education · Factoring · math

Factoring Quadratic Expressions

I have a very distinct memory of my students experiencing cognitive overload last year during our unit on factoring, so I knew I needed to make some changes if things were going to be different this year.  Much of the changes I have made to my approach to teaching this school year are a result of reading Craig Barton’s How I Wish I’d Taught Maths.  One big takeaway I had was to break down complex processes in to smaller steps.  While the process of factoring seems like one process to me, to a novice it requires about 4 or 5 different skills.  I decided to break the process down and teach each step separately before putting it all together as one process.  It went so much better than last year!  Here’s what I did:

Step 1:  What is a greatest common factor?  While some students might remember what a greatest common factor is from their elementary school days, I have learned that most students do not recall this term without at least doing a quick review.  So we take some time in class to practice finding the greatest common factor of two numbers – first by writing out a list of factors, then by using prime factorization.  We went back to our “birthday cake method” from simplifying radicals, which lessened the cognitive load for this process.

Step 2:  Sum and Product puzzles  Every year I find that this is actually the hardest part of the factoring process – finding two numbers that multiply to ac and add to b.  Even when students are able to make a list of factors, the biggest struggle comes with finding the sum of the factors, especially when negatives are involved.  So we spent an entire class just practicing this step.  In the past I would have considered this a waste of time, but now I know how important it is to master each step along the way.  The worksheet I used is not my own, so I won’t be sharing it here.  But a quick google search of “sum and product puzzles” will produce several good results.

sumandproduct

Step 3:  Factoring the Greatest Common Factor of a Polynomial  Once we had some practice finding the greatest common factor of two or three numbers, we moved on to factoring out the greatest common factor of a polynomial.  With my ELL class, this was done a few days after introducing GCF to allow time for practice and mastery.  For my non-ELL class who was more familiar with GCF, this was done on the same day.  Although this step combined several skills – identifying common variables, dividing by the GCF (prior knowledge of dividing by a monomial), and writing the answer in factored form – none of the skills proved to be too complex that they required separate attention.

Step 4:  Introduction to the “Box Method” with completed boxes  This was by far the best decision I made throughout the process.  I don’t remember the reason, but on the particular day I had planned to start teaching the box method, our school was running on a shortened schedule.  I knew it would be difficult to make any progress on teaching the box method in such a small amount of time, but I also wanted to continue building on our factoring momentum.  That’s when I decided to give students completed “boxes” so that we could practice the individual skill of factoring using the box method, but without having to complete the entire process from the beginning.  Although students struggled at first, by the end of the short class they were able to factor out the greatest common factors and divide to find the second binomial factor.

prefilledboxes

Step 5:  Factoring with the Box Method  Now that students had experienced success with the individual steps of factoring using the box method, it was time to put the steps together.  In this lesson, students were taught how to set up the “box” and the “X.”  While it was challenging, students were less overwhelmed because of their previous experiences with both components.  On the second day, to help scaffold the process, students were given the following template in a dry erase pocket.  Some students found it helpful, while others decided to use the blank side to create their own X-Boxes.

template

Step 6:  Combining GCF with the Box Method  The final step of this process was to have students factor trinomials with a GCF, and then use the box method.  Because we spent so much time finding greatest common factors throughout the process, this additional step didn’t overwhelm students either.

While breaking down the skill took a little more time and planning on my end, the results were so worth it!

20181101_111336.jpg
Click on the picture to download the question stack pdf

 

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s