Hook How could you calculate the measurements for a scale model of the Parthenon? Coach’s Commentary I chose this example because most students have had to build a scale model, so this makes a familiar context.

Objective In this lesson you will learn how to calculate actual lengths from a scale drawing by using ratios and proportions.

For example: # Apples : # Oranges Let’s Review Ratio – a comparison between two numbers. Example: Since there are 4 apples and 2 oranges, the ratio of the number of apples to the number of oranges is 4:2 or 2:1.

    Let’s Review Proportion – an equation that relates two equal fractions. Example: 5/6 = 25/30

B M L N A C A Common Mistake     A Common Mistake A common mistake is mixing up the order of one ratio. For example, triangles ABC and LMN are similar, so the ratios of their corresponding sides must be equal. Another way to say this is to say that their corresponding sides are proportional. We can see that the lengths of sides AB and LM correspond, so we can write the ratio 16/8. It is also true that the lengths of sides BC and MN correspond, but if we write the ratio 11/22, then we are comparing a fraction that reduces to 2 and a fraction that reduces to ½. Clearly, these are not equal fractions, so the proportion is not true. Coach’s Commentary The order in which the ratios are written is very important in correctly finding the missing measures in a scale drawing.

Core Lesson We can solve a proportion by equating fractions.         Core Lesson We can solve a proportion by equating fractions. For example, consider similar triangles ABC and LMN. Since angles B & M are the largest angles in the triangles, and angles C & N are the smallest, we know that side AB corresponds to LM, BC corresponds to MN, and AC corresponds to LN. Suppose we are given the side lengths shown, and we need to find the length of side LN. Sides BC and MN are the only pair of corresponding sides for which both lengths are known, so we will write the ratio 25/10. We must be careful to write the ratio of sides AC and LN in the correct order. Since we placed the large side in the numerator of the first ratio, we will write the ratio 30/LN. 25/10 reduces to 5/2, so 30/LN must also equal 5/2. 30 divides by 6 to produce 5, so LN must also divide by 6 to produce 2. This means that the length of side LN must be 12. Coach’s Commentary Establishing the importance of properly ordering the ratios will reduce a common source of student error.

Q T S R 6 ?     10 Y Z X W 15 Core Lesson Another way to solve a proportion is by cross multiplying. We begin with a pair of similar parallelograms. Sides QR and WX correspond, as do sides TQ and ZW. From the given measurements we can write the proportion 6/10 = TQ/15. Since 6 and 15 are the first and last values we say, they are called the “extremes.” Also, 10 and TQ are said in the middle, so they are called the “means.” The product of the extremes equals the product of the means, so 6*15 = 10*TQ, or 90 = 10*TQ. That tells us that TQ = 9. Notice that equating fractions would have been more difficult in this situation because 10 does not divide evenly into 15. The advantage to using cross multiplying is that it works even when one ratio is not multiplied by a round number to produce the other ratio. Coach’s Commentary Make sure to reinforce the importance of order when students write ratios.

A B = 7.1xA 7.1xB Core Lesson But why does cross multiplying work? Suppose we have a ratio made up of any two numbers A and B. An equivalent ratio can be formed only by multiplying the numerator and denominator by exactly the same number. Suppose we choose to multiply each by a number that is not round, like 7.1 This gives us 7.1A/7.1B. Multiplying the extremes results in a product of A(7.1B); multiplying the means gives us B(7.1A). Examine the products, and you will see that A, 7.1, and B are each present in both, therefore the two products are equal. This makes cross multiplication a very powerful technique because it works for solving any proportion, and many proportional relationships are related by numbers that do not come out round. Coach’s Commentary Research has shown that students who understand WHY the mathematics works will remember HOW to do it longer than those who simply memorize procedures without any understanding.

1 15 = H 45 105 ft 45 ft 225 ft Core Lesson Now, let’s return to our original question about the scale model of the Parthenon. The approximate dimensions of the Parthenon are shown on the diagram. (45 ft high, 225 ft long, 105 ft wide) We are to use a scale factor of 1 inch for approximately 15 feet. To find the height of the scale model, we need to compare a ratio of 1/15 to a ratio of L/45. Notice that we placed the model’s measurement in the numerator and the actual dimensions in the denominator of both fractions. Multiplying the extremes and the means, and solving the equation gives us a height of 3 inches on the scale model. Similar calculations could be used to find the other two dimensions.

Review In this lesson you have learned how to calculate actual lengths from a scale drawing by using ratios and proportions.

24 32 w 48 Guided Practice Find the value of w for these similar triangles.

Extension Activities For a struggling student who needs more practice: If the scale factor of a small figure to a large figure is 3 to 7, what will be the scale factor of the large figure to the small figure? Explain. 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 length of side DE. Answer: Length of DE = 21

Find the value of y. Answer: 48

2. If the scale factor of the large triangle to the small triangle is 9:5, find the length of AC. A B C L M N 2. If the scale factor of the large triangle to the small triangle is 9:5, find the length of AC. Answer: 54