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Finding Equations of Parallel and Perpendicular Lines
Before the
  • Assessment Task: Parallel and Perpendicular Lines (15 minutes)
    • Have 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 each student a copy of the assessment task Parallel and Perpendicular Lines.
      • Introduce the task briefly and help the class to understand the problem and its context.
        • Read carefully through the questions, and try to answer them as carefully as you can.
      • It is important that, as far as possible, students are allowed to answer the questions without your assistance.

      Advise your students that they should not worry too much if they cannot understand or do everything, because there will be lesson using a similar task that will help them. Explain that their goal is to be able to answer questions such as these by the end of the next lesson.

  • 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 the difficulties students will experience 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 will encourage students to compare their scores, and will 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 your own list of questions, based on your own 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, you could write a few questions that will help the majority of students. These can then be displayed on the board at the end of the lesson.

      Common Issues: Suggested Questions and Prompts:

      Student does not link the properties of a rectangle with slopes of lines.

      • For example: The student does not mention in Q1 that a rectangle has two pairs of parallel sides and that these pairs are perpendicular or connect this with the slope of the lines forming the sides.
      • What do you know about the sides of rectangles?
      • How is the property of being parallel [perpendicular] connected with slope?
      • Why is looking for slopes of pairs of parallel and perpendicular lines relevant?

      Student demonstrates limited understanding of the link between the slope and the form of the equation of a straight line.

      • For example: The student identifies slopes for equations in which y is given explicitly in terms of x, y = mx + b, but not for other equations.
      • Or: The student reads the number in front of x as if it were the slope in all equations.
      • How do you work out the slope of a line?
      • How do you work out if two lines are parallel from their slopes? Perpendicular?
      • Some equations are written with y isolated and others aren't. How does this affect how you calculate the slope?

      Student reasoning is insufficient.

      • For example: In Q1 the student does not explain how looking for parallel and perpendicular lines relates to the task of finding the sides of the rectangle.
      • Why is looking for slopes of pairs of parallel and perpendicular lines relevant?
      • How do you know that...? Explain your reasoning.

      Student does not identify relevant information from the equation.

      • For example: The student fails to identify the x- and/or y-intercepts in Q2.
      • What else can you figure out from the equations of the lines?
      • Where does a pair of lines intersect?
      • Where does a line intersect the x-axis?
      • Where does a line intersect the y-axis?
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