Scale Factors
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Scale Factors and Volume
Discovering the effect on the volume of a prism when its dimensions are multiplied by a scale factor
Find the volume of each prism 2.
1.
15cm 24m 14m
38cm
9cm
11m
V 1,848m
3
V 5,130cm
3
Objective •
Given a prism with its dimensions multiplied by a scale factor, you will find the NEW volume without using a formula.
Purpose •
•
• •
Architects need to know the effect of scale factors on volume when they make scale models of buildings. Gardeners need to know the effect of scale factors on volume when doubling the dimensions of a flowerbed in order to purchase the correct amount of soil. Knowing the effect of scale factors on volume is a 7th grade standard. There is one item on the CAHSEE related to scale factors and volume.
Suppose the length, width, and height of this container were doubled.
We call this “multiplying by a scale factor of 2.”
Suppose the length, width, and height of this container were tripled.
We call this “multiplying by a scale factor of 3.”
What is a scale factor? •
•
In your own words, give a description of a scale factor. Share your description with your partner.
How many cubes do you believe would fill this container if the length, width, and height were doubled?
Let’s Explore… We know that the volume of a 40 foot container is 2,752 ft3. What do you suppose would happen to the volume if we multiplied the length, width and height by a scale factor of 2? Or by a scale factor of 3? Or by a scale factor of 4?
First, let’s make a table
Original
x2 x3 x4
x5
Length
Width
Height
Volume
Volume compared to original
40
8
8.6
2,752 ft3
SAME
We will multiply our original dimensions by a scale factor of 2 and complete our table.
h 2
l 2
w 2
l 40 l 2 40 2 80
Let’s complete our table
Original
x2 x3 x4
x5
Length
Width
Height
Volume
Volume compared to original
40
8
8.6
2,752 ft3
SAME
80
We will multiply our original dimensions by a scale factor of 2 and complete our table.
h 2
l 2
w 2
w8 w 2 8 2 16
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
x3 x4
x5
We will multiply our original dimensions by a scale factor of 2 and complete our table.
h 2
l 2
w 2
h 8.6 h 2 8.6 2 17.2
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
x3 x4
x5
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016
x3
Find the volume V=(80)(16)(17.2) V=22,016
x4
x5
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3 x4
x5
Maybe you should22,016 It2,752 got divide. bigger! 3 8 or 2 What
happened??
Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.
h3
l 3
w 3
l 40 l 3 40 40 3 120
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
x4
x5
Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.
h3
l 3
w 3
w8 w 3 8 3 24
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
x4
x5
Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.
h3
l 3
w 3
h 8.6 h 3 8.6 3 25.8
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
x4
x5
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
74,304
x4
x5
Find the volume V=(120)(24)(25.8) V=74,304
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3 x4
x5
120
24
Need help?
25.8
Didn’t I 3 74,304 fttell 33 or 27 times you already? larger
74,304 2,752 What
3 27 or 3 happened??
Let’s complete our table Length
Do you see a Width Height pattern? Try completing the next row 8.6 on8your own.
Volume
Volume compared to original
2,752 ft3
SAME
Original
40
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
74,304 ft3
33 or 27 times larger
x4
x5
Let’s complete our table What canyou Maybe Length Width Height you should conclude? complete the last row.
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
74,304 ft3
33 or 27 times larger
x4
160
32
34.4
176,128 ft3
43 or 64 times larger
x5
Let’s complete our table Length
Last clue. Width Height What do you notice about these two 8 8.6 columns?
Volume
Volume compared to original
2,752 ft3
SAME
Original
40
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
74,304 ft3
33 or 27 times larger
x4
160
32
34.4
176,128 ft3
43 or 64 times larger
x5
200
40
43
344,000 ft3
53 or 125 times larger
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of 2, the original volume is multiplied by 23? or True
False
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of 3, the original volume is multiplied by 34? or True
False
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of 23, the original volume is multiplied by 232? or True
False
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of 54, the original volume is multiplied by 543? or True
False
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of x, the original volume is multiplied by x 3? or True
False
Let’s take a look at triangular prisms…
Will the same pattern work with triangular prisms?
