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1. Time and Distance Formulas Steve Tuckerman 11 th Grade Mathematics Education page

2. What are Time and Distance formulas? • Time and Distance Formulas relate time, distance, and speed. These relationships have many practical applications. Right now, Jeff Gordon is trying to decide how fast he has to go to pass Jimmie Johnson.

3. How to get around in this lesson • There will be a series of buttons to choose from. Forward and backward buttons are located on the bottom of the slide. • The home button returns you to the main menu • Quizzes have answer buttons. Click on the button that you think is the right answer. Answer buttons look like this:

4. Main Menu • Tutorial • Quiz One: Given rate and time, calculate distance. • Quiz Two: Given distance and time, calculate pace. • Quiz Three: Convert minutes per mile to miles per hour.

5. Why do you need to know the Time and Distance Formula? • To figure how long a trip will take • To see how far you can go in a set amount of time • To see how fast you took a trip • To compare different trips

6. Steps in solving Time and Distance Formulas • Step 1. Translate the question into mathematical terms. For example, if you are asked miles per hour, write the question as miles divided by hours.

7. Steps in solving Time and Distance Formulas • Step 2. Put everything in constant units. For example, if the question asks miles per hour, every time must be placed in terms of hours, and every distance must be placed in terms of miles.

8. Conversion Units • Remember: There are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day. • Remember: There are 12 inches in a foot, 3 feet in a yard, and 5280 feet in a mile.

9. Time Conversions • To convert minutes to hours, divide by 60. • To convert hours to minutes, multiply by 60 • To convert seconds to minutes, divide by 60 • To convert minutes to seconds, multiply by 60 • To convert days to hours, multiply by 24 • To convert hours to days, divide by 24

10. Measurement Conversions • To convert inches to feet, divide by 12 • To convert feet to inches, multiply by 12 • To convert feet to miles, divide by 5280 • To convert miles to feet, multiply by 5280

11. Step 3. Write the equation you want to solve. • Use the correct formula to solve it. • Be careful about what you multiply, and what you divide!

12. What are the basic formulas? • Distance = Rate * Time (d = r * t) • Rate = Distance / Time (r =d/ t) • Time = Distance / Rate (t = d / r) • Be sure you use the right formula!

13. To find distance • Distance = Rate * Time (d = r * t) For example, to find miles, multiply miles per hour (rate) times the number of hours. Miles = Miles x Hours Hours The hours cancel, you are left with miles.

14. To find the rate of speed • Rate = Distance / Time (r =d/ t) To find miles per hour, divide miles driven by the number of hours driven.

15. To find the time it takes to travel. • Time = Distance / Rate (t = d / r) Hours = Miles_____ Miles/Hour Remember, to divide fractions, flip and multiply Hours = Miles x Hours Miles Miles cancel, you are left with hours

16. Quiz 1 • How far can you go in 30 minutes at 55 miles per hour? Bowser wants to go skiing with his friends.

17. Step one • Put in constant units • Rate was given in miles per hour, so time must be in hours • Remember, to convert minutes to hours, divide by 60

18. Question 1What time should be used? • We are given 30 minutes. We want the answer in hours. Answer • A. 2 hours • B. 0.5 hours • C. 30 hours

19. Nice try. Bowser’s friends are still waiting. • Remember, to convert minutes to hours, divide by the number of minutes by 60. You divided by 15.

20. Nice try. Bowser’s friends are still waiting. You just changed minutes into hours.

21. Nice job! The time is one half hour, or 0.5 hours. Bowser goes skiing with his friends.

22. Question 2 • Now, how do you calculate how far you can go? • A. Distance = 0.5 hours / 55 miles per hour • B. Distance = 0.5 hours times 55 miles per hour • C. Distance = 55 miles / 0.5 hours Remember the basic formulas!

