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# Creating, critiquing and revising arguments: Ms. Hauser's Hexagon Train Task

After you've looked over the Hexagon Train Task for yourself, examine each of the student work samples below and think about the following:

1. What is each student's line of reasoning?
2. Which of these lines of reasoning are valid?
3. For those that are not valid, what aspects of the line of reasoning (argument) are mathematically sound and what aspects are not mathematically sound?

#### The Story Line—Ms. Hauser’s Hexagon Train Task: Eliciting and orchestrating students’ arguments

In this lesson, Ms. Hauser’s class works on the Hexagon Train Task, determining the perimeter of the 25th figure and justifying the results. She finds that some students are using a proportional-reasoning strategy (e.g., assuming that doubling the perimeter of the 5th figure yields the perimeter of the 10th figure), which overcounts the perimeter. After having students share this strategy publicly, Ms. Hauser orchestrates a conversation where students work through how the proportional-reasoning strategy overcounts and how to revise this strategy so that it accurately computes the perimeter of the 25th figure.

The Teacher, Class and Context

This cluster features Ms. Hauser’s 7th-grade pre-algebra class, during Ms. Hauser’s second year of teaching. There are 25 students in the class. The school is large, offering 15 sections of mathematics for 7th-graders, four of which are pre-algebra—the more advanced level. The school is located in a diverse community in New England. About 30% of the students receive free or reduced lunch. Relative to schools in the state with similar demographics, students at this school demonstrate higher levels of achievement on state standardized tests.

This lesson took place in December. Ms. Hauser noted some goals for the lesson:

“In this lesson I was setting the stage for our unit on pattern-seeking and variables. I was also working on having students justify their answers and listen to each other. Throughout the lesson, I expected justification to play a big part in students’ sense making, both in determining the perimeter of the 25th figure, and then ultimately in being able to describe the general case.”

Lesson Graph

Prior to this lesson, the students have worked a bit on patterns and generating new figures when given other figures in a pattern.

The next day, the class begins with the following warm-up.

The warm-up follows on the discussion students had about the proportional reasoning argument. After some discussion of this warm up, the teacher returns to the Hexagon task and the methods from the day before. They work through the initially incomplete methods, showing how for each “connector” (shared side) they have to subtract 2. Using this approach, they revise the methods from the previous day and calculate the perimeter of the 25th figure to be 102 cm.

The teacher then asks students to share different methods for how they found the perimeter of the 25th figure. Several approaches are shared at the board and explained. The students justify their approaches by explaining why they did the calculations they did, often connecting them to the diagram.

The teacher assigns the homework. They are to write up a “final copy of your solution” to the question: What’s the perimeter of the 25th figure? Students are reminded about a CLEAR solution, which is based on a rubric students have used when writing responses to open-ended problems.

C.L.E.A.R. Rubric

• Calculations - Show your work, including calculations, tables/charts, graphs, pictures, etc.
• Labels  - Be sure to tell what each calculation means. No floating numbers!
• Evidence - Be sure your calculations support the decision made and that you have provided evidence for all parts of the problem.