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Grade 7: Big Idea 1Develop an understanding of and apply proportionality, including similarity.
BIG IDEA 1• MA.7.A.1.1 Distinguish between situations that
are proportional or not proportional and use proportions to solve problems.
• MA.7.A.1.2 Solve percent problems, including problems involving discounts, simple interest, taxes, tips and percents of increase or decrease.
• MA.7.A.1.3 Solve problems involving similar figures.
BIG IDEA 1• MA.7.A.1.4 Graph proportional relationships
and identify the unit rate as the slope of the related linear function.
• MA.7.A.1.5 Distinguish direct variation from other relationships, including inverse variation.
• MA.7.A.1.6 Apply proportionality to measurement in multiple contexts, including scale drawings and constant speed.
MA.7.A.1.1
• MA.7.A.1.1 Distinguish between situations that are proportional or not proportional and use proportions to solve problems.
• INSERT TEST SPECS HERE
MA.7.A.1.2
• MA.7.A.1.2 Solve percent problems, including problems involving discounts, simple interest, taxes, tips and percents of increase or decrease.
• INSERT TEST SPECS HERE
MA.7.A.1.3
• MA.7.A.1.3 Solve problems involving similar figures.
• INSERT TEST SPECS HERE
Ratio of Similitude
Similar Figures
M.C. Escher
• Some artists use mathematics to help them design their creations.
In M.C. Escher’s Square Limit, the fish are arranged so that there are no gaps or overlapping pieces.
Square Limit by M.C. Escher• How are the fish in
the middle of the design and the surrounding fish alike?
• How are they different?
Square Limit by M.C. Escher
• Escher used a pattern of squares and triangles to create Square Limit.
These two triangles are similar.
Similar figures have the same shape but not necessarily the same size.
Similar Figures• For each part of one similar figure there
is a corresponding part on the other figure.
Segment AB corresponds to segment DE.
B
A C
D
E
F
Name another pair of corresponding segments.
Similar Figures
• Angle A corresponds to angle D.
B
A C
D
E
F
Name another pair of corresponding angles.
The triangle on the right is 20% larger than the one on the left. How are their angle measures related?
? ?
?
70o 70o
40o
Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees.If the size of theangles were increased,the sum would exceed180degrees.
70o 70o
40o
70o 70o
40o
70o 70o
40o
We can verify this fact by placing the smaller triangle inside the larger triangle.
70o 70o
40o
70o 70o
70o 70o
40o
The 40 degree angles are congruent.
70 70o7070o 70o
40o
40o
The 70 degree angles are congruent.
70 7070 70o 70o
40o
The other 70 degree angles are congruent. 40o
•Corresponding sides have lengths that are proportional.
• Corresponding angles are congruent.
Similar Figures
Similar Figures
A D
B C
3 cm
2 cm 2 cm
3 cm
W Z
X Y
9 cm
9 cm
6 cm6 cm
Corresponding sides:
AB corresponds to WX.
AD corresponds to WZ.
CD corresponds to YZ.
BC corresponds to XY.
Similar Figures
A D
B C
3 cm
2 cm 2 cm
3 cm
W Z
X Y
9 cm
9 cm
6 cm6 cm
Corresponding angles:
A corresponds to W.
B corresponds to X.
C corresponds to Y.D corresponds to Z.
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€
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€
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€
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Similar Figures
A D
B C
3 cm
2 cm 2 cm
3 cm
W Z
X Y
9 cm
9 cm
6 cm6 cm
€
AB
WX=AD
WZor
2
6=
3
9
€
AB
BC=WX
XYor
2
3=
6
9
Similar Figures
A D
B C
3 cm
2 cm 2 cm
3 cm
W Z
X Y
9 cm
9 cm
6 cm6 cm
If you cannot use corresponding side lengths to write a proportion, or if corresponding angles are not congruent, then the figures are not similar.
€
AB
WX=AD
WZor
2
6=
3
9
Missing Measures in Similar Figures
111
y
___ 100
200
____ =Write a proportion using corresponding side lengths.
The cross products are equal.200 • 111 = 100 • y
The two triangles are similar. Find the missing length y and the measure of D.
y is multiplied by 100.
Divide both sides by 100 to undo the multiplication.
The two triangles are similar. Find the missing length y.
€
100
111=
200
y
100y = 22,200
100y
100=
22,200
100y = 222
Angle D is congruent to angle C.
