Math Background

Lesson: Adding and Subtracting Fractions and Mixed Numbers
Introducing the Concept

Your students will need help learning the standard algorithms for adding and subtracting fractions. Relating the operations for fractions to operations with whole numbers will aid in this process as will providing students a visual model for the operations.

Materials: fraction strips with halves, thirds, fourths, sixths, and twelfths

Preparation: none

Prerequisite Skills and Background: Students should recognize that five-ninths means a unit has been divided into 9 equal parts and there are five one-ninth parts. That is, the denominator of a fraction tells you the size of the parts you have and the numerator tells you how many of those equal parts you have. Students should also know that addition can be modeled as the joining of sets and that subtraction can be modeled as the taking away of sets.

  • Ask: Who can tell me what three-fourths means?
    Students should offer various interpretations for three-fourths, such as the ratio of 3 items to 4 items, the division of 3 by 4, or 3 of 4 equal parts. It is this last interpretation that we want to stress in this lesson. We want students to think of three-fourths as three equal one-fourth parts.
  • Say: That's right; a fraction can mean different things. Today, however, I'd like us to think of fractions in this way: The denominator tells us how many equal parts a unit has been divided into and the numerator tells us how many of those parts we have.
  • Ask: Who can show us how to add 3 and 4?
    A student will probably suggest taking something like 3 counters and adding them to 4 more counters by joining the two sets. Discuss with the class what the student did in terms of joining one set to the other in order to add 3 + 4.
  • Say: So if addition is joining two sets of objects, and a fraction like three-fourths tells us how many one-fourth parts we have, who can tell me what the sum of five-ninths and two-ninths is?
    Students will say that the sum is seven-ninths, because taking five one-ninth parts and joining them with two more one-ninth parts will make seven one-ninth parts, or seven-ninths.
  • Say: We can add fractions that have the same denominators by simply adding, or joining, the numerators. Try the following problems at your desk: three-fifths + four-fifths and two-fourths + three-fourths.
    Have students volunteer to explain what they did. For example, when adding three-fifths + four-fifths, the students will get seven-fifths, which can be simplified to 1two-fifths. Similarly, two-fourths + three-fourths = five-fourths, or 1one-fourth.
  • Ask: Who can tell me how to subtract 5 from 8? (Students should say that you have 8 things and take 5 of them away.) Good. Keeping this and what we know about fractions in mind, let's look at how we would solve eight-ninthsfive-ninths.
    On the board, show the class that since you have eight one-ninth parts and you want to take away five one-ninth parts, you are left with three one-ninth parts, or three-ninths, which can be simplified to one-third by dividing both the numerator and denominator by 3. You might want to remind students about finding equivalent fractions, which they learned earlier.
  • Say: At your desks, do the following subtraction problems: three-fourthsone-fourth and five-eighthsthree-eighths.
    After students have had time to solve the two problems, have a volunteer come to the board and show how to do the problems. Make sure students simplify the answers to one-half and one-fourth.
  • Say: The process of adding mixed numbers when the fractions have common denominators is basically the same idea, except we can add the whole-number parts and then the fractional parts to find the answer. For example, if we want to add 1two-sevenths + 3four-sevenths, we could first add the whole-number parts 1 + 3 = 4 and then the fractional parts two-sevenths + four-sevenths = six-sevenths. Thus, 1two-sevenths + 3four-sevenths = 4six-sevenths.
  • Say: Now find the sum of 2three-fourths + 5two-fourths.
    After students have had a chance to solve the problem on their own, have a volunteer come to the board and explain how to do the problem. (7five-fourths, or 8one-fourth)
  • Ask: Who can summarize for us how to add fractions or mixed numbers with like denominators?
    Students should say that to add fractions with like denominators, all you need to do is add the numerators and simplify, and that to add mixed numbers with the same denominators for the fractions, simply add the whole-number parts and then add the fractions by adding the numerators and simplifying when possible.
  • Say: The process of subtracting mixed numbers when the fractions have common denominators is basically the same idea as subtracting fractions with a common denominator. We can subtract the whole-number parts and then subtract the fractional parts to find the answer. However, we may have to do some regrouping in order to subtract the fractional parts. Let's look at the following problem: 4five-sixths − 2one-sixth.
  • Say: We could first subtract one-sixth from five-sixths and get four-sixths, which can be simplified to two-thirds. We next subtract 2 from 4 to get 2, and our final answer is 2two-thirds.
  • Say: Let's try another problem together.
    Write this problem on the board: 4two-fifths − 1three-fifths.
  • Say: This one will involve some regrouping before we can subtract, since three-fifths > two-fifths.
    Now 4two-fifths = 3 + 1 + two-fifths = 3 + five-fifths + two-fifths = 3 + seven-fifths.
    Therefore, 4two-fifths − 1three-fifths becomes 3seven-fifths − 1three-fifths = 3 − 1 + seven-fifthsthree-fifths = 2 + four-fifths, or 2four-fifths.
    Write this on the board as you discuss it with the class.
  • Say: Now try these two problems at your desks and I'll walk around the room and help those who need it: 2two-thirds − 1one-third and 3two-sevenths − 1five-sevenths. Discuss with the class the answers 1one-third and 1four-sevenths and see if there are any questions.
  • Say: We have more work to do when adding or subtracting fractions or mixed numbers if the denominators are not the same. Since the parts are not the same size, we cannot just add or subtract the numerators. Instead we must first find equivalent fractions that have the same denominators. Please take out your fraction strips for thirds, fourths, sixths, and twelfths. Let's look at adding two-thirds + one-fourth.
  • Ask: How many twelfths are equivalent to two-thirds? (8) How many twelfths are equivalent to one-fourth? (3) Therefore, two-thirds + one-fourth can be written as eight-twelfths + three-twelfths and eight-twelfths + three-twelfths = eleven-twelfths. We can simply add the numerators, since the two fractions now have the same denominator. Use your fraction strips to help you add one-sixth + three-fourths.
    Give students some time to work on this at their desks, and then ask someone to do the problem on the board and explain his or her work.
  • Say: Now let's try a subtraction problem. On the board, write the problem 3one-half − 1two-thirds.
  • Ask: How can we find equivalent fractions? How many sixths are equivalent to one-half? (3) How many sixths are equivalent to two-thirds? (4)
    Write the following on the board: 3one-half − 1two-thirds = 3three-sixths − 1four-sixths.
  • Ask: What do we need to do next? Why?
    Students should say that you need to rewrite 3three-sixths as 2nine-sixths so that you can subtract the fractional parts. Ask students to finish the problem at their desks. (1five-sixths)

Houghton Mifflin Math Grade 6