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Estimations and Approximations: The Money Munchers
Before the
  • Assessment Task: The Money Munchers (15 minutes)
    • Have the students do this task, in class or for homework, a day or more before the lesson. This will give you an opportunity to assess their work, and to find out the kinds of difficulties students have with it. You will then be able to target your help more effectively in the follow-up lesson.

      • Give each student a copy of The Money Munchers.
      • Introduce the task briefly. Help the class to understand the problem and its context.
        • Today, you're going to work on a Fermi problem.
        • Fermi was a twentieth-century Italian physicist. He loved setting estimation problems for his colleagues and students.
        • A Fermi problem is a question for which you produce a rough but sensible estimate, without knowing exactly all the measures involved.
      • Ask students to read the scenario about Emily and the money carefully.

      If students in your class have literacy issues, it may help to read this information aloud.

      • Now explain what you are asking students to do.
        • I want you to work individually for 15 minutes.
        • Your work on this task will help me see how good you are at estimating quantities like length and using your estimates to calculate approximate solutions to problems.

      It is important that, as far as possible, students answer the questions without assistance. If students are struggling to get started, ask questions that help them understand what they are being asked to do, but do not do the problem for them. The first few questions on the Common Issues table were found to be helpful in trials of this lesson.

      Students should not worry too much if they cannot understand or do everything because,there will be a lesson using the same task, which should help them. Explain to students that by the end of the next lesson they should expect to answer questions such as these confidently. This is their goal.

  • 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 problem solving strategies.

      We suggest that you do not score students' work. The research shows that this will be counterproductive, as it will encourage students to compare scores, and distract their attention from what they can do to improve their mathematics.

      Instead, help students to make further progress by summarizing their difficulties as a list of questions. Some suggestions for these are given the Common Issues table on the next page. We suggest that you make a list of your own questions, based on your students' work, using the ideas on the following page. 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 does not identify missing information.

      • For example: The student says she cannot calculate an answer because the size of a bed/dollar bill is unknown.
      • In many problems, you have to find or estimate the information you need to solve the problem. How long do you think a dollar bill is? How could you make a good estimate?

      Student does not identify or justify the assumptions that shape the calculation strategy.

      • For example: The student does not say why he assumes there can be just ten piles of dollar bills.
      • Or: The student assumes that orientation and packing of the notes into the available space makes no difference.
      • You have assumed that... Explain why you have made this assumption.
      • Is it reasonable to assume that...?

      Student makes poor estimates.

      • For example: The student estimates the length of a bed as 5 feet or its width as 2 feet.
      • Or: The student estimates that a pile of ten one-dollar bills is an inch high.
      • How tall are you? How does this help you estimate the length of a bed?
      • Find a book. How many pages are there? How many sheets of paper are used to make the pages? How high is that pile of pages? How does this help you estimate the height of a pile of one hundred $1 bills?

      Student provides a poor explanation.

      • For example: The student writes calculations without showing which quantities the numbers refer to.
      • Or: The student makes estimates but does not justify them.
      • Your solution is difficult to follow. What does the number [...] stand for in this calculation? Explain what each number is in turn.
      • Imagine you have to explain this solution to another student. How could you make your solution easy to understand?

      Student makes inappropriate calculations and errors.

      • For example: The student multiplies rather than divides to find the number of dollar lengths that fit into the length of the case.
      • Or: The student makes an arithmetic error.
      • Explain what this calculation is for.
      • You are calculating the number of dollar lengths that fit into the length of the suitcase. Which operation do you need to use: add, subtract, multiply or divide? Why?
      • How can you check your calculations to make sure they are accurate?

      Student provides a complete and adequate solution.

      • How accurate are your assumptions?
      • Estimate how accurate your answer is, taking into account the accuracy of your assumptions.
      • Find another way of answering this problem with different plausible estimates and assumptions
      • Make up a new Fermi problem of your own and answer it.
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