5 ft
5 ft 4 ft
6 ft
14 ft
First, let’s make a table
Original
x2 x3 x4
x5
Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
6
4
14
168 ft3
SAME
We will multiply our original dimensions by a scale factor of 2 and complete our table.
2
5 ft
5 ft 4 ft
6 ft 2
14 ft
2
Let’s complete our table
Original
x2 x3 x4
x5
Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
6
4
14
168 ft3
SAME
12
8
28
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344
x3
Find the volume Area of the base = ½(12)(8) x4 Area of the base = 48 V=(48)(28) x5 V=1,344
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3 x4
x5
Make sure you check your work!
Don’t ask me! 1,344 168
What happened?? 8 or
23
We will multiply our original dimensions by a scale factor of 3 and complete our table.
3
5 ft
5 ft 4 ft
6 ft 3
14 ft
3
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
x4
x5
42
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
42
4,536
Find the volume Area of the base = ½(18)(12) Area of the base = 108 x5 V=(108)(42) V=4,536 x4
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
42
Sorry, I can’t or 27 times 4,536give ft3 you33any more help.larger
x3 x4
x5
18
12
4,536 168
What happened?? 27 or
33
Let’s complete our table By George, I you’veHeight Base ofthink Height Keepof the the gotofit!the going! prism triangle triangle
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
42
4,536 ft3
33 or 27 times larger
x4
x5
Let’s complete our table Go ahead and Basecomplete of Heightthe Height the lastofrow. the of the
Volume
Volume compared to original
triangle
triangle
prism
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
42
4,536 ft3
33 or 27 times larger
x4
24
16
56
10,752 ft3
43 or 64 times larger
x5
Let’s complete our table Did it work?
Base ofCanHeight we say Height the that ofthis the of the triangle triangle pattern is true prism
Volume
Volume compared to original
Original
6 for all prisms? 4 14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
42
4,536 ft3
33 or 27 times larger
x4
24
16
56
10,752 ft3
43 or 64 times larger
x5
30
20
70
21,000 ft3
53 or 125 times larger
True or False? •
The volume of a triangular prism is 100m3. If the dimensions are multiplied by a scale factor of 2, the new volume will be 800m3.
or True
False
True or False? •
The volume of a triangular prism is 310 ft3. If the dimensions are multiplied by a scale factor of 2, the new volume will be 620 ft3.
or True
False
True or False? •
The volume of a rectangular prism is 10 km3. If the dimensions are multiplied by a scale factor of 4, the new volume will be 270 km3.
or True
False
True or False? •
The volume of a rectangular prism is 21 yd3. If the dimensions are multiplied by a scale factor of 3, the new volume will be 567 yd3. or True
False
True or False? •
The volume of a rectangular prism is 50 m3. If the dimensions are multiplied by a scale factor of x, the new volume will be 50x3 m3. or True
False
On your own… •
1,080 ft3 •
54 cm3 •
3
Suppose the volume of a rectangular prism is 40 ft3. Find the new volume, if the dimensions are multiplied by a scale factor of 3. Suppose the dimensions of a triangular prism are multiplied by a scale factor of 2 and the NEW volume is 432 cm3, what was the original volume? Suppose the original volume of a triangular prism is 120 m3 and the NEW volume is 3,240 m3, what scale factor were the dimensions multiplied by?
Let’s wrap it up! Give a brief explanation of what would happen to the volume of the following prism if the dimensions were multiplied by a scale factor.
19ft
Discovering the effect on the volume of a prism when its dimensions are multiplied by a scale factor
Find the volume of each prism 2.
1.
15cm 24m 14m
38cm
9cm
11m
V 1,848m
3
V 5,130cm
3
Objective •
Given a prism with its dimensions multiplied by a scale factor, you will find the NEW volume without using a formula.
