We explain steps in great detail to students but often omit the underlying concept. The topic of adding or subtracting fractions with unlike denominators is an example of this.
The example above right is a short cut for what is shown above left. These short cuts, which math teachers love to use, add to the student’s confusion because these rules require the student to use rote memorization which does is not readily retained in the brain.
I suggest using what I call a meaning making approach. I present the student 2 slices of pizza (images courtesy of Pizza Fractions Game) and explain the following setting. “You and I both paid for pizza and this (below) is what we have left. You can have the pizza slice on the left and I will have the pizza slice on the right. Is that OK?” The student intuitively understands that it is not because the slices are different sizes. I then explain that when we add fractions we are adding pizza slices so the slices need to be the same.
I then cut the half slice into fourths and explain that all the slices are the same size so we can now add them. Then the multiplying the top and bottom by 2 makes more sense.