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: So, what we’re gonna look at now, is, go back to this idea of 0. What does 0 tell us about these three different plans? What did we learn about 0 with this table, with Student A’s table, that we didn’t necessarily learn with Student H’s table? So what are we learning about 0? Go ahead and talk to your shoulder partner, please, for a quick moment.
STUDENT: Ya vi que este no, no está bien (I saw that this one no, it’s no good)
STUDENT: No está bien. (It’s not good.)
STUDENT: Porque este está cobrando mas que tiene que cobrar. (Because this one is costing more than it has to cost.)
STUDENT: Si no fueran aqui, fueron 24. (If it weren’t here, they were 24.)
STUDENT: Oh, sí. (Oh, yes.)
CECILIO DIMAS: So, if 0 is important, I’d like you to tell me with thumbs-up under your chin, if 0 is not important, show me with thumbs-down. Okay. Hands down. Why is 0 important? Again, why is 0 important? Katie?
STUDENT: Because it’s showing you that with with On, with Online Flix, and Mail Flix, you still have to pay the monthly fee even though you’re not buying any movies.
CECILIO DIMAS: Okay. Could I have someone restate what Katie just shared? Hailey?
STUDENT: Like, even though, like it says for either one of them that you have to pay a fee, and you’re still paying it, but on Movie Busters, if you don’t get anything you’re not gonna have a fee to pay.
CECILIO DIMAS: Okay.
STUDENT: So it goes by what you rent.
CECILIO DIMAS: How would you, thank you. How would you help Student A make sense of the plans in this table? How would you help Student A make sense of it? Jessica?
STUDENT: You would get, I think, I would get rid of the 0. ‘Cause it’s kind of confusing. And then start off with 13. Instead of 12. So they don’t think that 12 plus 12 is 24. So I would get rid of the 0.
CECILIO DIMAS: Okay. What else could we do, what else could a person suggest to Student A to help them with their table? Kyle?
STUDENT: You could tell them that you don’t have to pay the first fee for every rental, you just have to pay $1.
CECILIO DIMAS: Okay, so, if you were to help them figure out that this was not 25, what would you tell them?
STUDENT: Uh, that you don’t pay the flat fee for every rental, you just pay $1.
CECILIO DIMAS: So what would it cost for 1 movie from Online Flix?
CECILIO DIMAS: And how did you get your 13?
STUDENT: Because 12 plus 1 is 13.
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.