Note to the teacher On this page, students will learn about the relationships between wheel diameter, circumference, revolutions and distance. They will also convert measurement units and use fractions and percentages. Students will use formulas relating the measurements to compute distance, diameter or wheel revolutions when given the other 2, and in some cases will convert units between the English and the metric systems. They will also have to find fractions or percentages of their results. Students will have to use fractions, decimals and percentages to make these calculations, and will also have to reconstruct and manipulate equations. While the worksheet is designed to help students learn the geometry of the circle and the relationship between wheel size, revolutions and distance, and is also designed to provide the information needed to convert units between English and metric systems, and may be completed by students with little background in these areas, the existing ability to multiply fractions, decimals and percentages, and the ability to manipulate equations, will be necessary to successfully complete the worksheet. Teachers may wish to review any or all of these skills depending on their students background. Note that these exercises are more challenging than the Wheels 3 and Wheels 4 exercises. In addition to the skills necessary for these pages, students will also be required to multiply by fractions and percentages to calculate answers. Note also that there are no instructions regarding rounding. The answers assume rounding to 2 digits beyond the decimal place, except known fractions. Teachers may wish to supply additional instructions. If they do not, students answers will vary slightly, according to what rounding conventions they use. Instructions Robot A and Robot B are being field tested. Use the formulas below to determine the answers from the information provided. Diameter π (pi) Inch Centimeters 1 1.125" x 3.14 = 3.53" 3 1 = 2.54 Revolutions Distance 3.53" x 1 = 2 3.53" 3. Robot A goes 3 wheel revolutions and has a wheel diameter of 1.75 inches. Robot B goes 3 wheel revolutions and has a wheel diameter 150% of the wheel diameter of Robot A. What is the wheel diameter of Robot B in inches? What distance does Robot A travel in inches? What percentage times the distance Robot B travels will equal the distance Robot A travels? 5.31
Mechanics Teacher After reading the instructions, students are expected to use the following procedure in the more complex problems: Reconstruct the equations provided as an example Enter the data provided into these equations Manipulate the equation if necessary Solve the equations for the missing variable(s) Multiply the result by a fraction or decimal Convert measurement units if necessary Approximate classroom time: 20-35 minutes depending on students background Students successfully completing the worksheet will be able to: 1. Describe the geometry of a circle 2. Describe the relationship between radius, diameter, circumference, revolutions and distance for a wheel 3. Calculate circumference from diameter 4. Calculate distance from wheel circumference and revolutions 5. Multiply decimals, fractions and percentages 6. Reconstruct equations relating diameter, circumference, revolutions and distance 7. Identify data provided in word problems 8. Manipulate these equations to solve for different variables 9. Convert between English and metric measurement units Standards addressed: Math Standards Numbers and Operations Algebra Geometry Measurement Problem Solving Connections Technology Standards The Nature of Technology Standards 1 Design Standards 8,9,10 Note: Workbook answers begin on the next page. 5.32
Instructions Robot A and Robot B are being field tested. Use the formulas below to determine the answers from the information provided. Diameter π (pi) Inch Centimeters 1 1.125" x 3.14 = 3.53" 3 1 = 2.54 Revolutions Distance 3.53" x 1 = 2 3.53" 1. Robot A goes a distance of 16.34 inches. Robot B goes 50% as far. How far did Robot B go in inches? Percentages are parts of 100, so 50 percent is equivalent to 50/100 or.5. Since Robot B went 50% as far as Robot A, to find its distance, simply multiply the distance Robot A went by.5. Distance of Robot B = 16.34 inches x.5 = 8.17 inches 2. Robot A has a wheel diameter of 3.25 inches. Robot B has a wheel diameter 7 /8 as large. What is the wheel diameter of Robot B in inches? The first thing we ll need to do is convert the fraction 7/8 into a decimal equivalent. We do this by dividing 7 by 8 and we get.875. Now, all we have to do is multiply Robot A s wheel diameter by the decimal equivalent and we ll have the diameter of Robot B s wheels. Robot B wheel diameter = 3.25 x.875 = 2.84 inches 5.33
