Number Sense – Division, Axes, Nerd Search

Wow. Quite a day. State testing (here, MCAS) is tomorrow. I believe teachers set the tone for kids, and I’m doing my best to make them feel calm and centered heading into tomorrow.

Number talk today:
Screen Shot 2014-05-07 at 7.36.59 PM
A few surprises:
How many student found the precise answer
How many of those students (almost all) expressed their answer as a decimal
How many of the students who chose to use an estimation strategy are my top students
How many students chose to think about it in terms of multiplying 9 by an unknown to get 1001, rather than division.

The last one is the most interesting to me. Division is defined (in rigorous mathematics, anyway) as no more than the inverse of multiplication). I think I’m happy that that is deep in the number sense of a number of my students. That said, division has got to be one of the basic procedures students do day in and day out that has the most possible representations and interpretations.  Need to find some more number talks to get at that ambiguity. Makes me think of this number talk, from Fawn, and how it could be adapted for a visualization of division.
Screen Shot 2014-05-07 at 7.43.52 PM

This question was the focus of class today
These 8th graders are having a really tough time with linear equations on axes with a scale other than 1 to 1. They love counting boxes to find slope, and get confused when that doesn’t work. And this is our fault — spending too much time on that procedure, and too many questions out of context (like the one above!) And it’s great that they can reliably find slope on a conventional coordinate plane, but when we think about concepts that students will apply in the real world, finding the slope on an Algebra textbook-style coordinate plane will probably never happen. The application of the principle of rate of change absolutely will. Which one are we preparing kids for? And what does preparing kids for the second one look like? That’s my tough question of the day. It’s a critical form of number sense, and I’m worried my students don’t have it and I don’t know how to teach it beyond saying here, think about this.

Nerd Search
Wrapped up class today by sharing Bill Amend’s Nerd Search from his Foxtrot comic.


Inspired by his talk at NCTM, I’ve been showing some of his nerdier comics to my students. I planned a bit of a shorter lesson today, and just planned to give the nerd search to a few students when they finished their problem set, but it ended up engaging a ton of students and inspiring a spirited discussion to end class. One student came up to me after class and asked how many she had to do (she’s a very conscientious student, but not usually especially motivated to work when she doesn’t need to). I told her it wasn’t homework, but she should try as many as she could if it made her happy. She said she wanted to get all of them, and made me explain the integral and sum. Then, feeling outworked by my students, I spent 20 minutes finding the square root of 375,559,383,241 by guess and check. I was successful! And grateful for the invention of calculators.

3 thoughts on “Number Sense – Division, Axes, Nerd Search

  1. howardat58

    Hi. You could probably save about 15 minutes by dividing your estimate into the big number and taking the average of the result and your estimate to get a new estimate, and repeating…..
    new estimate = (estimate + bignumber/estimate)/2

    1. dkane47 Post author

      Many students did! Although a number picked out 990 instead, which I thought was interesting.

      Part of my surprise was that they took that, and chose to express their answer as a decimal (11.22222..) rather than a fraction (111 2/9).


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