Once again I've found myself totally lost in a math problem. So thanks for your suggestions and solutions ...
In a previous assignment I learned that two expressions , 2(X+1) and 2X + 2 are equivalent because they gave the same number no matter what was inputted for X. Now 3 imaginary people have explanations and here is what I need to decide:
1) Decide which are correct (if any) and have a simple explanation why they are correct.
2) Which explanation is easiest to understand?
3) Which explanation is most convincing?
4) Adapt the explanation you understand best to explain why the expressions 3(X + 4) and 3X + 12 are equivalent.
[ Explanation One]
"We all know that 2A is twice A which is A + A. Think of 2(X+1) as being twice X + 1. In other words, it is equal to (X+1) + (X+1). And (X+1) +(X+1) is equal to 2X + 2."
[ Explanation Two]
"It works with numbers! Check it out! If X is 5, then 2(X+1) is 2(5+1) which the the same as 2*6 which is 12. And, well, 2X + 2 is 2*5 + 2, which is the same as 10 + , which is also 12. Wow!"
[ Explanation Three ]
"Multiplication is how you find the area of a rectangle, like length times width. Basically a product 'ab' can be thought of as the area of a rectangle with dimensions 'a' and 'b' like this:
"The product 2(X+1) can represent the area of a rectangle that is 2 units in one dimension and X + 1 units on the other. The length of X + 1 is like a segment of length X next to a segment of length 1. The picture is something like this:
"A simple diving the line shows that this figure can be thought of as two rectangles, with areas 2X and 2, put together. Because we are talking about the same area, 2(X+1) must equal 2X + 2.