solutions Page 25 of 42
T-3
Whole Class Activity 1:
Because each team in a company has responsibilities that extend to the whole
company, you will need to tie the small group work together in a cohesive way. The
whole class activity can be handled in a number of ways depending on teacher
preference. A chart is included to help the class build a comprehensive list of the
various outcomes. The class can then discuss the relationships in the chart as a whole
Keeping the information organized will help to reinforce the following ideas:
•
As the base side length is multiplied by x, the total edge length increases as a
linear function (12x).
•
As the base side length edge and height are both multiplied by x, the surface
area increases as a quadratic function (6x2).
•
As the base side length edge and height are both multiplied by x, the volume
increases as a cubic function (x3).
•
As the base side length edge and height are both multiplied by x, the diagonal
increases as a linear function ( x 3 ).
Here the particular rule can be an extension to the activity. Students should
leave with the knowledge that changing a particular dimension has implications beyond
simply multiplying by the change. This activity can be modified to the level of the class
as you discover the limits of what they see in the chart.
Teams could also be prompted to visit other teams to find the missing parts of the
chart rather than using the whole class discussion format. This would require
Designing Boxes
Teacher Solutions Page 26 of 42
conversation and discussion about the processes used. Because there are repeated
measurements, teams would have to verify their results as well.
Edge
Length
1cm
2 cm
3 cm
4 cm
5 cm
6 cm
7 cm
8 cm
9 cm
s
Total Edge
Length
12
cm
24
cm
36
cm
48
cm
60
cm
72
cm
84
cm
96
cm
108
cm
12s
Face Area
1
cm2
4
cm2
9
cm2
16
cm2
25
cm2
36
cm2
49
cm2
64
cm2
81
cm2
s2
Total
Surface
Area
6
cm2
24
cm2
54
cm2
96
cm2
150
cm2
216
cm2
294
cm2
384
cm2
486
cm2
6 s2
Volume
1
cm3
8
cm3
27
cm3
64
cm3
125
cm3
216
cm3
343
cm3
512
cm3
729
cm3
s3
Diagonal
1 3
cm
2 3
cm
3 3
cm
4 3
cm
5 3
cm
6 3
cm
7 3
8 3
cm
9 3
cm
s 3
cm
Acme Box Company Cube Compilation Sheet
Name: Teacher Resource
Look carefully at the table and look for patterns in the dimensions. Your work will
benefit from seeing how the length of the edge affects the various measurements. Use
the table to answer the questions below.
7. Describe the change in the Face Area as the Length of an Edge doubles.
The Face Area increases by a factor of 4 or 22
8. Describe the change in the Face Area as the Length of an Edge triples.
The Face Area increases by a factor of 9 or 32
9. Describe the change in the Face Area if the Length of an Edge were to be made
15 times longer?
The Face Area increases by a factor of 225 or 152
10. Describe the change in the Volume as the Length of an Edge doubles.
The Volume increases by a factor of 8 or 23
11. Describe the change in the Volume as the Length of an Edge triples.
The Volume increases by a factor of 27 or 33
Designing Boxes
Teacher Solutions Page 27 of 42
12. Describe the change in the Volume if the Length of an Edge were to be made 15
times larger?
The Volume increases by a factor of 3375 or 153
Designing Boxes
Teacher Solutions Page 28 of 42
T-4
This is the opportunity to split the lesson into two parts. After using the chart as a whole
class discussion activity, you can conclude the lesson for the day. The next day, can be
started with the students revisiting the chart and copying the information onto their own
table. They can then answer the accompanying questions which will help to reinforce
how changing a linear measurement affects surface area and volume.
T-5
Small group activity #2:
Start by explaining the workorder to the class. This is a standard component of
all business operations. Remind them that they have to be clear in their calculations
and results. They can be faced with changes and must roll with the changes.
Communication among team members is critical.
This activity asks teams to meet customer constraints and work backwards to
meet the customer’s needs. Let them puzzle these out. The teacher has significant
control over the difficulty and frustration level of the teams. Use the excel spreadsheet
calculator to fill out some customer request sheets prior to the activity. You can remove
various measurements to vary the level of the question. Be ready to become a fickle
customer and drop off a change in the order just as teams finish calculations. This can
be a very real experience in the frustration of meeting customer demands.
This activity could be adapted to be used as an assessment as well.
