Thinking Upstream with a Quadratics Menu

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.  I am imagining the reasons for this are three-fold: Instructional time. Developing these models is messy and takes time, especially with class sizes in the mid-to-high 30s. Accessibility. Developing these models isn’t always possible (i.e. can’t run trials and collect data on every …

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Desmos Art Project

[Post Updated June, 2018] This semester I gave my Grade 12s a term project to practice function transformations. I began by sourcing the #MTBoS to see who had ventured down this road before. Luckily, several had and they had great advice regarding how to structure the task. I use Desmos regularly in class, so it was not a huge stretch for them to pick up the tool. I did show them how to restrict domain and range (although most of them stuck exclusively to domain). I gave them the project as we began to talk about function transformations, and they …

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Connecting Quadratic Representations

I always introduce linear functions with the idea of a growing pattern. Students are asked to describe growth in patterns of coloured squares, predict the values of future stages, and design their own patterns that grow linearly. Fawn’s VisualPatterns is a perfect tool for this.While stumbling around Visual Patterns with my Grade 9s, we happened upon a pattern that was quadratic. The students asked to give it a try, but we couldn’t quite find a rule that worked at every stage. While I knew this would happen, the students showed a large amount of staying power with the task. The …

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On Collective Consciousness and Individual Epiphanies

I would like to begin with a conjecture: The amount of collective action in a learning system is inversely related to the possible degree of curricular specificity.  The mathematical action of a group of learners centred on a particular task gives rise to a unique way of being with the problem, but also reinvents the problem.In short, what emerges from collectivity is not tidy. How can I justify curating a collective of learners, when school is so interested in individuals?Learners commerce on a central path of mathematical learning while acting on a problem, but each take away personal, enacted knowings from …

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YouTube Relations

My goal this semester was to continue to improve my use of formative assessment (largely through the use of whiteboarding) and expand the role of Project-Based Learning in my classroom. Up to this point, I have developed a wide-scale PBL framework for an applied stream of math we have in the province called Workplace and Apprenticeship Math. Those specific topics lend themselves very well to the methodology; they are a natural fit for PBL. I am still looking for ways to branch the intangibles from PBL into a more abstract strand of mathematics–one that includes relations, exponents, functions, trig, etc. I …

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A Discussion on Slope

I have taught Grade 10 math more than any other class. I still have lessons that I created during internship that I use. Other sections of the curriculum I have perfected over the years. Today, I added another lesson to the list of those that I will do for a long time. This is my desperate attempt to describe and catalogue it. If I don’t do it now, it will filed as a good, but vague, memory. My goal was to introduce the idea of slope and be able to get numerical values for slopes from graphs. I also wanted …

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Using Real-Time Graphs

I have a class of grade nine students this semester that are part of a stretch program. This essentially means that they get 160 hours to complete a 120 hour course. The class is designed to accommodate the transition from elementary school (Grades 1-8) into high school (Grades 9-12) for those students who feel uncomfortable with their math ability. It also affords me a few extra days here and there to stress certain topics. One of my foci this semester has been pattern modeling. Essentially, we work with various patterns and develop generic rules to describe their behaviour. Linear relations will …

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Graphing Literacy

My school division has been pushing literacy for a few years now. The division priority has filtered its way down into many programs at the school level. As a basic premise, if students are exposed to literate people and perform literate activities, their skills will grow.  As the term is dissected, it seems that every stakeholder can find a way to skew the term to mean that their discipline is a crucial part of being literate. Reading and writing skills are an obvious avenue, but the ideas of technological and social literacy have emerged as important parts of every student’s …

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Mathematics for Bros

Before I begin, I would like to make sure that the title of this post was not misleading. If you are reading because you are fuming at the gender inequality reference in the title, please relax. I am in no way advocating that Mathematics is for Bros; the following post is a collection of the mathematical quips garnered from the “New York Times” bestseller, The Bro Code. It is a sacred cannon passed down from generation to generation of Bros designed to guide the lives and decisions of Bros worldwide. The book takes a humourous look at the superstitions of …

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The Mathematics of Laundry Soap

The grocery store is a brain workout for the mathematically inclined. Not only do the varying metric and imperial conversions tease out the micro-savings of bulk, but neon yellow discount signs encourage percentages and good ole’ multiplication tables. Often you find adults transfixed in a complex division trying to figure out which ham will be cheaper. Once that calculation is complete, they turn their attention to making sure the portion will be enough to feed their whole family. The sheer volume of available estimations overloads me; coupons just complicate the matter–significantly. When you add the typical male intolerance to shopping, …

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