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, 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. 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.