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
During the
  • Improve Individual Solutions to the Assessment Task (10 minutes)
      • Return the assessment task papers to the students, and hand out calculators.

      If you have not added questions to individual pieces of work, then write your list of questions on the board (excluding the ones for graphs and simultaneous equations). Students are to select questions appropriate to their own work and spend a few minutes answering them.

      • Introduce the "Improve Individual Solutions" part of the lesson.
        • Recall what we were looking at in the previous lesson. What was the task?
        • I have read your solutions, and I have some questions about your work.
        • I would like you to work on your own to answer my questions for about 10 minutes.
  • Collaborative Small-Group Work (10 minutes)
      • Organize the class into small groups of two or three students.
      • Provide a piece of paper to each group.
      • Introduce the Collaborative Small-Group Work.

        Ask students to try the task again, this time combining their ideas.

        • Put your own work aside until later in the lesson. I want you to work in groups now.
        • Your task is to produce a solution that is better than your individual solutions.
      • While students work in small groups, (1) note different student approaches to the task and (2) support student problem solving.

      Note different student approaches to the task—

      You can use this information to focus a whole-class discussion toward the end of the lesson. In particular, note any common mistakes.

      For example, are students consistently using all the constraints, or are they imposing unnecessary constraints?

      Also note whether students are using algebra, and, if so, how they are using it.

      Support student problem solving.—

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

      You may discover that some students experience some difficulty in keeping more than one constraint at a time in mind. In that case, you may ask them to consider these three questions:

      • If they were to make only small boomerangs, how much money would they make?
      • If they were to make two small boomerangs, how many large ones could they also make?
      • How much money would they make?

      For the first question, Cath's time is the limiting constraint, whereas in the second question, Phil's time is more significant.

      Students who organize their work into a table may choose to use column headings for "Time Needed for Phil" and "Time Needed for Cath," which they can use to check that both constraints have been met.

      To help students really struggling with the task, use the questions to support your own questioning. In particular, if students find it difficult to get started, these questions may be useful:

      • Try some examples. What happens if they make one large and three small boomerangs?
      • What would be sensible values to try? Why?
      • Can you organize the numbers of large and small boomerangs made in a systematic way?

      If the whole class is struggling on the same issue, write relevant questions on the board.

      You could also ask students who performed well on the assessment to help struggling students. If students are having difficulty making any progress at all, you could hand out two pieces of sample work to model problem-solving methods.

  • Collaborative Analysis of Sample Responses to Discuss (20 minutes)
      • After students have had sufficient time to attempt the assessment task, give each group of students a copy of each of the four Sample Responses to Discuss.
        • This task gives students the opportunity to evaluate a variety of possible approaches to the task without providing a complete solution strategy. Each of the sample responses poses specific questions for students to answer.
      • Introduce the "Collaborative Analysis of Sample Responses to Discuss."
        • Imagine you are the teacher and have to assess this work. Correct the work, and write comments on the accuracy and organization of each response.

      Each of the sample responses poses specific questions for students to answer. In addition to these, you could ask students to evaluate and compare responses.

      To help them do more than check to see if the answer is correct, you may wish to use the projector resource Evaluating Sample Responses to Discuss:

        • What do you like about the work?
        • How has each student organized the work?
        • What mistakes have been made?
        • What isn't clear?
        • What questions would you like to ask this student?
        • In what ways might the work be improved?

      You may decide there is not enough time for each group to work through all four samples. In that case, be selective about the ones that you hand out. For example, groups that have successfully completed the task using one method will benefit from using different approaches. Other groups that have struggled with a particular approach may benefit from seeing a student version of the same strategy.

      During the small-group work, support students as before. Note similarities and differences between the sample approaches and those approaches students took in the small-group work. Also check to see which methods students have difficulty understanding. This information can help you focus the next activity, a whole-class discussion.

  • Whole-Class Discussion: Comparing Different Approaches (10 minutes)
      • Organize a whole-class discussion to consider the different approaches used in the sample work.
      • Focus the discussion on those parts of the small-group tasks that students found most difficult. Ask students to compare the different solution methods.
        • Which approach did you like best? Why?
        • Which approach did you find the most difficult to understand?

      To critique the different strategies, use the questions on Evaluating Sample Responses to Discuss and the worksheets, Sample Responses to Discuss.

      Alex has realized that you have to take account of both constraints: Phil's time for making the boomerangs and Cath's time for decorating them. Alex has not examined different combinations of cases.

      Danny has found an effective way to organize his work, using a table. He has made some mistakes in this table, however. Part of the problem is that he loses track of the two constraints. It might have been helpful for him to include two additional columns headed: "Time needed (≤ 24 hours)" and "Total number made (≤ 10)." Then he could test each case and put a check mark if it satisfies both constraints.

      Jeremiah has tried an algebraic approach and has hit upon the correct solution. However, he has used equalities rather than inequalities. He needs to calculate the total profit to complete the question.

      Tanya has used a graphical approach, but her graph of 2x + 3y = 24 is inaccurate and should be redrawn. This graph is powerful in that it shows the entire feasible solution space—the integer points on the grid. She has not explained why her method will give the greatest profit.

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