Concentric Circles and Spirals

Following lots of interest in concentric circles in this setting, a girl brought in something that was a bit similar, but not quite the same

“From in my garden, next to my flowers. It is a spiral”

The practitioner brought the children together so they could think about things together. Because they had been so focussed on concentric circles, she first asked them

“How do we know if we have concentric circles?”

T: They are all inside each other
R: They are all separate circles
K: The little ones sit in the big ones

To help them consider this new shape, she then asked them

“What can you tell me about a spiral?”

T: It curls around
K: It’s a curved line
N: It’s a long line

The group then took some time to draw spirals and concentric circles, so they could consider further the shapes.

“My circles are all inside each other”

Another girl carefully draws a spiral, starting from the centre.

In the photo below, the boy on the left has drawn a spiral while his friend has drawn concentric circles.

As the group came back together again they shared their thoughts

R: I love spirals, them are my favourite. They go fast, round and round.
X: The spiral gets bigger. The concentric circles is getting bigger, but they are separate
T: The spiral is not separate - all one line. It goes in loads of circles.

The children have identified a fundamental mathematical difference between concentric circles and a spiral: the concentric circles are separate from one another, but the spiral is one long line. If you’re drawing concentric circles you need to lift your pencil off the paper between circles, but you can draw a spiral in one smooth motion. The practitioner reflects on how drawing the shapes emphasised this difference

“We had a mix of images, but one thing that was clear was that the spiral shape allowed for one continuous motion that got faster when the children were drawing, quickly becoming a cascade of overlapping lines.”

This links to an area of maths called topology, which is an area of geometry. In topology it doesn’t matter how far away things are or which way they bend, it matters whether they are connected are not. So, if you can smoothly change one shape to another without breaking or joining any lines, they are topologically the same. You can’t do this with spirals and concentric circles, even though they look quite similar, so topologically they are different. A spiral is just a long curved line, as the children point out, so it could be rearranged to make all sorts of other shapes (so long as it wasn’t broken or joined).

Concentric circles have quite a rigid structure, so it would be difficult to change them into something else that looked quite different (without breaking the topology ‘rules’), although on another occasion a boy did find something that was the same.

“Eeee the rectangles fit inside of one another too. I wonder what they are called - concentric rectangles?!”