RATIO AND PROPORTION

Wheels in Motion
(Student Copy)

### Activity 3: Finding Distance

•  two different sized bicycles
•  meter stick or metric tape measure
•  calculator

Part A

Make a small mark on the floor with the chalk or masking tape.  Place the bicycle so that the valve stem on the front tire is exactly lined up with the mark on the floor.

Push the bicycle forward so that its wheel makes one complete revolution.  That is, the valve stem on the tire will rotate exactly once.  Mark this spot and measure the distance (to the nearest centimeter) between this spot and the spot where you started.  Record the distance in the table below.

Now, measure the distance traveled for 2, 3, 4, and 5 revolutions to complete the table.

 Revolutions 1 2 3 4 5 Distance (cm)

Part B

Record how far you think the wheel travels in 6, 7, or 8 revolutions in the table below.  Check your predictions by measuring the actual distance traveled after 6, 7, and 8 revolutions.

 Revolutions 6 7 8 Predicted distance (cm) Actual distance (cm)

Part C

Predict how many times the wheel turns in 20 meters in the table below.  Write, in the space below, how you arrived at your prediction.

 Distance (meters) 20 40 60 80 100 Predicted revolutions

Now mark a distance of 20 meters.  Line up your valve stem with the start of the 20-meter path and push the bicycle to the 20-meter mark, counting the number of revolutions of the wheel.  How accurate was your prediction?

Complete the table above.  Make predictions (rather than do experiments).

Part D

Repeat Parts A through C with another bicycle with a different sized wheel.

### Activity 4: Comparing Distances

•  three different sized bicycles (including the two used for Activity 3)
•  meter stick or metric tape measure
•  graph paper ruled 4 squares to the inch
•  calculator
•  Activity 3 you previously completed
Part A

Using the data you collected in Activity 3 (Parts A & B only), make a graph of distance traveled (x-axis) in relation to the number of revolutions of the wheel (y-axis) for each bicycle.  Label the axes:  Distance Traveled (x-axis) and Revolutions (y-axis).  Draw a line for each bike's set of points.

Part B

Using the measuring tape, measure the outside diameter of the three tires of the bicycles.   Discuss with your group how the diameter of the tire is connected to the distance traveled in each wheel revolution.  (Recall the formulas discussed in Activity 1 on ratio, diameter, and circumference.)

 Diameter #1 (in cm) Diameter #2 (in cm) Diameter #3 (in cm)

Use the third bicycle wheel (not used in Activity 3).

• Discuss how you can find the distance this bicycle will travel for each revolution of its wheel without measuring.
• Using only your measurement of the tire diameter, construct a table for the third bicycle that shows the distance it will travel after 1, 2, 5, 8, and 10 revolutions of the wheel.
• Roll the bicycle wheel to check your predictions for 1, 5, and 10 revolutions.
• If your predictions were not accurate, explain in writing why they were incorrect.
Diameter  (cm)
 Revolutions 1 2 5 8 10 Distance (meters)

### Activity  5:  Pedaling Distances

•  bicycle with 10 or more gears
•  meter stick or metric tape measure
•  Reference Copy

Part A

In this activity, we are interested in the relation between the number of pedal revolutions and the distance traveled by the bicycle.

Carefully shift your bicycle into its lowest gear (smallest sprocket on the front, largest on the back).  Turn a pedal one full revolution and measure in centimeters how far the rear wheel turns.  (Watch the valve stem to see how many times the wheel turns.)  As a group, estimate how many meters you think the bicycle will travel after 5 and 10 revolutions of a pedal.  Record your predictions in the table below, then complete the measured distances portion of the table.

 Revolutions of the pedal 1 2 3 4 5 8 10 Predicted distance (m) x x x x x Measured distance (cm)

Were your predictions accurate?  If not, why not?

Now change gears.  Leave the front sprocket as it is but shift the gears so that the chain is on the second largest rear sprocket.  Measure the distance traveled after one revolution of the pedal.  Record that measurement in centimeters in the table below and estimate how many meters the wheel will travel after 5 and 10 revolutions.  Check your predictions by measuring to find the actual distances.

 Revolutions of the pedal 1 5 10 Predicted distance (cm) x Measured distance (cm)

Part B

The next table shows distance traveled for each of the 5 lowest gears on your bicycle.  If the bicycle you are looking at has more than five gears on the rear sprocket, only look at the largest (i.e., slowest) five.  Measure the distance traveled for one revolution of the pedal in centimeters and then estimate in meters the distance traveled for five and ten revolutions.  (You already have this information for the first and second gears!)

 Gear first second third fourth fifth one revolution (measured) five revolutions (predicted) ten recolutions (predicted)

How many revolutions of the pedal would be necessary for the bicycle to travel 100 meters or 1000 meters in first gear?  Discuss this question with your group and record your estimates below.  How many revolutions of the pedal would be necessary for the bicycle to travel 100 meters or 1000 meters in fifth gear?  Again, discuss and record your answers.

First Gear:
100 meters » revolutions
1000 meters »  revolutions

Fifth Gear:
100 meters »  revolutions
1000 meters »  revolutions

### Activity  6:  Gear Ratios

• bicycle with 10 or more gears
• calculator
• Reference Copy

Part A

Record the number of teeth on each sprocket of your bicycle.

 Rear Sprockets Front Sprockets

1) When the chain is on the smallest front sprocket and smallest rear sprocket, will the bicycle go faster or slower than when the chain is on the largest front sprocket and largest rear sprocket?

2) What do you think will happen to the speed of the bicycle when you keep the chain on the largest front sprocket but change the rear sprocket from the largest to a rear sprocket with half as many teeth?

Part B

We will now look at speed of a bicycle based on the gear being used.    Complete the table below for the lowest 12 gear combinations on your bicycle.  (If your bicycle has only 10 speeds, you do not need to record anything in the columns for gears 11 and 12).

To complete the third row of the table, divide the number of teeth in the front sprocket by the number of teeth in the rear sprocket.  Round your answer to two decimal places.  In the fourth row, write numbers 1 through 12:  1 for the smallest ratio (lowest gear) through 12 for the largest ratio (highest gear).

 Gear 1 2 3 4 5 6 Number of teeth in front sprocket Number of teeth in rear sprocket Row 1 divided by row 2 Gear (based on speed)

 Gear 7 8 9 10 11 12 Number of teeth in front sprocket Number of teeth in rear sprocket Row 1 divided by row 2 Gear (based on speed)

1. Why are the gear numbers in row 4 different from the gear numbers at the top of the table?  How could you design a bicycle so that the top and bottom rows of the table were the same?

2. About how many times faster is the fastest gear compared to the slowest gear?

Summary Questions

1.  Explain the following statement:  If your bicycle is in a gear where the chain is connecting a sprocket with 44 teeth in the front to one with 22 teeth in the back, it will travel at the same speed as if you had 50 teeth in the front and 25 in the back.

2.  Find additional combinations of gears that would be the same speed as having 44 teeth on the front sprocket and 22 teeth on the rear sprocket.

3.  Is it possible to have the same speed as above when the chain is on a front sprocket with 51 teeth?