**COMING SOON!!** To apply and develop students’ understanding of proportional reasoning, Katie’s 7^{th} grade students considered the following statement, “A survey of 100 students at Cougar Middle School found that 30 students liked Mrs. Anderson best and 70 liked some other teacher best.” She asked students to assess nine different statements to determine whether the proportional relationships exhibited were the same as those found in the target statement and use a graphic organizer to sort statements based on their accuracy and record their justification.

**Focus Practices:** Reasoning (MP2) and Argumentation (MP3)

The featured video involves two students, Ricardo and Varius, arguing about the accuracy of the following statement, "Out of 50 students, 30 like another teacher best." This statement is “not accurate” because 30 out of 50 teachers would mean that 60% of the students liked the “other teacher,” whereas the original statement involved 70%. The teacher chose these particular students because they knew that they had different answers and would be willing to discuss their reasoning to the whole-class.

See related material below for copies of the sorting statements and graphic organizer.

## Reasoning and Argumentation

**Summary of Student Claims**

Ricardo believed that “Out of 50 students, 30 like another teacher best” was *accurate*. To support his answer, Ricardo subtracted 30 from 50 to find that 20 students liked Mrs. Anderson best. Although this strategy was useful for determining the number of students who liked Mrs. Anderson, which was not included in the original response, he did not explain how and why the statements were aligned.

Varius believed that the statement was *inaccurate* because the total number of students differed from the original statement. To justify his answer, he added 50 to 30 to note that the total number of students was 80.

Rather than evaluating either student’s argument, Katie opened up the conversation to the class to understand whether they might have additional reasoning for why Ricardo’s answer was incorrect.

Ricardo wanted the opportunity to respond to Varius’ critique. Ricardo clarified his reasoning for how he determined that 20 students liked Mrs. Anderson best. Katie then asked Varius to repeat his reasoning and made explicit his disagreement, “So you are saying that when you add it together, it is not out of 100, it is out of 80?” Thus, providing focus for the target of argumentation.

Ricardo pointed out that the total number of students in the target statement was 50 rather than 80. During this moment, the bell rang. The teacher asked the two students to continue their conversation. The students were clearly engaged in the task, as students could be heard saying, “I like this argument” and “That was intense.”

**After-Class Discussion**

Katie and Ricardo asked questions to clarify their understanding of how Varius came up with 80 students (“Would it be 80 though? What are you adding together?”). Through the discussion, we realize that Varius actually meant that he added 50 to each number because he wanted to the total number of students to be 100 (“And if you add 50 to make it [total number of students] 100. And if you add 50 to that [30 students who prefer other teacher], it would be 80.”). Thus, by asking clarifying questions during argumentation, both students and teachers can develop a consensus understanding of each other’s ideas before you can engage in argumentation.

Now that he understood Varius’ idea, Ricardo asked him why he wanted to *add* rather than *multiply* the numbers by 2 to make the total numbers equivalent (“Why would you add these two? This would be more like 50 times 2”). Using Varius’ complaint that the total student numbers were not the same, Ricardo proposed an alternative way to make the statements equivalent. In doing so, Ricardo used proportional reasoning to assess the equivalence of these fractions.

Through this argument, Varius’ underlying reasoning beyond the “right answer” was made visible to the teacher (“Oh, I see what you’re doing. I see what you’re doing.”). She lent support to Ricardo’s methodology and asked both students to about what operation is needed to make ratios proportional to each other. Through this process, Ricardo now realized that the statements were not equivalent. However, now the students have a consensus understanding for why the statements were not equivalent. Thus, engaging students in argumentation provided a venue for students to make their thinking visible and use their ideas to defend mathematical claims. Students had the opportunity to assess the robustness of their understanding and make revisions to account for the critiques of others.

Attachment | Size |
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Statements | 15.56 KB |

Graphic Organizer | 14.82 KB |