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Materials Required

Estimated Time Needed

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

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Modeling Situations With Linear Equations
Before the
Lesson
  • Assessment Task: The Guitar Class (15 minutes)
    • Have the students do this 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 to find out the kinds of difficulties students have with it. Then you will be able to target your help more effectively in the follow-up lesson.

      • Give out The Guitar Class. Introduce the task briefly, and help the class to understand the problem and its context.
        • Read through the questions, and try to answer them as carefully as you can.
        • What does "profit" mean?
        • Don't worry too much if you can't understand and do everything.
        • I will teach a lesson with a task like this [tomorrow].
        • By the end of that lesson, your goal is to answer the questions with confidence.

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

  • Assessing Students' Responses
    • Collect students' responses to the task. Make some notes on what their work reveals about their current levels of understanding. The purpose of this is to forewarn you of issues that will arise during the lesson itself, so that you may prepare carefully.

      We suggest that you do not score students' work. Research suggests this will be counterproductive, as it encourages students to compare their scores and distracts 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 the table below. 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 that follow. 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 end of the lesson.


      Choosing the lesson task

      After writing your list of questions, use your assessment of students' current understanding to decide which task to use during the lesson.

      We have found that many students learn from the Making and Selling Candles task. However, if the majority of your students have answered most of the assessment task questions correctly, set the Rescue Helicopter task instead.

      Making and Selling Candles is a more structured task than Rescue Helicopter, so makes less demand on students' problem solving skills.


      Common Issues: Suggested Questions and Prompts:

      Student writes no equation or just a few numbers and letters (Q1).

      • What do you know from the question?
      • Suppose you have 10 students. How do you figure out how much profit the teacher would make?

      Student writes an equation with a particular value of n (Q1).

      • For example: The student substitutes n = 30.
      • Does the calculation method change as you vary n?
      • How can you write the calculation for any value of n?

      Student uses incorrect operation in equation (Q1).

      • For example: The student divides the cost by the number of students, and adds rather than subtracts c to get p = 70n + c.
      • How much money does each student pay? How much money do the students pay altogether?
      • Is the amount of money you have calculated before paying costs more or less than the profit?

      Student draws incorrect graph (Q2).

      • For example: The graph has negative slope, or is not a straight line.
      • What do you think happens to the amount of profit as the number of students increases?
      • What kind of function links n and p?
      • What graph would you expect from this equation?

      Student does not explain or misinterprets the significance of the x-intercept (Q2).

      • For example: The student does not link the answer back to the context. She just writes p = 5.5, and does not mention that this is the point at which the teacher begins to make a profit.
      • What does n stand for? Does your answer make sense?
      • Reread the first part of the sheet. What does p > 0 mean?

      Student uses incorrect operations in formulas in Q3, Q4.

      • For example: The student writes p = fn + c or f = pn − c.
      • What does "profit" mean?
      • What does f stand for?
      • Explain how you would calculate the profit in words.

      Student answers Q1-5 correctly.

      • Suppose you can make 100 candles from a kit costing $70. Use the profit you would expect to make to calculate the selling price. Graph this relationship.
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