Purpose •
•
• •
Architects need to know the effect of scale factors on volume when they make scale models of buildings. Gardeners need to know the effect of scale factors on volume when doubling the dimensions of a flowerbed in order to purchase the correct amount of soil. Knowing the effect of scale factors on volume is a 7th grade standard. There is one item on the CAHSEE related to scale factors and volume.
Suppose the length, width, and height of this container were doubled.
We call this “multiplying by a scale factor of 2.”
Suppose the length, width, and height of this container were tripled.
We call this “multiplying by a scale factor of 3.”
What is a scale factor? •
•
In your own words, give a description of a scale factor. Share your description with your partner.
How many cubes do you believe would fill this container if the length, width, and height were doubled?
Let’s Explore… We know that the volume of a 40 foot container is 2,752 ft3. What do you suppose would happen to the volume if we multiplied the length, width and height by a scale factor of 2? Or by a scale factor of 3? Or by a scale factor of 4?
First, let’s make a table
Original
x2 x3 x4
x5
Length
Width
Height
Volume
Volume compared to original
40
8
8.6
2,752 ft3
SAME
We will multiply our original dimensions by a scale factor of 2 and complete our table.
h 2
l 2
w 2
l 40 l 2 40 2 80
Let’s complete our table
Original
x2 x3 x4
x5
Length
Width
Height
Volume
Volume compared to original
40
8
8.6
2,752 ft3
SAME
80
We will multiply our original dimensions by a scale factor of 2 and complete our table.
h 2
l 2
w 2
w8 w 2 8 2 16
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
x3 x4
x5
We will multiply our original dimensions by a scale factor of 2 and complete our table.
h 2
l 2
w 2
h 8.6 h 2 8.6 2 17.2
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
x3 x4
x5
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016
x3
Find the volume V=(80)(16)(17.2) V=22,016
x4
x5
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3 x4
x5
Maybe you should22,016 It2,752 got divide. bigger! 3 8 or 2 What
happened??
Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.
h3
l 3
w 3
l 40 l 3 40 40 3 120
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
x4
x5
Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.
h3
l 3
w 3
w8 w 3 8 3 24
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
x4
x5
Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.
h3
l 3
w 3
h 8.6 h 3 8.6 3 25.8
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
x4
x5
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
74,304
x4
x5
Find the volume V=(120)(24)(25.8) V=74,304
Let’s complete our table Length
Width
Height
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3 x4
x5
120
24
Need help?
25.8
Didn’t I 3 74,304 fttell 33 or 27 times you already? larger
74,304 2,752 What
3 27 or 3 happened??
Let’s complete our table Length
Do you see a Width Height pattern? Try completing the next row 8.6 on8your own.
Volume
Volume compared to original
2,752 ft3
SAME
Original
40
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
74,304 ft3
33 or 27 times larger
x4
x5
Let’s complete our table What canyou Maybe Length Width Height you should conclude? complete the last row.
Volume
Volume compared to original
Original
40
8
8.6
2,752 ft3
SAME
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
74,304 ft3
33 or 27 times larger
x4
160
32
34.4
176,128 ft3
43 or 64 times larger
x5
Let’s complete our table Length
Last clue. Width Height What do you notice about these two 8 8.6 columns?
Volume
Volume compared to original
2,752 ft3
SAME
Original
40
x2
80
16
17.2
22,016 ft3
23 or 8 times larger
x3
120
24
25.8
74,304 ft3
33 or 27 times larger
x4
160
32
34.4
176,128 ft3
43 or 64 times larger
x5
200
40
43
344,000 ft3
53 or 125 times larger
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of 2, the original volume is multiplied by 23? or True
False
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of 3, the original volume is multiplied by 34? or True
False
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of 23, the original volume is multiplied by 232? or True
False
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of 54, the original volume is multiplied by 543? or True
False
True or False? •
Whenever the length, width, and height are multiplied by a scale factor of x, the original volume is multiplied by x 3? or True
False
Let’s take a look at triangular prisms…
Will the same pattern work with triangular prisms?
