The lesson study team had imagined that some correct mathematical statements about Student A’s work would be made and that some concern about zero would be raised. The team also had foreseen that students might forget the mathematical purpose of these tabular representations, specifically when the three plans would cost the same. Student A had created one vertical table depicting all three plans. The table did not match the plans, but was mathematically correct. Additionally, Student A did include zero in their table which provided the class the opportunity to think about the insight that is gained by including zero in a table. For this investigation, zero provided the students with the information on whether or not there was a monthly fee.
CECILIO DIMAS: I’m gonna ask that we go ahead and flip to the next table, which will be on blue paper. So I would like for you to take a look at this table and ask yourself, does this make mathematical sense? Why, or why not? Does this table make mathematical sense? Why or why not. Okay, you can take a moment now, and talk to your shoulder partner. If you’re a black card, I’d like for you to share first.
STUDENT: 12, so yeah, okay. This one’s wrong, and I think
STUDENT: and the rest make sense.
STUDENT: ...this one’s wrong, but these two make sense. So then, does that make sense? Because 18, 18, 18, 18, 18. And this one makes sense, because it says 3 dollars per month, so then there’d be 3,6,9,12,15,18, 21, 24. Then, the Online Flix is wrong, because it’s supposed to be
STUDENT: Yeah, it’s supposed to be 12, plus 1 for 1 rental. So 12, 13, 14, 15, 16.
STUDENT: And then right here, it says, 12, 25, 38.
CECILIO DIMAS: Boys and girls, I’m hearing some very interesting ideas, so I’d like for you to go ahead and share out. Does it make mathematical sense? Why or why not? Okay? Let’s start with Kyle.
STUDENT: I think it does make mathematical sense, because with 12, if you add 13 to that, it’s 25, and if you add 14 to 25, it’s 38, and 38 plus 15 is 51, and so on.
CECILIO DIMAS: Okay. Would someone like to add to what Kyle has stated? Or make a different statement? Does this make mathematical sense? Um, Debra?
STUDENT: It does, well, it does make mathematical sense, but it doesn’t follow the plan.
CECILIO DIMAS: Okay. So could you tell me which one does make mathematical sense and does follow the plan, and which one or ones don’t follow?
STUDENT: Movie Busters and Mail Flix follow the plan, and Online Flix doesn’t.
CECILIO DIMAS: Okay. Would you like to add to that, Victoria?
STUDENT: Um, actually, there’s a problem with it, ‘cause it says number of movies, zero? And it says Online Flix and Mail Flix, 18 and 12. But you’re really, not paying for anything right now, so why should you pay that much now without, without any movies right now?
CECILIO DIMAS: Okay, so you bring up an interesting point, Victoria, this idea of 0. So, we’ll move on to that in just a moment. Melanie, you had a comment about the previous question? Or about 0?
STUDENT: About the 0.
CECILIO DIMAS: Okay, do you want to talk about 0, then?
STUDENT: Yeah. I think that it does make sense because the 12, the Online Flix you have to pay 12 per month and so that would be, you would haven’t rent no movies, but you still have to pay for the month on each one.
CECILIO DIMAS: Okay, so if we rented 0 movies from Online Flix, you’re saying we would still have to pay the $12?
CECILIO DIMAS: Okay, even though I don’t have anything?
CECILIO DIMAS: Okay. Jocelyn?
STUDENT: Um, it doesn’t really make sense, because like Movie Busters, see how all the way down it adds 3?
CECILIO DIMAS: Uh huh?
STUDENT: Well, on Online Flix, when we were having our partner talking, we were looking at the 0, how it added 3 and how 18 added nothing, it stayed? Well, 12 plus 12 isn’t 25? And so on.
CECILIO DIMAS: Okay. Someone want to add to this conversation? Hailey?
STUDENT: Well, like, for Online Flix, for each movie, like 13 and 14 and so on and so on, like Online Flix, it says, uh, those, like other numbers? but if you take 13 and add it to 12, you get 25? And then from 25, if you add, like 13 more, don’t you get 38?
CECILIO DIMAS: Yes.
STUDENT: And it keeps going.
CECILIO DIMAS: So then we’re adding by what, here, Hailey?
CECILIO DIMAS: Okay. Danielle?
STUDENT: 12 plus 12 equals 24, but if you buy one movie, it equals 25.
CECILIO DIMAS: Okay. So the two flat fees of $12 plus the 1 makes the 25?
CECILIO DIMAS: Okay. Um. So is there-- would anyone else like to share about whether or not it makes mathematical sense? Jessica?
STUDENT: I think it does make mathematical sense because they all match the plan.
CECILIO DIMAS: Amir?
STUDENT: I think it does make mathematical sense, but I think there’s something that I don’t see really correctly. So 12 plus 12 equals 24. But why are we adding $1 to it?
CECILIO DIMAS: Could someone answer Amir’s question? Lisa?
STUDENT: Because you pay $1 per movie.
CECILIO DIMAS: Okay.
This documented lesson on cost-analysis and comparison of plans depicted on tables is one of three lessons being developed around students’ misconceptions and understanding in our lesson study process this school year. This lesson is focusing on using tables to understand a cost analysis situation and will be followed by a lesson using graphs in a cost analysis situation and a lesson using algebraic equations in a different cost analysis situation. Our goal is to then have students make all three representations for a new and different cost analysis situation and discuss the merit of each representation in that particular situation. We will then give the students the Mars task, Picking Apples for our third benchmark assessment to determine the effectiveness of our lesson study lessons. The majority of my regular math classes needed three days to complete the pre-re-engagement lesson and the re-engagement lesson focusing on Students H, A, E, and J.
Through these lessons we have been better able to understand the misconceptions that some students had when comparing the tables and/or reading tables in general. Some students noticed the multiplicative relationship and completed the table based on this understanding instead of looking at the relationship between variables which led them to then struggle to interpret the data that existed within the table that they had created.