Math Background

Fractions and Mixed Numbers: Overview

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 one-third and one-fourth, 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  one-third = four-twelfths and one-fourth = three-twelfths. So we can add the number of twelfths, four-twelfths + three-twelfths, and get seven-twelfths.

diagram

Thus, one-third + one-fourth = seven-twelfths

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, Factors and Fractions.) Let's look at adding 2 five-sixths and 1three-fourths. 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.

Addition problems

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 two-fifths from two-thirds, 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.

diagram

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: 4one-third − 1three-fourths. First, we need to rewrite the fractions as equivalent fractions with the same denominator. 4one-third = 4four-twelfths and 1three-fourths = 1nine-twelfths.

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

subtraction problem

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 three-fourths x two-thirds is to think of finding three-fourths of two-thirds. This will help students understand why the answer is less than two-thirds. Let's look at a visual model of three-fourths x two-thirds.

visual model

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 nine-twentieths by four-fifteenths we get Nine over twenty times four over fifteen equals three times three over two times two times five equals two times two over three times five,
and we can cancel the common factors 2 x 2 in both the numerator and denominator and a common factor of 3 from both, leaving three over five times five equals three over twenty-five

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 x 23, they have to multiply the 3 x7 and then 3 x 40. Next, they multiply 20 x7 and 20 x 40 and they add all four products. Similarly, when multiplying 2two-thirdsx 3one-fourth, students would have to multiply the 2 x3 and 2 x one-fourth and then multiply two-thirdsx 3 and two-thirdsx one-fourth. Fortunately, it is much simpler to change both numbers to improper fractions first and then multiply the fractions.

fraction equation

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.

three items in four groups

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

diagram

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 three-fourths × two-thirds.

fraction equation

Since two-thirds is a little less than three-fourths, the answer 11/8 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 3one-fourth divided by 1two-fifths.

fraction equation

Houghton Mifflin Math Grade 6