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Generalizing Patterns: Table Tiles
Before the
  • Assessment Task: Table Tiles (15 minutes)
    • Have the students do this task in class or for homework a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. Then you will be able to target your help more effectively in the follow-up lesson.

      • Give each student a copy of Table Tiles and a copy of the grid paper.
      • Introduce the task briefly and help the class to understand the problem and its context.
        • Spend 15 minutes on your own answering these questions.
        • Show your work on the worksheet and the grid paper
        • Don't worry if you can't do everything. There will be a lesson on this material [tomorrow] that will help you improve your work. Your goal is to be able to answer these questions with confidence by the end of that lesson.

      It is important that, as far as possible, students answer the questions without assistance.

      Students who sit together often produce similar answers so that when they come to compare their work, they have little to discuss. For this reason, we suggest that when students do the task individually you ask them to move to different seats. Then, at the beginning of the formative assessment lesson, allow them to return to their usual places. Experience has shown that this produces more profitable discussions.

  • Assessing Students' Responses
    • Collect students' responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches. The purpose of this is to forewarn you of the issues that will arise during the lesson, so that you may prepare carefully.

      We suggest that you do not score students' work. Research shows that this is counterproductive, as it encourages students to compare scores and distracts their attention from how they may improve their mathematics.

      Instead, help students to make further progress by asking questions that focus attention on aspects of their work. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit.

      We suggest that you write your own lists of questions, based on your own students' work, using the ideas below. You may choose to write questions on each student's work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the beginning of the lesson.

      Common Issues: Suggested Questions and Prompts:

      Student makes unintended assumptions

      • For example: The student has calculated the number of whole tiles required to cover the tabletop, assuming she can split tiles to make quarters and halves as needed.
      • Or: The student uses only quarter tiles to cover the tabletop.
      • Imagine you can buy tiles that are ready cut. You don't need to cut them up. How many of each type do you need?

      Student makes inaccurate drawing.

      • For example: The student divides the whole tabletop into "units" of a whole tile surrounded by four quarter tiles.
      • Or: The student draws freehand with a different number of half tiles along each side.
      • How would you describe how to draw a 30 cm by 30 cm tabletop?

      Student assumes proportionality.

      • For example: For Q1, the student writes "10 whole tiles, 8 half tiles, 8 quarter tiles." The student believes a tabletop with sides twice as long will need twice as many tiles of each type.
      • Read the rubric. Where does Maria use quarter tiles? Half tiles?
      • What happens to tiles in the middle of the diagram if you extend the size?

      Unsystematic work.

      • For example: The student draws seemingly unconnected examples, such as 10 cm by 10 cm or 40 cm by 40 cm.
      • Or: The student omits some diagrams, drawing tabletops that are 20 cm by 20 cm and 40 cm by 40 cm, but not 30 cm by 30 cm.
      • Which example will you draw next? Why?
      • What do you notice about the difference between the number of whole tiles in one tabletop and the next?
      • The sizes of square tabletops are all multiples of 10 cm. Do your diagrams show this?

      Student does not generalize.

      • For example: The student does not seem to know how to proceed with finding the quadratic expression.
      • Or the student identifies patterns in the numbers of different types of tiles but does not extend to the general case.
      • Can you describe a visual pattern in the number of whole tiles in consecutive diagrams?
      • How could I find out the number of tiles needed for a larger tabletop without having to continue the pattern?

      Student does not use algebra.

      • For example: The student shows awareness of how the number of whole tiles increases with dimensions, but links this to a specific example rather than identifying variables and forming an equation.
      • How can you write your answer using mathematical language?

      Student provides a recursive rule, not an explicit formula.

      • For example: The student provides a way to calculate the number of tiles in a tabletop the next size up from a given size, rather than a general formula for a tabletop of any size; for example, "next = now + 4", or "add two more whole tiles than you did last time."
      • Would your method be practical if I wanted to calculate the number of tiles in a 300 cm by 300 cm tabletop?

      Student writes incorrect formula.

      • For example: The student writes an incorrect formula such as 4x2 + 4x + 4 for the number of whole tiles, either using an incorrect algebraic structure or making a recording mistake.
      • Does your formula give the correct number of whole tiles in tabletops of different sizes?

      Student writes answers without explanation.

      • How could you explain how you reached your conclusions so that someone in another class understands?

      Student correctly identifies constant, linear, and quadratic sequences.

      • Think of another way of solving the problem. Is this method better or worse than your original one? Explain your answer.
      • Can you extend your solution to include rectangular tabletops that aren't squares?
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