We talked a little bit about equivalent fractions in the Week 2, modeling post, with the idea of using models, or representations, when adding or subtracting fractions, but we didn’t talk about how to find equivalent fractions, so we’ll go into that here in this quick post.
First off, when do we usually use equivalent fractions?
We often use equivalent fractions when:
- comparing fractions
- adding or subtracting fractions
- dividing fractions
- we can even use them when multiplying fractions, though it just creates more work:-)
- Multiply the denominators together (not a personal favorite)
- List the multiples of the denominators to find the least common multiple/least common denominator
- Use prime factorization
- Use the ladder method
Let’s look at each one, using the same fractions each time, just for comparison’s sake. We could be adding these, comparing these, or dividing these…..it doesn’t really matter; but we’ll choose adding for these examples
Then students must multiply 5/12 by 8/8 to get 40/96 and 7/8 by 12/12 to get 84/96.
The addition and simplifying are show in the image below.
- There’s a good chance that those same students having trouble finding LCM would have difficulty simplifying the larger numbers in their answer.
One benefit of using this method is that it helps students practice their multiplication facts. BUT, if students don’t know the multiplication facts, it might take a while to create the lists (they could use a multiplication chart to help them).
The multiple lists of 8 and 12 are below.
- For example, in the list of 8’s multiples, 24 is the 3rd multiple (8 x 3 is 24), so we’d multiply 7/8 by 3/3.
- 24 is the 2nd multiple in the 12’s list (12 x 2 = 24), so we’d multiply 5/12 by 2/2.
Students can use a factor tree or the ladder method to find the prime factorizations.
Benefits of this method:
- Students can see which prime factors the numbers have in common
- Students can see how both the common and ‘uncommon’ factors contribute to the LCM.
The prime factorizations for 12 and 8 are:
12: 2 • 2 • 3
8: 2 • 2 • 2
To find the LCD/LCM:
1) Identify the factors that 12 and 8 have in common (two 2s).
LCD/LCM: 2 • 2 • 2 • 3 = 24
And we end up with the same equivalent fractions as above (10/24 and 21/24).
To use the ladder method for LCD, we put both denominators into the ladder, side-by side, as shown in the diagram below.
- We identify a prime factor and then divide by that factor
- We identify a 2nd prime factor in the resulting quotients and divide again
- We continue identify and dividing by prime factors, until there are no common factors other than 1
(For a more detailed explanation about how the ladder method works, check out the posts listed below.)
- LCM/LCD: 2 • 2 • 2 • 3 = 24
To find the equivalent fractions, students use the factor at the bottom of the ladder that’s under the opposite number
- In this case, 3 is under 12 and isn’t a factor of 8, so 7/8 is multiplied by 3/3.
- 2 is under 8 and is an ‘extra’ 2 that’s not included in the factors of 12; so 5/12 is multiplied by 2/2.
If you want more info about the ladder method, I have a couple posts about it:
Help Your Middle School Math Students Find LCD When Adding and Subtracting Fractions
Using the Ladder Method in Middle School Math, for GCF, LCM, Factoring
In the image, for example, we might be adding 1/2 and 5/6.
- When we convert 1/2 to 3/6, the representation using the fraction strips helps students see that these fractions represent the same amount.
Or, if we were comparing 5/6 and 11/12, we could use the fraction strips to verify that 5/6 is equivalent to 10/12.
Do you use fraction strips or another method to model the equivalences?
Check out the course, Fractions: From Foundations to Operations.