5 ft
5 ft 4 ft
6 ft
14 ft
First, let’s make a table
Original
x2 x3 x4
x5
Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
6
4
14
168 ft3
SAME
We will multiply our original dimensions by a scale factor of 2 and complete our table.
2
5 ft
5 ft 4 ft
6 ft 2
14 ft
2
Let’s complete our table
Original
x2 x3 x4
x5
Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
6
4
14
168 ft3
SAME
12
8
28
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344
x3
Find the volume Area of the base = ½(12)(8) x4 Area of the base = 48 V=(48)(28) x5 V=1,344
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3 x4
x5
Make sure you check your work!
Don’t ask me! 1,344 168
What happened?? 8 or
23
We will multiply our original dimensions by a scale factor of 3 and complete our table.
3
5 ft
5 ft 4 ft
6 ft 3
14 ft
3
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
x4
x5
42
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
42
4,536
Find the volume Area of the base = ½(18)(12) Area of the base = 108 x5 V=(108)(42) V=4,536 x4
Let’s complete our table Base of the triangle
Height of the triangle
Height of the prism
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
42
Sorry, I can’t or 27 times 4,536give ft3 you33any more help.larger
x3 x4
x5
18
12
4,536 168
What happened?? 27 or
33
Let’s complete our table By George, I you’veHeight Base ofthink Height Keepof the the gotofit!the going! prism triangle triangle
Volume
Volume compared to original
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
42
4,536 ft3
33 or 27 times larger
x4
x5
Let’s complete our table Go ahead and Basecomplete of Heightthe Height the lastofrow. the of the
Volume
Volume compared to original
triangle
triangle
prism
Original
6
4
14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
42
4,536 ft3
33 or 27 times larger
x4
24
16
56
10,752 ft3
43 or 64 times larger
x5
Let’s complete our table Did it work?
Base ofCanHeight we say Height the that ofthis the of the triangle triangle pattern is true prism
Volume
Volume compared to original
Original
6 for all prisms? 4 14
168 ft3
SAME
x2
12
8
28
1,344 ft3
23 or 8 times larger
x3
18
12
42
4,536 ft3
33 or 27 times larger
x4
24
16
56
10,752 ft3
43 or 64 times larger
x5
30
20
70
21,000 ft3
53 or 125 times larger
True or False? •
The volume of a triangular prism is 100m3. If the dimensions are multiplied by a scale factor of 2, the new volume will be 800m3.
or True
False
True or False? •
The volume of a triangular prism is 310 ft3. If the dimensions are multiplied by a scale factor of 2, the new volume will be 620 ft3.
or True
False
True or False? •
The volume of a rectangular prism is 10 km3. If the dimensions are multiplied by a scale factor of 4, the new volume will be 270 km3.
or True
False
True or False? •
The volume of a rectangular prism is 21 yd3. If the dimensions are multiplied by a scale factor of 3, the new volume will be 567 yd3. or True
False
True or False? •
The volume of a rectangular prism is 50 m3. If the dimensions are multiplied by a scale factor of x, the new volume will be 50x3 m3. or True
False
On your own… •
1,080 ft3 •
54 cm3 •
3
Suppose the volume of a rectangular prism is 40 ft3. Find the new volume, if the dimensions are multiplied by a scale factor of 3. Suppose the dimensions of a triangular prism are multiplied by a scale factor of 2 and the NEW volume is 432 cm3, what was the original volume? Suppose the original volume of a triangular prism is 120 m3 and the NEW volume is 3,240 m3, what scale factor were the dimensions multiplied by?
Let’s wrap it up! Give a brief explanation of what would happen to the volume of the following prism if the dimensions were multiplied by a scale factor.
19ft
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