Developing depth
- Mr Bee
- Mar 3
- 3 min read
In Key Stage 2, we often introduce square numbers with a familiar line: “A square number is a number multiplied by itself.” It’s well‑intentioned, but it’s also a little clumsy. Children may memorise that 6 × 6 = 36, but memorisation alone doesn’t offer depth. It doesn’t help them see the structure beneath the surface.
A more meaningful explanation is that a square number is a number multiplied by the same factor. This small shift in language matters. It encourages pupils to think about the relationship between factors, not just repeat a phrase. But the real power comes when we look beyond definitions and explore the hidden patterns that sit in plain sight.
Because square numbers have a secret.
Every square number has an odd number of factors.
Most pupils will happily list factors of a number: 1 and the number itself, then factor pairs marching in from each end. They build factor bugs, arrays or lists. But here’s the moment depth appears: when children notice that for 36, the factor pairs meet perfectly in the middle at 6. That middle factor isn’t paired with anything else - because 36 is a square number.
Suddenly, a routine factor activity becomes something far richer.
And this is where deep mathematical understanding grows. It doesn’t grow from giving pupils harder numbers or more complex tasks. It grows from offering them new angles, new relationships, and new ways to see what they already know.
When learners spot that square numbers always have an odd number of factors, they are no longer simply recalling facts; they’re reasoning. They’re noticing structure. They’re beginning to generalise. And that shift is powerful.
Here’s why this matters.
1. It strengthens number sense
If children understand why square numbers behave differently, they can use this knowledge flexibly. When exploring unfamiliar numbers, they can spot patterns, make predictions, and test their ideas. This builds confidence far more effectively than rote recall.
2. It deepens understanding of multiplication and factors
Instead of viewing multiplication as repeated addition, pupils start to understand it as a relationship between numbers. They realise that factor pairs represent the dimensions of an array and that square arrays are unique because both dimensions are the same.
This builds the foundations for future concepts: area, prime factorisation, algebraic reasoning, and even early ideas linked to roots.
3. It shifts the focus from ‘knowing’ to ‘understanding’
A pupil might recite square numbers up to 12² without ever truly understanding what makes a square number special. But when they realise that these numbers have an odd factor structure, or that their arrays are perfectly symmetrical, or that they sit neatly on a diagonal when plotted visually - then they’re no longer just remembering. They’re seeing mathematics.
And when children see mathematics, they’re far more likely to retain it, enjoy it, and apply it with purpose.
In classrooms, depth doesn’t always come from adding more complexity. Sometimes, it comes from slowing down and shining a light on what is already there. In the case of square numbers, the richness was hiding all along.
So next time your class is listing the factors of 36 or building a factor bug, ask them what they notice. Ask them why the factor list doesn’t come in pairs. Ask them whether this happens with other numbers and which ones.
Because depth isn’t about doing more. It’s about seeing more. And once learners see what sits beneath the surface, mathematics becomes more than a subject to remember. It becomes something they understand.
Now... I'm off to wonder if this applies to cubed numbers - (it doesn't)
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