•Post a response to the following: 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 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.
(x^2-1)/(x-1)
(x-1)(x+1)/(x-1)
x+1
now subtract one :)
NOTE: we could not use x = 1 because that would give (x-1) = 0 in the denominator, undefined
Yes, I did reach the number 1 by following the example given. To see how this game works basically I needed to translate it into an algebraic expression (x^2-1)/(x-1)-x the expression needed to be simplified, factored using the difference of squares, I then cancel like terms, simplify, group like terms, and then combine those like terms.
To understand how the number game works, let's go through the steps:
1. Take any number (let's call it x).
2. Square that number: x^2.
3. Subtract 1: x^2 - 1.
4. Divide by one less than your original number: (x^2 - 1)/(x - 1).
5. Subtract your original number: (x^2 - 1)/(x - 1) - x.
The objective of the number game is to see if the final result always equals 1. If we simplify the expression (x^2 - 1)/(x - 1) - x, we can understand why it leads to the answer of 1.
To simplify the rational expression, we'll first find the common denominator. The denominator of (x^2 - 1)/(x - 1) is (x - 1), which has the same factors as (x - 1) - x. By multiplying (x - 1) - x by -1, we will get a common denominator:
(x^2 - 1)/(x - 1) - x = (x^2 - 1)/(x - 1) - (x(x - 1))/(x - 1)
Next, we can combine the fractions by subtracting the numerators:
(x^2 - 1 - x(x - 1))/(x - 1)
Simplifying further:
(x^2 - 1 - x^2 + x)/(x - 1) = (x - 1)/(x - 1) = 1
From this simplification, we can conclude that no matter what value is chosen for x (except for 1), the result of the number game will always be 1. This happens because the expression simplifies to 1.
Now, let's create a new number game using the rules of algebra. Let's call our variable "y".
1. Take any value for y (except for 0).
2. Cube that value: y^3.
3. Add 2y: y^3 + 2y.
4. Multiply by one less than your original value: (y^3 + 2y)(y - 1).
5. Subtract your original value: (y^3 + 2y)(y - 1) - y.
To solve this new number game, students would simplify the expression and see if it always leads to a specific result. In this case, the skill of simplifying rational expressions is not directly used because we are dealing with polynomial expressions. However, simplifying the expression by expanding and combining like terms will still be required.