Football scores problem

As a maths teacher, one of my aims is to get students to think about the world mathematically. So there aren’t many things more exciting than having a student come to me with a problem they noticed and are trying to solve themselves. Just for the fun of it. This is the story of one of those moments.

The other day I had a student stay back after school and told me of a problem he was going to figure out. He had noticed a pattern in the football scores he’d seen over the weekend, and wanted to know how many different ways that pattern was possible.

Now, unless you are from Australia, this going to take some explaining. In this part of the world, “football” refers to Australian rules football (which is not rugby, despite the fact that I’ve blogged about that before).

Credit: Tom Reynolds. Sourced from Wikipedia.

There are two ways to score in “Aussie Rules”:

  • A “goal”, which is worth 6 points.
  • A “behind”, which is worth 1 point.

For example, a team with 3 goals and 4 behinds has 22 points, which is usually reported as “3.4.22”.

My student had noticed that it is possible for the total score to be the product of the goals and behinds. For instance, 7.7.49 is a possible score, and 7 × 7 = 49.

His question was: how many scores like this are possible?

He’d already made some progress on the problem when he told me about it. He defined the problem as being the solution to 6a + b = ab, where a and b are both non-negative integers.

How awesome is that?

Now, this particular student is the type of kid who’ll go looking for problems like this, who just naturally love maths. But I’m wondering how I would go about using this in a whole class setting. How would I structure a lesson around this idea? What curriculum could it be fit into? This type of equation that only allows integer solutions was something I studied as an undergrad, but this seems simple enough for high school kids to get – one of them did pose the problem, after all.

We did manage to solve the problem. But I think I’m going to leave that for another blog post. I’ll give you a hint: 0.0.0 is also a solution. 😉