## Gearing Up

### Johnny WolfeSanta Rosa District Schools

#### Description

In mathematics, a ratio is a comparison of two numbers by division. A gear ratio can be expressed as a ratio to solve real-world problems.

#### Objectives

Understands that numbers can be represented in a variety of equivalent forms using integers, fractions, decimals, and percents, scientific notation, exponents, radicals, absolute value, or logarithms.

#### Materials

- Overhead transparencies (if examples are to be worked on overhead) for Gearing Up Examples (See Associated File)
- Gearing Up Examples (See Associated File)
- Gearing Up Worksheet (See Associated File)
- Gearing Up Checklist (See Associated File)

#### Preparations

1. Prepare transparencies (if teacher uses overhead for examples) for Gearing Up Examples. (See Associated File)
2. Have marking pens (for overhead).
3. Have Gearing Up Examples (See Associated File) prepared and ready to demonstrate to students.
4. Have enough copies of Gearing Up Worksheet (See Associated File) for each student.
5. Have enough copies of Gearing Up Checklist (See Associated File) for each student.

#### Procedures

PRIOR KNOWLEDGE: Students should be familiar with basic operation skills such as addition, subtraction, multiplication, division, exponents, fractions, decimals, solving equations, and the 4-step approach to problem solving.

NOTE: This lesson does not incorporate percents, scientific notation, exponents, radicals, absolute value, or logarithms.

1. Get students' attention by making the statement, “The pulling power that a truck generates is determined by, in large part, the gear ratio. Today we are going to set up ratios and proportions to solve problems as they do in the work place.”

2. Ask the students if they have ever been riding a bicycle on level ground and then had to go up a hill. Have one of the students describe the difficulty in pedaling up a hill. Have students discuss what happens when you change gears on a bicycle.

3. Go over the definition for the term ratio. (See Associated File, Gearing Up Examples)

4. Work example # 1 from Gearing Up Examples. (See Associated File) Answer student questions and comments.

5. Make the following statement, “The gear ratio is the number of teeth on the front sprocket divided by the number of teeth on the rear sprocket. When riding is easy the gear ratio is lower. When riding is hard, the gear ratio is higher. In other words, you change the gear ratio based on the difficulty of the ride. On a hill, for example, if the front sprocket gear of a mountain bike has 60 teeth and rear sprocket gear has 20 teeth, then the gear ratio would be 60:20 or 3:1. This means that the front sprocket revolves once for every time the rear sprocket revolves three times. On the other hand, if you are riding on a flat road, you could be using gears with the same number of teeth in both sprockets. For example, 30:30 would be a 1:1 gear ratio, common when riding fast.”

6. Work example # 2 from Gearing Up Examples. (See Associated File) Emphasize that the units should be the same in most cases! Answer student questions and comments.

7. Work example # 3 from Gearing Up Examples. (See Associated File) Answer student questions and comments.

8. Work example # 4 from Gearing Up Examples. (See Associated File) Point out again that the units should be the same in most cases! Answer student questions and comments.

9. Make the following analogy to the students, “One reason that engineers use gears is that sometimes they want something to spin slower or faster. For example, engineers designed eggbeaters to use a big gear next to two small gears to make the beaters spin faster. Every time you spin the handle around once, the beaters spin around many times.”

10. Go over the definition for the term proportion. (See Associated File, Gearing Up Examples)

11. Discuss the extremes and the means. (See Associated File, Gearing Up Examples)

12. Discuss the "Means-Extremes Property of Proportions." (See Associated File, Gearing Up Examples)

13. Work Example # 5 from Gearing Up Examples. (See Associated File) Answer student questions and comments.

14. Work Example # 6 from Gearing Up Examples. (See Associated File) Answer student questions and comments.

15. Work Example # 7 from Gearing Up Examples. (See Associated File) Explain the 4-step approach to problem solving. Answer student questions and comments.

16. Work Example #8 from Gearing Up Examples. (See Associated File) Explain the 4-step approach to problem solving. Answer student questions and comments.

17. Distribute the Gearing Up Worksheet. (See Associated File) This worksheet has 3 parts: writing ratios in simplest form; solving a proportion; and using the 4-step approach to problem solving.

18. Distribute the Gearing Up Checklist. (See Associated File) Describe what constitutes an A, B, C, D, and F in the Gearing Up Checklist.

19. The students write their responses on the worksheet.

20. The teacher moves from student to student observing the student's work and lending assistance.

#### Assessments

The student worksheet is collected and scored according to the Gearing Up Checklist. (See Associated File) Students scoring in the D or F range should be retaught and reassessed. Scores can be summative if students have received enough practice and feedback; otherwise, they should be formatively assessed.

#### Extensions

Have students take a bicycle (one with the ability to change gears) and record the teeth on the front gear and the teeth on the back gear. Then ride the bike up a steep hill and record the time and effort used in pedaling on the hill. Next have the students change to different ratios and ride up the hill (recording their results). Then have students draw a conclusion about gear ratios.