Additional Materials

Content Standard Strand

MATH: Algebra/Expressions & Equations

Materials Required

Estimated Time Needed

(Times are approximate and will depend on the needs of the students.)

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Optimization Problems: Boomerangs
Before the
  • Assessment Tasks: Boomerangs (15 minutes)
    • Have students do the assessment 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 find out the kinds of difficulties that 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 the assessment task, Boomerangs.
      • Introduce the assessment task briefly and help the class understand the problem and its context. You could show examples of boomerangs.
        • Boomerangs come from Australia, where they are used as weapons or for sport.
        • When thrown, they travel in a roughly elliptical path an return to the thrower.
        • Boomerangs are made in many different sizes.
        • Read through the assessment task and questions as carefully as you can. Show all your work so that I can understand your reasoning.
        • As well as trying to solve the problem, I want you to see if you can present your work in an organized and clear manner.

      It is important that students are allowed to answer the questions, as much as possible, without your assistance.

      Students who sit together often produce similar answers, and then when they 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 seats. Experience has shown that this produces more profitable discussions.

  • Assessing Students' Responses
      • Collect students' responses to the assessment task.

      Make notes on what students' work reveals about their current levels of understanding and their different problem-solving approaches. The purpose of making notes about students' work is to forewarn you of issues that will arise during the lesson itself so that you may prepare carefully.

      While a solution is provided for the optimization problem, we suggest that you do not score the student work. The research shows that this will be counterproductive as it will encourage students to compare their 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 series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this lesson unit.

      We suggest that you write a list of your own questions, based on your 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, just 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. If students have used graphs or simultaneous equations in their solutions, add the relevant questions to their work. You may also want to note students with a particular issue, so that you can ask them about their difficulties in the formative lesson.

      Common Issues: Suggested Questions and Prompts:

      Student has difficulty getting started.

      • What do you know?
      • What do you need to find out?

      Student makes an incorrect interpretation of the constraints and variables.

      • For example: The student has applied just one constraint, such as "Phil has only 24 hours to make the boomerangs" or "Cath can only make 10 boomerangs."
      • Or: The student has calculated the profit for making just one type of boomerang.
      • What figures in the task are fixed?
      • What can you vary?
      • What is the greatest number of small/large boomerangs they can make?
      • Have you used any unnecessary restrictions on the number of small and large boomerangs to be made?
      • Why can't they make 50 boomerangs?

      Student works unsystematically.

      • For example: The student shows three or four seemingly unconnected combinations, such as 5 small and 5 large boomerangs, then 10 large.
      • Can you organize the numbers of large and small boomerangs made in a systematic way?
      • What would be sensible values to try? Why?
      • How can you check that you remember all the constraints?
      • Do you cover all possible combinations? If not, why not?
      • How do you know for sure your answer is the best option?
      • Can you organize your work in a table?

      Student presents work poorly.

      • For example: The student presents the work as a series of unexplained numbers and/or calculations, or as a table without headings.
      • Or: The student circles numbers, and it is left to the reader to work out why this is the answer as opposed to any other combination.
      • Would someone unfamiliar with your type of solution easily understand your work?
      • Have you explained how you arrived at your answer?

      Student has technical difficulties when using graphs.

      • For example: Lines are plotted inaccurately, axes are not labeled, or the purpose of the graph is not explained.

      Student has technical difficulties when using simultaneous equations.

      • For example: A mistake is made when solving two correct simultaneous equations, or the correct solutions are obtained but the profit is not calculated.
      • Would someone unfamiliar with your type of solution easily understand your work?
      • How can you check your answer?
      • How do your answers help you solve the problem?

      Student produces a correct solution

      • Student needs an extension task.
      • Can you now use a different method? For example, a table or graph, or algebra?
      • Is this method better than your original one? Why?
      • If the problem investigated how many boomerangs can be made in a month rather than 24 hours, would any method(s) be preferable to others?
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