SWBAT recognize the difference between linear and "almost linear"

We can apply our understanding of linear functions to model things that are almost linear.

15 minutes

I like to begin this lesson by having students talk with a partner about the concept of "linearity." They are familiar with this term since this lesson follows our linear unit. I have found that having an open discussion about linearity is a vital first step in building conceptual understanding of linear regression: **Students need to understand linearity if they are going to talk about something that is almost linear**.

After they have had a few minutes to share their ideas on linearity, I will lead a brief class discussion where students can share their ideas. I expect students to volunteer concepts like:

- a proportional function
- some linear functions are not proportional.

This means I expect our conversation to focus mainly on ratios and slope. After our linear unit, my students understand the need for a linear function to be constant. They understand that a linear function must have some type of constant unit rate or slope. I like to make sketches on the board to review what these constant slopes look like in tables, graphs and equations.

Once I feel comfortable that everyone is thinking about linearity, I show them a video to introduce them to a context that we will use today to think a little more deeply about linearity:

(The video is also available in the presentation: mario 1.pptx. I will follow follow the presentation through all sections of the lesson.)

After the introductory video, I give my students a few moments to write down any questions they might have and then a minute or two to share their questions with each other. Then we talk as a group and I listen to their questions about the video. I like to quote them as they share questions (I type out their questions in a word document).

Invariably, we reach a question like "how many coins will Mario collect?" At this point,I like to focus specifically on the counter in the upper right corner by asking, "On what number will the counter end?" This is essentially the same question, but it is an interesting one as the counter goes from 0 - 99 and then starts over as Mario collects more and more coins.

I focus on this question because it allows me to extend from linearity into linear regression during the summary section of the lesson.

20 minutes

The end goal of the Introduction was to define a mathematical problem relevant to understanding Maria's actions in the video. Once students have helped me define a question around this problem (How many coins Mario will collect?), I ask them what they will need to know and do in order to solve the problem. This discussion brings up the concept that they need to know how many coins he starts with, how long he collects coins, and what his rate of collection is. Sometimes students also want to know if his rate is constant or not. Instead of answering this question directly, I ask the students to review the data I give them and see if his coin collection rate is constant. (Whether this comes up at this point or not, this topic will play a big role in the Summary discussion).

After discussing "needs to know" and "need to dos", I will have my students work with their partners to work together on a partner quiz. The Partner Quiz provides structured approach to the **How Many Coins?** problem.

**Teacher Note:** On this partner quiz, students work with a partner to to complete a class assignment and their collective score on the quiz is factored into each students class participation grade.

25 minutes

After the Partner Quiz, we rejoin our conversation by reviewing a full segment video (Mario Full Video 1 coin) to see where the counter actually ends. I want students to *see* that their answer was corrent (or incorrect). This self-evaluation provides a hook to motivate the students to talk in greater depth about the problem at hand.

After watching the full video, I usually ask several students to share his/her tables, graphs and equations describing Mario's coin collection. I bring up the question "Is Mario traveling at a constant rate?" again. When I do, I ask a student if he/she can confirm that Super Mario is or isn't moving at a constant rate. As needed, we return to the snapshots from the partner quiz. I expect my students to be able to demonstrate that he *seems* to be moving at a constant rate since the slope between any two points on the Partner Quiz is the same. At this point, our discussion focuses mostly on the concept of linearity and how the slope needs to be constant. My students are relying on their knowledge from our work with linear functions.

Once all of the students are settled on a strictly linear function, I will show them another clip of Mario. I will introduce them to the video by asking them, "What is weird about this video? watch closely." Then, I show them 27 seconds.mp4. The video starts off at normal speed, then speeds up and slows down around the 27 second time stamp. The ultimate idea is to notice that Mario collects many coins at 27 seconds. He starts at 106 coins and ends at 111 while the clock is showing 27:00.

I ask the class how it could be possible that Mario has 106, 107, 108, 109, 110 and 111 coins at 27 seconds. The answer is that he collects around 4 coins per second (as opposed to exactly 4). In case the first video was too difficult for some of my students to decipher, I plan to show an even more precise clip before getting too off track: 27 seconds zoom.mp4.

The idea (in terms of designing the lesson) is that I set up the snapshots on the Super Mario Partner Quiz to make Mario appear to collect coins at a constant rate. However, the actual situation is more complicated. This function is approximately constant. It seems that we can predict an approximate answer, but not a precise one.

Once students begin to grapple with this idea, I will bring the lesson to closure by introducing the idea of modeling a relationship with a linear function. This is just a gentle introduction, so the objective is to understand that something that is approximately constant can be approximated with a linear function. This is a big idea, so I like to leave it open at the end of our first lesson of the unit.

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