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## Tab Wrapper

### Goals:

Students will explore the concept of inverse variation by graphing the relationship between rate and time, using the formula:

Rate X Time = Distance.

### Prerequisites:

- Students must be able to solve one-step equations using multiplication and division.
- Students must be able to read a stop watch and tape measure.

- Students will be able to compute average time for specific variations in track conditions by calculating the mean for each variation.
- Students will be able to compute the rate of a car on a race track mathematically using the distance formula after measuring the distance and time.
- Students will be able to construct a graph showing the inverse relationship of time and rate.
- Students will be able to explain the inverse relationship between rate and time.

### Background:

Inverse relationship:

y = k/x

is a difficult concept for students to understand, and a new standard for 8^{th}grade. Students need a concrete representation in order to internalize this concept. The distance formula:

rt=d

is perfect to show that:

r=d/t

which is an example of inverse variation where the constant of variation is the distance.

The culminating graph will show that as time increases, rate decreases.

Students may need to practice using a stopwatch, since the time for each trial will be recorded in seconds. This may require a few minutes of time during a prior lesson to reach this skill level.

The teacher should determine the height to set up stations before class. One height should be as low as possible for the car to still roll to the end of the track. Another should be high enough to get the car to move quickly, but must be measurable. I use a set of shelves at the back of my classroom: the lowest level is one height, the second shelf is another, and the top ledge is a third. Since a total of 4 heights are required, I also use a tower of books that is between the other heights.

### Procedure:

- Put kits together. A kit contains 4 tracks and connectors, 1 car, at least 1 stop watch, a tape measure, and the Time Trial Worksheet and Time Trial Graph Worksheet
- Discuss the difference between American Standard Units and Metric units. Each group must decide whether they will be measuring in American Standard units or Metric units.
- Before class starts, divide students into workable groups of no more than 5 students.
*Group Work video*

- Determine and label stations.
- Have students determine roles: timer, recorder, starter . There can be 2 timers and 2 recorders for accuracy if groups are large.
- Give each group a kit.
- Have students assemble a track consisting of 3 pieces and measure the length. This should be recorded as “Length 1”.
- Have students set the height by taping their track to their station.
- Trials 1, 2, and 3 vary the height to increase or decrease the speed.
- Students release the car and time its descent with the stopwatch.
- Students will complete 5 trials at the first height before varying the height for trials 2 and 3.
- Trials 4, 5, 6 vary the length. See description on Trial Worksheet.
- When students have completed all six trials, they should average the 5 times to arrive at ONE average time for each of the six trials.
- Students will determine the rate for each trial using the distance formula and substituting their length and average time for each of the six trials. All instructions are printed on the worksheets.
- In order to show that rate varies inversely with time as a linear relationship, students will graph time and rate, with time as the independent variable (x axis) and rate as the dependent variable (y axis).

*Representing Data*

- To extend this lesson, students can convert their units from inches per second or centimeters per second to miles or kilometers per hour.

- Discussion can center around engineering applications where length and height will control or determine rate in the design of roller coasters, entrance and exit ramps on a highway, and handicapped accessibility ramps.

- Students can also use computer programs or graphing calculators to create the graph in addition to creating a graph by hand.

Kathy Zoda