Take any number (except for 1). Square that number and then subtract one. Divide by one less than your original number.

Now subtract your original number. Did you reached 1 for an answer? You should have. How does this number game work? Hint: Redo the number game using a variable instead of an actual number and rewrite the problem as one rational expression. How did the number game use the skill of simplifying rational expressions?

Create your own number game using the rules of algebra. State whether your number game uses the skill of simplifying rational expressions.

OK, look at the sequence of operations. Let x be the number, x≠1.

1) x→x^{2}-1

2) (x^{2}-1)/(x-1)

3) (x^{2}-1)/(x-1)-x

Let us simplify the expression in 3).

(x^{2}-1)/(x-1)-x=(x+1)(x-1)/(x-1)-x=x+1-x=1

We used the fact that x^{2}-1=(x+1)(x-1) to simplify the expression and prove that you will always get 1 no matter what x you take (except x=1, since you can't divide by zero!).

If you subtract the squares of two consecutive numbers, you will end up with their sum as a result. Example: 111^{2}-110^{2}=12321-12100=221; 110+111=221.