This post concerns using a pair of number lines to illustrate the successor identities. At lower grade levels, we can simply use a pair of number lines to “prove” the successor identities. We actually make them plausible, but at an introductory stage they can seem like a proof.
Right Zero Identity
x+0 = x
Right Successor Identity
x+y’ = (x+y)’
Here ‘ means successor. So 0′ = 1. 2′ = 3.
In number line terms, we have a first number line and the x is the position on the first number line of the first addend.
The y is the position on the second number line.
We take the 0 of the second number line and put it over the x on the first number line. Then x+0 means look under the 0 on the second number line. The result is x on the first number line. This is the zero identity. It is the right zero identity because 0 is on the right. This corresponds to the second number line.
Now we go to y on the second number line. The value under it on the first number line is x+y. This is the sum at the “hypothesis step” of induction. This mixes definition by induction and proof by induction, but we will ignore that here.
Now we go to y’ on the second number line. We look down below and we get a value x+y’. If we go to the x+y on the first number line under the y on the second number line, we can go one over to the right on the first number line. This is (x+y)’. So we get to the same point (x+y)’ in two different ways. One is by going to y’ on the second number line and down to the first number line. The second is go one over on the first number line from x+y to (x+y)’. These are the same point, so they are equal.
x+y’ = (x+y)’
This is “proof” by number line of the two right identities. We can call the right successor identity the right shift identity as well.
These identities can be taught at lower grade levels. They can be taught with numerical cases only not using letters. But they lead naturally into using letters for the numbers on each number line. This is a good way to get students used to thinking of letters for numbers concretely.
The x is the first addend on the first number line. The y is the second addend on the second number line. They have a concrete tangible meaning. So does x+y. so does y’. And so does (x+y)’. The prime notation also is concrete in the number line context. This lesson plan helps teach a lot of abstract math of both arithmetic and algebra using number lines.