23. Nice try. If distance equals rate times time, then you must multiply, not divide. • Distance = Rate * Time (d = r * t)

24. Question 3 • Well done. You have the formula correct. Distance equals rate times time, so distance equals 0.5 times 55, or 27.5 27.5 is in what units? • A. Hours • B. Miles • C. Speed

25. Nice try. Remember, we were talking about miles per hour, so the answer is in miles

26. Well done. You can go 27.5 miles in 30 minutes traveling at 55 miles per hour. • Are you ready for another quiz? If so, click the right arrow

27. Quiz 2 • Joan Benoit won the 1984 Olympic marathon in 2 hours, 24 minutes, and 52 seconds. If a marathon is 26.22 miles, what was her average pace in minutes per mile?

28. Question 1 • The first step is to put everything in constant units. If it took Joan 2 hours, 24 minutes, and 52 seconds to run her race, and we are interested in her pace in minutes per mile, what is our first step? • Place everything in terms of hours • Place everything in terms of minutes • Place everything in terms of seconds

29. Nice try. We want to know how many minutes it took Joan to run each mile, so we need to know how many minutes she was running.

30. Well done. Since we want to know Joan’s average pace in terms of minutes per mile, we need to put everything in terms of minutes.

31. Question 2 • We know that Joan’s time was 2 hours, 24 minutes, 52 seconds. What formula gives us the number of minutes?

32. Remember, to convert hours to minutes, multiply by 60. To convert seconds to minutes, divide by 60. Everything has to be in terms of minutes.

33. 2 times 60 plus 24 plus 52 • 2 times 60 plus 24 plus 52/60 • 2 times 60 plus 24 plus 52/100 Remember, you are converting 2 hours, 24 minutes, 52 seconds to minutes.

34. Nice try. Remember, we want the answer in minutes. To convert hours to minutes, multiply by 60. To convert seconds to minutes, divide by 60. You forgot to convert seconds to minutes.

35. Nice try. Remember, we want the answer in minutes. To convert hours to minutes, multiply by 60. To convert seconds to minutes, divide by 60. You divided seconds by 100.

36. Well done. The answer is 2 times 60 plus 24 plus 52/60. You multiply 2 hours times 60 minutes per hour, add the 24 minutes, and add 52 seconds / 60 seconds in a minute. All units are now in minutes. The correct answer is 144.87 minutes

37. Question 3 • What was Joan’s average pace per mile? Remember, she ran 26.22 miles in 144.87 minutes. • 144.87/26.22 = 5.525, or 5 minutes 31 seconds per mile • 26.22/144.87 = .18 minutes per mile

38. Nice try. Remember, we wanted Joan’s pace in minutes per mile. Therefore, you divide the number of minutes she was running by the number of miles she ran. You did it backwards.

39. Well done. Joan’s average pace per mile in the 1984 Olympic marathon was 5 minutes and 31 seconds per mile. How far can you run at that pace? Click the right arrow for the next quiz.

40. Quiz 3 • We know that Joan Benoit ran the 26.22 mile Olympic marathon at an average pace of 5 minutes and 31 seconds per mile. What was her speed in miles per hour?

41. Step one • First, we have to put everything in constant units. We are interested in miles per hour, so we need to convert minutes to hours.

42. Question 1How do you convert minutes to hours? • Multiply the number of minutes by 60 • Divide the number of minutes by 60 • Multiply by 24

43. Nice try. Remember, you have to divide the number of minutes by 60. You multiplied by 60.

44. Question 2 • Well done. There are 60 minutes in an hour, so you divide the number of minutes by 60. • How do you convert 5 minutes and 31 seconds into hours?

45. A. Divide 5.31 by 60 • B. First convert 31 seconds into minutes, add to 5, then divide by 60 • C. Multiply 5.31 times 24

46. Nice try. Remember, you need to convert everything into minutes, so you have to convert seconds into minutes. You forgot to make this conversion.

47. Nice try. Remember, you need to convert everything to minutes. You tried to convert to days.

48. Question 3 • Well done. First, convert the 31 seconds into minutes. 31/60 = .52. Now, add the .52 to 5 minutes to get 5.52. Then, divide by 60. The answer is .092. Joan ran each mile in .092 hours. • How do you convert one mile in .092 hours into miles per hour?

49. 1 mile divided by .092 hours equals x miles divided by one hour • Cross multiply: 1 mile times 1 hour = .092 times x miles • X miles = 1/.092 • X = 10.9 miles per hour.