Multiplication Metaphors 1

There are 5 metaphors for Multiplication as illustrated below. 

Join Several Equal Quantities – also called Repeated Addition

It is a common life experience to see an item arranged in groups of equal number. In such situations, multiplication is the short cut (by using multiplication facts) to find the number of items.

If in a market, fruits are arranged in collections of 6 and there are 8 such collections, the multiplication fact 6X8 gives the total number of fruits.

Repeated Addition needs to be qualified as Repeated Addition of Equal Quantities. I also feel that “join several equal quantities” is a better name for the idea since “repeated addition” is just a process.

There are 2 visualisations of 6 X 8. Apart from the above, it can also be visualised as 6 collections of 8 fruits each.

Calling a multiplication situation as "groups of equal groups" is a better visualisation of the metaphor.

Array

It is again a common life experience to see things arranged in an array. Examples are seats in an auditorium or formations for PT in school.

If an array has 6 rows and 8 columns (or 6 rows of s things in each row), the multiplication fact 6 X 8 gives the total number of things in that array.

An array is a concrete example of why 6 X 8 is the same as 8 X 6. In both cases, we are seeing the same array but from different perspectives. The perspective does not affect the total members in the array.

Area/ Volume

An array representation can be modified and viewed as an Area Model, where the product of the length & width of a rectangle gives its area. The advantage of an area model is that it can accommodate numbers other than whole numbers.

It can also be extended to a Volume model which involves multiplication of 3 numbers. A stack of cubic bricks arranged in a pattern of 3 (breadth) by 4 (length) by 5 (height) can be seen as containing 3 X 4 X 5 = 60 bricks. Like the Area model the Volume model also can handle numbers other than whole numbers.

Area Model Modified

The Area Model can be modified as a "Graphic Organizer" for multiplying algebraic expressions, which may contain "negative" terms.

The principle of visualising the "distributive property" is same but since it can involve negative quantities it cannot used to represent areas. We can now call it a graphic organizer which is an abstract model.

The Area model also develops into the operation of Integration in Calculus. In principle, integration divides the area under any curve into an infinite number of rectangles and finds the sum of these infinite areas.