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The lesson begins with a review of the foundational pieces of the lesson from the pre-assessment students did the day before: verbally describing and putting information into a table to determine when three plans for DVD rentals would cost the same amount. Students are then asked to state how they got started on their individual tables. After they share their strategies with their shoulder partners, students are given the objective of the day’s lesson. The focus of the lesson is to look at some different tabular representations. They are going to analyze each one to see if they can determine when the three different DVD plans cost the same, and they will look for similarities and differences between the tables. The order of the student work is purposeful. To make the lesson accessible to all students, the teacher chose student work that built upon the previous student to unfold a purposeful progression to tell the mathematical story about cost analysis.


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

Next Up:   Problem 1 - Part A
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CECILIO DIMAS: All righty! Good afternoon, boys and girls. On your desks you have a DVD plans packet, and your name, date and period number should be written down on it. I’m going to ask that you follow my lead with this packet. So please don’t flip ahead, or fill anything out, until I give you directions to do so. Okay? So yesterday, we started a conversation about the economic status of our world and how there’s been a lot of talk in the news and in the media about the financial markets here in the U.S. and around the world. We also then had a conversation about making responsible decisions when we’re spending our money, and one of the ways in which we can do that is to work with cost analysis. To really analyze the prices of items that we’re spending our money on. So I’d like for you to just think for a moment about the prompt that we worked with yesterday. What was the task I asked you to work with yesterday? That was your ticket out the door. So go ahead and think about it for a moment, go ahead and talk to your shoulder partner…When they’re, when they’re all the same amount, on a certain day, or a certain amount of DVD rentals. So could someone share out what the prompt was yesterday? What were you looking for? Amir?

STUDENT: We were trying to, we had this three services, that we could rent DVDs. But what you asked us was that to… try to figure out how many DVDs or how many … yeah. Would it take for us to have it the same… cost the same money.

CECILIO DIMAS: Okay. Thank you, Amir. Could I have someone restate what Amir just shared? Danielle?

STUDENT: At one point, all the services would be the same amount.


STUDENT: And we were supposed to make a table about it.

CECILIO DIMAS: Okay. So Danielle added this last piece, that we were supposed to make a table. In addition to a table, there was another representation that I asked you to use. What was that representation? Melanie?

STUDENT: Verbal representation?

CECILIO DIMAS: Verbal representation, thank you. So right now I’d like for you to take a moment to think about: How did you get started on your tabular representation, on your table, and on your verbal representation? So how did you get started? On the back side of your DVD plans packet is a pink sheet, go ahead and flip your packet over, and there’s a question: How did you get started? So I’d like for you to take a few moments right now to write down some ideas of how you got started on your table and on your verbal representation. Go ahead and do that now, please.

CECILIO DIMAS: So Tracy, here’s a helpful hint: When you’re talking about how you got started on your table, you can tell us how you set up your table. Okay? Sam? So I see that you’re actually starting to make a new table on your paper? If you, um, could you explain to me how you set it up? Okay. Daniel?¿Qué está aquí? (What is there?) Vas a poner la palabra si crees, alla, ¿como se dice… ayer? Con esa problema. ¿Que hiciste con su papel? (You’re going to put the words if you believe, how do you say, yesterday? With this problem. What did you do with your paper?)

STUDENT: Oh. Va a hacer… la… la tabla? (Oh. It’s going to make… the… the table?)


STUDENT: Y… (And…)

CECILIO DIMAS: ¿Cómo empezabas (How did you start?) …. Um, Meli, can you help me with, ‘How did you get started with the table?’

STUDENT: ¿Cómo empezaste con la tabla que hiciste ayer? (How did you start with the table that you made yesterday?)

CECILIO DIMAS: Y por eso puedes escribir en español, si puedas. (And so you can write in Spanish if you can.)

CECILIO DIMAS: Can I ask that you start wrapping up your thoughts? And boys and girls, even if you’re not done with your writing, I’m going to give you an opportunity to talk to your shoulder partner. If you are a red card, I’d like you to share first, and that’s where you can complete your thoughts.

STUDENT: I started with the table, I knew that one of the movie rentals was 18 per month and unlimited, so I knew that if I wanted them all equal together, I would have to get 18 dollars for all of them. I kept multiplying, I kept multiplying, I kept adding each one by its own number every time. Then the sixth time I did it, all three of the movie rentals were 18 dollars, so, that’s how I got that. Then on my verbal representation, representation I just put the chart with the words, so I just told them how I, practically what I just told you.

CECILIO DIMAS: So now I’m going to give you the opportunity to share out, we’ll have two or three responses, share how they set up their table, how they got started? Charles?

STUDENT: I started, I knew that one of the movies was 18 per month no matter what, so if I wanted them all together, I’d have to have 18 dollars for all of them. So I got the other two, and I kept adding them by itself until I got 18 dollars for both of them, and that’s how I set up my chart.

CECILIO DIMAS: Thank you, Charles. Would someone else like to share how they got started? Some of the ideas I heard some of you talking about were that you got started by setting up your table by business. So that was one thing that some students did. And another thing that students did is they organized it by money. That was part of how they set up their table. Is there any other way that students set up their table or got started? Okay. We’ll continue on, then.

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.