Additional Materials

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Solving Linear Equations in Two Variables
During the
  • Individual Work: Cash Registers (10 minutes)
      • Give each student the task sheet, Cash Registers.
      • Help students to understand the problem, and explain the context of the task briefly.
        • Spend 10 minutes on your own answering these questions.
        • What does 'simultaneously' mean?
        • Show all your work on the sheet.

      Students who sit together often produce similar answers and, when they come to compare their work, they have little to discuss.

      For this reason we suggest that, when students do this task individually, you ask them to move to different seats. Then, for the collaborative task, allow them to return to their usual places. Experience has shown that this produces more profitable discussions.

  • Collaborative Small Group Work: Cash Registers (10 minutes)
      • Organize the class into small groups of two or three students and hand out a fresh sheet of paper to each group.
      • Students should now have another go at the task, but this time they will combine their ideas.
        • I want you now to work together in your groups.
        • Your task is to produce an answer together that is better than your individual ones.

      Throughout this activity, encourage students to articulate their reasoning, justify their choices mathematically, and question the choices put forward by others.

      As students work you have two tasks: to note student approaches to their work and to support their thinking.

      Note student approaches to their work.

      • How do students choose to tackle this task?
      • Notice the variety in approaches.
      • Notice any common errors.

      You can use this information to focus your questioning in the whole-class discussion towards the end of the lesson.

      Support student thinking.

      Try not to make suggestions that prompt students towards a particular answer. Instead, ask questions to help students to clarify their thinking.

      You may find that some students interpret the letters as "quarters" and "dollars," rather than the number of quarters and number of dollars. For example, they may say things like:

      • "3x = y means three times as many quarters as dollars."
      • "4x + y = 70 means 4 quarters plus dollars equals 70."
      • "There is $70 in the till."

      The following questions and prompts may be helpful for both students struggling with the task and those making quick progress:

        • What do the letters x and y represent?
        • Replace x and y in this equation by words, and now say what the equation means.
        • Are there more dollar bills or more quarters in the cash register? How do you know?
        • Do you have any values for x and y that work for the first equation? How can you check to see if they also work for the second one? If these don't fit, what other values for x and y can you use?
        • Why have you chosen these values for x and y?
        • Suppose there are 5 quarters in the drawers of the cash register, so x = 5. From the first equation, how many dollar bills are there? [15] From the second equation, how many dollar bills are there? [50] There cannot be both 15 and 50 dollar bills!
        • Can you find a value for x that will give the same answer in both cases?
        • How can you check that your answer is right?
        • Can you use these equations to calculate the amount of money in the cash register?

      If the whole class is struggling on the same issue, you may want to write a couple of questions on the board and organize a brief whole-class discussion. You could also ask students who performed well in the assessment to help struggling students.

  • Collaborative Analysis of Sample Student Work (15 minutes)
      • When all groups have made a reasonable attempt, ask them to put their work to one side.
      • Give each group copies of the Sample Student Work. This task will give students the opportunity to discuss and evaluate possible approaches to the task without providing a complete solution strategy.
      • Ideally, all groups will review all four pieces of work. However, if you are running out of time, choose just two solutions for all groups to analyze, using what you have learned during the lesson about what students find most difficult.
      • Encourage students to think more deeply using the following questions. (These are reproduced on the  projector resource, Assessing Sample Student Work.)
        • You are the teacher and have to assess this work.
        • Correct the work, and write comments on the accuracy and organization of each response.
          • What do you like about the work?
          • What method did the student use? Is it clear? Is it accurate? Is it efficient?
          • What errors did the student make?
          • How might the work be improved?
      • During this small group work, support the students as before. Also, check to see which of the explanations students find more difficult to understand. Note similarities and differences between the sample approaches and those the students used in group work.
  • Whole-Class Discussion: Comparing Different Approaches (15 minutes)
      • Hold a whole-class discussion to consider the different approaches used in the sample work. Focus the discussion on those parts of the task that the students found difficult.
      • Ask representatives from each group to explain and critique one student's method from the Sample Student Work. During the discussion you may find it helpful to use the projector resources, which show the different sample solutions.
        • Which approach did you like best? Why?
        • Which approach did you find most difficult to understand? Why?
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