Dividing -24 into -2 groups

My younger son had a little trouble with this problem today:

This is problem 8 from the 1991 AJHME, which you can find here:

The 1991 AJHME on Art of Problem Solving’s website

When we sat down to talk through the problem I asked him how you could make a fraction large? His answer was that you wanted to divide by 1, and he thought that 8 divided by 1 would lead to the largest value here.

However, he knew that 8 was not the correct answer and he then wondered what would happen if you divided a negative number by a negative number – specifically -24 divided by -2.

I asked him what division meant in this case and he said “you’d have to divide -24 into -2 groups.”

Interesting.

I thought it would be fun to show him an old video from 2013 – in this video he and I were talking about division and specifically the definition of division:

Of course, we talked about division in many different ways over the course of that year:

and looking at some of these old videos helped me get a better understanding of today’s problem.

Thinking about the importance of understanding the definition of division reminded me a little of the first couple of minutes of Jacob Lurie’s Breakthrough prize talk (specifically from about 1:00 to 3:00):

As with Lurie’s point about an imaginary number of apples, dividing -24 into -2 groups is hard to understand. Though the definition of division is a little abstract, I think it is useful for kids to see and will help them down the road as they try to understand more difficult problems.