Where do odd numbers hide?

In grades 1–2, we began with a simple, playful question:

“Where do odd numbers hide?”

Children set to work building odd numbers using link cubes, alternating colours to form L patterns —1 cube, then 3, then 5, and so on. Soon the tables were covered with bright, striped L towers of odd-number models.

Then something magical happened.

Using nothing but their eyes and their sense of pattern, the children started arranging the models to fit like spoons. As the shapes clicked together, they noticed a surprising fit.

The odd numbers formed perfect squares.

“Odd numbers hide in squares!” a few children exclaimed, wide‑eyed.

We explored this discovery together:

• The first odd number makes a 1 × 1 square.
• The first two odd numbers make a 2 × 2 square.
• The first three odd numbers make a 3 × 3 square.
• The first four odd numbers make a 4 × 4 square.
• The first five odd numbers make a 5 × 5 square.
• The first ten make a 10 × 10 square.

The pattern was astonishingly simple—and beautifully powerful.

Next, we counted the total number of blocks:

• The first odd number needs 1 block.
• The first two need 4 blocks.
• The first three need 9 blocks.
• The first four need 16 blocks.
• The first five need 25 blocks.
• The first ten need 100 blocks.

And so, through touch and sight and play, children rediscovered one of the most elegant ideas in mathematics:

The sum of the first n odd numbers is n².

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Where Do Even Numbers Hide?

Naturally, someone asked,
“If odd numbers hide in squares… where do even numbers hide?”

This question launched a new wave of building, arranging, testing, and noticing—another journey of mathematical discovery.

Where Do Natural Numbers Hide?

Working in pairs, students built two identical sets of the first four natural numbers.

They experimented with combining these sets—and to their delight, discovered that they could be arranged to form the exact shapes made by the first four even numbers.

In their hands, the natural numbers revealed their secret structure.

And then the room exploded with creativity.

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Math Art (or “Mart”)

Students began rearranging number models into patterns, pictures, and playful designs.
Mathematical structure became artistic structure.
The classroom became a gallery of joyful “mart”.

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Visiting Their Buddy Class

These younger students brought their discoveries to their older buddy class in grade 6, 7 or 8.

With great pride, they demonstrated how odd numbers form squares, how natural numbers fit together, and how patterns emerge when you simply look closely enough.

In return, the older students showed them how these same physical models can be used to derive algebraic formulas for the sums of odd, even, and natural numbers.

For both groups, the models became bridges between arithmetic, geometry, and algebra.

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A Surprise in Grade 11

They were stunned:

“We teach these concepts in grade 11.”

Then added:

“But I’ve never seen these physical representations.”

And with that one sentence, the room fell quiet.
What else might be possible if students across the grades were engaged in seeing and understanding mathematics in such powerful and accessible ways?

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Seeing-As

The Canadian poet‑philosopher Jan Zwicky writes that metaphors and representations invite us to “look at things like this.”

These classroom models did exactly that.

They became tools for seeing-as—
seeing number as shape,
seeing pattern as structure,
seeing mathematics as something you can hold, build, rotate, and understand.

They turned ideas into objects, and objects into insight.

And the children, without fear or hesitation, stepped right into the heart of mathematical thinking.

So, where do numbers hide when they play hide and seek?

Well… numbers aren’t exactly subtle.
They think they are—but they’re about as sneaky as a giraffe in sunglasses.

Odd numbers, for example, are convinced they’re invisible if they stand inside perfect squares.
You can still see them peeking out of the corners, counting themselves over and over:
“1, 3, 5… shh!! Don’t give us away!”

Even numbers prefer rectangles—“more elbow room,” they say.
They line up neatly inside the sides, trying to look very geometric and official,
but they giggle whenever someone walks by yelling, “Ready or not!”

And natural numbers?
Oh, they’re mischievous!
They hide inside the even numbers, who find this extremely annoying.

What do you see?

How many odd numbers are hiding in the blue square?

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Odds are tricky

It started during a math‑for‑teachers course that the group eventually nicknamed the math therapy course.

In one of the sessions, what began as a routine investigation of odd an even numbers turned into a moment of clarity—one of those small mathematical awakenings that makes everyone sit back and say, “Wait… why didn’t anyone show it to me like this before?”

The lyrics in the song below are based on the teachers’ comments—collected, treasured, and performed through song and animation—evidence of their journey through the puzzling, delightful world of odd and even numbers.

ODDS and EVENS. A song based on comments in a math-for-teachers course. Lyrics by George Gadanidis. Music and performance by Ian Parliament and Ryan Casselman.

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