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Generalizing Patterns: Table Tiles
During the
  • Improve Individual Solutions to the Assessment Task (10 minutes)
      • Return your students' work on the Table Tiles problem.
      • Ask students to re-read both the Table Tiles problem and their solutions. If you have not added questions to students' work, write a short list of your most common questions on the board. Students can then select a few questions appropriate to their own work and begin answering them.
        • Recall what we were working on previously. What was the task?

        Draw students' attention to the questions you have written.

        • I have read your solutions and have some questions about your work.
        • I would like you to work on your own for ten minutes to answer my questions.
  • Collaborative Small Group Work (15 minutes)
      • Organize the students into small groups of two or three.

      In trials, teachers found keeping groups small helped more students play an active role.

      • Give each group a new sheet of grid paper.
      • Students should now work together to produce a joint solution.
        • Put your solutions aside until later in the lesson. I want you to work in groups now.
        • Your task is to work together to produce a solution that is better than your individual solutions.

      You have two tasks during small-group work: to note different student approaches to the task, and to support student problem solving.

      Note different student approaches to the task.

      Notice how students work on finding the quadratic function for the number of whole tiles.

      Notice also whether and when students introduce algebra. If they do use algebra, note the different formulations of the functions they produce, including incorrect versions, for use in whole-class discussion. You can use this information to focus the whole-class plenary discussion towards the end of the lesson.

      Support student problem solving.

      Try not to make suggestions that move students towards a particular approach to this task. Instead, ask questions to help students clarify their thinking.

      If several students in the class are struggling with the same issue, you could write a relevant question on the board.

      You might also ask a student who has performed well on one part of the task to help a student struggling with that part of the task. The following questions and prompts have been found the most helpful in trials:

        • What information have you been given?
        • What do you need to find out?
        • What changes between these diagrams? What stays the same?
        • What if I wanted to know the thousandth example?
        • How will you write down your pattern?
        • Why do you think your conjecture might be true?
      • In trials, teachers expressed surprise at the degree of difficulty some students experienced in drawing the tabletops. If this issue arises in your class, help the student to focus his or her attention on different parts of the tabletop, how they align with the grid, and then get them to draw the whole diagram from those pieces.
        • Where are the half tiles? Whole tiles?
        • How do the whole and half tiles fit together?

      You may find that some students do not work systematically when drawing tabletops and organizing their data.

        • What sizes of tabletop might Maria make? Which ones is it useful for you to draw? Why?
        • What can you do to organize your data?
      • If students have found formulas, focus their attention on improving explanations, exploring alternative methods, and showing the equivalence of different equations.
        • How can you be sure your explanation works in all cases?
        • Ask another group if your argument makes sense.
        • Which is the formula you prefer? Why?
        • Show me that these two expressions are equivalent?

      Students may justify their formulas by drawing another example to see if the generalization fits a new case, reasoning inductively. Some stronger explanations are shown in the Sample Responses to Discuss.

  • Collaborative Analysis of Sample Responses to Discuss (15 minutes)
      • Give each small group of students a copy of the Sample Responses to Discuss.

      These are three of the common problem-solving approaches taken by students in trials.

      • Display the following questions on the board or project slide P-1 Students Responses to Discuss.
        • Describe the problem-solving approach the student used.
        • You might, for example:
          • Describe the way the student has colored the pattern of tiles.
          • Describe what the student did to calculate a sequence of numbers.
        • Explain what the student could do to complete his or her solution.

      This analysis task will give students an opportunity to evaluate a variety of alternative approaches to the task without providing a complete solution strategy.

      • During small-group work, support student thinking as before. Also, check to see which of the explanations students find more difficult to understand.

      Identify one or two of these approaches to discuss in the plenary discussion.

      Note similarities and differences between the sample approaches and those the students took in small-group work.

  • Whole-Class Discussion Comparing Different Approaches (20 minutes)
      • Organize a whole-class discussion to consider different approaches to the task.

      The intention is for you to focus on getting students to understand the methods of working out the answers, rather than either numerical or algebraic solutions. Focus your discussion on parts of the two small-group tasks students found difficult.

      • You may find it helpful to display slide P-2, Grid Paper, or slide P-3, Tabletops.
        • Let's stop and talk about different approaches.
      • Ask the students to compare the different solution methods.
        • Which approach did you like best? Why?
        • Which approach did you findmost difficult to understand?
        • Sami, your group used that method. Can you explain that for us?
        • Which method would work best for the thousandth tabletop?

      Below, we have given details of some discussions that emerged in trial lessons.

      • Some students found the work on quadratic expressions very difficult. If your students have this problem, you might focus on Gianna's method from the Sample Responses to Discuss (slide P-4).
        • Describe Gianna's pattern in the whole tiles in the 30 cm by 30 cm tabletop.
        • How would you describe her pattern in the next size tabletop?
        • Using Gianna's pattern, how many whole tiles would there be in any tabletop?
      • If students have found different algebraic formulations for the number of half and whole tiles, it might help to write a variety of their expressions on the board.
      • Ask students to link different variables and manipulate algebraic expressions to identify errors and show equivalences.
        • Which of these formulas would you use to find the number of half tiles?
        • Which are quadratic?
        • Are there any formulas that are equivalent?
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