Designing Boxes
Teacher Solutions Page 29 of 42
This is an example of how the excel calculator works. To produce an unlimited number
of solutions to the small group activity #2, simply change the values of A, B, and C to
generate outputs. The teacher can control the difficulty of the experience by assigning
various combinations to force groups to solve for different attributes.
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol. scalar
1
2
3
0.5
3
4
12
6
9
1.5
8
12
2
24
36
6
13
26
39
6.5
396
1512
3348
108
1
4
9
0.25
144
1152
3888
18
1
8
27
0.125
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
1
2
3
0.5
6
12
18
3
b
c
d
S.A.
S.A. scalar
Vol
Vol. scalar
8
9
16
24
4
18
27
4.5
13.45362
26.90725
40.36087
6.726812
972
3744
8316
261
1
4
9
0.25
432
3456
11664
54
1
8
27
0.125
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol. scalar
1
2
3
0.5
3
9
10
6
9
1.5
18
27
4.5
20
30
5
13.78405
27.5681
41.35215
6.892024
540
2088
4644
144
1
4
9
0.25
270
2160
7290
33.75
1
8
27
0.125
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol. scalar
1
2
3
0.5
6
2
12
12
18
3
4
6
1
24
36
6
13.56466
27.12932
40.69398
6.78233
648
2448
5400
180
1
4
9
0.25
144
1152
3888
18
1
8
27
0.125
T-6
Assessment solutions are included below.
Designing Boxes
Teacher Solutions Page 30 of 42
Name: Solutions
Area, Perimeter, Volume and Pythagorean Theorem Quiz
1. Find the perimeter of a right triangle with legs measuring 10 inches and 24
inches
o a. 34 inches
o b. 60 inches
o c. 120 inches
o d. 240 inches
2. The side lengths of the base of right prism are doubled while the height is not
changed. Which of the following best describes the result on the volume of the
prism
o a. The volume remains the same
o b. The volume doubles
o c. The volume triples
o d. The volume quadruples
3. A square pasture is bordered on one side by a stream and on the other three
sides by a fence. If the fence is 204 feet long, what would be the area of the
pasture?
o a. 408 square feet
o b. 2,601 square feet
o c. 4,624 square feet
Designing Boxes
Teacher Solutions Page 31 of 42
o d. 10,404 square feet
Designing Boxes
Teacher Solutions Page 32 of 42
4. A cylinder has radius of 6 cm and height of 8 cm what would be its volume?
o a. 48 cubic cm
o b. 96 cubic cm
o c. 226 cubic cm
o d. 905 cubic cm
5. A cylindrical tank has diameter of 6 m and height of 10 m what would be its
surface area?
o a. 60 square m
o b. 90 square m
o c. 245 square m
o d. 528 square m
6. If the volume of a new container is 8 times larger than a previous container, by
how much has each dimension increased?
o a. 2 times
o b. 3 times
o c. 4 times
o d. 8 times
Designing Boxes
Teacher Solutions Page 33 of 42
7. Because of a change in her company’s best selling product, Korie has been
given the job of redesigning the packaging for the latest product upgrade. The
marketing department has told her that the package needs to be larger to catch
the customers’ eye on the shelf. She decides to double each edge length. The
volume of the old container was 27 cubic inches.
What is the volume of the new package?
Show your work using words, numbers and/or diagrams.
Volume was 27 cubic inches so possible container would be 3X3X3.
If she double them then it would be 6X6X6 for a total of 216 cubic inches.
OR
As edge lengths all increase by a factor of 2, the volume increases by 23
So, new volume is 8X27 = 216 cubic inches.
The volume of the new package is ________216______ cubic inches.
Designing Boxes
Teacher Solutions Page 34 of 42
8. Harold is the marketing manager for a major toy company. His company is ready
to release a new toy. Harold’s research tells him that more people will buy the
toy if the front of the box has a volume of 54 cubic inches. The depth of the box
Harold designs is the same as one of the edge lengths of the front of the box. All
of Harold’s measurements are whole numbers (no fractions or decimals). As
always, Harold designs a box that sells well.
What is the surface area of the box he designs?
Show your work using words, numbers and/or diagrams.
Surface area is 54 square inches and two of the dimensions are equal, so I need a
perfect square that divides 54. That number is 9. Thus, the equal edges are 3
inches each and the third is 54 divided by 9 or 6.
