Our STEM projects involve plenty of data collection, regression, analysis, and other math we call ‘statistical Algebra 2.’ This post is about how we’re studying ‘theoretical Algebra 2’ – the math you took in school. Beautiful to some and boring to others, it’s the math often left out of project-based learning.
This week, parallel to our STEM work, I gave students this assignment – a list of “challenges” – to complete by next Friday using Desmos. Our goal, explained at our kitchen table with an old TV, was for them to study math with a shifted focus. We will emphasize accurate vocabulary, the connection between a function and its graph, and explain why we know something is true. They will tinker in Desmos to create their own functions that satisfy the conditions I’ve laid out for them. They will “defend algebraically” some of their creations; it means they’ll do the Algebra out in a manner more consistent with a typical course. Honors students are expected to do this more often. They will answer questions about their functions in an effort to incorporate topics like domain and range, which, surprisingly, already came up in STEM class this week. In this Introductory Level, I want them playing with the structure of lines, parabolas, and inequalities.
I will share two stories from today that exceeded my expectations. This morning, Sofia was working on a challenge where she needed to create a function with a vertex in a particular quadrant. When I walked over to her table, I saw an almost complete version of this:
As Sofia tinkered with her quadratic, she noticed patterns and wrote them down. She eventually created what’s essentially a guide to quadratic transformations. There’s a bit in here even I forgot. With some questioning, she eventually realized the need to consider negative coefficients. I also prodded her to clean up language – vertical and horizontal replaced conversational descriptors – and this was the result. Sofia: “I didn’t do this because I had to – I did it because it helped me understand.” She snapped a pic on her way out. If we let students tinker, they will ask questions and find answers for them.
During lunch block, Dylan was struggling with a tougher challenge: create an inequality whose solutions are always positive. He began not even remembering what an inequality was (and “always positive” is intentionally ambiguous). I drew some graphs where an equation would yield one, two, and three solutions. I drew a one-variable inequality (x > 2) and he gave all positive solutions. We spent about three minutes on all of this. I then challenged him to find a solution to the challenge with two variables. He spent 10 minutes graphing lines, a quadratic, and even a hyperbola on accident before hitting a circle:
He wanted to move it into the first quadrant. By this point, Sarah (STEM / Env. Chem) had dropped in and added a few of her own probing questions. Last year, I would have told him the answer. Today, I suggested he Google circle transformations. He found an eight-minute video, skipped to the part he needed, and ended up with something like this:
It took him a half hour and he stumbled on ellipses and hyperbolas, but he got there. He won’t forget how.
We screened Most Likely to Succeed this week and in the film there’s a multi-year study where students sat for their final exam a second time three months later. A B+ average dropped to an F. ‘Covering all the content’ doesn’t result in long-term retention. Sofia and Dylan learned something today that they’ll remember much longer than if they were told.
Hard to believe it’s the sixth day of school. I’m excited to see what else they figure out.