We're in the earlier stages of a unit on graphing, and I want my students to get used to the idea that they'll need to interpret & analyze figures, diagrams, etc. and make predictions or generalizations based on solid mathematical reasoning. This is a skill that I believe is crucial for a student's success in Pre-Calculus and Calculus (and beyond).
This week, my Advanced Pre-Calc students did a one-period math task I developed called, "Mario: Life or Death?" It wasn't anything terribly complex or flashy, but we did get at a few important ideas about graphing, symmetry, and making predictions from what we know conceptually rather than numerically.
(Also, after I came up with the idea for a Mario-based problem, I found out about this Mario problem from Nora Oswald that's way, way better than what I did. I was actually inspired to follow a similar format after finding it.)
I was taking (or trying to take) a 3-Act Math approach with this math task. It definitely still needs some tweaking all around, but considering I'd never tried anything of the kind before, I was pretty pleased. (I don't know how truly 3-Act-ish this task actually is, but, eh.)
I played this video for the students:
The students also had the first two pages of this worksheet (I purposefully withheld pages 3 and 4 until later).
As students watched the video, I had them think about what questions came to mind. They came up with some pretty good ones:
- Will Mario make it to the other side?
- When should Mario jump to make it to the other platform?
- What is Mario doing wrong that's not letting him jump across the cliff?
- When does Mario reach the apex of his jump?
- How can we calculate what height he needs to jump to make the distance?
- Why doesn't he use the chain to make it across?
- Did the player hold down the A button long enough?
"Will Mario make it across on the 3rd jump?"
Working On the Problem
Next, the students were asked to come up with a few ideas. What information did they think would be needed to solve the problem? I gave them a little time to think of a few ideas on their own, and then they shared with each other. A couple of examples from their work:
With some group discussion, we pulled together some important concepts to think about: the distance between platforms, the height of Mario's jump, and in a few cases, the ideas of "arcs," "parabolas," and "symmetry."
I had the students explore these ideas further by having them create sketches of what they thought the path of Mario's jump looked like:
We talked about the shape of Mario's jump. The words "semicircle" and "arc" came up a lot. One or two students already knew the word "parabola."
I asked the students to think about what they knew about arcs; what is their structure? How do they behave? Particularly, I asked them if they've ever seen any "lopsided" arcs. The students said no; arcs are made up of "mirror images," or are "symmetrical." I asked them what they meant by that. They replied that an "arc" can be cut into two halves that are mirror images of each other; particularly, the arc is cut at the "highest point" into two congruent halves.
The students were asked to draw the path of Mario's jump again, this time on a grid. They were also asked to take the observations from their discussion into account:
After looking at the gridded drawings, the students had a few moments to think about how their observations about Mario's jump could be useful in predicting the outcome. They wrote their thoughts down:
After giving them time to think about how to use their observations to predict whether or not Mario would make the jump, I gave them pages 3 and 4 of the worksheet. Using the graphs of Mario's jumps, I asked the students to make their predictions and to give their mathematical reasoning:
Everyone shared their predictions and gave their reasoning. The consensus was that Mario would make his jump, but just barely.
The Thrilling Conclusion
After everyone made their predictions, the moment of truth arrived. I played the solution video:
So the students had predicted correctly! They seemed a bit amazed at how accurate their predictions were -- not only had they figured Mario would make his jump, but many of them said specifically that Mario would barely make his jump, which is what happened. They had made a mathematical prediction using conceptual observations about the path of Mario's jump, and very minimal use of numbers or calculations.
To wrap things up, we tied our discussion back to the idea of symmetry. The type of symmetry used in this problem was just one type of symmetry (even symmetry, and arguably symmetry about the y-axis). We talked about other types of symmetry besides the type used in this problem. The students took down some notes for their problem sets they were working on from our textbook.
I hope that the students came away from the lesson with a bit of a deeper understanding of symmetry and how to use it to make predictions. In the larger picture, I hope students started to see the usefulness of applying what they know and observe in order to gleam new information. As I've said before, there's so much more to math than just calculations and number-crunching.
This was totally the highlight of my week. However, I know that this task is far from perfect. It could certainly go deeper and touch on quadratics, graphing equations, and so forth. I'm almost certain this task could be made into something richer than its current form. I'll definitely be taking another look at it after some time passes and I'm free enough to sit down and make revisions.
But in the meantime, I welcome any and all feedback/comments/insults from any readers out there. Don't be shy!