If we are thinking of numbers as corresponding to geometric lengths (as measured in terms of some chosen unit), then after defining addition by placing lengths end to end, it is natural to ask what is the geometric definition of multiplication – or in other words, how to construct the line of length ab from lines of length a and b. If a is a whole number this can just be done by repeated addition (ie putting a copies of b end to end), and for fractional numbers of form a=n/m by first cutting b into m equal pieces and then adding together n copies of one such piece, but we are looking for a geometrical interpretation that works even when a is not rational.
The first thing most people think of when asked for a geometric example of multiplication is the area formula for a rectangle.
Indeed it is clear from arranging tiles or blocks (as we probably all did as children) that a rectangle of sides m units and n units can be exactly filled by m rows of n squares and so that the total area is mn square units – at least when m and n are whole numbers. And putting the rows together end to end gives us a total length of mn linear units, which fits in with the notion of multiplication as repeated addition (ie starting with 0 and successively adding copies of the multiplicand to the running total).
The following worksheet shows how this whole-number case might be introduced to students:
Worksheet #1 – the case of whole number factors
[basically same as existing Example&Exercise#1&2 except that in Example#2 I would explicitly draw the 1×12 rectangle and not refer to counting, and in Exercise#2 I might go with 5(2) and 2(5) – (to show how the model supports commutativity without taking too much space and stamina)]
Most people don’t find it surprising that the result applies more generally and that the area of a rectangle of length a units and width b units is given by ab square units even when a and b are not whole numbers. For the case where a and b are rational it is not hard to demonstrate that the usual definition of the product (in terms of dividing into parts according to the denominator and then adding together a number of such groups according to the numerator) does give the area, and conversely that if we use the width of the matching rectangle of height one as the definition of the product then that gives the same rule as the more common definition. The student can also see how, in the case of rational numbers a and b, to rearrange the rectangle of height a and width b into a rectangle of height 1 and width ab. (We will address the question of how to deal with the case when a and b are not rational after the worksheet on rationals.)
Worksheet#2 – working with fractional factors
[basically same as Eg&Ex#3..6 except that I’d include the business of converting to a rectangle of height 1 and use less algebra and maybe have the students do some of it by physically cutting and rearranging.]
Thus we can convince students that the idea that ab is the width of a rectangle of height one whose area is the same as that of the rectangle of height a and width b certainly agrees with the dividing and repeated adding approach when a is rational. But in order to extend that idea to the general case when a and b are not rational we need a way of constructing the rectangle of height one with the same area as the axb one without breaking the original up into little rectangles. How can we do this?
Worksheet#3 – the general case
[First introduce the idea of shear preserving area, then use it to construct the 1xab rectangle from the axb one (by leaving the corner 1xb triangles undistorted and shearing the middle parallelogram)]
But at this point you may have noticed that the length of the unit height rectangle whose area matched the one of height a units and width b units was just given by drawing the line from the top left corner parallel to the one at height a on the left down to the bottom right corner. If the number of units in this width is our definition of the product of the numbers a and b then maybe we never needed areas in order to find it. Our next module will look at the idea of defining multiplication directly in terms of parallel lines rather than by using areas.