If angle C = 70°, then angle D = 70° .
The two triangles are similar. Find the measure of D.
€
∠
Try This
Write a proportion using corresponding side lengths.
The two triangles are similar. Find the missing length y and the measure of B.
Divide both sides by 50 to undo the multiplication.
A B60 m 120 m
50 m 100 m
y52 m
65°
45°
€
50
100=
52
y
5,200 = 50y
5,200
50=
50y
50104 = y
€
∠
A B60 m 120 m
50 m 100 m
y52 m
65°
45°
Try This
The two triangles are similar. Find the missing length y and the measure of B.
Angle B is congruent to angle A.
If angle A = 65°, then angle B = 65°
€
∠
This reduction is similar to a picture that Katie painted. The height of the actual painting is 54 centimeters. What is the width of the actual painting?
Reduced
2
3
Actual
54
w
Using Proportions with Similar Figures
Write a proportion.
The cross products are equal.
w is multiplied by 2.
Divide both sides by 2 to undo the multiplication.
Using Proportions with Similar Figures
Reduced
2
3
Actual
54
w
€
2 cm
54 cm=
3 cm
w cm
€
54 • 3 = 2 • w
162 = 2w
81 = w
Try these 5 problems.
These two triangles are similar.
1. Find the missing length x.
2. Find the measure of J.
3. Find the missing length y.
4. Find the measure of P.
5. Susan is making a wood deck from blueprints
for an 8 ft x 10 ft deck. However, she is going
to increase its size proportionally. If the length
is to be 15 ft, what will the width be?
36.9°
30 in.
4 in.
90°
12 ft
€
∠€
∠
Find the length of the missing side.
30
40
50
6
8
?
This looks messy. Let’s translate the two triangles.
30
40
50
6
8
?
Now “things” are easier to see.
30
40
50
8
?
6
The common factor between these triangles is 5.
30
40
50
8
?
6
So the length of the missing side
is…?
That’s right! It’s ten!
30
40
50
8
10
6
Similarity is used to answer real life questions.
• Suppose that you wanted to find the height of this tree.
Unfortunately all that you have is a tape
measure, and you are too short to reach the
top of the tree.
You can measure the length of the tree’s shadow.
10 feet
Then, measure the length of your shadow.
10 feet 2 feet
If you know how tall you are, then you can determine how tall
the tree is.
10 feet 2 feet6 ft
The tree must be 30 ft tall. Boy, that’s a tall tree!
10 feet 2 feet6 ft
1) Determine the missing side of the triangle.
3
4
5
12
9
?
1) Determine the missing side of the triangle.
3
4
5
12
9
15
2) Determine the missing side of the triangle.
6
4
6 36 36
?
2) Determine the missing side of the triangle.
6
4
6 36 36
24
3) Determine the missing sides of the triangle.
39
24
33?
8
?
3) Determine the missing sides of the triangle.
39
24
3313
8
11
4) Determine the height of the lighthouse.
2.5
8
10
?
4) Determine the height of the lighthouse.
2.5
8
10
32
5) Determine the height of the car.
5
3
12
?
5) Determine the height of the car.
5
3
12
7.2
MA.7.A.1.4
• MA.7.A.1.4 Graph proportional relationships and identify the unit rate as the slope of the related linear function..
• INSERT TEST SPECS HERE
MA.7.A.1.5
• MA.7.A.1.5 Distinguish direct variation from other relationships, including inverse variation.
• INSERT TEST SPECS HERE
MA.7.A.1.6
• MA.7.A.1.6 Apply proportionality to measurement in multiple contexts, including scale drawings and constant speed.
• INSERT TEST SPECS HERE
A mural of a dog was painted on a wall. The enlarged dog was 45 ft. tall. If the average height for this breed of dog is 3 ft., what is the scale factor of this enlargement? Can you express this scale in more than one way
Imagine that you need to make a drawing of yourself (standing) to fit completely on an 8.5-by-11-in. sheet of paper. Determine the scale factor, allowing no more than an inch of border at the top and bottom of the page. How long will your arms be in the drawing?
Cartoon Blow Up
Reflection Questions
Often, tasks are selected based on the amount of time it should take to complete them rather than on the quality of the activity itself. True investigations consume larger chunks of time because they require the use of physical materials and require a higher level of thinking on the part of the students. Children need time to put the pieces together and to communicate their findings. Share your thoughts on time vs. task.
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