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Sorting Equations and Identities
During the
Lesson
  • Whole-Class Introduction (10 minutes)
      • Use slide P-1 of the projector resource in this introduction.
      • Give each student a mini-whiteboard, a pen, and an eraser.

      • Write the following equation on the board:

        (x + 2)(y + 2) = xy + 4

        • Is this equation "always true," "never true," or "sometimes true"? [Write "always," "never," or "sometimes" on your whiteboard.]
      • Typically, most students will begin by saying this is never true.

        • Can you show me values for x and y that make the equation false?
        • Can you show me values for x and y that make the equation true?
      • Hold a discussion about the students' responses, asking students to provide values for x and y to support their response.
        • Can the values of x and y be the same number? Can you figure out one?
      • This misconception needs to be explicitly addressed. Some students may assume that because x and y are different letters, they should take different values.

        Students may spot that the equation is true when x = y = 0.

      • If students are struggling to find any values of x and y for which the equation is true, drawing an area diagram may be helpful (Slide P-1)

        • For these two area diagrams to be equal, what are the values of x and y?
      • For the area diagrams to be equal, 2y must equal 0, and 2x must equal 0. This is true when x and y are both equal to 0.

      • When students are comfortable that when x = y = 0 the equation is true, ask them to summarize their findings.
        • We have found values of x and y that make the equation false and values of x and y that make the equation true. Is the equation always, sometimes, or never true? [Equation is sometimes true.]
      • Next, ask students:
        • Are x = 0 and y = 0 the only values that make the equation true? How could we find out?
      • Using an algebraic approach here might be helpful, as we are unable to describe a negative area. The following method may be appropriate:
      • (x + 2)(y + 2) = xy + 2x + 2y + 4

        • We want to know when this is the same as xy + 4, which must be when 2x + 2y = 0, i.e. when x + y = 0 or when x = −y. We can, therefore, conclude that the equation is true when x = −y.
      • Now write this equation on the board:

        (x + 2)(x − 2) = x2 + 4

        • How about this equation? Is it "always true," "never true," or "sometimes true"?
      • Students will probably find value for x for which the equation is false.

      • After a discussion of a couple of these examples, encourage students to justify their conclusions.
        • Give me a value of x that will make the equation false/true. And another. [There are no values of x that will make the equation true.]
        • Do you think the equation is never true? Convince me. [Students should simplify the left side of the equation to x2 − 4. The equation is never true because 4 does not equal −4.]
      • After a few minutes, ask one or two students to explain their answers. Encourage other students to challenge their reasoning. In this activity, students use the term identity.
        • If an equation is always true, we say it is an "identity."
      • Teachers may be accustomed to varying uses of the term "identity." While this is not the main focus of this activity, for the purpose of the lesson, the term "identity" is used to describe equations that are always true.

  • Collaborative Activity: Always, Sometimes, or Never True? (30 minutes)
      • Ask students to work in groups of two or three.
      • Give each group Card Set A: Always, Sometimes, or Never True?, a large sheet of paper, a marker pen, and a glue stick.
      • Ask students to divide their large sheet of paper into three columns and head respective columns with the words: Always True, Sometimes True, and Never True.
      • You may want to use slide P-2 of the projector resource to display the following instructions:
        • You are now going to consider whether the equations on your desk are Always, Sometimes, or Never True.
        • In your groups, take turns to place a card in a column and justify your answer to your partner.
        • If you think the equation is sometimes true, you will need to find values of x for which it is true and values of x for which it is not true.
        • If you think the equation is always true or never true, you will need to explain how we can be sure that this is the case. Remember, showing it is true, or never true, for just a few values is not sufficient.
        • Another member of the group should then either explain their reasoning again in his or her own words or challenge the reasons you gave.
        • It is important that everyone in the group understands the categorization of each card.
        • When the entire group agrees, glue the card onto your poster. Write the reason for your choice of category next to the card.
        • It does not matter if you do not manage to place all of the cards. It is more important that everyone in the group understands the categorization of each card.

      The purpose of this structured work is to encourage each student to engage with their partner's explanations and to take responsibility for their partner's understanding.

      While students work in small groups, you have two tasks: to make note of student approaches to the task, and to support student reasoning.

      Make a note of student approaches to the task.

      Listen and watch students carefully. In particular, listen to see whether students are addressing the difficulties they experienced in the assessment. You can use this information to focus the whole-class discussion toward the end of the lesson.

      Support student reasoning.

      Use the questions in the Common Issues table to help address misconceptions.

      Encourage students to explain their reasoning carefully.

        • You have shown the statement is true for this specific value of x. Now convince me it is always true for every number!
        • Can you use algebra to justify your decision for this card?
        • Can you draw a diagram to explain your categorization for this card?
        • (Card 8) Can you sketch a graph to show why x2 = 2x has only two solutions?
        • (Card 9) Draw an area diagram to show that (x + 3)2means something different from x2 + 32.
        • (Card 11) Can you draw an area diagram to show why (3x)2 is always equal to 9x2?

      If some students try to solve the equations by algebraic manipulation, they may notice that while sometimes this gives them possible solutions, sometimes they just get 0 = 0. These are, of course, the identities. Equations that have no solutions give absurdities, such as 1 = 2.

      • If students finish the task quickly, ask them to create new examples.
        • Can you make up an identity? And another one?
        • Can you make up an equation that has two solutions?
        • Can you make up an equation that has no solutions and shows a common algebraic mistake? [e.g., 3(x + 4) = 3x + 4]

  • Whole-Class Discussion (20 minutes)
      • Organize a whole-class discussion about different methods of justification used for two or three equations.
      • Ask each group to choose an equation from their poster that meets some given criteria. For example:
        • Show me an equation that has no solutions.
        • Show me an equation that has just one solution. Write this solution on your mini-whiteboard.
        • Show me an equation that has two solutions. What are they?
        • Show me an equation that has an infinite number of solutions.
        • Show me an identity.
      • You may find that numerous different equations are displayed in response to a given criterion. If more than one group shows the same equation, ask each of these groups of students to give a justification of their thinking.
      • Then ask other students to contribute ideas of alternative approaches and their views on which reasoning method was easier to follow.
      • It is important that students consider a variety of methods, and begin to develop a repertoire of approaches.

        • Why did you put this equation in this column? How else can you explain that decision?
        • Can anyone improve this explanation?
        • Which explanation do you prefer? Why?
      • Draw out issues you have noticed as students worked on the activity.

      Make specific reference to the misconceptions you noticed during the collaborative activity.

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