The lesson study team had anticipated that students would make correct mathematical statements about Student H’s work and have some concern regarding the third table. Additionally, the team had expected that some students would present the argument that table #3 is about months and cumulative costs for that particular DVD plan. The team had agreed that any incorrect responses would be managed by asking the class if we were all in agreement by using hand signaling and ask if someone would be willing to defend a particular answer to provide further clarification. Student H had created three different horizontal tables that were all mathematically correct. However, the third table did not match the DVD plans. Additionally, Student H did not label all three of the tables, thereby allowing room for confusion and an inability to accurately respond to the original prompt. The notion that mathematical comparisons in this situation can only be made with like units is a big mathematical idea in this particular case or context. Likewise, the otion that just because there is a correct mathematical pattern doesn’t mean that the table is correct for this context. This too, is a big mathematical idea for our students.
CECILIO DIMAS: So let’s look at Movie Buster. Does Movie Buster make mathematical sense? Can we have someone raise their hand and tell us why? Erin?
STUDENT: Yes, because it’s for, for the 1,2,3,4,5,6,7, 8, movies, then it’s $3 per movie.
CECILIO DIMAS: Okay. And so the pattern here is growing by what?
STUDENT: By multiples of 3.
CECILIO DIMAS: Multiples of 3. Okay. Online Flix, does this make mathematical sense? Could someone tell us why here? Chad?
STUDENT: Because it says it’s 12 bucks a month, and then plus 1 dollar every rental,
CECILIO DIMAS: Okay. Does this match the plan? Does what’s shown here on the table, for Online Flix, does it match the plan?
CECILIO DIMAS: Yeah. Could someone explain to us how it matches the plan? Melanie?
STUDENT: Well, the 13, they charged, for Online Flix, they charge 13 dollars, no. I’m not sure.
CECILIO DIMAS: Want some more think time?
CECILIO DIMAS: Okay. Danielle?
STUDENT: Online Flix charges 12 dollars per month for... just for being in the membership in the Online Flix, and 1 dollar per DVD, So, 12 plus 13 equals 1, and if you were to buy 1 DVD, it... yeah. It would cost...
CECILIO DIMAS: Okay. And how is Mail Flix set up? How is Mail Flix set up here? Amir?
STUDENT: Every movie, costs, uh, from what I could see over there, I see every DVD that’s going up there is 18 dollar-- multiples of 18 dollars added to it. So that gives me an idea that a movie should be 18 dollars. But then, I think, how much would they, would they even charge you per month?
CECILIO DIMAS: Okay, so you have questions about...
CECILIO DIMAS: How this is represented. Jessica, I noticed that you had your hand up, earlier, too?
STUDENT: Instead of, ‘cause, the rentals, it doesn’t matter just, for the month. For 18 dollars for a month.
CECILIO DIMAS: Okay. So, can you explain that a little bit more in how this is set up here?
STUDENT: Um, the multiples of 18?
CECILIO DIMAS: Okay. Charles?
STUDENT: It doesn’t match the plan. What’s going on is, it’s 18, it’s 18 times 1, equals 18, 2 DVDs, that’d be plus another 18, that’d be 36, then plus 3 DVDs that’d be 54, then so on.
CECILIO DIMAS: Okay. So I’m gonna go back to the question that I posed earlier. What is the purpose of us creating these tables? Why are we creating these 3 tables? Why did Student H do this? What are we trying to find out? Kyle?
STUDENT: So you can compare the plans easily?
CECILIO DIMAS: Okay, Danielle, would you like to add to that?
STUDENT: See which plan is cheaper?
CECILIO DIMAS: Okay, so we can compare the 3 plans to find which one, when theyt match up, and then we’re also looking to see which one’s cheaper. Okay. Was Student H confused?
CECILIO DIMAS: What was, um, what do you think helped Student H get confused? What was something that Student H did or did not do, that allowed them, or led them to confusion? Talk to your shoulder partners for a brief moment. If you’re a red card, I want you to go first.
STUDENT: .. and then, um, they got it confused, because it was unlimited rental. They read, they read the plan wrong.
STUDENT: Okay, so. Student H, he got confused ‘cause for 1 month, Online Flix is 12 dollars, and he’s saying every month he bought 1 extra movie, so for the first month he bought 1 movie, and the second month he bought 2, and so on, and that’s how I think he got confused on that.
CECILIO DIMAS: Finish up your thought process, but I’d like to go ahead and start getting... hearing some of the things you’re saying regarding these three tables. Melanie?
STUDENT: I think Student H got mixed up on the Mail Flix, ‘cause I think he thought they charged you $18 for each movie you rent
CECILIO DIMAS: Okay, so then, Melanie, when you looked at Mail Flix, did you use the same label that was here for the first table, of movies and money? So you said that 1 movie would be $18, and 2 movies would be $36? So you’re using the label of movies and money? Okay. Erin?
STUDENT: We, um, Student H thought it was made in months.
CECILIO DIMAS: So it’s possible that Student H thought that this was the number of months and that was the cost after that number of months? Okay. Seneca?
STUDENT: I think that Student H got mixed up with Mail Flix, because in all the other ones you have to add another to itself, and I think that maybe he or she thought that you would add another set.
CECILIO DIMAS: Okay. So, Amir, what did you want to add to that?
STUDENT: I wanted to add, for the Movie Buster and Online Flix, something else, ‘cause when you get number 5 over there, it’s 15 and 16, because it goes up the same amount, and number 7 for Movie Buster, it costs more money. But if you look at Online Flix, the first, for 1 DVD costs $13, and then 2 DVDs $14. And then for Movie Buster, it’s $6. So I found out that if you want to, like, buy more, rent more DVDs from Movie Buster, Like 7 DVDs, it would cost more than Online Flix. But if you just wanted to rent more DVDs from Online Flix, it would cost less than Movie Busters.
CECILIO DIMAS: So you’re focusing on doing some cost analysis with those two companies right there?
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.