In a previous lesson, you watched a video of an 8^th grade classroom in which students were learning to make conjectures about functions. In this activity, you will read three excerpts from a transcript of that lesson. In each excerpt, you will be prompted to notice one of the Fundamentals of Learning (FOLs): Participating and Contributing, Meaning Making, and Managing Learning.

View the video that has been transcribed below:

My name is Audra McPhillips, I'm a math coach for the West Warwick public schools and I taught an 8th grade lesson at Dearing Middle School today on conjecturing about functions.

If you remember yesterday we were looking at a set of different functions and we were trying to come up with different conjectures…

Teacher: The heart of the lesson, 8th graders are expected to be able to not only understand what functions are but be able to represent them in variety of ways.

Teacher: For a couple of days before this lesson the students had examined a particular set of functions.

Teacher: So we started the lesson today by having them share their thinking with the class

Student: So we proved that by making….like making a picture of the dots

Teacher: What up there convinces you?

Student: Oh, well, there’s the dot in the middle that stands for the extra 1 and then….

Teacher: I think more importantly this lesson and this series of lessons is really getting them to move beyond the set of problems that’s provided to start to notice important math structures and to really make a conjecture.

Student: I think it should be the number of minutes and the Y should be the total

Teacher: When they get to the point where they’re making a conjecture they’re really engaging in deep math reasoning, which is what the common core is looking for.

Teacher: I want you to take a look at these three different patterns, OK? There are three of them. There’s pattern A, pattern B, and pattern C.

Teacher: I chose the three functions that we used for this lesson because all three of them had the same rate of change. That’s a structure, that’s a pattern that I want them to recognize. I also chose those three because they all had a different Y intercept. They all started in a different place and I knew that students would struggle with that a little bit because they hadn’t had a chance to think about what happens when the beginning is not clear.

Teacher: So I first asked them to take some individual time to look at that set of functions and just see what they noticed.

Teacher: What did you notice about this set before we get started solving them? Matthew?

Student: They all start at different times.

Teacher: They all start at different times. Tell me more about that.

Student: Like one will start at the beginning and then one will start at one minute and then [noise] one doesn’t have anything at all

Teacher: That’s good stuff, Matthew. Chloe, what’d you notice?

Student: In pattern A, at the beginning, they didn’t ….at one minute they had one dot in each arm and pattern B at one minute they had two dots in each arm, and in pattern C they had no arms at one minute

Teacher: And Michael, one more. Do you want to add to that?

Student: They all have the same number of arms.

Teacher: They all have the same number of arms. Ohhhhhhhh

Teacher: In this particular case there's a lot of reasoning going on. They’re looking for repeated reasoning, they’re looking for that structure, they're reasoning abstractly and quantitatively. All of those reasoning math practices that are difficult to get in kind of fall together when you have them work on conjectures

View the video that has been transcribed below:

Teacher: I want you to try to make a conjecture. So the end goal is to kind of analyze this whole set of functions and think about is there a conjecture you can make. Can you think of a math statement you can make that kind of goes beyond this set that will always hold true, and remember some of the things we talked about that make a strong conjecture. So if you think you’re onto something jot some notes in your conjecture paper while you’re finishing up your work and we’ll see if anybody kind of gets there today.

Student: So first of all we need a conclusion….

Student: Because to get Y we need the equation….

Teacher: Once you get your graph done color code everything so it matches so the other people will be able to understand your thinking, OK? That was smart to go there first, I like it.

Teacher: Well I definitely really steered students towards using color. Students need to understand this algebra but they also need to kind of have that picture in their mind to back it up. So the color really [weird] helps them to kind of develop that conceptual understanding of how all of those representations fit together so they’ll be able to apply that later on.

Teacher: We know one minute, two minutes, three minutes, what do we not know that you want to know? We don’t the beginning, right. So let’s do that

Teacher: Mike, Tony and Eric, they were struggling with that last growing pattern. It was difficult for them because there was no beginning provided, they were going to have to kind of think about a beginning and that beginning was going to get into some negative numbers.

Teacher: So from here to here how much do we grow?

Student: So there’d be like negative three dots

Teacher: Ohhh, where’d that come from, that was pretty impressive. Go ahead, tell me what you’re thinking, Michael

Student: If you would subtract 4 from what ___ you have a negative 3

Teacher: Right, so tell them why though, because I don’t know that they’re following that. So what did you notice when I said what happens from here to here?

Student: That it adds 4

Teacher: Adds 4, adds 4, adds 4. So you’re saying if we go this way we’re adding 4 so….

Student: …if we go this way you’re subtracting 4

Teacher: So to get from 9 to 5 you….

Student: Subtract 4

Teacher: To get from 5 to 1 you….

Student: So he’s saying that the beginning must have …

Student: A negative three dots….

Teacher: Which is kind of weird to think about but I think mathematically it makes a lot of sense. Does that help….

Teacher: He was recognizing that inverse which is really powerful. He was able to justify his thinking and explain it to the rest of the group and just another ah-ha moment that was kind of exciting.

View the video that has been transcribed below:

Teacher: I’ve noticed some people already starting to notice some important math structures, some things that might get to the conjecture point. I don’t think we’re going to get to the point where you can all make a very solid conjecture that we can try to reason about. I think we’ll save that for tomorrow where we kind of get to the final conjecture

Teacher: At the end of the lesson after students had worked on the problems they worked on an exit slip for me so I could see where their thinking was at the end of the lesson, and then I asked them what they noticed and I was excited with some of the results.

Teacher: If you had to pick two or three math practices that you think you engaged in the most today, that were the most helpful to you while you did this work, go ahead and chat with each other….

Teacher: I think it's really important with the common core standards since the standards for mathematical practice are tools for the students. These are tools that they need to kind of hone throughout their K-12 experience. If they can do these eight things, they're going to be successful with any kinds of problems. So I want them to really stop and think about their own thinking, what strategies were you using, which math practices were you engaging in that allowed you to be successful with this lesson today.

Teacher: Michael’s group, I heard you guys talking about something good. What was it?

Student: Look for and make use of structure

Teacher: So go ahead and find the yellow one guys, look for and make use of structure. So Michael, why was structure important in this lesson? What was your group discussing?

Student: We were discussing on how to find the beginning and um….like constant rate of change for each problem

Teacher: So Michael is saying we had to look for important math structures while we were building this function. We had to look for things like rate of change and Y intercept and beginning, and so he said they were examining all those structures to kind of make connections. So yeah, I think people who make good conjectures use structure really well.

Teacher: This is the kind of lesson that generally takes two days. They’re solving these problems but you know they're solving them as a tool to kind of go beyond and they were already getting to that point today which was good because that means tomorrow they’ll be able to test it with a wide range of items, continue to color code their work so other people can understand it and most importantly really try to get down to the math of why is this conjecture seeming to be true.