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.
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.
Many teachers tell me that it is their creativity that limits their ability to be adaptive in the classroom. Somehow the “reform” movement (or should I say re-movement) has pigeon-holed itself into a connotation where high-energy teachers give vague tasks to groups of interested students. Out of all this, curricular outcomes explode in no particular order. This can’t be further from the truth. In my view, the biggest steps toward changing student learning is changing teacher perception.
I have spent the better part of 2 weeks going over various mathematical relationships in my Grade 10 class. They have been represented as tables of values, arrow diagrams, and sets of ordered pairs. Relationships, both qualitative and quantitative, have been defined, analyzed, and graphed. My focus on graphical literacy has been previously detailed on the blog. See this link for details.