Teepees for Factoring

One thing that's been very shocking to me as I've made the transition from college instructor to high school teacher is the lack of number sense that many of my [Algebra II] students possess. Estimation skills, knowledge of times tables 0-12, and the recognition of a negative sign are so sorely lacking.  On a daily basis, I get told that a negative plus a negative is a positive, because, "two negatives make a positive, Mrs. Peterson."  Students also regularly explain to me that when we multiply a negative times a positive, the product will take the sign of the larger factor.

I haven't figured out how to break these bad habits.  But I'm working on it.

Needless to say, the thought of teaching factoring was a bit daunting.  How can I teach them to un-distribute, if they can't distribute correctly in the first place?

And then a couple colleagues of mine introduced me to the x-method, or what I call the teepee method.  This may be old news to many, but I had never seen it before, and I found that it was just the bit of organization some of my students needed in order to factor trinomials.

So, let's say we want to factor x^2-9x+20.  We create the following teepee:

Then we find two numbers that multiply to be the top number and add to be the bottom number:

And, viola!  Then we can factor the original trinomial: (x-4)(x-5).

This by no means solves all my problems.  How can we find those numbers in the first place if we don't know how to multiply?  However, it is a nice little organizer for those students who are visual learners.

I can't take any of the credit for this visual organizer.  I'm just passing along what I learned from my wonderful department.  But, in the words of LeVar Burton, "Don't take my word for it."  Here's what some of my students wrote when I asked them to choose their favorite form of factoring from the ones we had discussed so far (GCF, difference of squares, and trinomial factorization) and tell me why...

“Trinomials are my favorite because I like to make lil x’s and then put the numbers inside that would make the others true.”

“My favorite factoring exercise is trinomial factoring because it really makes you think.  The x’s really help too.”

“The trinomial factoring is my favorite because it’s easy to use the teepee.”