# Composing and Decomposing Fractions

Not too long ago I wrote about the power of part/whole thinking, and how an understanding that begins in Kindergarten has tons of applications throughout the elementary curriculum. Here’s yet another one. Take a look at this fraction addition problem.

Think about how we would normally teach students to solve that problem. They’d add the two fractions to get a sum of seven-fifths (Hopefully! How many would say seven-tenths?). Recognizing that seven-fifths is greater than a whole, they would proceed to apply a procedure for converting an improper fraction to a mixed number.

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Now, look at this solution.

What you see above sounds more like,ย I know I need two more fifths to add to three-fifths to make a whole. I can decompose four-fifths into two-fifths and two-fifths. Once I make a whole from three-fifths, I end up with one and two-fifths.ย That’s some deep understanding of fractions!

Let’s take a look at the standards, both the Common Core State Standards (CCSSM) and the Texas Essential Knowledge and Skills (TEKS).

CCSSM 4.NF.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

TEKS 4.3(A) Represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers ad b>0, including a>b. [Note: this standard states that unit fractions are being added. Example: 3/8 = 1/8 + 1/8 + 1/8]

TEKS 4.3(B) Decompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations. [Note: this standard goes beyond unit fractions. Example: 5/8 = 3/8 + 2/8; 5/8 = 1/8 + 4/8; etc.]

As I stated earlier, a common error students make when adding fractions is that they add both the numerators and denominators. So when adding fifths to fifths, they somehow magically become tenths. The standards listed above are written to prevent that very error. In other words, they are designed to help students understand that when you add fifths together, you still have fifths!

Here’s an idea for providing students with practice composing and decomposing fractions. Give students a problem, like the one shown below, with two missing addends. Challenge them to use fraction tiles to find as many solutions as possible.

Once students have had ample practice choosing their own fractions for the addends, they can practice finding a missing addend.

Be sure to provide manipulative (concrete) support for this activity. I like to have students always put out the whole tile when working with fraction tiles, just to reinforce that benchmark of one. Students can record their work with a model (representational) and completed equation (abstract). Voila! All three stages of CRA in one activity.

You can grab your free cards for composing and decomposing fractions here.

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1. Katie says:

Thanks for this! I will be starting my CCSS unit on fractions after Christmas…and just reading the standards makes my head hurt! I like the way you explained this one.

1. Donna Boucher says:

Ha ha! I totally understand. Glad this helps. ๐

2. Dondee says:

Thank you so much for this!

1. Donna Boucher says:

You’re welcome!

3. Cindy in Wisconsin says:

Thanks so much! I know this will be a help to my students; I look forward to using it!

1. Donna Boucher says:

My pleasure, Cindy! ๐

4. Rhonda Griswold says:

This is just the latest download I have and I will use for summer school. Glad to have the freebie and I appreciate all the work you do. thanks for making it possible to not have to reinvent the wheel so to speak.

1. Donna Boucher says:

I need a t-shirt: “Will work for sweet comments!”. Thanks so much, Rhonda. ๐

5. I’m looking for a concrete way to introduce and work with composing and decomposing fractions. Any suggestions?

6. We have been sharing this activity with teachers in our Fractions training (I am a K-5 math consultant) this year! We model it with connecting cubes, a different color for each addend. With one missing addend, sometimes we play a version of “I wish I had”… I wish I had 7/8. Right now I have 3/8 (built with 3 cubes, each with a unit of 1/8). What do I have behind my back? (4/8)

The equations with two missing addends have been especially powerful in generating great mathematical discourse. When my 10/12 looks different than your 10/12, we have lots to talk about!

Thanks for this awesome activity!

1. Donna Boucher says:

Love it, Chrissy! I like the idea of using linking cubes. Thanks for sharing.

7. Maria says:

For Texas, which TEKS does this activity cover? You have so many awesome activities and sometimes with the TEKS I get confused.
Thanks,
maria

1. Donna Boucher says:

Maria, it is 4.3(B), Decompose a fractions in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations.

8. Penny says:

Donna,
I absolutely LOVE your work and insight into all the various areas you share with us! I always say “I want to be like you when I grow up!!” You’ve helped me and my teachers understand the TEKS / SE’s … thanks for that!

1. Elizabeth says:

Hello- Love your work. Iโm having trouble finding the link to the free resources.

1. Donna Boucher says:

Sorry for the confusion! Look for the link at the end of the post.

9. Kyla McGinley says:

I love how you pose the decomposing fraction problem, giving students the opportunity to come up with many different solutions, and using fraction tiles for hands-on learning. I look forward to using this with my students – thank you!

10. Taylor Rawlings says:

I love the way you proposed the problem! It presents the idea to find all possible answers while using fraction tiles to provide hands-on learning. This is my next standard to tackle, and I cannot wait to see the endless ideas my students explore!

1. Donna Boucher says:

That’s so exciting! I’m sure your students will soar!

11. Stephanie Puckett says:

Thank you for such an amazing site with so many useful resources!
This is a teacher in training’s dream come true ๐
I wanted to mention it might be worth figuring out how to add affiliate links for Canadian versions of the sites you’ve got as affiliates. I found the item I was looking at on amazon.ca, but that means you miss out on those benefits.

Thank you again!

1. Donna Boucher says:

Thanks for the idea. I’m glad you found what you were looking for!