*Find the greatest common factor, least common multiple, factor these trinomials, collect and simplify like terms, the swimming pool has a width of 5x + 1 and a length of x + 2*… YAWN.

**The Challenge**

## It Begins with a Question…

*almost none*. The question sparked a great dialogue between us about our approach to teaching the same content in our respective classrooms. It really made me think. I realized that while I still dreaded teaching polynomials, I had found a way to improve the way I taught it from first semester that required less rote work and more thinking.

One thing that has not changed, however, is that I avoid teaching FOIL method like the plague. It only works for expanding binomials and does not apply for polynomials with more than two terms. After I read this article I was convinced I would never need FOIL in my classroom:

http://www.makesenseofmath.com/2016/11/why-i-will-never-teach-foil.html

For a good laugh:

https://saravanderwerf.com/2017/04/01/why-ive-started-teaching-the-foil-method-again/

__Some things that came up in our discussion__:

- Algebra tiles – benefits and fall backs
- Picture talks
- Factoring method (criss cross or sum-product?)
- WODB
- Progress checks
- Taboo
- Human Bingo

## 1. Algebra Tiles

Nevertheless, we spent a few classes examining algebra tiles and their usefulness. Rather than approach it from the typical standpoint of using algebra tiles as a manipulative, I wanted students to see the link between the algebraic and pictorial representations of polynomials. This took work and was not as straightforward as it seemed. A big takeaway for me was that students gained much more out of the experience when they were able to physically manipulate the tiles and arrange them into their “factored forms.” Last semester, I “taught” algebra tiles by merely showing them examples and drawing them on the board. It took a bit more prep, but this semester I printed eight sets of tiles (positive and negative) in my classroom and had students manipulate them instead.

If we were to spend any more time on the unit, or if this was a lower grade level, as an enrichment activity I would have students discuss the limitations of algebra tiles and look for ways to address them.

**2**. Picture Talks

What are my photo prompts, you ask?

__Goals for students:__

- Make observations and ask questions
- Use math vocabulary
- Share ideas with their peers

I like this activity because it is easy to differentiate and works well as a “minds on” for any topic. Asking students a general question like what they notice/wonder means that lower ability students can comment on ANY aspect of the photo (e.g. “there are blue and green rectangles”) while higher ability students can be pushed towards making observations based on any mathematical patterns or relationships they observe (e.g. “the green tiles represent positive polynomials and red tiles are negative”).

## 3. Factoring Method – Criss Cross or Sum Product?

- Common factors
- Ex: 12xy + 3x

- Trinomials with leading coefficient of 1
- Ex: x^2 + 4x + 4

- Factoring when the GCF is a binomial
- Ex: x(x + 1) – 2(x + 1)

- Trinomials with leading coefficient not equal to 1
- Ex: 3x^2 + 8x + 4

That, together with a quick exercise on sum/products, helped me push students towards seeing the relationship between the factored form of a trinomial, and the sum/product method.

- Focus is on noticing patterns
- Less trial and error work
- Allows them to answer questions like this:

**Which One Doesn’t Belong? (WODB)**

__Benefits:__

- Everyone speaks (I always encourage full sentences and proper use of math vocabulary)
- More than one argument may arise for each expression

__Sample answers:__

*“27x^2 doesn’t belong because it is the only expression that has a coefficient with a perfect cube”
“45x^2 doesn’t belong because it is the only expression that has a coefficient with 5 as one of its prime factors”
*

More WODB prompts can be found here.

## Taboo

__How it works__: One student is chosen to stand/sit at the front of class facing the audience, they are in the “hot seat”. Behind them, a vocabulary term is shown for the rest of class to see. Students in the audience must help the student in the hot seat guess the vocabulary word by miming, explaining the definition, or giving examples. They may not use any part of the word in their explanation.

__Modifications:__Differentiate by giving students the option of bringing a “cheat sheet” of vocabulary terms with them. Prepare students for the activity by giving them cross word or fill in the blank exercise reviewing the vocabulary words for the unit. An “expert round” can include vocabulary not on the cheat sheet. “Challenge round” can be facing a peer or the teacher. Can play in teams or as a class.

## Human Bingo

__Benefits:__

- Gets students moving
- Students can pick the question they want to answer
- Fun review activity for a test or quiz!