Polynomials

Oh Polynomials. My least favourite unit by far in the Foundations of Math and Pre-Calculus 10 course I am teaching. 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

How can I frame a boring, completely algorithmic and skill-based unit into something that’s relevant and meaningful for my students? I am borrowing Dan Meyer’s definition for relevance here.

It Begins with a Question… 

A colleague asked me today, “How much time do you have for homework at the end of class?” This was a surprising question to me, and as I thought back over the 10 day unit, my answer was 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:

  1. Algebra tiles – benefits and fall backs
  2. Picture talks
  3. Factoring method (criss cross or sum-product?)
  4. WODB
  5. Progress checks
  6. Taboo
  7. Human Bingo

1. Algebra Tiles 

Definitely a hate-hate relationship. As a math teacher, I am obligated to entertain this idea and I do admit it has its benefits, especially in lower level math classes when students are initially being exposed to distributive property and the like. The problem, however, was that my students were already armed with the skills and knowledge of multiplying and factoring polynomials. Moreover, the limitations of using tiles far exceeded the benefits, in my opinion. Algebra tiles do NOT work for: polynomials higher than degree two, multiplying more than two polynomials, and multiplying polynomials with more than three terms. This meant that it took more effort for students to understand how and why it works.

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.

2Picture Talks

I like to use Sarah VanDerWerf’s Stand and Talks as a format for students to discuss picture prompts. I find that the buy in for engagement is much higher when the prompt is linked to physical movement. My favourite questions for photo prompts are: “What do you notice?” and “What do you wonder?”

What are my photo prompts, you ask?

That’s right. Algebra tiles.

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?

I’ve had the great fortune of only having one prep and a spare block this semester (for friends and readers who don’t teach, that’s teacher jargon for FREE TIME, kinda. The details are not important). Anyway, I’ve been making drop-in’s to my fellow colleagues classrooms with my new-found “free time” and one thing I picked up was the importance of proper SEQUENCING. For instance, a natural progression for factoring trinomials might look as follows:

  • 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.

I prefer this method over the traditional “criss cross” method for a few reasons

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

Which One Doesn’t Belong? (WODB)

Fantastic activity for building up thinking skills and vocabulary. Each student picks one of the expressions and must argue why that one doesn’t belong.

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 

chapter_5_human_bingo.docx
File Size: 87 kb
File Type: docx

Download File


This activity was shared by a good colleague of mine. To get BINGO, students must find one “expert” in the classroom to answer each question on the bingo card until all the questions have been answered. The student who answers the question must sign their name. A student may not be asked more than once to answer the same Bingo card.

Benefits:

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