CHAPTER 18Proportional Reasoning

RATIOS There are several types of ratios

Part-to-Part Ratios Example: There are nine boys to every 7 girls. A ratio of 9/7. Note here that

this is not a fraction.

Example: The slope of a line is a part-to-part ratio. Rise to run. Not a fraction here either.

Part-Whole Ratios Example: Suppose that nine girls are wearing dresses Elsa costumes for

Halloween out of 16 total neighborhood girls.

Ratios as Quotients Example: Suppose you have 12 dollars and you are going to share equally

among 5 friends. Each friend gets 12/5 dollars or 2 and 2/5 dollars = $2.40.

Ratios as Rates Example: You travel 200 miles in 4 hours means you traveled an average of

200/4 = 50 miles per hour.

TWO INTERPRETATIONS FOR RATIOS Multiplicative Comparison

Suppose that Sam is 6 feet tall while Sarah is 5 feet tall. The ratio of these two people is 6 to 5 or 5 to 6.

Sam is six-fifths as tall as Sarah.

Alternatively, Sarah is five-sixths as tall as Sam.

Composed Units If you can buy 3 bananas for $1.00, then you can buy 6 bananas for

$2.00, 12 for $4.00, etc. Each of these is a unit composed of the original ratio. This is an example of iterating using proportional reasoning.

READ the section on “Ratios Compared to Fractions!”

PRACTICE PROBLEMS 1. On graph paper or the peg boards, make a square with a triangle on top (like a house

picture). Draw another version in the same way where the ratio of corresponding parts (sides) is 1:2. How do you know the ratios are correct? How do the areas of the pieces compare? Now try with 2:3.

2. Sue and Julie were running equally fast around a track. Sue started first. When Sue had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run? What is the ratio here?

3. Sue and Julie started running around a track at the same time. When Sue had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?

4. At the midway point of the basketball season, you must recommend the best free-throw shooter for the all-star game. Here are the statistics: Ben: 8 out of 11, John: 15 out of 19, Brad: 22 out of 29, Sam: 33 out of 41. Make sure you can operate without a calculator.

5. The Science Club has four separate rectangular plots for experiments with plants (in feet): 1 x 4, 7 x 10, 17 x 20, and 27 x 30. Which rectangular plot is most square? How does this connect to ratios?

6. Plot the points (1,2), (2, 4) and (3,6). Discuss the slope of the line in relation to the ratios of the sides of the three rectangles each with the bottom left corner at (0,0) and the top right corner one at each of the three points given.