Comparing Scaling & Shape Using a Copying Machine

1. Comparing Scaling & Shape Using a Copying Machine AISD Math Workshops All http://generative.edb.utexas.edu/materials/aisd.htm >to be posted soon with links to others<

2. 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.”

3. Scaling & “Same to Same” Comparisons s1 s1’ • 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 ato 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. s4’ s4 s2 s2’ s3’ s3 A B a1 a1’ a1 a1’ a2 a2’ A B

4. Using SAME to SAME to Decide What 78% Means 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) This Table Compares Image(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) 100% 50% s’ (cm) This Table Compares Image(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) s (cm) 100% 50% a’ (cm^2) * - If working with partner, at least first 3 should be your values, then get at least 3 more from your partner a (cm^2)

5. Using SAME to SAME to Decide Which is WhichFrom Copier: 25%, 50%, 64%, 73%, 78%, 100%, 121% & 129% Post Question: I think Compared to A, image _____ is scaled by ______ % Compared to A image _____ is scaled by ______ % This Table Compares Image(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) 100% 50% s’ (cm) from image ____ s (cm) from image A This Table Compares Image(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) 100% s’ (cm) from image ____ 50% s (cm) from image A * - If working with partner, at least first 3 should be your values, then get at least 3 more from your partner

6. SAME to DIFFERENT comparisons and SHAPE Post Questions: [1] For a Given Shape or Figure, what pattern stays the Same across 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”? ** Circle Corresponding Shapes or Parts on Each Image ** For a Shape This Table Compares Image(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) 100% 50% d (cm) For a New Shape This Table Compares Image(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) s (cm) 100% 50% d (cm) * - If working with partner, at least first 3 should be your values, then get at least 3 more from your partner s (cm)

7. SAME to [SAME or DIFFERENT] Comparison Questions: 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) 100% 50% __ ( ) 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) __ ( ) 100% 50% __ ( ) __ ( )

8. A B

9. E Q

10. Z R

11. I W

12. originals

13. A 100 % Height 3.75 inches B 78 % Height 2.295 (2.3) inches

14. E 73 % Height 2.7375 inches R 64 % Height 2.4 inches

15. I 50 % Height 1.875 inches Q 25% Height 0.9375 inches

16. Z 121 % Height 4.5375 inches W 129 % Height 4.8375 inches

17. 10X10 10X10

18. 5X5 5X5

19. 20X20 20X20

21. 100 % Height 3.75 inches 100 % Height 3.75 inches