Attempting to understand fractions as numbers, and that arithmetical operations on fractions follow naturally from arithmetical ops on whole numbers, we started to familiarize ourselves with arithmetic on the number line. Picking up from where I left off in this previous post, we remind ourselves that the critical key is not to be counting the Fence Posts, but the jumps. We could explain 7 - 4 by jumps, rather than counting the hash marks on the number line. Once that is crystal clear in our minds, we can then abstract away from the jumps, and think instead of motion on the number line as continuous motions, where we moved along the number line progressively:
This idea is a kind of intermediate step. For now, let's think of this motion to facilitate us thinking about the number of Unit Segments we've moved. That is, instead of thinking of addition in terms of jumps, we can think of addition as a continuous motion whose number of Unit Segments corresponds to the number. What's a Unit Segment? It's the length of the segment between 0 and 1.
We add 4 and 3 by starting at Zero and moving 4 units to the right. Then we move 3 units to the right. We get 7. Graphically, we see that on the number line, we reached 7 by moving right for 7 segments. So 7 is made up of 7 unit segments, just as it was reached by 7 jumps.
Subtraction, then, means moving the correct number of unit segments to the left. 7 - 3 means moving 7 units to the right, and then moving 3 units to the left. The number we reach, 4, is made up of 4 unit segments, just as it is 4 jumps from zero.
Another way to look at this is that the number 7 can be represented EITHER by the position of the point on the number line, OR by the length of the segment from 0 to 7. They are equivalent representations (isomorphisms, actually.) (It's not clear to me that this is the most child-friendly way to think about it. Someone with 4th graders needs to find out if it's useful to present this to children--I think it's easier to think of the jumps being of the unit length for now, and we'll adapt the lengths of the jumps.) Wu puts this as "we identify this standard representation of the number with its right endpoint" of the segment that starts at Zero. But if this is too hard, the jumps still work: the number is represented as the location on the number line that we end up at after we've completed all jumps.
We are now ready for fractions. No, really, we are! We begin with a specific fraction, and we work up to a generalization.
The fraction 1/3 has a numerator and a denominator. We will show how it is a fraction of our unit segment, the segment from 0 to 1. We now partition our unit segment into thirds. 1/3 is then symbolized by starting at 0 and moving to the right one partition of one segment:
The position of the right endpoint of that segment is the location of 1/3 on the number line.
But just as we could partition the unit segment into thirds, we can partition all of the segments between consecutive integers into thirds. The hash marks of these partitions are the fractions whose denominators equal 3. Again, this is saying "we identify this standard representation of the number with its right endpoint." These thirds function as our new "units", and we could make jumps on them just as before. The place we land on our jump is the number.
So, now that we know how to imagine the number line in thirds, we can generalize:
The fraction m/n is the point on the number line, when we partition each unit into equal nths, and then make m jumps to the right of Zero on those new nth-hash marks, or equivalently, partition each unit into n equal segments, and then move m nth-segments to the right.
There you have it--a definition of a fraction! Now, an important point glossed over in these examples is that a point on the number line is unique. Its representation, however, can be varied. This is worth stressing: a number, be it whole or fraction, corresponds to the unique location on the number line (or, as Wu would put it, to the unique line segment whose right endpoint ends at that location.)
With whole numbers, most people don't think that's very interesting. But with fractions, there are many representations for the same number. This can be confusing. The best way to clarify it is to say: they are still the same number-it's just a different way of reading the same number. This leads to a clear understanding of equivalent fractions.
Consider the fraction 4/3. we will show that (5 x 4)/(5 x 3) = 4/3 as follows: First, locate 4/3 by its unique spot on the number line. We do this by breaking each unit into thirds, and then jumping 4 such 3rd-units to the right:
Now we partition each 3rd-unit into 5ths. Doing so immediately gives us 15-ths for each unit. Now we count how many 15ths our point 4/3 is: the answer is 20. We could do this for any partition--7ths, 162nds, etc. It doesn't matter. No matter how you partition into equal bits, you haven't moved off of your point on the number line. Nothing about repartitioning changed your location.
Similarly, improper fractions are a breeze: we can see that 10/3 is 10 jumps to the right on the 3rd-hash-marks. It's also 3 jumps on the Unit Hash marks and 1 jump on the thirds: 3 and 1/3. Since the same point on the number line can be read in either fashion, they must be the same number, because a number is defined by its unique point on the number line.
We'll pick up with common denominators next.