## Tuesday, January 10, 2017

### Running, rates, rounding

My running session this morning gave me an idea for a kind of 3-act math discussion with J1 and J2. I will discuss this with them when they come back from camp and see what they think. I expect the last questions will be hard for them and I would like to see how much progress they can make working together.

# First Act

Today, I went running and recorded some information on my GPS. For five laps, I ran moderately fast. Here is the data:
Time           Rate         Distance
3:00          12.7 kph         635 m
3:00          12.9 kph         647 m
3:00          12.6 kph         633 m
3:00          12.7 kph         637 m
3:00          12.8 kph         645 m

What do you notice?
What do you wonder?

# Second Act

My target was actually to run 12 kph for each of these three minute segments. After the first lap, I knew that I could run more slowly and still hit my target. I wondered, how much less than 635m could I run and still hit my target?
If I compare two laps, both rates and distances, can I figure out the distance I get for each 0.1 kph? Is there another way to calculate the difference in distances for each 0.1 kph?

# Third act

For some reason, this made me think about rounding that J1 had recently been studying. He is a bit disturbed about what to do with values that are halfway between the rounded levels, for example whether 15 should round up or down to the nearest ten. Since this investigation of running data involved calculations with measured values and rounding, I though it would be instructive to explore a couple of calculations:

• I have two distances, rounded to the nearest 10 cm of 20 cm and 10 cm. What is a reasonable range for the difference of those distances?
• My GPS measured a time of 3 minutes (3:00, rounded to the nearest second) and speed of 12 kph (12.0 kph rounded to the nearest tenth of a kilometer per hour). What distance did I run? What is a reasonable range for that distance?