After several posts where the entire point was to highlight situations where the whole is more than the sum of the parts, I thought it would be interesting to bring in a contrast. In this post, we will look at situations where it is really arbitrary what we call the whole.
In each row, we have a collection of blue squares: the same collection of blue squares in each row. You probably have already figured out how many blue squares are in the top row. If you counted 8 blue squares, your count matches mine. So far, that was pretty simple, right?
What you and I did with the top row – and usually we do so without giving it much explicit thought – is that we picked a unit and then counted, and expressed the count in terms of the unit. We picked a single blue square as our unit, and counted eight of those.
The middle row suggests a different unit, a different whole. Imagine the blue squares are high-end candies, and the yellow indicates a wrapper: the package in which the candies are sold. Two candies go in the package. The package is now our unit, the whole. How many units are there in the middle row?
In the bottom row, the yellow indicates a wrapper that holds four of the candies. The four-candy package is now our unit, the whole. How many units are there in the bottom row?
What we did here has a close association with the operation of division. You may have answered the question of how many units there were using division, or you may have used multiplication, or you may have used repeated subtraction. Each of these ways of thinking are justifiable. However, I’m going to use the vocabulary and notation of division for this.
Each row of the above table show a particular way in which we divide up the candies in packages. The first, second and fourth row correspond to the top, middle and bottom row of the picture above. For example, the second row indicates that 8 candies, when packaged two candies in a package, make 8/2 units, which we can see is exactly 4 units. Though normally we consider “4” the answer and “8/2” the problem, I invite you to look at “8/2” as a way to describe a situation. You could say that “8/2” is a notation for describing parts and whole. In this view, “8/2” says there are 8 parts, and that 2 parts make a whole.
We showed yellow wrappers as a visual aid, but we can practice seeing wholes even without any visual aids. In the middle row we only showed a wrapper for the first two candies, but you could easily imagine three more packages of 2 candies each. Similarly, we could imagine a unit of 5 candies, and “see” the 5 candies that make up a whole, and “see” the second whole for which we only have 3 candies, with two candies missing. We’d say there are “8/5” wholes, or equivalently, we could say there is 1 whole and an additional “3/5” whole.
What I’m primarily interested here, more than the mechanics of dealing with the division notation or the fractions, is the possibility of a shift of perspective, in which we can look at the same top row of 8 candies, and go back and forth between different conceptions of what the whole is. Can you look at the top row of 8 candies, and see a whole of 10 candies? Can you look at the same row and see a whole of 4 candies? Can you look at that same row and see a whole of half-a-candy? We’re talking strictly about different ways of seeing. We’re not doing anything in particular, just seeing.
To practice the ability to see different wholes, let’s look at the top row of 8 candies and attempt to see that row as 24 wholes. Could you do that? How many candies did you imagine there being in a package for there to be 24 packages?