**Fraction Series, Week 4: Equivalent Fractions**

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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 didnโt talk about * how* to find equivalent fractions, so weโll 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:-)

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- 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

Letโs look at each one, using the same fractions each time, just for comparisonโs sake. We could be adding these, comparing these, or dividing theseโฆ..it doesnโt really matter; but weโll choose adding for these examples

**Method 1:**Letโs start with

**multiplying the denominators together**, 12 x 8. This gives us a common denominator of 96.

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 imageย below.

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- There’s a good chance that those same students having trouble finding LCM would have difficulty simplifying the larger numbers in their answer.

**Method 2: Find LCM/LCD by listing the multiples.**ย

One benefit of using this method is that it helps students practice their multiplication facts. BUT, if students donโt know the multiplication facts, it might take a while to create the lists (they could use a multiplication chart to help them).

โThe multiple lists of 8 and 12 are below.

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- For example, in the list of 8’s multiples, 24 is the 3rd multiple (8 x 3 is 24), so weโd multiply 7/8 by 3/3.
- 24 is the 2nd multiple in the 12’s list (12 x 2 = 24), so weโd multiply 5/12 by 2/2.

**Method 3: Find LCM/LCD using prime factorization**

Students can use a factor tree or the

**ladder method**to find the

**prime factorizations.**

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Benefits of this method:

- Students can see which prime factors the numbers have in common
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- Students can see how both the common and โuncommonโ factors contribute to the LCM.

The prime factorizations for 12 and 8 are:

ย ย ย 12: 2 โข 2 โข 3

ย ย ย 8: 2 โข 2 โข 2

**To find the LCD/LCM:**

โย ย 1)ย Identify the factors that 12 and 8 have in common (two 2s).โ

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**LCD/LCM: 2 โข 2 โข 2 โข 3 = 24**

And we end up with the same equivalent fractions as above (10/24 and 21/24).

**Method 4: Find the LCM/LCD using the ladder method**

To use the ladder method for LCD, we put both denominators into the ladder, side-by side, as shown in the diagram below.

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- We identify a prime factorย and then divide by that factor
- We identifyย a 2nd prime factor in the resulting quotientsย 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.)

**outside**of the ladder.

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**LCM/LCD:ย 2****โข****ย 2****โข****ย 2****โข****ย 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.ย
- 2 is under 8 and is an ‘extra’ 2 that’s not included in the factors of 12;ย so 5/12 is multiplied by 2/2.

โโIf you want more info about theย 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

**Using Representations for Equivalent Fractions**

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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 fractionsย 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,ย

**Fractions: From Foundations to Operations.**