# Mod Clocks

Sometimes math sneaks up on my kids and me while we are playing (like when we made potholders), and sometimes I force it a little. This is a time when I forced it a little. Because who would ever think of making a clock with only 2 or 3 or 5 numbers on it? I wouldn’t. I thought of it because I was meandering through the postings of Henri Picciotto, my former department head and philosopher king of mathematics. If you’re interested in the article that inspired me, read Operation Sense, Tool-Based Pedagogy, Curricular Breadth: A Proposal. If you just want to start playing, read on.

After reading the article, I was curious what the 7-year-old would do with some of the activities discussed. So, I pulled out our trusty Boogie Board (which makes anything you write or draw more fun because…well…it has the word boogie in it), and I drew a circle. In the 12 o’clock position I wrote “0”, and in the 6 o’clock position I wrote “1.” Then I told the kid that this was a really funny counting clock. It only has one hand (and I drew the hand.)  The hand starts at 0. If the hand doesn’t move at all (input 0), then you get 0 (output 0.) If the hand moves 1 unit (input 1), then you get 1 (output 1.) But, get this, if the hand moves 2 units (input 2), you get 0 (output 0) again !

Here’s what it looks like visually, and with inputs/outputs written down:

The 7-year-old quickly connected it to the function machine game we had played a few weeks ago. I actually hadn’t expected that. So, I drew some more mod clocks for him, and let him guess the outputs based on the inputs I gave him.

And then something pulled him away. Maybe it was the prospect of playing LEGO with his brother. More likely it was the concern that his brother was already playing with his LEGO collection. In retrospect, if I still had his attention, I would have asked the 7-year-old to make up a clock of his own and quiz me. I would have given a couple right answers and a couple wrong answers so he’d have the pleasure of correcting me. If the 4-year-old had not already been pilfering his brother’s LEGO toys, I would have involved him, too, by asking him to draw crazy clocks of his own.

(Mathematically… for a 7-year-old, this mod clock activity is just fun. It has mathematical value at this age as a game to puzzle through and another way to make connections back to the idea of input/output and what happens to change one thing into the other. You can challenge older kids with inputs that would be unreasonable to count out empirically (like 72 or 2017), so that they have to find an algorithm or a formula for the mod clock function. If your older child is working with long division, you can (or they will) make the connection to the remainder in a division problem. If you’d like a quick lesson or refresher on this connection, you can check out Khan Academy’s explanation of modular arithmetic.)