McNugget Numbers in Grade 3

I was lucky enough to be a guest in @EmmPenn32 grade 3 class.

Like any good guest, I had to BYOM (Bring Your Own Math).

I decided to bring McNugget numbers as the day’s math. This choice was made, because I find it interesting. I wanted to know what grade 3s would think about this interesting number. (43, is the significant number, in McNuggetology, the biggest integer that CAN’T be made out of packs of 6, 9, and 20).

Numberphile’s fun video on the topic is here:

The question we posed was basically this:

McNuggets only come in packs of 6, 9, and 20. Can I get exactly 43 McNuggets.

Canadians need to suspend their disbelief: the 9 pack changed to a 10 pack, as I found out when I went to get empty boxes (at which point the girl behind the counter asked the manager, who just waved her hands at this odd request). And now, they have 4 packs that they use in Happy Meals. In the real world, the mathematics of McNuggets are now boring. In the mathematical world, we can do whatever we like with McNuggets.

This gave us the perfect chance to say to kids, “this is our mathematical world”. We WANT kids to create mathematical worlds. Indeed, one kid wanted to say they would get 44 and eat 1. This solution is not allowed in our particular mathematical world.

The extensions we used to tease out the structure of the problem were:

Can you get exactly 44 McNuggets?

What’s a number much bigger than 44 that you can make with packs of 6, 9, and 20.

This problem was enough to keep kids going for close to 2 hours, with a number talk and some talk about multiplication before.

Kids patiently worked out different combinations. We noticed that they assumed 43 would be a possible combination. I think we are more used to proving things true, than not true.

At one point, @EmmPenn32 prompted a kid to use a number line to start showing combinations. We realized the hundreds square would help too.

We consolidated the problem by circling numbers we knew would work. 6, for example, obviously would work. 8 would not. Kids gave us all the ones they had, like 60, and 72, and 102, that they knew worked. We circled them.

It was here that I had my own magic moment. I realized that just by going back to proven cases, and then jumping forward by 6, 9, or 20, we would be able to get all combinations up to infinity. I couldn’t quite think of a way to convey that to the kids on the spot, but think about: 1, 111, 456, 505, 999 McNuggets. 1 111 111 McNuggets. Odd looking primes like 201 would have their own McNugget expansion. Try this, and blow kids’ minds.

Better yet, make your own context. Make a problem with packages of, say, 6, 7, 12. These numbers are called Frobenius numbers. What’s the Frobenius number for 6, 7, and 12?

Play. Play with numbers, and let kids wonder. That’s what I say. It’s not REALLY about Chicken McNuggets- it’s about combinations, primes, composites, factors, the structure of integers, and play. It’s always about play.