I teach Kindergarten and they think circles have sides even though I told them a side is like a straight line... The kindergarteners insist that they see a tiny straight line in the circle..how can I teach it so they understand?

Some argue that a circle has an infinite number of sides but that's not what your Ks are saying...they describe 'seeing' straight lines. Explain to them that circles are round...don't talk about sides in relation to circles at this age. Is it possible that the kids who are arguing with you have seen drawings of circles illustating diameters or radii? Perhaps that's why they are 'seeing' straight sides?

The circle cannot have sides because it doesn't have vertices (angles). While to the eye it may seem that the line is straight between two close points on the circle, the curve is actually so slight that it becomes hard for the eye to see even though the curve is there. Unless the student has a concept of the horizon looking flat but actually being curved, you might have to go with the approach that great mathematicians have proven that there is actually a continuous curve even though it looks a bit different to our eyes. This is one that they have to trust you on until they get older and can prove it with higher math. I would definitely give kudos to this kid for really thinking about it. I can easily see if he puts two points close together that our tools for creating the circle (print or pencil) would make it seem to be a straight line. It is going to be hard to get this kid to understand that a circle edge actually consists of points so small on a continuous curve that we can't see them. Until you measure to that level, someone might actually believe the points have a straight line between them. Visual perception is at play here - just like a pencil put in a glass of water looks bent even though it isn't. Sometimes our eyes and brains see something that isn't quite accurate.

I would give them each a piece of yarn. Have them pull it taut and tell them it is a straight line. Have them get in groups of four to connect their straight lines to make a rectangle with their yarn. Then have them pull away from the group and use just their yarn to join the ends into a circle. After doing several activities with this, tell them to show you a straight line, or side, and then show you a circle. Then ask them if a circle has sides. Hopefully this kinesthetic and visual approach will cement the idea in their minds.

I say it has a curved line and only straight lines are sides. I also make an anchor chart for sides, vertices, and where we see it in real life. When I put 0 - 0 I point out that even the zero looks like a circle to help them remember.

(Cute, Caesar!) Hm. You might need to explain the concept of the radius. Each radius of a circle - and a circle has an infinite number of radii - is exactly the same length; this can be demonstrated by having pairs of kids draw a big circle using two pencils (or a whiteboard marker and a magnet, if you want) connected by a loop of string: the string's length doesn't change. It helps to mark the center of the circle first. Then have them place two points fairly close together on the circle and connect those points to the center and to each other, making a long skinny isosceles triangle. They can use the pencils and string to verify that each of the long sides is the same length: each is a radius of the circle. If they then draw another radius that goes through the middle of the short side, they should see that THAT radius extends slightly but perceptibly beyond the short side of the triangle: the distance from the center of the circle to the middle of the short side is thus not equal to a radius, so a circle is NOT in fact exactly a set of really skinny triangles.

In high school, the beauty of geometry, like debate, is in the definitions. So, no, a circle does not have "sides" because, as 'Daisy mentioned, a side is a line segment with two clearly defined endpoints acting as vertices. I wonder if this would work with your little ones: Get a printed picture of a circle, and put it up on the screen. Illustrate that the line is curved, not straight. Then zoom in... still curved. Zoom in a bit more... still curved. Mention that no matter how much you zoom, even if you go well beyond the capability of the machine, the points won't form a straight line.

I think this child is using the two points on the edge but his tools make it seem that very close points produce straight lines between the point, but very, very small straight lines. Funny thing is, some highly educated engineers still question the continuous curve and wonder if the point distance gets so small it eventually is straight lines, but they are wrong. I'm thinking that this kid is thinking beyond the other kids and noticing that by his eyes and tools available that two very close points will look like they are connected by a very small straight line.

Yeah, actually it's kind of like Trapezoidal Rule in Calc... fit enough teeny tiny little trapezoids under a curve, and your combined area will approach the area under the curve. The difference, of course, is that no matter how skinny your trapezoid is, there's still a little room between the curve and the leg of the trapezoid.

The 2D Shapes song on youtube describes circles and having one continuous side. Just another way to explain it.

But that contradicts the defintion of a side as being composed of a straight line segment with endpoints. I'm glad I teach high school... I don't envy you guys having to explain this stuff to kids

I think we are getting way too complicated for kindergarteners...just define a circle as round and polygon shapes as having straight sides...when in doubt, check your curriculum standards. Don't argue with k kids...you are the authority.

You're right, of course. But that's why I love HS Geometry... the "arguments" are such fun. I wish I had thought of Trapezoidal rule when I taught geometry last year, I would have brought all this up when we first spoke about arcs and circles.