Why do you need a common denominator to add fractions?

I can only think to explain this with an example: Consider that 1 foot = 12 inches. So each inch is 1/12 of a foot. If you want to add 2 feet and 3 inches, you would not come up with 2 + 3 = 5. You need to convert so they are some common unit, before adding. so 2 feet = 24 inches (or 24/12 feet), and 3 inches = 3/12 feet. Then you can add them and get 27/12 feet.

Another example. Suppose you have a Pizza that has been sliced into 12 slices, so each piece is 1/12 of the whole. As soon as you open the box, some friends come over and grab some slices, and you notice that 1/2 the pizza has been taken. You go ahead and take 2 slices (2/12). So the question could be how many pieces have been taken (addition), or how many pieces are left (subtraction).

Well if you had counted when your friends were grabbing slices, then you could just subtract the actual slices, so this is what we are doing with fractions. Everything must be in common. So the 1/2 that was taken first must be converted to be in common with the 2/12 that you took. So you could simplify your 2/12 to 1/6, then convert 1/2 to sixths = 3/6. 1/6 + 3/6 = 4/6, which simplifies to 2/3. And 1 - 2/3 = 3/3 -2/3 = 1/3. Then if you want to know how many pieces were taken: (2/3)*12 = 24/3 = 8 slices.