On our way to fractions: the number line

Prof. Wu's document Critical Concepts for Understanding Fractions tries to clarify operations on fractions by use of the number line. Wu stresses that fractions should be defined succinctly as numbers (rather than as pieces of wholes, or operations, etc.) To show that what students already know about whole number guides them to understand what is true of fractions, he shows the relationship between arithemetic on the number line for whole numbers and arithmetic on the number line for fractions.

That means we need to already have mastery of the number line for whole numbers in order to make fractions clear. That mastery means fluency showing addition, subtraction, multiplication and division with remainder on the number line, as well as flency with properties of commutativity, associativity and distribution. The number line should reinforce all of what we already know about whole number arithmetic. It should not be new--we should use our mastery of whole number arithmetic to arrive at mastery of the number line, and then we'll use mastery of the number line for whole numbers to achieve understanding of fractions.

So let's begin:
The standard number line for non-negative numbers is a ray, beginning at Zero and extending to infinity. We show a piece of it here:

By convention, this ray extends to the right. The non negative integers, or whole numbers, are shown at the has marks. The space between 0 and 1 is a fixed length. This length is the same length as between 1 and 2, and any two consecutive numbers. We don't care what the length is, but we do like to recognize that it represents a UNIT. Later, this will matter more, but for now, just keep clear that the actual distance between the 0 and 1 is arbitrary; the issue is that that spacing is consistent for each consecutive number.

The biggest confusion with the number line is that it starts at Zero. This is probably odd for kids because we don't hold up our first finger and say "Zero!" But here, the First Tick Mark is the Zero. So we have to learn to go from counting on our fingers, where we sort of assume "zero means nothing", including no place/position, to an abstraction where Zero exists.

On the number line, Zero is our starting point. This is actually how we counted on our fingers, but we didn't think about it very often.

Now, we use Zero, the place, on our number line as our starting point, and count by keeping track of MOTIONS, or JUMPS from the starting point to the next tick mark. We don't count the fence posts; we count the motions from fence post to fence post.

Practicing with the number line until it's ALWAYS clear that there's 1 more fence post than motions between fence posts is critical.

Addition, then, looks like this: jumps to the right. (Again, this is convention.)
We add 4 and 3 by starting at Zero and making 4 jumps to the right. The first jump takes us to 1, then fourth to 4.

Then we make 3 more jumps to the right. The first of these takes us to 5. The last takes us to 7. We end up at 7.


Commutativity of addition should be obvious now: whether you made 4 jumps to the right and then 3 to the right, or 3 to the right and 4 to the right, you always ended up at 7. Associativity should be just as clear: 3 + (4 + 5) is the same as (3 + 4) + 5 for the same reason.

Subtraction is then defined as well. Subtracting a number from another number, you jump to the left (by convention.) 7 - 3: We start at zero, and take 7 jumps to the right to arrive at 7. Then to subtract 3 FROM 7, we make 3 jumps to the left. We end at 4.

Multiplication then is just a set of grouping of jumps. 4 X 10 is 4 Tens. That means, you learn to make a set of ten jumps, and then you call that grouping something like "a ten jump". And now you make a total, from 0, of 4 of these grouped jumps.

(Exercise: How do you do division with remainder? Answer: For X divided by Y, you start at zero and make grouped jumps of length Y. When you reach X, you stop. You count the number of grouped jumps you were able to complete, and the remainder is the number of singleton jumps left to reach X.)

The critical key no matter what the operations is to remember not to be counting the Fence Posts, or the hash marks between the numbers (inclusive or exclusively), just the jumps.

If the jumps are clear, then the next thing to realize was that you could have visualized those motions between numbers not as jumps but as continuous motions, where we moved along the number line progressively:


This is the simplest way to connect it back to fractions, because when we look at fractions, we now will consider the space between the hash marks.