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The class does not get to Student E in this lesson. However, they begin with Student E the following school day and continue the class discussion about whether or not the tables make mathematical sense and if the tables match the original plans, therefore making an evidence-based conclusion about the number of DVD rentals needed in order to have all three plans cost the same amount of money. The class also continued to compare all student work together to look for similarities and differences.

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7th & 8th Grade Math - Comparing Linear Functions
Cecilio Dimas , Ida Price Middle School, Cambrian School District, San José, California

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CECILIO DIMAS: So boys and girls, I have some clarifying questions. I would like to hear some of the changes that were made to the tables. What changes did you make to your tables? And tell us why you made those changes. Sam?

STUDENT: I changed Online Flix because it’s supposed to be \$12 per month and it says add a dollar, a dollar per rental, so when it was 12, I added a dollar, so it was 13, 14, 15, 16, 17, 18, 19, 20. So then... I got my answer.

CECILIO DIMAS: Okay, so for 1 DVD, how much would I pay at Online Flix?

STUDENT: A dollar.

CECILIO DIMAS: Altogether. What would I pay for 1 DVD? Do you want some assistance? Omar?

STUDENT: 13.

CECILIO DIMAS: 13? And if I rented 2 DVDs, what would it cost me? Emily?

STUDENT: 14.

CECILIO DIMAS: And if I rented 3 DVDs? Sergio?

STUDENT: 15.

CECILIO DIMAS: If I rented... 6 DVDs? Amir?

STUDENT: 16.

CECILIO DIMAS: For 6 DVDs?

STUDENT: Uh, 17.

CECILIO DIMAS: Well, let’s break it down into pieces. We have our flat fee of how much money?

STUDENT: Uh, 12.

CECILIO DIMAS: Plus 6 DVDs?

STUDENT: Uh, 18 dollars.

CECILIO DIMAS: So \$18? Okay. And what about 8 DVDs? How much would you spend there? Victoria?

STUDENT: 20.

CECILIO DIMAS: 20. Okay. Any other changes or corrections to Student A? Erin?

STUDENT: Up at the top I wrote the word “price” because at first I was a little confused about...

CECILIO DIMAS: Okay, so you wrote “price?” So this is the cost, the price? Okay. Let’s go ahead and flip to the next page, for Student E, and we probably actually need to go ahead and stop here, we have one more minute before the period is over. So I’m gonna ask that you make sure that your name is written on your DVD plans packet, and you can leave it on top of your desk, I would also like to see Charles, Tricia, Danielle, and Kyle after class for a few moments, I want to thank everyone for your participation today in our taping, and I look forward to seeing you again tomorrow. Keep in mind that Bridge to Algebra Page 9 is due tomorrow. 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.