[3ACTS] Split Time

Here I am tinkering with Google Maps again.

This is the kind of application of proportional reasoning you can find in abundance on 101questions. What’s remarkable about it is the e-mail I received that kicked it off:

My workouts during the indoor season are based on 200 meter split times (since most indoor tracks are 200 meters around), but our local track is only 160 meters around. So if I wanted to be running a 35 second 200, what would I have to run 160 in?

You have here a math teacher who applied proportional reasoning to his own life, who recognized what he was doing, and who then took steps to reconfigure that experience into a task so that his students could experience and resolve the same dilemma.

Math teachers use math. Our challenge is to preserve those experiences for our students.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I’ve done a problem very similar. If I ran a half marathon at this pace, how long will it take me to do a 10-miler? And what is that pace? Totally real life situation for me, and something I’ve been calculating the last couple of weeks!

  2. A related problem came up as our school’s track was being resurfaced this past year. Have students determine where the starting lines need to be placed for the 400 meters (one lap), if all runners run in their lane the entire way around. What information do we need: How wide is each lane? Should we measure from the inside of a line, or from the center?

    There’s a lot of neat track and field math to be done. About 5 years ago, the national track and field federation decreased the angle for measuring the sector lines for discus from 60 degrees to 34.92 degrees. What proportion of landing area was lost in this transition?

    And am I the only one impressed that Dan runs 75 second quarters?

  3. I gave a question like this on a recent learning target quiz to assess students’ abilities to write and solve a proportion. I hadn’t talked to them much about the power of making an estimate before they began “doing the math” of the problem. Some students incorrectly set up the proportion or solved in incorrectly, and came up with a time that was less for more distance. They didn’t realize they had made an error because they had nothing to compare their answer with to check for reasonableness. Won’t make that mistake again.

  4. Tried this problem out with my son at home yesterday. He totally loved working through this. When he came home from his cross country meet today, he looked at his time for the 3km race and began talking about how it compares to the 4.5km trail he trains on. Real life application at work. love it

  5. You could do a straight forward solution using ratio or proportion but the result would not be a true representation of what should happen in reality. In this case of changing 200m to 160m outside influences on the runner may not change things much but the logic that you can think proportionately and have accuracy would not be totally correct. Things like runner fatigue over time and distance will change outcomes. For example it is fairly easy to run 100m in 12 seconds but not many can run the proportional time of 48s for 400m. I think to be truly proportional in workload and give the same exercise benefits the times for the 160m split would need to be faster than the time given using a proportion. We could use math to calculate that real time as well. Could we create a formula to calculate proportional workloads over different distances so the runner gains same benefit? What factors or variables would we include?

  6. thanks for this. I’m very interested in proportional reasoning, and there’s lots of potential here to elicit interesting ideas from students. Additive thinkers might think it’s impossible: 40 fewer meters means 40 fewer seconds and that can’t happen. Others might say “Why don’t you just figure out where to stop so that you run 200 m?” or “Why don’t you run for 35 seconds and see how far you go?”
    With the plethora of free GPS apps, students can actually try this out, which is great. PE teachers would probably love to collaborate on this.
    The major discussion I would want to have with my middle schoolers is “How can both the distance AND the time be different, but the rate of speed is the same?” I’m always curious how students come to believe this. Many middle schoolers think doubling a chocolate chip cookie recipe doubles the chocolatiness