I’ve been on my travels recently in the United States. I was staying with some friends in Arizona and they had this flag on the wall. It’s an old flag with 48 stars neatly arranged in 6 x 8 matrix. It was from the days before Hawaii and Alaska joined the Union, sometime before 1959.

Now, I’m sure that all American school children know most of this stuff already, but it got me thinking about the US flag and specifically the arrangement of the stars and it gave me an idea for a lesson.

So, I know that there are now 50 stars for the 50 states of the USA. But how are they arranged? And how could we describe this mathematically? I might start by showing students this zoomed-in portion to give them a hint.

It is, of course a 5×4 matrix inside a 6×5 matrix.

Was there a 49? Yes, briefly when Alaska joined in 1959. But what did 49 look like? Clearly it’s a square number, but they decided that to keep the aspect ratio looking more like a rectangle they offset the rows like this.

There were actually lots of different versions of the stars and stripes detailed on this wikipedia page:

It might be better to start with a simpler example from the early days when there were only 13 states (2×2 + 3×3). What other numbers can be represented as a sum of two square numbers in this way?

Is there a better way of doing this one? (It’s 36 stars, maybe 6×4 & 4×3)

You might want to give this table as a handout (or here as a pdf). Get students to work in pairs to come up with their own designs and compare those to the ones actually used.

And then, what if we go beyond 50? I don’t want to think too much about the political implications of such of move and who might end up as the 51st state, so let’s stick to this as a mathematical exercise!