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: Again, going back to our original prompt. When will all three plans cost the same amount of money? Does this, would this table, if there wasn’t the confusion with Online Flix, help us answer that question? Would we be able to answer the question, um, of when the three plans would cost the same? If yes, thumbs-up, if no, thumbs-down. Okay. Hands down. Could someone explain to us why you would be able to answer the prompt of when the three plans would cost the same, if we could correct the confusion here with Online Flix? Erin?
STUDENT: Well, because then you could compare the prices, and eventually meet up with one of them for the prices, which is 18.
CECILIO DIMAS: Okay. Could I have someone restate what Erin said, or share their own idea of why this is, why we’d be able to compare them. Charles?
STUDENT: Well, if like, out of those they can be even, I’d first look at Mail Flix, because if that’s always 18, that means that you’re gonna need to look for 18 matching, so I’d go down to the other ones, and look ‘til they had 18.
CECILIO DIMAS: Okay. Thank you, Charles. So what I’m going to ask of you right now, is, I’m gonna ask that you make any mathematical corrections or additions to Student A’s paper, and make sure at the bottom of the page that you give reasons why you made those additions or changes.
CECILIO DIMAS: So, Jessica, are you gonna make any changes to Student A’s work?
STUDENT: I’m gonna shift all the numbers downwards so I can get rid of the 0.
CECILIO DIMAS: Keep the 0. Keep the 0. And Jessica, see that if you keep the 0, it then tells the reader of your table whether or not you’re gonna have to pay if you don’t rent any movies. So we can see here with Movie Buster, that if you don’t rent anything, you don’t pay anything. Where with Online Flix and Mail Flix, if you don’t rent anything, you still have a fee to pay. So we want to
STUDENT: Isn’t that kind of like a rip-off?
CECILIO DIMAS: That would be a rip-off, and that’s why if we don’t rent any movies, that’s also something for us to consider, with these two plans. Okay.
CECILIO DIMAS: Amir, what changes have you made for Student A?
STUDENT: No changes
CECILIO DIMAS: Why not?
STUDENT: ‘Cause I think they should keep the 0, ‘cause maybe the Student A would want to go to rent a DVD and they would find out that every month a DVD would be 12, um, dollars, for the Online Flix, and..
CECILIO DIMAS: So I’m gonna stop you for a moment, Amir. Online Flix costs $12 a month plus $1 a movie. So if you rent 1 movie, how much money is that gonna cost you?
STUDENT: Uh, $25. Oh! But, if you wanna rent a movie and one month...
CECILIO DIMAS: Again, this is number of movies, so 1 movie is gonna cost you $1 plus a $12 flat fee. So how much will 1 movie cost you all together?
STUDENT: Um... 25 bucks?
CECILIO DIMAS: You have your flat fee of $12 plus $1 for that movie. So what’s that going to be?
STUDENT: I don’t understand.
CECILIO DIMAS: So, if you rent 1 DVD, for Online Flix you pay $12 a month, even if you don’t rent anything. Then you add a dollar for each movie that you rent. So you pay your $12, and then you rent 1 movie, so that’s an extra dollar. So you pay 12 plus the 1.
STUDENT: Oh. $13.
CECILIO DIMAS: So for then 2 DVDs, how much are you gonna spend?
STUDENT: Um...for 2 DVDs... $13?
CECILIO DIMAS: So you have your $12 dollars plus the $2 that you spent for the 2 movies. So that’s 12 plus 2.
STUDENT: 12 plus 2 equals $14.
CECILIO DIMAS: And what about 3 DVDs?
STUDENT: 12 plus 3.
CECILIO DIMAS: Which would be how much money?
STUDENT: Uh, 16, I mean 15 dollars.
CECILIO DIMAS: So do you need to make some changes?
STUDENT: Yes I do.
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.