Connecting Middle Grades to Advanced Placement* Mathematics
A Resource and Strategy Guide
Unit Dog
Updated: 01/27/11
Objective:
Students will investigate the resulting effects on surface area and volume when dimensions of a
shape are changed proportionally.
Connections to Previous Learning:
Students should be able to calculate surface area and volume of a rectangular prism and draw nets
for 3-dimensional figures.
Connections to AP*:
AP Calculus Topic: Areas and Volumes
Materials:
Student activity pages, 13 snap cubes for each student, grid paper copied on cardstock, tape, scissors,
a sample unit dog made from snap cubes, a sample unit dog made from grid paper, and a large copy
of the chart provided in step 6 of the activity
Teacher Notes:
The Unit Dog lesson is teacher-led. The activity pages can be given to students who finish early or
as an additional assignment. Organize the students into small groups and lead the students through
the following steps.
Step 1:
Demonstrate how to make a unit dog from snap cubes then have each student build their
own dog, using their 13 snap cubes. Note: Larger dogs should not be built with snap
cubes. They are heavy and very unstable.
Step 2:
Have each student determine the surface area and volume of their unit dog. Generally
students do not correctly determine surface area if they try to count. Give them time to
think about a good procedure. Once students begin to have the correct answer, have them
share their process with the rest of the class.
Step 3:
Have each student draw front, side and top views of the unit dog.
Step 4:
Show the students how to draw the nets and construct a unit dog from the grid paper.
(Before class begins, construct a unit dog to use as a model.) Help the students realize
they are to construct one net for the dog’s body and that the other 5 nets (head and legs)
are the same net. Do not ask the students to make one net for the entire body.
*Advanced Placement and AP are registered trademarks of the College Entrance Examination Board.
The College Board was not involved in the production of this product.
®
Copyright © 2008 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
1
Teacher Notes
Step 5: In order to create a variety of different-sized dogs, assign groups to build particular dogs
from grid paper nets. A group of two should construct a dog with dimensions that are
double those of the unit dog. A group of three students should construct a dog with
dimensions triple those of the unit dog. Four students can do quadruple and five students
can do quintuple. Each group should calculate surface area and volume for their particular
dog. If there are two groups for each size dog, then the groups can verify the answers of the
other group.
Step 6: Once all of the dogs are built, have students post their measurements on the large copy of
the chart shown below. Display all of the dogs.
Dimensions
scaled by a
factor of . . .
1 (the unit dog)
2
3
4
5
Surface area
of the dog
Area scaled by
a factor of
...
1
Volume of dog
Volume scaled
by a factor of
...
1
Step 7: As the students fill in the factors by which surface area and volume have increased, ask
them how they determined the factors and how they used the factors in the calculations.
Step 8: Have the students predict the surface areas and volumes for dogs with dimensions six, seven
and ten times those of the unit dog. Then ask them how they determined the values.
Step 9: When the unit dog activity is completed, the practice problems and/or the state assessment
problems on the next two pages may be assigned for extra practice.
®
Copyright © 2008 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
2
Student Activity
Unit Dog
Practice Problems:
1. The flower pots shown below are similar. If the smaller pot has a volume of 100 cubic inches,
what is the volume of the larger pot?
2.
Myrna wants to construct a box that will hold 64 times as much as a similar box. By what
factor should she multiply the dimensions of the smaller box in order to determine the
dimensions of her new, larger box?
3.
Quincy works in the customer service department of a store and is responsible for ordering
supplies for the gift wrap department. The smallest box requires 88 square inches of wrapping
paper. How much wrapping paper would be needed for a box having dimensions that are three
times those of the smallest box?
4.
A delivery service charges a fee based upon the volume of the box to be delivered. If they
charge $3 for a 6 inch by 6 inch by 6 inch box, then how much would you expect them to
charge for a box that is 1 foot by 1 foot by 1 foot?
5.
An architect is working on a scale model home for a client. The linear dimensions of the scaled
1
model will be
the size of the linear dimensions of the actual house. If he uses 2 square feet
20
of wallpaper for the kitchen in the scale model, how much would he need for the kitchen in the
actual house?
®
Copyright © 2008 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
3
Student Activity
State Assessment Practice:
1. A shipping company sells two types of cartons that are shaped like rectangular prisms.
The larger carton has a volume of 720 cubic inches. The smaller carton has dimensions that are
half the size of the larger carton. What is the volume, in cubic inches, of the smaller carton?
A. 90 in.3
B. 120 in.3
C. 240 in.3
D. 360 in.3
2. An ice-cream carton has a volume of 64 fluid ounces. A second ice-cream carton has
dimensions that are ¾ the size of the larger carton. Which is closest to the volume of the smaller
carton?
A. 20 fl oz
B. 27 fl oz
C. 36 fl oz
D. 48 fl oz
3. The radius of the larger sphere shown below was multiplied by a factor of
1
to produce the
2
smaller sphere.
Radius =
1
r
2
Radius = r
How does the surface area of the smaller sphere compare to the surface area of the
larger sphere?
1
A. The surface area of the smaller sphere is as large.
2
1
B. The surface area of the smaller sphere is as large.

1
as large.
4
1
D. The surface area of the smaller sphere is as large.
8
C. The surface area of the smaller sphere is
* Items taken from TAKS (Texas Assessment of Knowledge and Skills) Information Booklet, Mathematics Grades 8 – 11,
Texas Education Agency, Student Assessment Division, 2002
®
Copyright © 2008 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
4
Connecting Middle Grades to Advanced Placement* Mathematics
A Resource and Strategy Guide
Unit Dog
Answers:
6.
Dimensions
scaled by a
factor of . . .
1 (the unit dog)
2
3
4
5
Surface area
of the dog
54
216
486
864
1350
Area scaled by
a factor of
...
1
4
9
16
25
Volume of dog
13
104
351
832
1625
Volume scaled
by a factor of
...
1
8
27
64
125
7. The area measures increase by squaring the linear factor. The linear factor is always multiplied
by itself in the area calculation. The volume measures increase by cubing the linear factor. The
linear factor is always used as a factor three times in the volume calculation.
8. A dog having dimensions six times those of the unit dog will have a surface area of 1944 square
units and a volume of 2808 cubic units. A dog with dimensions seven times those of the unit
dog will have a surface area of 2646 square units and a volume of 4459 cubic units. A dog with
dimensions ten times as big would have a surface area of 5400 square units and a volume of
13,000 cubic units.
9. See answers below.
Practice Problems:
1. Since the flower pots are similar the ratio between similar dimensions is the same.
2.
3.
4.
5.
20in.
 2.
10 in.
The volume of the larger flower is (100 in 3 )(2  2  2)  (100 in 3 )(2)3  800in 3
Since 64  43 each dimension must be multiplied by a factor of 4.
The wrapping paper is covering the surface of the box and each dimension of the surface of the
box is multiplied by a factor of 3, the surface area of the box is multiplied by a factor of 32  9 .
Quincy will need 88 square inches  9  720 square inches of wrapping paper.
The dimensions of the larger box are multiplied by a factor of 2. The delivery service would
charge $3  (2)3  $24 for the larger box.
The dimensions of the house are 20 times as large as that of the model; therefore, the surface
area of the wall will be (20) 2 time the surface area of the walls of the model. The architect will
need (20) 2  2 square feet  800square feet of
wallpaper for the kitchen in the actual house.
State Assessment Practice:
®
1. A
2. B
3. C
Copyright © 2008 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
5
Answers
®
Copyright © 2008 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
6