When I was in school, I was the kid who did not “click” with the idea of comparing fractions. To this day if you tell me to *use the butterfly method* to compare fractions I immediately start to feel anxious.

*Am I supposed to cross and multiply from top to bottom or bottom to top? Does it matter? (It doesn’t) Why does this work? And why is there a procedure for adding and subtracting fractions also called the butterfly method? Make it make sense!! *

## A Students Who is Unable to Compare Fractions May Not Understand Fractions

If a student is struggling to compare fractions, ask yourself these questions:

- Given a fraction, could the student draw a representation of that fraction? Do they truly know what the fraction
*means*? - Could your student represent a fraction as a sum of unit pieces?
- Can your student describe what the numerator of a fraction tells you about the fraction?
- Can your student describe what the denominator of a fraction tells you about the fraction?
- Can your student describe what happens to a fraction as the denominator gets larger or smaller?

Each of these pieces are foundational to comparing fractions. If your students don’t understand what a fraction is at its core, they will likely have a great deal of difficulty comparing fractions together!

If your students does have these understandings around fractions, move ahead to troubleshoot your students’ reasoning strategies.

## Comparing Fractions Should Be An Exercise in Reasoning

When our students are first learning to compare fractions, there are many fractions that can be compared through simple reasoning alone.

- Are the denominators the same? If the size of the pieces is the same we need to attend to the numerators to compare– how many of those equal sized pieces do we have?
- Are the numerators the same? Then the denominator and the size of the pieces is the information we need to consider to compare.
- Can we compare our fractions to nearby benchmark numbers? Is one fraction less than one half but the other is more than one half? We can use this information to compare! Is one fraction nearer to the whole than the other? Using benchmarks can help to reason and compare fractions.

## How To Build Understanding to Compare Fractions

- Add context! A story problem about a tray of brownies that is divided in half (Woah! each friend is going to have a REALLY BIG TREAT!) but then divided again and again and again to accomodate more friends will drive home the lesson that a larger denominator indicates a smaller number of pieces. Get out a piece of brown construction paper to stand in as your tray of brownies and have fun “slicing” up the brownies and recording the resulting fractions with your students.
- Use pattern blocks. Asking your students to build a variety of fractions using pattern blocks will help them to physically compare the size of the pieces. Be sure to ask your students to draw and label their fractions as they compare so that you are linking this concrete and hands-on activity with the more abstract written fractions.

- Use fraction bars to compare. Fraction bars are often labeled which make them an ideal tool for fraction comparison. As your students are using the materials they will get comfortable looking for a “larger” piece when searching for a half and a “smaller” piece when searching for an eighth. Even this simple act help your students build spatial relationships and number sense around fractions which will positively impact their ability to compare!
- Drawing to compare fractions is a controversial suggestion. Students in 3rd or 4th grade don’t necessarily have the motor skills and spatial planning skills necessary to draw accurate fractions. This leads to incorrect comparisons and leads our students to draw incorrect conclusions while comparing fractions. So how do we move from the hands-on word to drawn representational models given this challenge?

Start with a template, giving your students rectangles to draw in to indicate the whole will get their drawings off to a strong start. Next, provide your students with a hands-on tool to use alongside their drawings such as fraction bars or paper fraction strips. This way, your students will have a correct model to use alongside their drawings so that they can be sure they are accurate in their representations.

## Comparing Fractions Troubleshooting

In summary, if your students are struggling to compare fractions FIRST check in with their foundational fraction understandings.

- Given a fraction, could the student draw a representation of that fraction? Do they truly know what the fraction
*means*? - Could your student represent a fraction as a sum of unit pieces?
- Can your student describe what the numerator of a fraction tells you about the fraction?
- Can your student describe what the denominator of a fraction tells you about the fraction?
- Can your student describe what happens to a fraction as the denominator gets larger or smaller?

Next, move to teaching strategies that promote reasoning strategies to compare fractions.

- Are the denominators the same? How can this help us compare?
- Are the numerators the same? How can this help us compare?
- Can we compare our fractions to nearby benchmark numbers?

And finally, promote hands-on practice that helps your students build their spatial awareness of fractions and boost their fraction number sense!