A little background about myself:
I am currently in my 13th year of teaching mathematics. All 13 years have been with the Wapakoneta
City Schools. Throughout most of those years, I have taught some level of Algebra.
I made my first OCTM conference presentation last year in Toledo. The topic was Discrete vs.
Continuous Graphs.
The materials from my 2011 presentation and this year’s presentation are available on the website of
Wapakoneta City Schools.
http://highschool.wapak.org/Students/TeacherWebsites/MrsBecker
If you have any questions, comments or suggestions, please feel free to send me an email at
[email protected] or fill out a comment card before you leave.
Thanks for attending my session and I hope that you will leave with something that can help your
students become better mathematical thinkers and problem solvers.
Scale Factor Investigation
Investigation Objective: Understand how a scale factor affects the perimeter and area of a figure.
Extension Objective: Understand how a scale factor affects the volume of a figure.
Materials needed for Teacher:
Circle cutouts with diameters of 8 inches and 16 inches
Materials needed for Students:
Copies of “Scale Factor Investigation – Area and Perimeter Worksheet”
Copies of “Scale Factor Investigation – Questions”
Copies of “Scale Factor Investigation – Grid Paper”
Extension Materials needed for Teacher:
Cubes
Extension Materials for Students:
Copies of “Scale Factor Investigation – Extension Worksheet”
Copies of “Scale Factor Investigation – Isometric Dot Paper”
OPENING QUESTION:
Jack and Becky walk into the Pizza Palace to get some pizza after seeing a movie. There is a large sign
above the counter. “Save Big! Don’t buy two 8-inch pizzas at $7 each when you can buy one 16-inch
pizza for $14.” Jack’s not impressed. He doesn’t think you are saving anything. Becky is thrilled because
she thinks it’s a great deal. Who do you agree with and why?
Becker
Scale Factor Investigation – Area and Perimeter Worksheet
Name: ______________________
1.
Rectangle A has a base of 2 and a height of 3. Draw Rectangle A on the grid paper.
2.
Calculate the perimeter and area of Rectangle A. Record these in Table 1.
3.
Rectangle B is created from Rectangle A by using a scale factor of 2. (This means that the
dimensions of Rectangle A were multiplied by 2 to get Rectangle B’s dimensions.)
What are the base and height of Rectangle B? Base = _____ Height = _____
4.
Draw Rectangle B on the grid paper.
5.
Calculate the perimeter and area of Rectangle B. Record these in Table 1.
6.
How many Rectangle A’s fit inside Rectangle B? Record this in Table 1.
7.
Rectangle C is created from Rectangle A by using a scale factor of 3.
What are the base and height of Rectangle C? Base = _____ Height = _____
8.
Draw Rectangle C on the grid paper.
9.
Calculate the perimeter and area of Rectangle C. Record these in Table 1.
10.
How many Rectangle A’s fit inside Rectangle C? Record this in Table 1.
11.
Rectangle D is created from Rectangle A by using a scale factor of 4.
What are the base and height of Rectangle D? Base = _____ Height = _____
12.
Draw Rectangle D on the grid paper.
13.
Calculate the perimeter and area of Rectangle D. Record these in Table 1.
14.
How many Rectangle A’s fit inside Rectangle D? Record this in Table 1.
TABLE 1
Rectangle
Perimeter
Area
A
B
C
D
15.
Number of Rectangle
A’s that fit inside this
Rectangle
1
Use TABLE 1 to help you fill in TABLE 2.
TABLE 2
Rectangle
Scale Factor Used
What can you multiply
Rectangle A’s perimeter
by to get the perimeter
of the new rectangle?
What can you multiply
Rectangle A’s area by to
get the area of the new
rectangle?
B
C
D
Page 1
Becker
Scale Factor Investigation – Extension
Name: ______________________
1.
Look at Table 2. How can I find the “new” perimeters by using the scale factor and the “original”
perimeter? (The Perimeter of A is considered the “original” perimeter.) Write a formula using
words.
