## Fractions and Mixed Numbers: When Students Ask

• Why do I need like denominators to add and subtract fractions and mixed numbers?
Relate the need for finding common denominators to the idea of adding whole numbers. Students know they have to add ones to ones, tens to tens, hundreds to hundreds, and so on. For example, to find the sum of 25 + 43, you add numbers that have common factors.

The mathematical property that is being used here is the Distributive Property. The Distributive Property states ab + ac = a(b + c). The left side of this equation states that when two numbers have a common factor, in this case a, you can factor that number out and add what is left inside, which is the right side of the equation. In the case of fractions with like denominators, we have:

As you can see, to go from the second step to the third step you factor out and add 3 and 2 to get x 5, or .

• Why is the product of two fractions less than one a number less than either fraction?
There are many ways to help students understand this. One way is to have them think of x as of . Thus, you are finding of , so the answer will be less than . Another way to understand this is to think of as 1 ÷ 3, and when you divide by a whole number the answer is always less than the dividend. Another way would be to have students observe the patterns in the multiplication shown below.

They can see that when they multiplied by a number less than one, the product was a number less than the other factor.

• Why is it that when you divide a whole number by a whole number, the quotient is smaller than the dividend, but when you divide a whole number by a number less than one, the quotient is larger than the dividend?
Show students the following pattern to dividing by smaller and smaller numbers.

For the pattern to continue, the quotient must be twice 8. One way to see this clearly is to read the final division problem as “How many halves in 8?” (16) Continue with 8 ÷ . (32)