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Evaluating Statements about Length and Area
During the
  • Interactive Whole-Class Introduction (20 minutes)
    • This activity is best introduced with a whole-class discussion, in which you model the reasoning required for the lesson's activities.

      • Give each student a piece of plain paper.
      • Write this statement on the board or use Slide P-1 of the projector resource.
        • Is this statement always, sometimes, or never true?
        • If you think the statement is always true or never true, then how would you convince someone else?
        • If you think the statement is sometimes true, would you be able to identify all the cases of a quadrilateral where it is true/not true?
      • Allow students a few minutes to think about this as individuals.

      It is important that they understand the problem and spend some time thinking about what would need to be included in their solutions.

      • After a few minutes of working individually, give students a couple of minutes to discuss their initial ideas in pairs.
      • Bring the class together and discuss the statement.
        • What kinds of quadrilaterals could we look at?

      Students may not have considered all possible quadrilaterals or may have misconceptions about some of the properties of quadrilaterals, so this could form part of your discussions.

        • Can anyone suggest a type of quadrilateral for which the result is true?
        • Can anyone suggest a type of quadrilateral for which the result is false?

      Students may suggest squares, rectangles, or parallelograms as quadrilaterals for which the result is true; and kites, trapezoids, and so on, as quadrilaterals for which the result is false. They are likely to have come to their conclusions by calculating the areas for a few different-sized quadrilaterals. It is important that students move beyond this and look at the general case. They should be encouraged to justify their conjectures.

        • OK, it seems like this result works for some quadrilaterals but not for others. So it is sometimes true. How do we show that it will always work/not work for a specific quadrilateral?
        • For example, in a rectangle, how could you convince me that the four triangles will always have the same area?

      Students may use a method similar to this one:

      To show that the two pairs of triangles have the same area, we use the area formula ½ x base x height:

      Area of triangle A = ½ × a × ½b = ¼ab

      Area of triangle B = ½ × b × ½a = ¼ab

      We can, therefore, conclude that this statement is true for all rectangles.

      • You may now want to ask:
        • This method has assumed that the two diagonals bisect each other. How do we know this? Show me. [Students could then prove that two opposite triangles are congruent (using Angle, Side, Angle).]

      Throughout this lesson, it is important that students recognize that there are different methods which can be used to solve a problem. During the whole-class discussion, they will consider which methods they prefer as well as spending some time in the next activity evaluating different approaches to a problem.

  • Individual Task: Student Work: Diagonals of a Quadrilateral (10 minutes)
      • Give each student a copy of the sheet, Student Work 2: Diagonals of a Quadrilateral containing sample work from two students.
      • The sample solutions allow students to evaluate two different strategies used to justify or disprove the statement.
        • I want you to assess this work. Write comments on the sheet.
      • To encourage students to write more than simple comments, such as "great work" or "incorrect," ask students questions like these.
        • What do you like about the work?
        • Has this student made any incorrect assumptions?
        • Is the work accurate?
        • How can the explanation be improved?

      The sample work focuses on justifying/disproving the conjecture for two types of quadrilaterals; kites and parallelograms. The aim of this activity is to encourage students to consider what makes a good explanation, as well as addressing some common misconceptions concerning length and area.

  • Whole-Class Discussion: Student Work: Diagonals of a Quadrilateral (10 minutes)
      • Once students have assessed the sample work, discuss the different approaches used.
      • You may find it helpful to display the projector resources P-2, Student Work 1 and P-3, Student Work 2.

      Student Work 1

      In this example, the areas of the two triangles A and B have been calculated correctly. The work would benefit from more explanation.

      The student has concluded the statement is never true after considering only one specific example. This is incorrect; for example, the statement is true for a rhombus.

      This strategy can help students get started on the task, and it can be useful to show that a given statement is sometimes true.

        • When is this method useful? [It shows the statement is sometimes true.]
        • Show me a kite that makes the statement false.

      Students often mistakenly think this method can be used to show a statement is always true and always false.

      Student Work 2

      In this example, the student has indicated lines of equal length. Some justification for this is needed.

      The student has made the incorrect assumption that the diagonals are always equal. The student has then incorrectly concluded that all the small triangles are congruent.

        • What mistake has this student made?
        • Is the statement always, sometimes, or never true? Show me.

      In this case, the student's explanation is inaccurate.

      In the parallelogram below, all the white triangles are congruent, and all the grey triangles are congruent.

      This congruency can be proved using Side, Side, Side (SSS) or Angle, Side, Angle (ASA).

      The diagonals of the larger parallelogram bisect each other. This means the small grey quadrilateral is a parallelogram. Opposite sides of the grey parallelogram are equal.

      Angle a = Angle b and Angle c = Angle d (the diagonal of the parallelogram cuts two parallel lines.) The two grey triangles have one common side.

      Therefore, the four triangles consisting of one grey and one white triangle are all equal in area. Hence, the statement is true.

      This example shows how students can add construction lines in order to prove or disprove a statement.

  • Working in Pairs (30 minutes)
      • Ask students to work in pairs or small groups.
      • Give each pair Card Set A: Always, Sometimes, or Never True?, a large sheet of paper for making a poster, and a glue stick.
        • You are going to choose one statement to work on. Decide if it is always, sometimes, or never true.
        • First, you will need to make sure that you fully understand the problem. It may help to try a few examples to begin with before justifying or disproving the conjectures.
        • If the conjecture is sometimes true, you will need to explain in which cases it is true/false.
      • Students who you think may struggle should be guided towards statement Card A.
      • If students are successful in classifying their chosen statement, they should be encouraged to consider alternative methods before moving on to a second statement card as appropriate.
      • Card F may provide more of a challenge for students who are progressing well.

      You have two tasks during the small-group work: to note different student approaches to the task and to support student reasoning.

      Note different student approaches to the task.

      Listen and watch students carefully. Note different student approaches to the task and any common mistakes. You can then use this information to focus a whole-class discussion towards the end of the lesson.

      Support student reasoning.

      Try to avoid explaining things to students. Instead, encourage them to explain to one another and to you.

      If you find one student has produced a solution for a particular statement, challenge another student in the group to provide an explanation.

        • John thinks this statement is sometimes true. Sharon, why do you believe John thinks this?

      If you find students have difficulty articulating their solutions, the cards from Card Set B: Some Hints can be used to support your own questioning of students or can be handed out to students who are struggling.

      If the whole class is struggling on the same issue, you may want to write a couple of questions on the board and hold an interim, whole-class discussion.

  • Whole-Class Discussion (20 minutes)
      • There will not be time to go through every statement card (and it is likely that some cards will not have been attempted by any of the student pairs), but a discussion about the chosen category for at least one of the statement cards is essential, focusing on reasoning and the different methods of justification employed by the students.
      • It may be helpful to do a quick survey of the statement cards chosen by students and select one which has been attempted by more than one group.
      • For the chosen statement, ask each group that has attempted it to describe to the rest of the class their method for categorizing the statement as always, sometimes, or never true.
      • Then ask other students their views on which reasoning method is easiest to follow, as well as contributing ideas of alternative approaches. It is important that students consider a variety of methods and begin to develop a repertoire of approaches.
        • How else could we explain our reasoning for categorizing this statement as always/sometimes/never true?
        • Which explanation did you prefer? Why?
        • Is their explanation sufficient? How could we improve it?
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