Mechanics Teacher 3. Robot A goes 3 wheel revolutions and has a wheel diameter of 1.75 inches. Robot B goes 3 wheel revolutions and has a wheel diameter 150% of the wheel diameter of Robot A. What is the wheel diameter of Robot B in inches? Remember what we said, earlier, about percentages? We ll do the same thing with this calculation. 150% is the same as 150/100 or 1.5. So the wheel diameter for Robot B is 1.5 times the wheel diameter for Robot A Robot B diameter = 1.75 x 1.5 = 2.625 inches. Now that we know the diameter of Robot B, it s easy to calculate its circumference., which is the distance the robot will travel in one wheel revolution, is equal to π x D. of Robot B = 3.14 x 2.625" = 8.24". Since the wheel made 3 revolutions, it would travel 3 circumferences in distance: distance Robot B travels = 3 x 8.24" = 24.72". What distance does Robot A travel in inches? To find the distance Robot A traveled, we ll do the exact same calculations we did on part B, above; the only difference is the diameter of the wheels on Robot A. The wheel diameter for Robot A is only 1.75 inches, so: circumference of Robot A = 3.14 x 1.75" = 5.50". And since the wheel made 3 revolutions, it, too, would travel 3 circumferences in distance. Distance Robot A travels = 3 x 5.50" = 16.50". What percentage times the distance Robot B travels will equal the distance Robot A travels? We can find the ratio of the distance Robot B traveled to the distance Robot A traveled simply by dividing the distance Robot B traveled, 24.75" by the distance Robot A traveled, 16.50". Ratio = 24.72" 16.50" = 1.50. Converting the ratio of 1.50 to percentage, we multiply by 100 and get 150%. Did you notice that the ratio of the distances was the same as the ratio of the wheel diameters? That s because to calculate the distance traveled, we multiplied each diameter by the same numbers, and when we calculate the ratios of distance traveled, all those numbers, like the number of wheel revolutions and π were in both the numerator and denominator and so they cancelled each other out. 5.34
4. Robot A travels 25.25 centimeters and goes 4.75 wheel revolutions. Robot B has a wheel diameter 5 /8 as large as Robot A and also goes 4.75 wheel revolutions. What is the wheel diameter of Robot A in centimeters? The first thing we re going to want to do is find out how far Robot A traveled in one wheel revolution, because that s the same as the circumference. To find out how far it went in one revolution, just divide the total distance traveled by the number of wheel revolutions: circumference = 25.25 cm 4.75 = 5.32 cm. Once we know the circumference, it s easy to find the diameter. Because we know that C = π x D, we can calculate the diameter, D = C/π. Diameter of Robot A = 5.32 cm / 3.14 = 1.69 cm. What is the wheel diameter of Robot B in centimeters? Since the wheel diameter of Robot B is 5 /8 the diameter of Robot A, all we need to do is multiply the wheel diameter of Robot A by 5/8. Actually we ll multiply it by the decimal equivalent of 5 /8 or.625: diameter of Robot B = 1.69 cm x.625 = 1.06 cm. Now that we know the diameter of Robot B s wheels, we can easily find the circumference of the wheels, the distance the robot will travel in one wheel revolution., as we know, is equal to π x D. The circumference of robot B=3.14 x 1.06 cm=3.33 cm. Since the Robot makes 4.75 wheel revolutions, it will travel 4.75 times the circumference: distance traveled = 3.33 cm x 4.75 = 15.82 cm. But the question isn t how far would the robot travel, in centimeters; it was how far would the robot travel in inches. To get that answer, we ll just divide the answer by the conversion factor of 2.54 cm/inch: distance traveled = 15.82 cm 2.54 cm/inch = 6.23 inches. What is the ratio of the distance Robot A travels to the distance Robot B travels? Remember what we said in the last part of question 3, earlier? The ratio or the fraction of the distances traveled will be the same as the ratio or fraction of the diameters, as long as the number of wheel revolutions is the same. We know that Robot B has a wheel diameter 5 /8 as large as Robot A. Therefore, the ratio of the wheel diameter of Robot A to the wheel diameter of Robot B is 8:5. So the ratio of the distance Robot A travels to the distance Robot B travels would be 8:5. 5.35