More about Constant Speed

10 min

Narrative

In this Warm-up, students calculate the two unit rates associated with a ratio relating time and distance. They connect these unit rates to the terms “speed” and “pace.” They learn that speed describes distance traveled per 1 unit of time and pace describes time elapsed per 1 unit of distance.

To find the time it took to run 1 mile, students may divide 75 minutes directly by 12. They may also find it more incrementally, by finding the time it took to run one or more intermediate distances, with or without using a table or a double number line diagram. (For example, they may divide both 75 and 12 by 3 to find the time to run 4 miles, and then divide that by 4 to find the time to run 1 mile). 

Likewise, to find the distance run in 1 minute, students may divide 12 miles by 75 and express it as 1275\frac{12}{75} or 0.16, or they may reason indirectly. (For example, they may divide both 12 and 75 by 3 to find the distance run in 25 minutes, and then divide that by 25 to find the distance run per minute.)

Monitor for different ways of reasoning, and select students with varying approaches to share later.

Launch

Arrange students in groups of 2. Give students 3 minutes of quiet think time, followed by time to share with a partner and for a whole-class discussion.

Student Task

While training for a race, Andre’s dad ran 12 miles in 75 minutes on a treadmill. If he runs at that rate:

  1. How long would it take him to run 1 mile? Show your reasoning.
  2. How far could he run in 1 minute? Show your reasoning.

Sample Response

  1. 6.25 minutes. Sample reasoning:
    • 75÷12=6.2575 \div 12 = 6.25
    • Using a table:
    distance (miles) time (minutes)
    12 75
    4 25
    2 12.5
    1 6.25
  2. 0.16 mile. Sample reasoning:
    • 12÷75=0.1612 \div 75 = 0.16
    • 1275=425=16100\frac{12}{75} = \frac{4}{25} = \frac{16}{100}, which is 0.16
    • Using a table:
    distance (miles) time (minutes)
    12 75
    1215\frac{12}{15} or 45\frac{4}{5} or 0.8 5
    425\frac{4}{25} or 0.16 1

Synthesis

Select students with different strategies to share with the class. Record their methods, and display them for all to see. If the strategies of dividing 75 by 12 for the first question and dividing 12 by 75 for the second question are missing, demonstrate them and add them to the display.

If not already mentioned by students, highlight that 6.25 minutes per mile and 0.16 mile per minute are two unit rates associated with the 12-to-75 ratio of distance in miles to time in minutes.

Then, introduce the distinction between speed and pace:

  • When we find the number of miles per minute or meters per second that an object is moving, we are finding the speed of the object. The unit rate 0.16 mile per minute is the speed of running. 
  • When we find the number of minutes per mile of seconds per meter, we are finding the pace of the object. The unit rate 6.25 minutes per mile is the pace of running. 

If time permits, consider asking students:

  • “Which unit rate—speed or pace—would you choose to find how long it would take Andre’s father to run 8 miles? Why?” (Pace, because it tells how far he runs in 1 mile, so multiplying it by 8 would give us the time to run 8 miles. Speed, because dividing 8 by 0.16 tells us many groups of 0.16 mile are in 8, which gives the number of minutes.)
  • “Which unit rate—speed or pace—would you choose to find how many miles Andre’s father could run in 30 minutes? Why?” (Speed, because it tells us how far he could run in 1 minute, so multiplying by 30 would give the distance run. Pace, because dividing 30 by 6.25 tells us how many groups of 6.25 minutes are in 30, which gives us the number of miles.)
     
Standards
Addressing
  • 6.RP.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
  • 6.RP.A.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

25 min

20 min