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Solving Linear Equations in Two Variables
Before the
  • Assessment Task: Notebooks and Pens (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 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 Notebooks and Pens.
      • 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.
        • Show all your work, so that I can understand your reasoning.

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

      Students should not worry too much if they cannot understand or do everything because there will be a lesson using a similar 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. The purpose of doing 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. The research shows that 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 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, 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.

      Common Issues: Suggested Questions and Prompts:

      Student assumes that the letter stands for an object, not a number.

      • For example: The student says that the statements are correct.
      • Or: The student realizes the equations are incorrect, but is unable to explain why.
      • What does the letter p represent?
      • Write the equation as a sentence. Does your sentence match what Dan/Emma said?
      • If n = 3, what would p equal in the first equation? Which is greater: n or p?
      • Are there more notebooks than pens? How do you know?

      Student only uses one equation.

      • For example: The student finds a value or values for n and p that fits one equation but not the other, such as n = 1 and p = 4 for the first equation.
      • For this equation, is there another pair of values for n and p? And another? How do you know which value is correct?
      • How can you check that your values for n and p work for both equations?

      Student produces unsystematic guess and check work.

      • For example: The student works out three or four seemingly unconnected combinations of values for n and p.
      • What is a sensible value to try for n (or p)? Why?
      • Can you organize your work in a table?

      Student provides poor explanation.

      • For example: The student presents the work as a series of unexplained numbers and/or calculations.
      • Would someone unfamiliar with your type of solution easily understand your work?
      • Have you explained how you arrived at your answer?

      Student makes algebraic mistakes

      • For example: The student makes a mistake when manipulating the algebra in the equations.
      • How can you check that your answer is correct?

      Student solves the two equations correctly.

      • Student needs an extension task.
      • Can you now use a different method, for example, a table, a graph, or algebra?
      • Is this method better than your original one? Why?
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