Changing Conceptions of Shapes: 4.25

Indicator of Progress
Teaching Strategies

Supporting materials

Indicator of Progress

Students classify quadrilaterals with reference to a definition or a key property.

Previously, they will have classified quadrilaterals and other shapes by noticing features.

This is the third phase in how geometric shapes are perceived. It is a substantial reconceptualisation, which will take place over several years, beginning from this level.

Illustration 1: Changing conceptions of shapes

Students' understanding of the names of geometric shapes passes through three phases.

Phase 1
Very young students recognise a shape from its appearance alone. For example, a triangle is seen as one iconic entity and may not even be known to have three sides. A shape is a triangle just because it looks like a triangle. There is a gradual transition away from this conception to the next.

Phase 2
In the next phase, students perceive features of the shape. For example, they will see that an isosceles triangle has two equal sides, two equal angles and a line of symmetry. A student will say this is an isosceles triangle because it has these features. This phase is typical of upper primary students.

Phase 3
At this level students are working towards the third phase where they classify shapes using a definition. Not all the properties are mentioned in the definition; other properties follow from the definition. In the third phase, they see an isosceles triangle as being defined to be any triangle with at least two equal sides. From this definition, it follows by logical deduction that there are at least two equal angles and at least one line of symmetry.

Illustration 2

Examples of the types of tasks that would be illustrative of classifying shapes and verbalising features aligned from the Mathematics Online Interview:

Question 55 (a - c) - Identifying triangles and explain features (all triangles)
Question 58 - Triads - use 'prototype' or 'properties' strategy with cards 1, 2 and 3

Teaching Strategies

These activities give students the opportunity to discuss the various features of shapes (in this case quadrilaterals) and then classify them according to definitions. This contrast is necessary to move them from a focus on what shapes look like and a list of their features (second phase above) to an appreciation of the need to use a definition (third phase above).

Activity 1: Classifying quadrilaterals according to features is a classification activity of quadrilaterals designed to highlight their features (length and orientation of sides and size of angles).
Activity 2: Classifying quadrilaterals according to definitions is a first activity for students to work strictly from a definition.

These are activities designed to introduce some preliminary ideas about mathematical definitions. Students will make observations from a set of examples. At this level, it is not expected that student prove that the features follow from the definitions.

Activity 1: Classifying quadrilaterals according to features

This activity uses the Quadrilaterals sheet (PDF - 25Kb). You might choose to copy the sheet for each student or laminate sheets for repeated use. You might need to explain the symbols for 'equal' or 'parallel'. The shapes are labelled A - X to facilitate discussion.

Students cut up the sheet so that they can arrange the quadrilaterals into groups according to various criteria suggested by the teacher or the students.

For example, on the basis of the marked properties, the quadrilaterals can be sorted according to:

number of equal sides
number of pairs of equal opposite sides
number of pairs of parallel sides
number of right angles

For example, the following Venn diagram shows the sets of quadrilaterals with 0, 1 and 2 pairs of parallel sides.

Venn diagram showing the set of quadrilaterals

The sorting activity should be followed up with discussion where students justify why they have placed the shapes in certain sets and subsets. Questions to discuss could include:

Can a quadrilateral have one angle that is greater than 180°? (Yes)
Can a quadrilateral have exactly three right angles? (No, as the angle sum is 360°, the fourth angle would also be 90°)
If the angles of a quadrilateral are all equal, are the sides all equal? (Not necessarily - many rectangles are like this)
Can a quadrilateral have two obtuse angles? (Yes - L, P, Q and R are examples)
Can a quadrilateral have four acute angles? (No)
Can a quadrilateral have three acute angles? (Yes)
If both pairs of opposite sides are parallel, are they also equal length? (At this stage, it is only intended that student observe this from the examples: they do not have to prove it)
If both pairs of opposite sides are equal length, are they also parallel? (Yes)

Activity 2: Classifying quadrilaterals according to definitions

Divide students into groups and allocate each group one of the definitions from the table below. The task for students is to:

select all the quadrilaterals from the Quadrilaterals sheet (PDF - 25Kb) that fit their definition
make a list of their common features
give their set of shapes a good name, and
display their set for the subsequent discussion

Definition 1	Is a quadrilateral with all angles 90°.
Definition 2	Is a quadrilateral with both pairs of opposite sides parallel.
Definition 3	Is a quadrilateral with all sides equal.
Definition 4	Is a rectangle with all sides equal.
Definition 5	Is a quadrilateral with one pair of opposite sides parallel.
Definition 6	Is a quadrilateral with two pairs of adjacent sides of equal length.
Definition 7	Is a quadrilateral with at least one acute angle.
Definition 8	Is a quadrilateral with equal diagonals.

Examples of important points for a discussion:

Definition 1 students might want to name their shapes as 'rectangles and squares'.Take this opportunity to point out that a square is a special rectangle.
There will be other examples of this; e.g. Definition 3 students may initially want to specially identify the squares in their set, not including them in the set of rhombuses.
Students may be surprised that Definition 3 quadrilaterals (chosen on equal sides) also have parallel sides. This highlights that not all the features of shapes have to be specified in a definition.
Definition 8 and Definition 1 result in the same sets, which highlights that there are alternative definitions.
Definition 7 results in the set of non-rectangles, the complement of Definition 8 and Definition 1 .

Using the formal geometric terms, students could now answer the following:

Is a square a rhombus? (Yes, it is a special rhombus)
Is a square a rectangle? (Yes, it is a special rectangle)
Every square is a rhombus, but is every rhombus a square? (No)
Every square is a rectangle, but is every rectangle a square? (No)

Answers:

Definition 1	Rectangle	Is a quadrilateral with all angles 90 degrees
Definition 2	Parallelogram	Is a quadrilateral with both pairs of opposite sides parallel.
Definition 3	Rhombus	Is a quadrilateral with all sides equal
Definition 4	Square	Is a rectangle with all sides equal
Definition 5	Trapezium	Is a quadrilateral with one pair of opposite sides parallel.
Definition 6	Kite	Is a quadrilateral with two pairs of adjacent sides of equal length
Definition 7	All but rectangles	Is a quadrilateral with at least one acute angle
Definition 8	Rectangles	Is a quadrilateral with equal diagonals

The above ideas could also be used with triangles.

Mathematics Developmental Continuum P-10