Wednesday, March 25, 2015

The high chair for learning inequalities (also, a broken calculator)

who: J2 and J3
when: at lunch
where: local Japanese restaurant

Who is taller

While eating lunch today, we found a good excuse to talk about (mathematical) inequality. Next to our table were two spare chairs, a kid high chair and a standard adult chair. The natural questions:

  • if J2 sits in the high chair and J3 in the adult chair, who will be higher? 
  • Are you sure and why do you think so? 
  • What if you switch with J2 in the high chair and J3 in the adult chair? 
  • How confident are you of the answer now?

In the course of the conversation, they talked about who is taller standing (J2) and which chair has a higher seat. It made intuitive sense to them that the taller person in the higher seat would end up higher. Still, it was good to test:

For the question about switching seats, they weren't sure, but thought J2 would still be taller (he was). Finally, I asked J2 if this would always be the case: if he sat on a lower seat, would he still be taller? After a minute's reflection, he said it could be either of them. Could they happen to end up the same height?  Also, yes!

With these simple props, it ended up being a surprisingly good conversation.

A broken calculator

After reading Mike Lawler's post about of Dan Finkel's Broken Calculator puzzle, I had to share it with J2. He was asleep at the time, so I made my own in pencilcode (a souped up version here). This morning, after breakfast, I showed it to J2, gave him the back story. We briefly talked about square roots to remind him, and then he was hooked.

You can see his current progress here, working toward finding a way to get every integer from 0 to 109:

Mike's post and videos are very good, so I only want to make a couple points to complement his discussion:

  1. Playing with the calculator first made the problem much more accessible. For J2, it helped him see that the +5 and +7 buttons could only make the value larger. It also helped him recognize that he needed square numbers for his square root and to strategize about how to make them. Finally, it led him to discover the trick for making 1.
  2. Making other numbers than 2 became a very natural extension that he asked on his own. At first, he started recording (or having me record) the numbers he had made on a paper, then I added the table to our program to keep track automatically.
  3. He had fun the rest of the day asking other people, mostly his mother, if they could figure out how to make 2.
  4. It was also very easy to extend this by asking about other combinations than +5 and +7. We played with a +6 and +7 version that is, conceptually the same, but practically much more difficult since you lose the ones-digit preservation.
For anyone who wants to sneak in some calculation practice, this served that purpose, too. Why, you might ask? Even though he could always see an answer by pressing the button, there was a cost if he pressed the wrong one because then he would have to go through his sequence again. As a result, he would pre-calculate each operation to make sure it was taking him along the right path.

Finally, this same framework could be used easily with other operations. In particular, for kids who aren't yet ready for square roots, the reduction button could be division (e.g., divide by 4) or even subtraction (e.g., subtract 19).


  1. I love these sort of conversations. Your "who's taller?" discussion reminded me of a similar conversation I once had with my daugher

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