Comparing Scaling & Shape Using a Copying Machine

Comparing Scaling & Shape Using a Copying

Machine

AISD Math Workshops Allhttp://generative.edb.utexas.edu/

materials/aisd.htm>to be posted soon with links to

others<

How Does SCALING Work on a Copy Machine?

• Most copy machines allow the user to SCALE (make bigger or smaller) the copy.• For example, in the image above, which numbers make the copy smaller and

which make the copy bigger? What does 100% mean?• Area or Sides? – If we focus on 78%, does this refer to AREA or the LENGTHS of

the SIDES? What do you think? • For the following example, can you make sense of each person’s idea. Then,

which ideas do you agree/disagree with most?

From discussions overheard in faculty copy rooms:

Person 1: “Material costs like paper and ink are generally related to AREA, so area is what matters most and so the 78% is what happens to the area.”

Person 2: “Sides are what matter. If you want 50% you want the sides to be half as long.”

Person 3: “Hey, it doesn’t matter. Bigger is bigger and smaller is smaller. 50% or 78% or whatever means basically the same thing for both.”

Person 4: “I don’t know what it is but I’m pretty sure that they can’t be both … they can’t both be 50% at the same time … so it’s either Area or Length.”

Scaling & “Same to Same” Comparisons

• To help decide what the 78% means we will need to compare Length of Sides (s) Before (100%) to Sides After (s’) being reduced (78%) and Areas (a) Before to Areas (a’) after. Whichever is scaled by 78% (0.78) will help us decide.

• Because we are comparing s to s’ and a to a’, we’ll call this a “SAME TO SAME” comparison.

• Note: A convention is to use an apostrophe for the corresponding values. So we’re comparing s to s’ and a to a’. Another convention is to use small letters for measured or measurable values.

• Us centimeters for all values.• To get area, we can use the

formula length * width = area or overlay with cm*cm graph paper and count or estimate the number of squares.

A B

A B

s1

a1

a1’

a1a1’

s1’

s2 s2’

s3 s3’

s4 s4’

a2 a2’

Using SAME to SAME to Decide What 78% Means

# s (cm) s’ (cm)1(all)

2*

3

4

5

6

7

8

9

This Table ComparesImage(s) ___A to B_____ using (check one) _X_ Same to Same (s to s’ or a to a’) OR ___ Same to Different (s to d or d to s for each)

s (cm)

s’ (c

m)

100%

50%

# a (cm^2) a’ (cm^2)

1(all)

2*

3

4

5

6

7

8

9

a (cm^2)

a’ (c

m^2

)

100%

50%

This Table ComparesImage(s) ___A to B_____ using (check one) _X_ Same to Same (s to s’ or a to a’) OR ___ Same to Different (s to d or d to s)

* - If working with partner, at least first 3 should be your values, then get at least 3 more from your partner

Pre-questions[1] What do the 100% and 50% lines mean?[2] Where would a 75% line go? 78% line?Post Question: I think 78% refers to scaling ___ sides ___area (check either or both)

Using SAME to SAME to Decide Which is WhichFrom Copier: 25%, 50%, 64%, 73%, 78%, 100%, 121% & 129%

# s (cm) s’ (cm)1(all)

2*

3

4

5

6

7

8

9

This Table ComparesImage(s) ___A to _____ using (check one) _X_ Same to Same (s to s’ or a to a’) OR ___ Same to Different (s to d or d to s for each)

s (cm) from image A

s’ (c

m) f

rom

imag

e __

__

100%

50%

This Table ComparesImage(s) ___A to ______ using (check one) _X_ Same to Same (s to s’ or a to a’) OR ___ Same to Different (s to d or d to s)


Post Question: I think

Compared to A, image _____ is scaled by ______ %

Compared to A image _____ is scaled by ______ %

# s (cm) s’ (cm)

1(all)

2*

3

4

5

6

7

8

9

100%

50%

s (cm) from image A

s’ (c

m) f

rom

imag

e __

__

SAME to DIFFERENT comparisons and SHAPE

# s (cm) d (cm)A

B

E

Q

R

Z

W

For a Shape This Table ComparesImage(s) ___ALL_____ using (check one) ___ Same to Same (s to s’ or a to a’) OR _X_ Same to Different (s to d or d to s for each)

s (cm)

d (c

m)

50%

# s (cm) d (cm)

1(all)

2*

3

4

5

6

7

8

9

s (cm)

d (c

m)

100%

50%

For a New Shape This Table ComparesImage(s) ___All_____ using (check one) ___ Same to Same (s to s’ or a to a’) OR _X_ Same to Different (s to d or d to s)


Post Questions: [1] For a Given Shape or Figure, what pattern stays the Sameacross ALL the Scaled Images?[2] If the comparisons of “same to different” do not have this pattern, what can we conclude about the Shapes or Figures[3] When might you use SAME to SAME comparisons and when might you use SAME to DIFFERENT comparisons?[4] A comparison like “there are 3 boys for every 2 girls “ is most like which of these comparisons? How is this comparison like or not like “shape”?

100%

** Circle Corresponding Shapes or Parts on Each Image **

SAME to [SAME or DIFFERENT] Comparison

# __ ( ) __ ( )

This Table Compares_______ ___________ using (check one) ___ Same to Same (s to s’ or a to a’) OR ___ Same to Different (s to d or d to s for each)

__ ( )

__ (

)

50%

# __ ( ) __ ( )

Questions:

100%

This Table Compares_______ ___________ using (check one) ___ Same to Same (s to s’ or a to a’) OR ___ Same to Different (s to d or d to s for each)

__ (

)

50%

100%

__ ( )

originals

100 %Height 3.75 inches

78 %Height 2.295 (2.3) inches

A

B



E

R


25%Height 0.9375 inches

I

Q



Z

W

10X10

10X10

5X5

5X5

20X20

20X20

Comparing Scaling & Shape Using a Copying Machine

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