This is a presentation of a visual approach to fraction arithmetic, using an area model on grid paper. It need not replace other approaches: it can complement them, or precede them, or follow them. I do not propose this as a way to get students to be quick and accurate in their calculations. Rather, it is an attempt to have them develop some understanding through reflection and discussion.
Addition and Subtraction
Let's say we want to add 2/3 and 1/5. We will use grid paper to make it easier to visualize what is going on. If one unit was defined as a 3 by 5 rectangle, we could easily represent either fraction in it:
But now we see that the sum has to be 13/15. The same figure helps us see that the difference has to be 7/15.
The key is to have students decide on the appropriate rectangle for a given sum or difference. What is it about a 3 by 5 rectangle that made it useful for this situation?
For a problem like 2/3 + 1/6, using a 3 by 6 rectangle would work, but it is more elegant (or saves paper, for the conservation-minded) to use a 1 by 6 rectangle, which will work for both fractions. For a given addition or subtraction, what is the best choice of a rectangle to represent one unit?
For multiplication, once again, we choose the rectangle with the most convenient dimensions.
Let's say we'd like to multiply 2/3 by 1/5. This time, we will represent these fractions one-dimensionally using line segments on the sides of the rectangle. Again, a 3 by 5 rectangle will come in handy:
The product is 2/15. This method would still work for fractions greater than 1. Here is 4/3 times 1/5:
It is of course important to remember that each rectangle is one unit, so that the answer here is 4/15.
Unsurprisingly, division is more difficult. Our strategy will be to observe what happens when using the area model to divide by smaller and smaller numbers. We will arrange the divisions in a manner similar to the layout for long division. For example, 6 divided by 2 equals 3 would look like this, with 6 as the area of the rectangle, the divisor 2 as the height, and the quotient 3 as the width:
All the rectangles on the rest of this page have area 6. These represent 6 divided by 6, 3, 2, and 1:
As the divisor gets smaller, the quotient gets bigger. (In fact, when we divide the divisor by n, we multiply the quotient by n.) Continuing that process, we represent the division of 6 by 1, ½, and ⅓:
In order for the area to still be 6, we see that when we divide 6 by 1/n, the quotient must be 6 times n. To divide by a unit fraction, multiply by its reciprocal.
But what if we are dividing by, say, ⅔? Since we are multiplying the divisor by 2 (as compared to 6 divided by ⅓,) we have to divide the quotient by 2:
So to divide 6 by m/n, we multiplied by n, and divided by m, which amounts to multiplying by n/m.
In short: to divide by a fraction, multiply by its reciprocal.