Fractions and Mixed Numbers

Working with fractions can be difficult for students, because they may have a hard time relating their understanding of operations with whole numbers to operations with fractions. Students have learned that addition can be thought of as joining two or more groups, or sets, to form a new set. They have also learned to add units of like quantities, such as tens to tens, ones to ones, and so on. Addition of fractions should be an extension of these concepts. So, when students are trying to add and , they will learn that they need to add "like" quantities. To do this, they learn that they need to rewrite the fractions with common denominators or "parts equal in size" when adding or subtracting fractions. Note in the diagram below that   = and = . So we can add the number of twelfths, + , and get .

Thus, + =

Understanding the need to use common denominators will help students remember to look for them when adding fractions or mixed numbers. After students have a visual concept for adding fractions, the next step is to have them practice adding fractions by using an algorithm. Remind students that in order to add fractions, they need to find a common denominator (a common multiple for the denominators). Some students will find it easier to add fractions if they use any common denominator. Other students may find it easier to use the least common denominator (by finding the least common multiple of the denominators). (See Grade 6, Chapter 3.) Let's look at adding 2 and 1. We can add the whole-number parts and then add the fractions. Both ways to do this are shown below. One shows using 24, the product of 6 and 4, as the common denominator. The other shows using the least common denominator, 12. Note that 12 is the LCM of 6 and 4.

The idea of "taking away" is one way of thinking about subtraction with whole numbers. When students subtracted whole numbers, they took away things that were alike. Students subtracted tens from tens and ones from ones. Similarly, when they want to subtract from , they need to find equivalent fractions with a common denominator, as shown below, and then subtract or take away quantities that are alike—in this case, fifteenths.

Once students have a good understanding of this visual model for the subtraction of fractions, they are ready to use an algorithm to subtract fractions. If the fractions to be subtracted do not already have a common denominator, then students should rewrite the given fractions as equivalent fractions with a common denominator as they did when adding fractions with unlike denominators.

When subtracting mixed numbers, it may be necessary to regroup in order to subtract the fractional parts. Relate this concept to the need to regroup when subtracting whole numbers, as when subtracting 28 from 72. Let's look at the following subtraction problem involving mixed numbers: 4 – 1. First, we need to rewrite the fractions as equivalent fractions with the same denominator. 4 = 4 and 1 = 1.

In this case, the LCD is the same as the common denominator found by multiplying the two denominators. Because you can't subtract from , you need to rename or regroup 4 as 3 so that we can easily subtract the two numbers.

Multiplication of fractions is probably the easiest of the fraction operations to perform. However, not many students have a good visual model for multiplying fractions. One way to think of multiplying is to think of finding of . This will help students understand why the answer is less than . Let's look at a visual model of .

Thus, the number of parts that are double shaded is 6, which is the product of the two numerators. The number of equal pieces is 12, which is the product of the two denominators. When having students multiply two fractions, it is a good idea to have them cancel any common factors before multiplying the two fractions, by writing each number in prime factorization form. This may be easier than having to simplify the fractions after multiplying. For example, when multiplying by we get ,
and we can cancel the common factors 2 2 in both the numerator and denominator and a common factor of 3 from both, leaving

A mistake that students often make when multiplying mixed numbers is to multiply the whole numbers and multiply the fractions and then add them. This results from a misuse of the Distributive Property. Remind students that when they multiply 47 23, they have to multiply the 3 7 and then 3 40. Next, they multiply 20 7 and 20 40 and they add all four products. Similarly, when multiplying 2 3, students would have to multiply the 2 3 and 2 and then multiply 3 and . Fortunately, it is much simpler to change both numbers to improper fractions first and then multiply the fractions.

The algorithm for the division of fractions is relatively easy to learn. In order to help students understand it, relate it to the division of whole numbers. In 12 ÷ 4, we see how many of 12 items can go in 4 groups or sets. We can see from the diagram below that there are 3 items in each of 4 groups. Thus, 12 ÷ 4 = 3.

Similarly, to solve ÷ , we want to count how many sixths are in . In the diagram below, we see that there are 4 sixths in .

Providing students with a visual model to show the division of two fractions will help them better estimate the answer to a division problem and help them recognize the reasonableness of an answer. The procedure for dividing two fractions is relatively simple. You need only find the reciprocal of the divisor and multiply it by the dividend. Let's look at ÷ .

Since is a little less than , the answer 1 seems reasonable.
To divide two mixed numbers, students should first change the mixed numbers to improper fractions and then follow the same procedure for dividing two fractions as described above, namely multiply the dividend by the reciprocal of the divisor. Let's look at 3 divided by 1.