Central principles

Transitivity

The mathematical relationship ‘is a multiple of’ applied to numbers possesses what is called the transitive property. This means that if A is a multiple of B and B is a multiple of C then it follows that A is a multiple of C. This is illustrated in general terms in the figure below.

How does transitivity apply to measurement?

The principle of transitivity is shown in the diagram above. If we know that A is related to B (indicated by an arrow) and B is related to C, the question is whether A is related to C as a logical consequence. With some relations (such as ‘is a factor of’) it does follow logically and we can draw in the arrow connecting A to C. .

We can now see that whenever we compare and order three or more objects (or events) using a measuring attribute such as their lengths, their masses, their capacities or the length of time (for events), then we are again making use of a transitive relationship.

The arrow used in the Figure above could represent any one of the measuring relationships used to compare two objects or events, such as: ‘is longer than’, ‘is lighter than’, ‘holds more than’ or ‘takes less time than’. In each case, because A is related to B and B is related to C, then it follows logically that A is related to C. This principle is fundamental to ordering a set of more than two objects or events: once we know A is greater than B and B is greater than C, for example, it is this principle which allows us not to have to check A against C. Grasping this is a significant step in the development of a pupil’s understanding of measuring.

The transitive property of measurement can be expressed formally using inequality signs as follows:

If A > B and B > C then A > C.

If A < B and B < C then A < C.