I am a big fan of helping students to use STRATEGIES when approaching math rather than mindlessly following steps in an algorithm. Personally, I rarely follow order of operations when simplifying a math expression. Having number sense means you look at the math task and THINK about it and find ways to break apart and regroup in more efficient ways when it can be done. However, I have never done this with fractions.
Decomposing Numbers
Standard algorithm requires carrying.
Adding 19 and 23 using the standard algorithm is definitely slower. when I see this problem I want to add one and subtract one.
I was able to EASILY make this transformation in my head. The resulting expression is significantly easier to add than the first expression.
What About Fractions?
But what about looking at fractions in this manner. I saw this tweet by Jennifer BayWilliams with an example of how a student added mixed fractions. I had an epiphany: How have I never thought of doing this? It is almost like I needed permission that I can rethink how I approach fractions.
Thinking First
The challenge with teaching students only the standard algorithm… and shudder marking them down if they do not do it exactly how we showed them… is that students are not being asked to think. They are being asked to REMEMBER.
[Tweet]The first approach to a math problem should be THINKING about it. Not competing with a calculator. [/tweet]
Fractions are part of a whole. How do I get a whole?
Using the commutative property I can organize the fractions and the whole numbers seperately. THINKING about the fractions I can ask, “how do I make a whole?” This helps me to think about the concept of fractions and not just the steps.
Sometimes the standard algorithm is faster and sometimes it is not. Students should stop and THINK about math problems before just diving into the algorithm. Helping them to be more flexible with numbers and to apply strategies when appropriate will help them to be more confident and proficient at mathematics.
Q5:What do you notice about this student’s work? What ideas about teaching come to mind? #ElemMathChat pic.twitter.com/x3fyTJi5w3
— Jennifer BayWilliams (@JBayWilliams) November 4, 2022
Graspable Math Whiteboard
To create the sample math problems I used the Graspable Math Whiteboard.

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Decompose Fractions
Instead of diving into the steps of adding fractions have students look at how they can break apart the fractions to make it easier to approach the math problem. Have students decompose fractions as another way of thinking about the numbers.