Counter-Examples - Progression Points

Dimension

Level

Progression Point

Working Mathematically

1.5

• Formulation and testing of conjectures using models that involve, for example, objects, patterns, shapes and numbers

2.0 Standard

...Students make and test simple conjectures by finding examples, counter-examples and special cases and informally decide whether a conjecture is likely to be true.

2.75

• Use of materials and models to illustrate and test generalisations

3.0 Standard

… Students test the truth of mathematical statements and generalisations. For example, in:

• number (which shapes can be easily used to show fractions)
• computations (whether products will be odd or even, the patterns of remainders from division)
• number patterns (the patterns of ones digits of multiples, terminating or repeating decimals resulting from division)
• shape properties (which shapes have symmetry, which solids can be stacked)
• transformations (the effects of slides, reflections and turns on a shape)
• measurement (the relationship between size and capacity of a container).

3.25

• Search for counter-examples in an attempt to disprove a conjecture

3.5

• Development and testing of conjectures with the aid of a calculator; for example, divisibility tests

4.0 Standard

… Students develop and test conjectures.

They understand that a few successful examples are not sufficient proof and recognise that a single counter-example is sufficient to invalidate a conjecture. For example, in:

• number (all numbers can be shown as a rectangular array)
• computations (multiplication leads to a larger number)
• number patterns ( the next number in the sequence 2, 4, 6 … must be 8)
• shape properties (all parallelograms are rectangles)
• chance (a six is harder to roll on die than a one).

4.25

• Use of technology to extend their own ability to make and test conjectures