Addition and Subtraction of Fractions Part 3 – Unrelated Fractions

Addition and subtraction of unrelated fractions refers to adding and subtracting fractions that have different denominators that are not multiples of each other (please refer to our previous posts on like fractions and related fractions). In dealing with unrelated fractions, these are some of the problems we observed:

  • Students learn tricks like the “butterfly method“, without any strong conceptual understanding.
  • Teachers teach conceptual understanding, but are undermined by parents’ intervention at home who teach their kids to use these tricks.
  • Teachers do not teach these tricks in classroom, but have no idea how to show the operations pictorially to help the students gain conceptual understanding.

In this blog post, we discuss the pedagogical sequence teachers can take in order for students to have a good conceptual understanding of the topic.

Same-sized equal parts

The fundamental concept in dealing with addition and subtraction of unrelated fractions, is to ensure that students rewrite the fractions involved as same-sized equal parts, before adding or subtracting. For example,

1/4 + 1/3 = 3/12 + 4/12

This requires the student to find a number that both denominators 3 and 4, go into (or a number that is both divisible by 3 and 4).

Students should also understand that the common denominator they are seeking need not be the product of the original denominators. For example, we like to have students work out the same problem using two different denominators:

Method 1: 1/6 + 1/4 = 2/12 + 3/12 = 5/12

Method 2: 1/6 + 1/4 = 4/24 + 6/24 = 10/24 = 5/12

Then ask the students why both methods give the same answers (because 4/24 and 2/12 are equivalent, etc).

Instruction Sequence

Sometimes, different types of fractions (unit fractions, non-unit fractions, mixed numbers) are presented without any logical order or plan of instruction. This creates confusion in the kids as they are suddenly overwhelmed by the different fractions and the “methods” required to handle them.

Here is our recommended sequence of instruction and example bar models for each step, that will help build conceptual understanding in students sequentially, minimizing their confusion and anxiety.

1. Adding unit fractions, sum less than 1

E.g. 1/6 + 1/4

example_unit_fractions

2. Adding non-unit fractions, sum less than 1

E.g. 2/3 + 1/4

Screenshot 2016-06-30 14.51.10

3. Adding non-unit fractions, sum more than 1

E.g. 1/2 + 2/3

Screenshot 2016-06-30 14.51.34

4. Subtracting unit fractions

E.g. 1/2 – 1/3

Screenshot 2016-06-30 14.51.43

5. Subtracting non-unit fractions

E.g. 3/4 – 2/3

Screenshot 2016-06-30 14.52.06

6. Mixed numbers without regroup

E.g. 3 1/3 + 2 1/4

Sometimes students like to write all fractions in improper form before adding/subtracting, e.g.

3 1/3 + 2 1/4 = 10/3 + 9/4 = 40/12 + 27/12 = 67/12 = 5 7/12.

This is unnecessary and introduce more chance for errors. Instead, the easier way is to do:

Screenshot 2016-06-30 14.52.15

7. Mixed numbers with regroup

E.g. 2 1/4 – 1 1/3

Screenshot 2016-06-30 15.19.52

8. Adding and Subtracting three or more fractions

E.g. 1/4 + 1/3 +1/5

Screenshot 2016-06-30 14.52.48

This is an excellent example to counter the argument that the use of tricks such as the butterfly method is the way students should learn addition and subtraction of unrelated fractions. These method generally do not work when the number of fractions to add or subtract is more than two. See our previous post on why we don’t like to teach the use of the butterfly method.

Conclusion

Addition and subtraction of unrelated fractions is arguably one of the harder concepts to master. This is exacerbated when different types of fractions (unit fractions, non-unit fractions, mixed numbers) are presented without any logical order or plan of instruction. This creates confusion in the kids as they are suddenly overwhelmed by the different fractions and the “methods” required to handle them. Instead, we can introduce the topics through progressively more challenging examples to build a strong foundation in the students’ learning of the topic.


This article is part of a series of blog posts on Fractions:


More Fraction Resources

For more fraction resources, refer to our main fractions page.

4 thoughts on “Addition and Subtraction of Fractions Part 3 – Unrelated Fractions”

  1. Thanks for denouncing the butterfly method!! How do “we” get teachers to teach for understanding and not just the right answer?

  2. Thank you so much for sharing this post on how to add add subtract fractions. You have explained it in a very detailed manner. I have been searching reading a lot of ways to add and subtract fractions and encountered only two ways. First is you used the LCM. This is best for smaller fractions. The other is to use butterfly method where you have to cross multiply the fractions. This method is best for bigger fractions and those that has no LCM. Again thanks for sharing for this valuable information.

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