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. You reached 1 for an answer, didn’t you? 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 and post it for your classmates to solve. Be sure to think about values that may not work. State whether your number game uses the skill of simplifying rational expressions.

Follow the rules:

(n^2 - 1)/(n-1) - n = ?

start by factoring n^2-1.

The number game you described involves a series of mathematical operations that ultimately leads to the answer of 1. Let's go through it step by step using a variable instead of an actual number.

1. Start with any number (let's call it x).
2. Square that number, meaning multiplying it by itself: x^2.
3. Subtract 1 from the squared number: x^2 - 1.
4. Divide the result by one less than your original number (x - 1): (x^2 - 1) / (x - 1).
5. Finally, subtract your original number: (x^2 - 1) / (x - 1) - x.

Now, let's see how this series of operations leads to the answer of 1. If we simplify the expression (x^2 - 1) / (x - 1) - x, we get:

(x^2 - 1) / (x - 1) - x
= [(x + 1)(x - 1)] / (x - 1) - x
= (x + 1) - x
= 1.

As you can see, the result simplifies to 1 regardless of the value of x (as long as x is not equal to 1).

This number game involves the skill of simplifying rational expressions because we had to simplify the expression (x^2 - 1) / (x - 1) in order to reach the final answer of 1.

Now, I will create my own number game using the rules of algebra:

1. Start with any variable (let's call it y).
2. Square the variable: y^2.
3. Add 2 times the variable: y^2 + 2y.
4. Multiply the result by the variable itself: y(y^2 + 2y).
5. Finally, subtract 3 times the original variable: y(y^2 + 2y) - 3y.

The challenge for your classmates is to simplify the expression y(y^2 + 2y) - 3y as much as possible. This number game also involves the skill of simplifying rational expressions.