RATIO AND PROPORTION

Wheels in Motion
(Teacher Copy)

Activity  3:  Finding Distance

Students will measure distances traveled by revolutions of a bicycle tire and make predictions based on their measurements.  Also use the student copy.

To understand the relationship between wheel circumference and distance traveled for bicycles. Introduction: Have you ever noticed that when you are bicycling, the distance you travel is related to the number of revolutions the wheels make?

Answers will vary.  Sample answers are based on a 21-speed, 26-inch (tire size) mountain bike.

Part A

 Revolutions 1 2 3 4 5 Distance (cm) 202 cm 404 cm 606 cm 808 cm 1010 cm

Part B

 Revolutions 6 7 8 Predicted distance (cm) Answers and methods of prediction will vary Actual distance (cm) 1212 cm 1414 cm 1616 cm

Part C

One way to predict the number of revolutions of the bicycle wheel in 20 meters is to divide the number of centimeters (e.g., 2000 for 20 m) by the number of centimeters traveled in one revolution.  Another method is to first divide the number of centimeters per revolution (obtained from the table in Part A) by 1000 to get the number of meters per revolution and then divide 20 meters by the number just calculated for meters per revolution.

 Distance (meters) 20 40 60 80 100 Predicted revolutions 9.9 19.8 29.7 39.6 49.5.

Closing Discussion

Discuss the following situation as a class or in small groups:  "Once while riding on my bicycle along a path, I crossed a strip of wet paint about 6 inches wide.  After riding a short time in a straight line, I looked back at the marks on the pavement left by the wet paint picked up on my tires.  What did I see?"  There will be a series of six-inch marks, with half made by the front wheel and half made by the rear wheel.  The distance between the "front wheel" marks will be the circumference of the front wheel.  The distance between the "rear wheel" marks will be the same.  Distance between front and rear wheel marks will depend on the frame size of the bicycle.

Activity 4:  Comparing Distances

Students will graph the relationship between distance traveled and number of bicycle wheel revolutions, and calculate the relationship between diameter and distance traveled.

To make accurate predictions of distance traveled from tire size and number of wheel revolutions on a bicycle.

Introduction:

Remind students of the relationship they found between circumference and diameter (Activity 2).  Do you think the relationship between distance traveled and bicycle wheel revolutions is linear (i.e., plotted points lie on a line)?  Yes.  Does the relationship depend on how fast you go?  No.  Would you rather have large or small wheels on your bicycle?  Speed is dependent on gears and pedaling speed, rather than wheel size.

Part A

The graphs should be drawn with even increments along the x- & y-axes.  The lines drawn through the points should show a linear relationship between distance traveled and wheel revolutions.
Part B
Because diameter is measured in centimeters, distance traveled (in meters) in one revolution is 2¹d/1000, where d is the measured diameter.
Sample answer for a 26" mountain bike tire (69-centimeter /27-inch outside diameter):
 Revolutions 1 2 5 8 10 Distance (meters) 0.43 m 0.86 m 1.29 m 1.72 m 2.15 m

Activity  5:  Pedaling Distances

Students will make predictions about distances traveled and measure actual distances based on revolutions of the bicycle pedals.

To understand the relationships between gear size and distance traveled for various gears on a bicycle.

Introduction:

Have you every thought about how much you have to pedal to get where you are going?  Have ever realized that the gear you are in determines how much you have to pedal?  In this activity, we will look at the relationship between the number of times you pedal and how far you travel.

Sample answers are based on a 26-inch tire.

Part A

 Revolutions of the pedal 1 2 3 4 5 8 10 Predicted distance (m) x x x x 10 m x 20 m Measured distance (cm) 202 cm 404 cm 606 cm 808 cm 1010 cm 1616 cm 2020 cm

 Revolutions of the pedal 1 5 10 Predicted distance (cm) x 11 m 22 m Measured distance (cm) 224 cm 1120 cm 2240 cm

Part B

 Gear first second third fourth fifth one revolution (measured) 202 cm 224 cm 263 cm 309 cm 354 cm five revolutions (predicted) 10 m 11 m 12.5 m 15 m 7.5 m ten recolutions (predicted) 20 m 22 m 25 m 30 m 35 m

First Gear:
100 meters » 50 revolutions
1000 meters » 500 revolutions

Fifth Gear:
100 meters » 29 or 30 revolutions
1000 meters » 290 to 300 revolutions

Activity  6:  Gear Ratios

Students will calculate gear ratios and relative bicycle speed.  Also use pages S6 and R2.

To understand the relationships between gear size and speed for various gears on a bicycle.

Introduction:

Have you ever realized that the more you pedal, the more your wheels turn, and the farther you go?  [Discuss how the gears on a bicycle are arranged, lowest to highest (see the Reference Sheet).]  In this activity, you will look at the relationship between the number of times you pedal and how far you travel.

Sample answers are based on a 21-speed, 26-inch (tire size) mountain bike.

Part A

Rear Sprockets:     14     16     18     21     24     28

Front Sprockets:     28     38     48

1. In general, the largest front sprocket with the largest rear sprocket will be a higher gear than smallest with smallest, and the bicycle will go faster in a higher gear.   However, it will depend on the ratios of the teeth on the front sprocket to the teeth on the rear sprocket; the highest ratio will go the fastest.
2. Students should say something like the following:  The gear will be higher, or the speed of the bicycle will be twice as great after the change.  The speed will only double, however, if you continue to pedal at the same rate.  Most people pedal slower when they shift to a higher gear.

Part B

 Gear 1 2 3 4 5 6 7 8 9 10 11 12 Number of teeth in front sprocket 28 28 28 28 28 28 38 38 38 38 38 38 Number of teeth in rear sprocket 28 24 21 18 16 14 28 24 21 18 16 14 Row 1 divided by row 2 1 1.17 1.33 1.56 1.75 2 1.36 1.58 1.81 2.11 2.38 2.71 Gear (based on speed) 1 2 3 5 7 9 4 6 8 10 11 12

1. If the ratios did not overlap, the bicycle rider would have to shift the rear sprocket to the opposite end to make a smooth transition between gears.  To design a bicycle as described, the front sprockets could be made with a greater difference in number of teeth between them, or the rear sprockets could be made with a smaller number of teeth between them.
2. The answer will depend on the ratios found.  In general, the speed of the highest to lowest gear is found by dividing the largest ratio by the smallest ratio.

Summary Questions

1.  The ratio of the two pairs of gears are in the same ratio, 2:1.
2.  The back gear must have exactly half as many teeth as the front gear.
3.  No, because the rear sprocket would need 25 1/2 teeth.
Closing Discussion
• Mountain bikes and bikes for use in hilly areas often have three front sprockets, the smallest having only 28 teeth.  Why would you want a gear this small on the front?  For use in going up very steep hills.
• A common set of rear sprockets (called the freewheel) on a racing bicycle has gears with 13, 14, 15, 16, 17, and 18 teeth.  The front sprockets (called the chainwheels) often have 44 and 52 teeth.  Would this be a good combination to use in a race over hilly or flat roads?  Why do you think racers use this combination of gears?  These gears could be used for going on roads that are flat or downhill.
Further Exploration
You may wish to visit a bicycle shop and get a gear chart which shows how the number of teeth in the sprockets relate to the distance traveled by the bicycle.