When students enter the room, I play the following clip: Mario Star. A per our routine for these lessons, I ask students to write down any questions they have (individually). When I see that the writing has slowed, I ask students to turn-and-talk to share their ideas and questions.
For the class discussion, I ask them "what do you think we will count in this video?" Depending on the response that I hear, I might drill a bit more:
My goal is to build consensus around the idea that we are dealing with a scenario that can be modeled with several straight lines, as opposed to one.
Once we agree that the problem can be modeled with several different lines, I will give my students a variety of snapshots to sort. I print them out like a deck of photo cards and I ask the class to identify times when Mario is transitioning from not gaining lives to gaining lives.
For example, this shot shows the starting number of lives Mario has, a critical piece of information for their investigation:
Students spend the next segment of the class sorting the cards, setting up a reasonable graph of time versus lives, and then drawing appropriate lines of best fit. I ask them to write equations for each line they draw.
To launch the summary conversation, I play the following video, which shows an estimated graph of Mario's progress: Mario with grid.mp4
To make this graph, I simply used key time stamps, like at 0 seconds we know he had 161 coins and then at about 6 seconds we know he started to gain lives until he dies at 14.7 seconds and then starts over.
By connecting the key points between 6 and 14.7 seconds, we get an average rate or slope at which he gains lives per second.
However, there are many time frames not represented. So I ask students to write some of the other time life points on sticky notes. We then plot these notes on top of the grid image in the video. This helps students see that our piecewise graph is really a scatter plot with several linear functions to model its behavior.
I also use this summary as an opportunity to review how to write the equation of a line and to help students realize that this piecewise function is made from 5 lines and to see which are parallel and why. My goal is to help students to explain their observations in the context of the problem. For example, "the line at the start and the end have a slope of 0 since Mario isn't collecting any coins over the passing time." Likewise, "the lines with a positive slope are the same because Mario gains lives at an equal rate." Although the rate at which Mario gains lives may not be perfectly equal (especially since we have established in early lessons that the rotation of the coins along with Mario's varying speed can really change the function), but I want students associate the slope with the context of this problem and articulate the meaning of the linear functions in terms of the Mario and the lives he is gaining.