Developing fraction reasoning through hands-on learning
In the final episode of the AAMT x We Support Hands-On Learning video series, Allan Dougan and Professor Catherine Attard turn their attention to ordering and sorting fractions, and why it matters more than you might think.
Why ordering fractions matters
Ordering fractions is how children begin to understand fractions as quantities with size and magnitude, rather than just names or symbols.
As Allan explains, being able to compare and order fractions is foundational to proportional reasoning, which flows on into decimals, percentages, measurement, graphing, and probability. “So much of it relies on having a really good understanding of which fraction is bigger, which fraction is smaller, how to combine them and so on.”
Understanding fraction size
One of the misconceptions that comes up again and again is that the bigger the denominator, the bigger the fraction. It makes intuitive sense to children – nine is bigger than three as a whole number, so ninths must be bigger than thirds, right?
Wrong. This is exactly the kind of thinking that hands-on materials can untangle in a way that explanation alone simply can’t.
Allan and Catherine are also clear that before children touch any materials, they should be given time to reason first. What do you think? Is 2/3 bigger than 1/2? Why? Getting children to articulate their thinking is how you understand where they actually are.
And as Catherine points out, younger students often need to be given the language of reasoning explicitly. They can’t be expected to have it already. Asking “but why do you say that?” and “how can we understand that?”. Modelling language is really important in mathematics.
Hands-on maths: Comparing fractions with materials
It’s really important that children can actually see, feel and compare different fractions. Here are some of the hands-on materials Allan and Catherine recommend:
Fraction bricks that are colour-coded and labelled let children physically pick up, hold, and compare fractions side by side. Placing a half next to a third, with the whole as a reference point, makes it immediately clear which is larger. There’s no ambiguity. The size is right there.
Fraction bars offer a slightly more representational step up from the bricks, while still being hands-on and concrete. A useful activity: place a selection of fraction bars randomly on the desk, then ask students to order them from smallest to largest in their books or on their whiteboards.
Teacher insight: Using number lines to develop magnitude
Placing fractions on a number line is conceptually different from comparing them with bricks or bars. It’s not about size relative to a whole. It’s about where fractions live on a continuous scale, and that’s a significant shift in thinking.
A simple starting activity: draw a number line from zero to one, hand students labelled fraction pieces and ask them to place each one where they think it belongs. Then ask the class whether they agree and why. Where does a quarter sit? Where does a third go in relation to it?
One important thing to keep in mind is that number lines for fractions don’t have to run from zero to one. Allan encourages teachers to explore zero to two, or even zero to ten. One of the most persistent misconceptions around fractions is that they only exist between zero and one. Improper fractions and mixed numbers have a place on the number line too and showing that explicitly helps break down the idea that fractions are somehow separate from the rest of the number system.
The number line also opens up a natural conversation about equivalence. Two quarters and one half land in exactly the same spot. Mark them both. Then do the same with decimal and percentage equivalents. All of a sudden, connections that usually feel abstract become visible.