Grade 7: Big Idea 1 Develop 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.

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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.

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MA.7.A.1.3 MA.7.A.1.3 Solve problems involving similar figures.

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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 AC D E F Name another pair of corresponding segments.

Similar Figures Angle A corresponds to angle D. B AC 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? ?? ? 70 o 40 o

Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees. If the size of the angles were increased, the sum would exceed 180 degrees. 70 o 40 o 70 o 40 o

70 o 40 o We can verify this fact by placing the smaller triangle inside the larger triangle. 70 o 40 o

70 o 40 o The 40 degree angles are congruent.

70 70 o o 40 o The 70 degree angles are congruent.

70 70 o 40 o The other 70 degree angles are congruent. 40 o

Corresponding sides have lengths that are proportional. Corresponding angles are congruent. Similar Figures

AD BC 3 cm 2 cm 3 cm WZ X Y 9 cm 6 cm Corresponding sides: AB corresponds to WX. AD corresponds to WZ. CD corresponds to YZ. BC corresponds to XY.

Similar Figures AD BC 3 cm 2 cm 3 cm WZ X Y 9 cm 6 cm Corresponding angles: A corresponds to W. B corresponds to X. C corresponds to Y. D corresponds to Z.

Similar Figures AD BC 3 cm 2 cm 3 cm WZ X Y 9 cm 6 cm

Similar Figures AD BC 3 cm 2 cm 3 cm WZ X Y 9 cm 6 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.

Missing Measures in Similar Figures 111 y ___ ____ = Write a proportion using corresponding side lengths. The cross products are equal = 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.

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 B 60 m 120 m 50 m 100 m y 52 m 65° 45°

A B 60 m120 m 50 m 100 m y 52 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

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 ?

This looks messy. Let’s translate the two triangles ?

Now “things” are easier to see ? 6

The common factor between these triangles is ? 6

So the length of the missing side is…?

That’s right! It’s ten!

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 feet 6 ft

The tree must be 30 ft tall. Boy, that’s a tall tree! 10 feet 2 feet 6 ft

1) Determine the missing side of the triangle ? 

1) Determine the missing side of the triangle 

2) Determine the missing side of the triangle  ?

2) Determine the missing side of the triangle  24

3) Determine the missing sides of the triangle ? 8  ?

3) Determine the missing sides of the triangle  11

4) Determine the height of the lighthouse  ?

4) Determine the height of the lighthouse  32

5) Determine the height of the car  ?

5) Determine the height of the car  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..

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MA.7.A.1.5 MA.7.A.1.5 Distinguish direct variation from other relationships, including inverse variation.

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MA.7.A.1.6 MA.7.A.1.6 Apply proportionality to measurement in multiple contexts, including scale drawings and constant speed.

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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.