Friday, December 9, 2011

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!

1. Your post inspired me to share my own factoring method:

http://misscalculate.blogspot.com/2011/12/factoring-ax2-bx-c.html

2. Hey there, we've posted something similar to this method to our MPI blog:

http://patternsinpractice.wordpress.com/2011/04/25/factoring/

It's the method used in CME Project, and it does a pretty good job of justifying each step in the process. For example, the "multiply a and c" step happens naturally in the math instead of being something to memorize. Good luck!

3. I struggled with teaching factoring, and stumbled upon your post this past spring. I used this method to help students who came in for extra help on factoring ax2 + bx + c, and it was a hit!! I saved your post in my bookmarks, and I am making a note to teach your method the first time around this year! Thanks!!

4. We use this method as well and call it "slide and divide." Love it!

5. 15 years of teaching maths and I never knew! Thank you for sharing.

6. Oh my goodness! I just stumbled upon your blog through Pinterest and I just want to say THANK YOU for posting this! I have been teaching high school math for 10 years and I've never seen this method before. My students are VERY appreciative! :)

7. Neat method! I teach algebra at a community college, and I am going to give this method s try.