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Estimations and Approximations: The Money Munchers
During the
  • Introduction and Individual Work (10 minutes)
      • Return their solutions to The Money Munchers to students.
      • If you have chosen not to write questions on individual student papers, display your list of questions on the board.
        • [Last lesson] you worked on Estimations and Approximations. Do you recall what the task was about?
        • I have read your solutions, and have some questions I'd like you to think about.
        • Work individually for 10 minutes, answering my questions to improve your work.
  • Collaborative Small-Group Work (20 minutes)
      • Organize students into groups of two or three.
      • Give each group a new copy of the task sheet The Money Munchers and a large sheet of poster paper.
        • I'd like you to put your solutions to one side now.
        • Start afresh on the same Fermi problem. I want you to work together in your groups to produce a better solution together than you each did individually.
        • I'd like you to make your solution into a poster. Make sure you write down all your reasoning and label everything clearly.
        • To begin, I'd like you to take turns in your groups to share your assumptions. Think about when you were working alone. What extra information did you need to identify to solve this problem?
      • You have two roles while students are working: to find out about student methods, and to support student problem solving.

      Find out about student methods.

      Listen, and watch, to find out about the assumptions students make about the context, and about the quantities they identify in their rudimentary mathematical models.

      Note students' estimates of quantities such as the length of a bed and the height of a dollar bill. Do students explain and justify their estimations? If so, do they write their reasons down?

      Notice whether students are naming and writing down the quantities with which they are working, and if they draw diagrams and label them.

      Do they notice when different units of measurement arise, such as feet and inches in lengths, and if so, do they convert between them?

      Do they make sense of packing dollars into the shape of the mattress or the dimensions of the case, or do they work with area/volume only?

      Do they justify their calculation methods to each other?

      Do they check their solutions to see if they make sense in the context of the problem?

      Support student problem solving.

      Try not to prompt students into using a particular problem-solving method, and try not to point out the difficulties with their chosen methods to them. Instead, ask questions to prompt students to justify and evaluate their own solution strategies.

      The questions in the Common Issues table were found to be useful in trials of this lesson.

      Prompt students to write their solutions so that other groups can understand what is written.

      If any group finishes their solution, ask them to consider the accuracy of their solution, and then to develop a solution using a different method.

  • Collaborative Analysis of Sample Student Responses (10 minutes)
      • Give a copy of the three Sample Student Responses to each small group of students.
      • Ask students to read the solutions and to answer the questions together.
        • Looking at your posters, I can see you have used a range of different methods to solve this problem.
        • I'm giving you some work produced by students from another class on this same problem.
        • I would like you to answer these questions about each student's work:
          • Read through the solution, and make sense of how the student is solving the problem.
          • Figure out what assumptions the student makes.
          • Are the estimates reasonable or way off the mark?
          • Figure out how the student calculates an answer.
          • Then decide what is good about the solution and how you might improve it.

      The instructions for this task are reproduced on slide P-2 Analyzing Sample Student Responses.

      • During small-group work, support students as they work. If students find it difficult to get started, suggest they read the solution aloud, slowly.
        • Stop after one sentence and check that everyone understands which numbers stand for which quantities.
        • Explain what the calculation is. What assumptions are being made? Why is Mattie calculating that at this point? How does it help him?

      Encourage students to write their reasoning in full.

  • Whole-Class Discussion (10 minutes)
      • Organize a whole-class discussion of the The Money Munchers.
      • Focus the discussion on the methods students have seen and used during the lesson rather than discussing who has the "best" or a "correct" solution.
      • In particular, ask students to discuss the strengths and weaknesses of the different approximation methods seen in the Sample Student Responses: Mattie's shape-packing approach, Idora's division of one area by another, or Stephan's approach of working backwards.

        • Is there only one reasonable estimate for the width/length of a bed?
        • Can we say which of the sample estimates is best? Which are good enough? How would we decide?
        • Is it important to make the bed comfortable by providing equal layers?
        • How could you improve Mattie's [Idora's/Stephan's] solution?
      • Ask students to contribute with reference to their own posters.

      Try to avoid making evaluative comments yourself. Instead, encourage students to respond to other students' explanations.

      Try to help students understand that different adequate solutions can arise from very different, but still reasonable, assumptions.

      • If you have time, begin to address the issue of accuracy in estimation.

      None of the respondents make a serious attempt to estimate the accuracy of their answers, taking into account the uncertainties in their assumptions.

      The thickness of a pile of dollar bills, in particular, is difficult to estimate. It might be much greater than the same number of pages in a book or sheets in a new pack of paper.

      Gaps between piles are another source of inaccuracy. Overall, their estimates when done correctly are probably accurate to ±20%.

      This is fine for the mattress but might cause problems with the suitcase.

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