# Mathematical Cogwheels

This is a sub-page of our page on Interactive Learning Objects.

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Experiment here with the mathematical cogwheels. You will be able to verify that:

• The number of pairs that are output in each complete run of the wheels
is precisely lcm(m, n) = the least common multiple of the numbers m and n.

• The number of start positions of the left wheel
in order to generate all possible combinations (= pairs) of the two numbers
is precisely gcf(m, n) = the greatest common factor of the numbers m and n.

Here is a pictorial description of these facts.

The Mathematical cogwheels provide a way to experience the ‘action’ (= effect) of these important concepts ( lcm and gcf) without having to talk about them explicitly. This direct cognitive experience of the actions of the concepts involved makes it possible to give the students a chance to describe the resulting structures as they experience them in a first encounter, without introducing the definitions of the underlying concepts in advance.

Hence it becomes possible to engage the students in the crucially important activity of mathematics design, by encouraging them to represent the structures that they experience in a way that is as effective and efficient as possible.

This represents a form of “mathematical didactics” that is based on:

providing cognitive contact with the underlying mathematical structures, and
representing these structures in an increasingly effective and efficient manner.

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