Factoring Woes

My students are doing okay with factoring overall, but it has been a struggle.

This is the graphic organizer I ended up using this year:

Students find it helpful, which is a good thing. I like/hate it.

I want them to be able to factor without it.

Here are some things I am thinking of changing for next year:

1.  Fewer methods:  Reduce the number of factoring strategies, so students have less to sort through. I am thinking I could teach trinomials with ax^2 first, and then apply that method to trinomials where a=1. Students should be able to adapt to the simpler situation, and they'll have one less method to remember.

2.  Figure out how to connect the type of polynomial with the name of the method and how that method is completed:  I am currently expecting students to see a trinomial, then identify that it has a leading coefficient, then recognize that they should use the "airplane" method, and then remember how to do the airplane method. So complicated! It is kind of amazing that any of them can do this at all.

One of my students pointed out that the arrows I am drawing for the "airplane" method resemble a trident. What if I renamed the "airplane" method the "trident" method? That seems like a better connection between the original expression and what you do with it. (Trinomial = Trident method?).

3.  School-wide consistency:  There are only three math teachers in our school. Why haven't we done this already? No idea. We definitely need to get together and agree on an approach to factoring so that students aren't seeing a completely new process from year to year.

4.  Find a hook:  I haven't figured out how to motivate factoring beyond, "You are going to need to use this all kinds of ways later this year and next year".

All things to keep in mind for next time . . .

Unless . . . Is there a magic factoring wand that I don't know about?

Airplane Method for Factoring

I mentioned the "airplane" method for factoring in a recent post. Someone asked me what that was, so I thought I'd share.

I have seen a lot of methods for factoring a quadratic with a leading coefficient. Out of the ones I've tried, this is my favorite. The analogy to an airplane is a bit of a stretch, but students seem to remember it pretty well. So I'll take it.

I should also mention that, before I show this to students, I always spend some time letting them work on these by trial and error. I figure a process like this is worthless if they don't actually understand what they are doing. Once I feel like students understand the concept but they are still struggling to get every problem to work, I show them this. We treat it like a shortcut, and boy do they appreciate it!

Here is an example:

First, my students know they will need two binomials, so I start with two sets of parenthesis. Then I put the leading coefficient in each parenthesis. Hopefully, the students have a problem with this. We talk about why it is a problem, and I promise them that we will get rid of the extra 2 before we're all done.

Then, multiply a and c. (See the airplane wings? Use your imagination.)

Look for two numbers with product ac and sum b. (Propeller?  I know this is really a stretch.)

Put those numbers in the parenthesis.

Divide the extra 2. (The landing? Maybe.) It is pretty cheesy, but when students are having trouble I can say something like "you forgot the landing", and they know what I mean.

Done.

For something like this, you may need to divide both binomials. I point out how dividing by 3 and by 2 is the same as dividing by 6. We just choose the division that will keep integers.

Happy factoring!

Figuring Out Factoring

I have been thinking a lot about factoring lately. My algebra 2 students really struggle with it, and we have only factored quadratics (no sum/difference of cubes or grouping yet). I am worried because our first unit after winter break is rational expressions/equations. This unit is challenging when you CAN factor well, and almost impossible if you can't.

I am sure it will be necessary to take a few days to work on factoring before we start rationals. I need to review quadratics, teach cubes and grouping, and put them all together. I am working on how to structure that.

I have been doing some reading and searching for ideas. It seems that almost every method out there requires students to memorize some sort of process. (Except for this, and I am not sure my students are ready for that). As far as algorithms go, I am happy with what I've been using.

There are two issues that I want to address:

1. Helping students make a connection between what they SEE (trinomial, difference of squares, four terms, cubes, etc.) and what they DO (write two binomials using your preferred method, grouping, use a formula, or whatever).

2. Helping students get through problems where more than one method is required. (Like factoring out a GCF and then factoring the trinomial, or using grouping and then recognizing that one of the binomials is a difference of squares).

Here is a flow-chart I sketched out this morning:

I am picturing this on the white board, with an answer box next to it. You could 'drop' a polynomial into the top box and it comes out factored. If there was a GCF, you could put it in the answer box and then examine the remaining part and possibly drop it into another box. You could then take the pieces that come out of the second box and either add them to the answer box or drop them into another box. This all seems a little elementary, but I want it to be very obvious and very clear . . .

I am still refining this and hope to use it for the beginning of second semester. I will write about the final product. In the mean time, suggestions welcome. :)