SWBAT explain why a function may be close to, but not precisely linear.

Linear functions can be modeled with a basic understanding of slope and intercept.

15 minutes

Unlike the first two Mario lessons, this one presents a new phenomenon. In this lesson students will attempt to explain some truly *bizarre* behavior on Mario's part.

When my students enter class, I begin the lesson by showing the new Mario clip:

I will only show a few moments of the video. Then, I will ask my students to write their questions and share with their questions with a partner. After a few minutes we will share with the class through a process in which I type their ideas in Microsoft Word document and quote them. (So, if Bobby asks, "Is Mario's speed consistent?" I type his question with his name next to it.)

After all students have had a chance to share their questions, I filter and answer some questions to reach the primary question for this lesson:

**How many coins does Mario collect in the video?**

This whole introduction is represented in the PowerPoint presentation.pptx.

20 minutes

Once students know they are focusing their investigation on the number of coins Super Mario has collected in the video, I ask them, "what do you need to know to solve this problem?" Much like the last problem, I expect that they will nominate his rate (or speed) in relation to time and the number of coins he has collected.

For this lesson, I explain that I put the coins "right next to each other, leaving no space for variation." I ask them to predict if they expect more consistent (linear) results. When a student makes a prediction, I will also ask them *why* they feel the way they do. Typically, students might argue that "we already saw Mario's speed slow down over time, so the distance between pairs of coins is irrelevant." In other words, the distance between the coins won't change the fact that Mario is getting slower over time.

To keep the discussion going, I point out that Mario is now "big" and ask if they think that is a new variable. This typically brings out some "what if" questions:

- What if being big does change his speed?
- What if it makes Mario faster? slower? more consistent? less consistent?

I ask them what they expect and why.

Then, I hand them a series of time shots and/or a spreadsheet of the data: mario 2 coin count.xlsx. I sometimes give different groups different numbers and then share the variety of the answers at the end. This makes for a great discussion, but does create a few more things for me to manage at this part of class.

**Teacher Note**: In today's lesson I highlighted the time stamps I like to use. I also included all the snapshots to help you mix up this part of the lesson to best help your class. Please note that all of the snapshots will not appear in the viewer, but you can access them individually by scrolling down the resources on the main lesson page.

20 minutes

Before summarizing this investigation, I generally want all of the students to reach a reasonable range of coins that they believe Mario will collect. To begin this summary, I show them the answer using this video clip: Mario 2 Coins.mp4.

I play the whole clip to add suspense. Students always celebrate the result simply because they *see* whether or not they got it. Then, we discuss how they modeled the collection and decided upon a prediction. We compare equations, tables, and graphs before discussing the bizarre behavior of "big" Super Mario.

When I pull up the data table from the Partner work session, we discuss the amazing fact that Mario is sometimes speeding up and at other times slowing down. There seems to be no consistent pattern in his speed. I like to give students a minute or so to come up with a theory as to why this might be happening. Students like to attribute the discrepancy to human error, in this case it would be the person designing the flash software that governs how Mario will run over time.

I present an alternate theory using this presentation: summary. I show them this presentation so they can reflect on the complexity of this problem. The coins are "next" to each other, but they are spinning. And this seems to have an impact on when he reaches the coins. Also, how do we know when Mario reaches the coin? At what moment does this happen? I show students individual time frames to help them see that even milliseconds leave moments of time out. It is possible that Mario is grabbing these coins between the time frames, which means that Mario might be grabbing some of the coin before or after we take a measurement.