Homeschool Math: How to Model Algebraic Equations with the Decanomial Square

Math can start getting really abstract when a kid begins algebra, so it's nice to take the time for plenty of hands-on modeling to reinforce the fact that those equations? They're all REAL numbers!

Well, until they aren't. That will be a whole new level of fun!

As always, the kids start by building the decanomial square. The first few times they did this, it took them a while to figure out where everything goes, but now they're old hands at it and can do it seamlessly. They don't know this, but they've internalized the visual patterning of the multiplication table, mwa-ha-ha!



You probably can't read the little squares, but we've labeled the rows and columns of the decanomial square. The row and column for 1 is a, 2 is b, 3 is c, etc.

I then gave the kids graph paper, had them frame out a 10x10 grid, and label every piece of the decanomial square with its designation, by row then column. The little 1x1 piece is a squared. The 1x2 piece is ab. the 10x1 piece is ja. The 10x4 piece is jd. The 10x10 piece is j squared.



After Will looked at her finished chart, she said, "It's like chess notation!"

And that's why they call it algebraic notation! Just another reason why chess is such a great game.

Now that the kids have a reference for all of their labels, they can begin to make equations using the notation:

Notice that Will's top equation, (a+b)b=cb is incorrect. She reworked it when I pointed it out. Below that, Syd is modeling an equation that she wants to write. It will end up reading (cb)b=fb. Is she correct?

 This turned out to be an excellent hands-on exercise. Syd, a fifth grader, happily wrote and solved algebraic equations. Will, a seventh grader, happily multiplied polynomials. They're both internalizing big, intimidating concepts before they can even start to be intimidated by them.

Hopefully, that intimidation will never make itself known!