Comparing Scaling & Shape
Using a Copying Machine
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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
s1
•
s1’
s4’
s4
s2
s2’
s3
s3’
A
B
a1
•
a1 a1’
•
a2’
•
•
a1’
a2
A
B
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.
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%
#
s (cm)
s’ (cm)
1(all)
2*
s’ (cm)
3
50%
4
5
6
7
8
9
This Table Compares
Image(s) ___A to B_____ using (check one)
s (cm)
_X_ Same to Same (s to s’ or a to a’) OR
100% ___ Same to Different (s to d or d to s)
#
a (cm^2)
a’ (cm^2)
1(all)
2*
a’ (cm^2)
3
50%
4
5
6
7
8
9
a (cm^2)
* - If working with partner, at least first 3 should be
your values, then get at least 3 more from your partner
Using SAME to SAME to Decide Which is Which
From Copier: 25%, 50%, 64%, 73%, 78%, 100%, 121% & 129%
Post Question: I think
This Table Compares
Compared to A, image _____ is scaled by ______ %
Image(s) ___A to _____ using (check one)
Compared to A image _____ is scaled by ______ %
_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%
#
s (cm)
s’ (cm)
s’ (cm) from image ____
1(all)
2*
3
50%
4
5
6
7
8
9
This Table Compares
Image(s) ___A to ______ using (check one)
s (cm) from image A
_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 ____
#
s (cm)
s’ (cm)
1(all)
2*
3
50%
4
5
6
7
8
9
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
SAME to DIFFERENT comparisons and SHAPE
Post Questions:
** Circle Corresponding Shapes or Parts on Each Image **
[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
(check one)
[3] When might you use SAME to SAME comparisons and when might you use
SAME to DIFFERENT comparisons?
___ Same to Same (s to s’ or a to a’) OR
[4] A comparison like “there are 3 boys for every 2 girls “ is most like which of
_X_ Same to Different (s to d or d to s for each)
these comparisons? How is this comparison like or not like “shape”?
For a Shape This Table Compares
Image(s) ___ALL_____ using
100%
#
s (cm)
d (cm)
A
B
d (cm)
E
50%
Q
R
Z
W
For a New Shape This Table Compares
Image(s) ___All_____ using (check one)
s (cm)
___ Same to Same (s to s’ or a to a’) OR
100% _X_ Same to Different (s to d or d to s)
#
s (cm)
d (cm)
1(all)
2*
d (cm)
3
50%
4
5
6
7
8
9
s (cm)
* - If working with partner, at least first 3 should be
your values, then get at least 3 more from your partner
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
100% ___ Same to Different (s to d or d to s for each)
__ ( )
#
50%
__ ( )
__ (
)
__ (
)
A
B
E
Q
R
Z
I
W
originals
A
B
100 %
Height
3.75
inches
78 %
Height
2.295
(2.3)
inches
E
73 %
Height
2.7375
inches
R
64 %
Height
2.4
inches
I
50 %
Height
1.875
inches
Q
25%
Height
0.9375
inches
Z
121 %
Height
4.5375
inches
W
129 %
Height
4.8375
inches
10X10
10X10
5X5
5X5
20X20
20X20
100 %
Height
3.75
inches
100 %
Height
3.75
inches