Additional Materials
Materials Required
Estimated Time Needed
(Times are approximate and will depend on the needs of the students.)
This activity is best introduced with a whole-class discussion, in which you model the reasoning required for the lesson's activities.
It is important that they understand the problem and spend some time thinking about what would need to be included in their solutions.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.