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.
My first attempt at animating patterns was published on this blog in 2013. I suppose you can consider this post a long-overdue extension of the thinking there, however with a much-needed bump in production quality. In those old days, I hunched over a whiteboard with a collection of square tiles, creating six-second loops on the (now defunct) social media platform, Vine. Now, thanks largely to Berkeley Everett and his crash course on how to make animations in Keynote, the process has become much more streamlined.
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.
After an emoji was named 2015 Oxford Dictionary word of the year, I am holding out hope for next years’ candidate:
1. to transform the condition, nature, or character of a classroom activity using Desmos.
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.