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## Trinomial Factoring Match

Fractions, factors, and functions.

A large portion of my career to date has been spent musing over how to engineer classroom environments that infuse meaning into these three mathematical structures. When it comes to polynomial factoring, the area model has provided the most success. After connecting 2-digit by 2-digit multiplication, the area model becomes a beautiful visual to make sense of the “adds to ___; multiplies to ___” phrase that echoes around the room.

But we don’t keep the area model around forever. Once we’ve used the model to build meaning, we mobilize that understanding in more symbolic situations in a careful, deliberate march toward mathematical abstraction.

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## WODB: Polynomial Functions

If you haven’t experienced the conversation stemming from Which One Doesn’t Belong? activities, you are missing out.

As far as I can decipher (#MTBoS feel free to correct me), this all began with Christopher Danielson’s Shape Book centered around this structure.

From there, a crew of tweeps (headed up by Mary Bourassa) established WODB.ca (YES! Canadian) to curate a collection of problems of this format.

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Every year, my students study the general characteristics of polynomial functions. We investigate the various shapes of various functions and slowly shift parameters to watch changes in the graphs. Eventually, we deduce the roles of the constant term, leading coefficient, and degree.

It should be noted that Desmos makes this process much easier than years previous. Just set up the generic polynomial, add sliders, set specific ones to play (depending on what you want to investigate), and have students discuss in groups.

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## All Factors Considered

I have only been teaching for 2 years, but am already beginning to encounter the recursive nature of the profession. I have had several repeat classes in my 4 semesters of teaching, and they require the achievement of the same outcomes. This does not bother me, in general, because I am excited to see the improvement in my teaching. There is one unit, however, that has already frustrated me. Its ability to sabotage creative exploits is unrivalled throughout the mathematics curriculum; I am speaking of the unit on polynomial factoring.

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## Manipulative Revelation

I completed school before manipulatives were in vogue. I am still not sure that they are today (where I teach). I know that my department’s manipulatives are locked up in a cupboard. In this Potter-like clandestine state, I didn’t even learn of their existence until the end of the year. I was moving classrooms, and found a pile of algebra tiles that the previous teacher had left behind. I didn’t discover that I had manipulatives available to me until, ironically, I inquired where I could dispose of this rather large supply of algebra tiles. When I opened the doors of the cupboard, my eyes were bombarded with a vibrant display of primary colours; it is the bright reds, blues, and yellows that initially deter high school students from using these instruments. It creates an aura of immaturity and frivolity. They are coloured in such a way that one may expect students to pack their algebra tiles up neatly and proceed to recess or nap time. Kindergarten students play with blocks; algebra deals with “big-kid’ stuff–no use for toys.