I found this at a conference somewhere and it is the only way I've ever introduced factoring x^2 + bx + c.

It was called diamond math but I think it makes more sense to call it x-factor, mostly because it

*is*a type of factoring.

I give this to students at the start of class and tell them I have a puzzle for them to figure out. I tell them the first two are done as examples for them and I want them to figure out the rest.

That's it.

Then....we wait. There are always a few who figure it pretty quickly. I make them work 3-4 in a row to prove to me they know what's going on. I don't let students help each other or tell the answers.

Then there are a few who immediately start complaining...these are usually the students who get good grades but are not used to actually thinking.

After a few minutes/complaints I tell them to figure out a pattern with the numbers that can be repeated for every x.

This seems to help some. I let this drag out. I act very unhelpful. We wait some more. I try to hold out until every person has figured it out. If I can tell some people are getting verrrrry frustrated, I go to them one-on-one and try to prompt them with questions only.

From there, I give them a quadratic expression like x^2 + 6x + 8. I tell them that the c always goes on top and the b always goes on bottom and that we are always looking for the left and right numbers. I show them how to write the answer (x + 4)(x + 2) and then we practice.

A lot. A lot a lot.

Tip: Give students expressions with variables other than x to make sure they realize that the answer is written with the variable from the problem, not always an x.

See this post for my INB pages on x-factoring.

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