A recent online conversation got me thinking about the factoring of trinomials.
To start with, I would like to step back, and think about why this topic is prominent in the teaching of algebra. In the age of computer algebra systems (CAS), factoring trinomials is not an important skill, except of course for the purposes of school math. Most of the trinomials we ask students to factor are created specifically so that students can actually factor them. It is a curious, circular situation, and we’re back to asking why we do this.
As I see it, it’s a little bit like multi-digit arithmetic: we can do it electronically, but there is something to be learned by doing at least some of that “manually” in various ways, so as to develop a feel for the underlying math. In this case, the underlying math is the distributive property. A student who cannot factor anything does not really understand the distributive property, which is a foundational concept. Without it, symbol sense is out of reach, as is any work in a quantitative discipline.
In short: factoring, as a skill, is not as important as it used to be. But as a concept, it is an essential component of symbol sense. What are the implications for the classroom? During my decades teaching high school math, even before the wide availability of CAS, I worked under the assumption that speed and accuracy in computation could no longer be a priority. Instead I was aiming for lasting understanding. Here is, more or less, the approach I developed for this topic.
- Introduce the Lab Gear manipulatives (watch this 4-minute video if you’re not familiar with those manipulatives).
- Given a set of blocks, have students make a Lab Gear rectangle, and write the resulting “length times width equals area” identities.
- Without naming it, this is factoring. Do it many times. Occasionally have students sketch the solution.
- Once the stage has been set that way, use some sample rectangles to introduce and explain the distributive law.
- Make more rectangles, but now start from knowing the dimensions.
- Move towards abstraction by having students sketch solutions without using actual, physical blocks — e.g. in homework.
- Introduce “the box” to multiply polynomials.
- Do this many times. Note: “the box” makes it possible to use negative numbers, which are awkward to deal with when using manipulatives.
- Use “the box” to factor polynomials.
- Review all this in an all-class discussion using this applet.
This last step is an opportunity to discuss the “sum and product” strategy for factoring. Doing this at the end of this sequence guarantees understanding of why it works, and makes it hard to forget. Some time later you might revisit all this, and make a connection with graphing, with the help of this applet:
As to whether to tell students to start with “what numbers add up to b” vs. “what numbers multiply to c”, I suggest that there is no universal answer to this. It depends on many things: how good are the students at finding the factors of c? how large is b? are negative numbers involved? I suggest that a better choice is to have that discussion with the class: what are the advantages of each approach? which do you prefer and why?
I do not claim that this is the only or the best way to get at trinomial factoring. But I am certain that it is preferable to limiting oneself to manipulating algebraic expressions with no reference to anything concrete or visual. Illustrating the symbol manipulations with algebra tiles is a good thing, but what I outlined in this post goes beyond merely illustrating. We start with hands-on factoring puzzles, before discussing or even mentioning the distributive law. The whole sequence builds gradually on a visual approach that makes more sense to more students, and at the same time offers a geometric representation for those who are comfortable and quick with symbol manipulation. It can also be extended to three dimensions, and it is good preparation for completing the square.
I present this blog post’s approach in a 7-minute video, part of a 12-part series about the Lab Gear.
The Lab Gear and the Lab Gear books are available from Didax.
There is much more information on the Lab Gear home page on my website.
See also: Algebra Manipulatives — Comparison and History, a rather technical article.
— Henri
Relevant blog posts: