Open Response Today

Gave my students this task today. I think I stole it from the NCTM website, where it was referenced on a bog somewhere. I forget, can’t give proper credit.

We’re in the middle of our Pythagorean Theorem unit. My students are pretty good at finding different sides of right triangles and we’ve talked a little about Pythagorean triples.

Today, we looked at points in the coordinate plane. I didn’t introduce the distance formula, although a number of students gravitated in that direction–we just put triangles on the coordinate plane and found distances a few different ways.

After doing some questions as a class, I gave students the task. They were working in partners (new pairs this week, with mixed success), and had about 15 minutes to work through the question before we looked at some different examples of the distance formula.


  • The task was too hard for them. They struggled, the vast majority persevered, and everyone asked great questions.
  • Lots of kids had a big a-ha moment when they put the right triangle in the diagram.
  • They loved using their intuition with a complex question like this. Lots of connections with actual birds, dogs, and roads.
  • They did a great job of figuring out that the square root of 65 is between 8 and 9, and then applying that.
  • Partners worked well together. Makes me think that management issues in the past have come more from boring tasks than I thought.


  • A bunch of kids got caught up in arguing about their guesses without doing any calculation–trying to avoid the calculation until I stopped by to prod them.
  • I shouldn’t have included the scale factor, and used speed in blocks per hour or blocks per minute instead. I wanted to be faithful to the task, but a bunch of bright kids got stuck thinking about the conversions too early, and a bunch of strugglers ignored the conversion altogether.
  • Intuition around relative speed was pretty weak. Too many kids focused only on the distance, and I didn’t see a ton of learning on that point.
  • Explanations were, in general, poor. I don’t want to read too much into this, because an active, partner activity like this doesn’t lend itself to great written work, but it’s a big area for growth with my students.

Overall, it felt great. My biggest takeaway is that I want to ask my kids more often to make an estimate or a guess, and then follow up to find out if they were right. I haven’t done that enough this year, and I saw its power today.

2 thoughts on “Open Response Today

  1. Fawn Nguyen

    Hi Dylan. I appreciate your writing the wins and losses of this lesson. I think it’s worrisome if we didn’t see how a lesson could be improved in some shape or form.

    Nice “meeting” you at GlobalMath tonight. We were wondering what your twitter handle is.


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