Power and the Unit

Power and Units

A wonderful benefit of emphasizing power is that it also facilitates important conversations regarding the unit. How many teachers have been faced with this question: Why do we need common denominators when I add and subtract but not when we multiply?  What is the answer? Just because? No, the answer is that multiplication is a more powerful operation and deriving a common denominator is built into the algorithm for this more powerful operation.

Addition and subtraction are the weakest operations.  Note that addition and subtraction are merely counting up things that already have a common unit. The algorithms for addition and subtraction will not generate the needed common unit size – a common unit size is presumed for these weak operations.  The algorithms therefore also must presume common units. So the mathematician has to do everything by hand including finding the common denominator or unit BEFORE they engage with this operation or algorithm.

Multiplication goes ‘up a notch’ in power and is not merely unit added to unit, but is groups of units. This more complex operation does not demand like units throughout and in fact the algorithm clumps units into groups.  So the algorithm is likewise more powerful as it reflects a step up from merely counting up like units.

Note that, when I multiply ½ by ¾, we create a common unit when we multiply the denominators as part of the operation.    (Go ahead and try it – think through what is actually happening when I ‘multiply the denominators).

The greater power of multiplication also makes multiplying numbers written in decimal form less cumbersome than adding or subtracting them. With addition and subtraction I need to carefully keep track of the unit size of each digit. Lining up the decimal point is not actually required, but it facilitates this keeping track of my like units.  With multiplication, however, the mathematician is freed up by this more powerful operation and need only consider the placement of the decimal point once all calculations have been exacted. This mathematical logic supports the teacher in making an important connection for the student:  When we add decimals we need to keep track of the units because we are merely counting up like units.  When we multiply with decimals we are clumping groups of units so we can consider what size clumps we ended up with after we perform our calculations.

With division the explanation is slightly different but equally powerful:  When we divide by any number other than 1 we are effectively changing the unit.   Try this:  What does it mean to divide by ½?  Watch these three videos to better understand the unique power of division and its impact on units. 

Division is the ultimate in proportional reasoning!  While we can think of division as repeated subtraction it is, indeed,  much more powerful than merely rapidly counting down, as in subtraction.

Different operations have a different impact on the unit because of their power. Keeping track of units is critical in developing and keeping track of meaning as we use mathematics to model situations.