1. MTH 231 Section 7.3 Proportional Reasoning

2. Overview In grades K – 4, a main focus is the development of the additive principles of arithmetic. In the upper elementary grades, students should now see that multiplicative relations are essential to understanding: 1.How relative quantities can be compared; 2.How changes in quantities can be measured by a rate.

3. Common Core SMP (Standards for Mathematical Practice) 4 states: “In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community.”

4. An Example Suppose 6 new students are added to a history class of 18 students and 6 new students are added to a PE class of 24 students. Additive reasoning: both classes underwent the same change. Proportional reasoning: the history class underwent a greater change because ¼ of the students (6 out of 24) are new, compared to 1/5 of the students (6 out of 30) in the PE class.

5. Ratio A ratio is a comparison of two quantities. “the ratio of a to b” can be written in at least three ways: a to b a:b

6. Applications of Ratio 1.A ratio measures the relative size of different parts.

7. Applications of Ratio 2. A ratio measures the relative size of a part to a whole. Example: if a basketball player attempts 40 free throws and makes 31 of them, then the ratio of free throws made to free throws attempted is

8. Applications of Ratio 3. A ratio measures the rate of change in one quantity with respect to a corresponding change in a second quantity. Example: slope of a line

9. Unit Rate A unit rate is the amount of change in one quantity compared to a change of 1 unit of a second quantity. For example, 20 mpg (miles per gallon) means that, for every 1 gallon of gas consumed a car will travel 20 miles.

10. Ratios in Simplest Form Expressing a ratio in simplest form is analogous to reducing a fraction to lowest terms:

11. Proportion A proportion is the equality of two ratios. Let a,b,c, and d be non-zero real numbers. if and only if

12. Verifying Proportions To verify if two ratios form a proportion, find and compare the cross products (the bottom value of one proportion times the top value of the other proportion).

13. Solving Proportions 1.Find the cross products. One of them will include the unknown variable. 2.Simplify each side, then divide both sides by the coefficient of the unknown variable.