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Finding Equations of Parallel and Perpendicular Lines
During the
  • Interactive Introduction (5-10 minutes)
      • These introductory questions are designed to give you a greater insight into students' misconceptions and are not intended to form a whole-class discussion.

        The questioning of students should be fast paced and should act as a means of stimulating students to think about the equations of parallel and perpendicular lines.

        • Today, we're going to be looking at slopes and the equations of parallel and perpendicular lines.
        • I am going to ask you a series of questions. I would like you to write your answers on your whiteboards.
        • You may wish to draw a diagram if that helps.
      • Give each student a mini-whiteboard, pen, and eraser.
        • What is the slope of the line joining the points (2,5) and (7,15)?
          • [2]
        • Show me the coordinates of two points for which the slope of the line between them is 3.
          • [E.g. (1,4) and (3,10)]
        • Give me an equation of a line with a slope of 3.
          • [E.g. y = 3x + 1]
        • Keeping your equation on your whiteboard, write it in a different way.
          • [E.g. y − 3x = 1]
        • What is the y-intercept of the line y = 3x + 7?
          • [(0, 7)]
        • What is the x-intercept of the line 2y = 3x − 6?
          • [(2, 0)]
        • Show me the equation of a line that is parallel to y = 2x + 4.
          • [E.g. y = 2x − 6]
        • What would be the slope of a line perpendicular to y = 2x + 4?
          • [−½]
        • Show me the equations of two lines that are perpendicular to each other.
          • [E.g. y = 2x + 4 and y = −½x + 4]

      During this activity, you will be able to identify common misconceptions within the group; this will help you to target groups of students effectively during the collaborative group work.

  • Collaborative Group Work: Matching Task (20 minutes)
      • Ask students to work in pairs or groups of three.
      • Give each small group a copy of the Card Set: Equations (already cut up) and two copies of the Properties sheet.
      • Introduce the activity:
        • Find two Equations cards to match each of the Properties.
        • You may want to spend some time thinking about the equations first.
        • It might be helpful to figure out some extra information and write it on the cards.
        • Once you have found two Equations cards that match a Property, explain to your partners how you came to your decision.
        • If your partners agree, they should explain their reasoning in their own words.
        • If they disagree, they should explain why they think you are incorrect.
        • In your group you need to be able to agree on and explain the placement of every card.

      The purpose of this structured group work is to make students engage with each other's explanations and take responsibility for each other's understanding.

      • You have two tasks during the small-group work: to note aspects of the task that students find difficult, and to support students' reasoning.

      Note aspects of the task that students find difficult.

      For example, students may begin by working with the mathematics they understand best, and get stuck on later categorizations.

      You can use information about particular difficulties to focus whole-class discussions towards the end of the lesson.

      Support students' reasoning.

      Try not to make suggestions that move students towards a particular categorization. Instead, ask questions to help students to reason together.

        • How can you determine the slope for any equation? Is there a form of the equation that makes this easier?
        • How do you find the x-intercept from a written equation?
        • If you cannot place all the cards, you may need to rethink some of your categorizations.
      • If you find one student has produced a solution for a particular card or match, challenge the other students in the group to provide an explanation.
        • Jenny matched these cards parallel to each other. Jonathon, why does Jenny think these lines are parallel?
      • If several students in the class are struggling with the same issue, you could write a relevant question on the board.
      • A few minutes before the end of the activity, ask one student from each group to write the equations they have succeeded in categorizing onto one copy of the Properties sheet.
      • Once they have done this, ask each small group of students to join with another group, taking with them their Properties sheet with equations written on them.
      • It may be advisable to group students who have displayed a similar level of competence on the task, thus allowing for a richer discussion.
      • Give each group a glue stick.
        • In your new groups, decide whether or not you agree with each other's answers.
        • If your new partners disagree with your answer, explain your reasoning to them, and let them explain why they disagree.
        • Once you are all comfortable with your answers and can explain your reasoning, glue the Equation cards into place on the blank copy of the properties sheet.
      • If students finish early, you may want to ask them to invent an additional equation for each category, using the blank cards.
      • Alternatively, they could create a new heading for the empty box on the Properties sheet, and write a pair of possible equations for their own choice of category.
  • Whole-Class Discussion (15 minutes)
      • Organize a whole-class discussion.
      • First spend about five minutes considering categorizations and any discrepancies found among the groups.
        • After you changed groups, did your new partners disagree with any of your answers? Give me an example of an equation or pair of equations on which you disagreed. What was the misconception?
        • Can you explain your reasoning for the final categorization?
        • Did any other group have different reasoning or categorize those equations under a different heading?
      • Then spend about ten minutes generalizing the mathematics students worked on during the lesson.
      • Display slide P-1, Lines and Rectangles. Tell students that the line segment SP has equation y = 2x + 3.
        • In your groups, I want you to decide on a possible equation for each of the line segments PQ, QR, and RS.
        • Once you are all in agreement, write your equations on your whiteboards and show them to me.
      • Give students a couple of minutes to come up with a possible solution.
      • Then select an equation from each group and discuss whether or not it is a correct possible solution, focusing on why this is the case.
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