2.
Look at Table 2. How can I find the “new” areas by using the scale factor and the “original” area?
(The Area of A is considered the “original” area.) Write a formula using words.
3.
If a scale factor of 5 is used to change the dimensions of Rectangle A to get Rectangle E, what do
you think Rectangle E’s perimeter would be? Explain or show work.
4.
If a scale factor of 5 is used to change the dimensions of Rectangle A to get Rectangle E, what do
you think Rectangle E’s area would be? Explain or show work.
5.
If the dimensions of a rectangle with a perimeter of 16 inches and an area of 15 square inches are
changed by using a scale factor of 2, what would the new perimeter and area be? Explain or show
work.
6.
At the Pizza Palace, the diameter of the 8-inch pizza was doubled to create the 16-inch pizza. If
you were the owner of the Pizza Palace and charged $7 for an 8-inch pizza, what should you
charge for a 16-inch pizza? Explain how you determined the price for the 16-inch pizza.
Page 2
Becker
Scale Factor Investigation – Extension
Name: ______________________
Extension:
1.
Rectangular Prism X has dimensions of one by one by two and has a volume of 2 cubic units.
Prism X has been drawn for you on the isometric dot paper.
2.
A scale factor of 2 is used to change the dimensions of Prism X to make Prism Y. What are the
dimensions of Prism Y? _____ by _____by _____ Draw Prism Y on the isometric dot paper.
3.
What would Prism Y’s volume be? Record this in Table 3.
4.
A scale factor of 3 is used to change the dimensions of Prism X to make Prism Z. What are the
dimensions of Prism Z? _____ by _____by _____ Draw Prism Z on the isometric dot paper.
5.
What would Cube Z’s volume be? Record this in Table 3.
6.
Complete Table 3.
TABLE 3
Rectangular Prism
Scale
Factor Used
X
Y
Z
1
2
3
Volume of Prism
2
How many Prism
X’s can fit inside
this new prism?
1
What can you multiply Prism
X’s volume by to get the
volume of the new prism?
--------------------------
1.
Look at Table 3. How can I find the “new” volumes by using the scale factor and the “original”
volume? (The Volume of X is considered the “original” volume.) Write a formula using words.
2.
If a scale factor of 4 is used to change the dimensions of Prism X to get a new prism, what do you
think the new prism’s volume would be? Explain or show work.
3.
If a rectangular prism is 3 by 2 by 4. What is its volume?
4.
If the dimensions of the rectangular prism above are doubled, what would the volume of the new
prism be? Explain or show work.
5.
The volume of a certain rectangular prism is 20 cubic units. If a scale factor of 3 was used to
change its dimensions, what would the new volume be? Explain or show work.
Page 3
Becker
Scale Factor Investigation – Grid Paper
Page 4
Name: ______________________
Rectangle A
Rectangle B
Rectangle C
Rectangle D
Becker
Scale Factor Investigation – Isometric Dot Paper
Name: ______________________
Prism X
Prism Y
Prism Z
Extra
Explanation:
Page 5
Becker
EXPLANATION
Why is the perimeter’s change equal to the scale factor?
Perimeter  3x  5x  3x  5x
3
(factor out an x)
3x
Perimeter  x(3  5  3  5)
5
The original perimeter is being multiplied by x.
5x
Why is the area’s change equal to the (scale factor)2?
Area  3x  5x
(rearrange using the commutative property)
Area  x  x  3  5
3
(rearrange using the associative property)
3x
Area  x 2 (3  5)
5
5x
The original area is being multiplied by x2 .
What is the volume’s change equal to the (scale factor)3?
Volume  3x  5x  2 x
(rearrange using the commutative property)
3
2
5
Volume  x  x  x  3  5  2
3x
2x
5x
(rearrange using the associative property)
Volume  x 3 (3  5  2)
The original volume is being multiplied by x 3 .