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

Student Reasoning about the 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?


Student Work Sample #1

Student Work Sample #2

Student Work Sample #3

Student Work Sample #4  


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.

0 – 11

Students do warm-up; teacher checks homework; they discuss the warm-up problems.

HEXAGON TASK

 

11 – 15

Teacher hands out and projects the Hexagon Train Task. Students read the task, then teacher checks for understanding about the problem. Teacher also reminds class about good groupwork and what they have agreed is important. There is a list written on the board:

Focus
Cooperation
Participation
Strong communication
Include everyone

 

15 – 27

Video Clip 1 

Students work with one or two partners on the Hexagon Train Task. Teacher circulates, prompting students to share their ideas with their partner(s) and listening to students’ ideas. Teacher’s interaction with one group is featured in Part 1. She elicits some incomplete arguments and a common misconception about approaching this task. One student has done 25 x 4, counting 4 sides per hexagon. Another student, Amanda, has used a proportional-reasoning method: She has doubled the string of 4 hexagons, with perimeter 18, to find that the perimeter of 8 hexagons is 36. Using this, she has scaled up to find 114 as the perimeter of Figure 25.

 

27 – 30:30

Teacher selects two students of the students featured in Part 1 to share their solutions. Both solutions are incomplete. Teacher reminds students of their audience roles: to ask questions, to understand and make suggestions.

30:30 – 33:30

Kylan shares his ideas with the class. With teacher prompting, he explains why he thinks it’s not complete. Another student suggests that Kylan add 2.

33:30 - 35:00

Video Clip 2a 

 

Jahlil shares his method of adding “the tops” and “the bottoms” and then each side, or 50 + 50 + 1 + 1. Teacher involves other students to be sure they understand Jahlil’s method.

35 – 37:30

Video Clip 2b 

Robert shares his method, another proportional reasoning method that overcounts. It produces the same value (114) as Amanda's method (in Part 1), but the specifics are different. Teacher facilitates class discussion to be sure Robert's methods are understood.  

37:30 – 38:30

Teacher has Amanda explain her method – already on the board – to the class.

38:30 - 46

Video Clip 3 

Students offer initial arguments (“I don’t think that will work…”) to try to explain why Robert’s and Amanda’s methods might not work, identifying that these methods overcount the perimeter. Teacher facilitates a class discussion trying to sort out the issue of whether and how some sides counted as part of Amanda’s or Robert’s methods are not part of the perimeter of the 25th figure.

46-47

Teacher closes – giving feedback on what was good and what needs a little more work (patience and respect).


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

  Student Example 

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.
  • Answer the question -Answer the question asked using a complete sentence.
  • Reasons why - Identify your procedure and explain why you did what you did or what it means.

The following day, Ms. Hauser has students swap solutions and briefly peer review for CLEAR. She collects their work. The teacher then poses the question of how they would find the perimeter of the 100th figure. They use several different methods to figure this out, and students are prompted to explain where their values are coming from. Ms. Hauser also extends the discussion asking, “How would you describe that for any figure?” or “How would you describe that for n?” Students explain for their own method what they would do. About halfway through class, the teacher transitions them to other pattern tasks. She hands out a “Patterns Packet” and they begin working through several of those problems.