Happy Town Smile City Funville Scale in miles: 1 inch = Approx. 15 miles Funville Gladberg Happy Town Smile City Hook What is a scale factor on a map and how do you compute one? Coach’s Commentary I chose this example because it gives students a familiar context in which to learn about scale factors.

Objective In this lesson you will learn how to determine the scale factor between two similar figures by applying ratios.

For example: # Apples : # Oranges or 2 : 1 Let’s Review Ratio – a comparison between two numbers. Example: # Apples : # Oranges

For example: # Apples : # Oranges A Common Mistake A common mistake is reversing the order of the ratio. For example, using the same basket of fruit, someone might say that the ratio of the number of apples to the number of oranges is 2 to 4, or 1 to 2. However, since this would tell us that there are twice as many oranges as apples, which is incorrect. Coach’s Commentary The order of the ratio is very important as students move into finding dimensions of similar figures.

8 16 = 1 2 16 8 = 2 1 A B C L M N 11 22 = 1 2 22 11 = 2 1 12 24 = 1 2 24 12 = 2 1 Core Lesson Vocabulary: Similar figures – two figures are similar if the lengths of every pair of corresponding sides are in the same ratio. Corresponding sides are those that go together. We can identify pairs of corresponding sides by looking for pairs of congruent corresponding angles, and then matching the sides up. For example, similar Triangles ABC and LMN are shown. Notice that the largest angle of Triangle ABC is B and the largest angle of Triangle LMN is M. Also, the smallest angles are C and N. That shows us that side AB corresponds to side LM, side BC corresponds to MN, and side AC corresponds to LN. Now we see the lengths of the sides of Triangle ABC, and the length of side LM. We only need one pair of corresponding sides to find the scale factor between the two triangles. Since 16 is twice as large as 8, the lengths of sides of Triangle ABC must all be twice as large as their corresponding sides on Triangle LMN. Using this reasoning, how long must side MN be? If you said 11, you are correct because 22 is twice as large as 11. How long must side LN be? Now, let’s begin with the smaller triangle – Triangle LMN. This time, the ratio of sides LM to AB is 8/16, or ½. This means that all corresponding sides must be in the same 1 to 2 ratio because the sides of Triangle LMN are half as long as the sides of Triangle ABC. We can use this fact to find the lengths of sides MN and LN as we did before, but this time the order of the ratios will be reversed. How long will side MN be? Since MN is half as long as BC, we know that the length of MN is 11. Using this same reasoning, how long will side LN be? Coach’s Commentary Establishing the importance of proper naming will form good habits for the later study of more advanced geometry, including proofs.

= 6 4 = 4 6 = 2 3 = 3 2 6 Y Z X W Q T S R 4 Core Lesson = 6 4 = 4 6 = 2 3 = 3 2 Core Lesson The Scale Factor is the ratio of the lengths of corresponding sides in similar figures. Consider these two similar parallelograms. Since sides QR and TS correspond to sides WX and ZY, we form the ratio of their lengths to determine the scale factor. We can now see that the scale factor of Parallelogram QRST to Parallelogram WXYZ is 4/6, which reduces to 2/3. This tells us that each side of QRST is 2/3 as long as the corresponding side of WXYZ. If we want to find the scale factor of Parallelogram WXYZ to Parallelogram QRST, we must write the ratio of side lengths in the order 6/4, which reduces to 3/2. This tells us that each side of WXYZ is 3/2 as long (or 1-1/2 times as long) as the corresponding side of QRST. Coach’s Commentary Make sure to reinforce the importance of order when students write ratios.

Smile City Happy Town Funville Scale in miles: 1 inch = Approx. 15 miles Funville Gladberg Happy Town Smile City Core Lesson Now, let’s return to our original question about the scale factor on a map. The number of inches on the map corresponds to the actual number of miles between two locations on a map. Suppose that there are three inches between Happy Town and Smile City on our map, while the actual driving distance between the two locations is approximately 45 miles. This gives us a scale factor of 3 inches for every approximately 45 miles. The ratio 3/45 is equivalent to 1/15, so each 1 inch on the map represents approximately 15 actual miles.

Review In this lesson you have learned how to determine the scale factor between two similar figures by applying ratios.

25 40 10 16 Guided Practice Find the scale factor of the small triangle to the large triangle. (Mention congruent corresponding angles when determining pairs of proportional corresponding sides.)

Extension Activities For a struggling student who needs more practice: Describe in words what it means when a diagram is drawn in a 1:10 scale. Coach’s Commentary Explaining why this happens will help students to solidify their understanding.

D E F P Q R Extension Activities For a student who gets it and is ready to be challenged further: Find the scale factor of triangle DEF to triangle PQR. Answer: Length of PR = 18; scale factor = 7/3

Find the scale factor of the small rectangle to the large rectangle. Answer: 3/8

A B C L M N 2. Find the scale factor of the large triangle to the small triangle. Answer: 9/4