So the box is 3X3X6 and its surface area is:
Sides:
3X6X4 = 72 square inches
Top and Bottom: 3X3X2 = 18 square inches
Then the total surface area is 90 square inches
The surface area of the new package is _______90_____ cubic inches.
Designing Boxes
Teacher Solutions Page 35 of 42
9. Jerry works for a shipping company whose customer has requested a square
based container with a volume of 96 cubic feet. The dimensions of the container
must be whole number values (no fractions or decimals) per the customer’s
request. As always, Jerry makes his customer happy.
What are the dimensions of the container he designs?
Show your work using words, numbers and/or diagrams.
Designing Boxes
Teacher Solutions Page 36 of 42
I need a perfect square that evenly divides 96.
That number is 16. So two of the dimensions are 4 feet and 4 feet.
Then the third dimension is 96 divided by 16 or 6.
The dimensions of the container are _____4X4X6________ feet.
Designing Boxes
Teacher Solutions Page 37 of 42
10. As the product manager for a packaging company, Chuck is responsible for
creating the package that best fits his customers’ needs. His customer has
decided to change their packaging for the summer season. Their current
package holds a pen that reaches diagonally through the center of a cube
measuring 8 inches on an edge. Chuck is given the job of creating a cylindrical
prism container that has equal diameter and height that will, as close as possible,
hold the same pen. Chuck also needs to calculate and compare the cost of the
package options. The material used to build the cube cost $0.35 per square inch
and the material for the cylinder costs $0.32 per square inch. Determine which
package costs more to build.
8”
8”
x”
x”
8”
What is the diameter and height of the cylinder? (round your answer to two
decimal places)
Identify the more expensive option and the cost to build it.
Helpful ideas:
Circumference of a circle: C = πd
Area of a circle:
A = πr2
Show your work using words, numbers and/or diagrams.
Designing Boxes
Teacher Solutions Page 38 of 42
Designing Boxes
Teacher Solutions Page 39 of 42
Designing Boxes
Teacher Solutions Page 40 of 42
8”
x”
8”
x”
8”
First, find diagonal of cube: 8sqrt3 ≅ 13.86
Then in the cylinder the diagonal is 13.86 and since diameter and height are
equal, use Pythagorean Theorem to solve for diameter and height with equal leg
lengths (alternately use 45-45-90 relationship).
Thus we get dimension of cylinder of 9.80 inches diameter and height
Then find S.A. of cube: 64x6 = 384 square inches.
Use formulas on front to find S.A. of cylinder 9.8x30.78 + 2x4.92 π = 444 square
inches
Now cost to produce Cube is 384x0.35 = $134.40
Cost to produce Cylinder is 444x0.32 = $142.08
So the cylinder costs more to produce .
The diameter and height of the cylinder is ________9.80________ inches.
The __________cylinder ____ is more expensive to build.
It costs _______$142.08____
Designing Boxes
Teacher Solutions Page 41 of 42
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol.
scalar
1
2
3
0.5
3
4
12
6
9
1.5
8
12
2
24
36
6
13
26
39
6.5
396
1512
3348
108
1
4
9
0.25
144
1152
3888
18
1
8
27
0.125
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol.
scalar
1
2
3
0.5
6
8
9
12
18
3
16
24
4
18
27
4.5
13.45362
26.90725
40.36087
6.726812
972
3744
8316
261
1
4
9
0.25
432
3456
11664
54
1
8
27
0.125
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
b
c
d
S.A.
S.A. scalar
Vol
Vol.
scalar
3
9
10
6
9
1.5
18
27
4.5
20
30
5
13.78405
27.5681
41.35215
6.892024
540
2088
4644
144
1
4
9
0.25
270
2160
7290
33.75
1
8
27
0.125
Scalar
a
1
2
3
0.5
Use this table to calculate values for the solutions of Group Activity #2
Change the dimensions of a, b, c in the first row to modify table
double sides
triple sides
halve sides
Designing Boxes
Scalar
a
b
c
d
S.A.
S.A. scalar
Vol
Vol.
scalar
1
2
3
0.5
6
2
12
12
18
3
4
6
1
24
36
6
13.56466
27.12932
40.69398
6.78233
648
2448
5400
180
1
4
9
0.25
144
1152
3888
18
1
8
27
0.125
Teacher Solutions Page 42 of 42