Quadratics feel important. This impression is no doubt influenced by the boated importance placed on calculus in secondary school. They represent the giant leap from linearity and pave the way for more elaborate functions; therefore, I often find myself musing on ways to have students meaningfully interact with the topic. Once the structure of the function is established, I’ve played around with interesting ways to help students visualize quadratic growth, connect that growth to the Cartesian plane, and build these functions to specifications; however, my introduction to quadratics in vertex-graphing form has always been a series of “What happens to the graph when I change the ___ value?” questions. These aren’t bad questions (and a quick setup of Desmos sliders helps visualize the effects), but they don’t exactly build up understanding from experience. Such was my introductory quadratics lesson for years, lukewarm but lacking the epiphany to address it.
There is too much to like about Desmos. Really, though. The pace of innovation is gross. I am the first to admit that my sophistication with the platform is lagging behind the possibilities. I have never dabbled in Computation Layer, and I haven’t played with the Geometry. Part of my problem is the core team and the army of fellows are so darn accommodating with any questions.
One of my favourite activities remains the Marbleslides.1 They set a beautiful stage for students to stretch their imagination, and I have not yet met an activity that sponsors a need domain and range in a more organic fashion. I have used them with all secondary grade levels, and they will be a regular part of the weekly work for my undergraduate students in their mathematics methods course this Winter.
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.
[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 had 3.5 months to complete it. They complained, but the results were fantastic. (…bunch of drama queens).
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 pattern growth was an engaging hook. After a conversation about what made this pattern ugly (the non-constant growth), we looked at the growing square.
I would like to begin with a conjecture:
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.
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.
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 to introduce the four types of slope: positive, negative, zero, and undefined.
The class began with a quick discussion on how rate of change relates to slope. I handed out student whiteboards at the beginning and drew four lines GeoGebra.
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 modelling. Essentially, we work with various patterns and develop generic rules to describe their behaviour. Linear relations will be our finish line, but I am making sure to provide ample concreteness before abstracting into notations.
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.