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I teach university courses in mathematical problem solving at St. Francis Xavier University during my Summer break. The classes involve initiating numerous problem solving episodes and then interrogating and filtering our collective experience through the lens of current theory in the field. This structure provides plenty of opportunity to workshop new ways to launch tasks, and this year, I began experimenting with a new sort of launch routine that had pleasant results. This post is an attempt to reflect on why that may have been the case.

First, however, you must indulge me by responding to a prompt in five seconds or less.

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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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The best thing about online communities (IMO), is the emergence of artefacts from the collected actions of many people. The online math education community (known as the MTBoS) has seen many of these collections throughout the years, most of which are aimed at supporting imaginative mathematics instruction in grade school. Personally, I have felt the community around Fraction Talks explode right under my nose, and it has been a joy to see how the prompts have sponsored amazing student reasoning. A few months ago, I had another idea for a task structure–that I dubbed #MenuMath–and began to collect examples from engaged math teachers. Since then, the collection has grown and become bilingual thanks to the translation work of Joce Dagenais. I love hearing about student and teacher creations, and you are encouraged to submit menus via my contact page if you feel inspired to do so.

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## Re-Constructing Shapes

For the first time in a decade, I am not reconvening with a high school staff to begin preparations for the school year. (I’m preparing to work with pre-service teachers on a university campus). It feels weird–very weird. It is a day that I look forward to because optimism is a constant across the building. Staff feels fresh, materials are crisp, and possibilities are endless. It sadly belies what’s to come.

Bummer, right?

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Much of what appears in mathematics textbooks is what I like to call, downstream thinking. Downstream thinking usually involves two features that set the stage for learners. First, it provides a context (however doctored or engineered–often referred to as “pseudo-context”). Second, the problem provides a pre-packaged algebraic model that is assumed to have arisen from that context.

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Exactly one month ago, fellow Saskatchewan mathematics teacher Ilona Vashchyshyn tweeted about an area task that she used in her class. Long story short, it captured the imagination of Math Ed Twitter like elegant tasks have a tendency of doing.

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## Counting Factors with Grade 7/8s

One of the great parts of my job as a split classroom teacher and division consultant is that I get to spend time in classrooms from grades six to twelve. This means I often need to be in one head space to teach my own Grade 12s and then switch gears to act with younger mathematicians. It also means that the classroom experiences are sporadic and involve teachers working in several different places in several different curricula.

On this particular occasion, I was working with a 7/8 split class that had just finished a unit on perfect squares and divisibility rules, and we wanted an activity that could serve as a sort of review of divisibility rules but also reveal something cool about perfect squares. I thought about the locker problem, but it doesn’t require students to factor in order to see the pattern. Instead, I took some of the underlying mathematical principles (namely: that perfect squares have an odd number of factors) and wrapped it in a structure suited for a Friday afternoon.

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## A Twist on Ordering Decimals

Every time I teach a unit on fractions, there are many students who insist that they’d rather use decimals, and I don’t blame them. The obvious parallels to the whole numbers make decimals a “friendly” extension from the integers into the rational numbers.1 Many of the things school math asks kids to do with rational numbers can be easily transferred into decimals with minimal stress on the algorithms. Such is not the case with fractions. Take addition for example.

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## Problem(s) with Triangles

My provincial curriculum scatters trigonometry throughout several high school courses. Right-angled trig appears first as an isolated experience at the Grade 10 level. From there, the two pathways in Grade 11 cover the Sine and Cosine laws, but only one stream (Pre-calculus) continues into the idea of the unit circle and eventually the connections between the side ratios of right-angled triangles, the unit circle, the wave functions, and trigonometric identities. Since trig is doled out in piecemeal portions each semester, I often find that the hidden beauty of trig is masked by things like SOH CAH TOA. (Or, if you dare to place special triangles on the unit circle, SYR CXR TYX 1).

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## Shoe Sale Remake

I transferred schools at the end of last year, so for the first time in seven years, every one of my students I meet on the first day of school will be a stranger. This means that the first hour I have with each of the four classes is not only their introduction to the course, but also their introduction to me. It won’t take long for them to make an impression of me, of mathematics, of their classmates, and how I expect us all to co-exist for the next five months or so.

I have written on first day tasks before, but not for several years. In that post I identify four “attributes” of an effective first-day task*. One of the tasks I settled on for this year was The Shoe Sale task from Peter Liljedahl. (Other bloggers have documented work with the problem as well: e.g. Fawn Nguyen and Chris Hunter).