Circles

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RSD

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So do circles have one or zero sides, because it is just one continual line so is that one side or is it refered to as something else?
 
That is a very intuitive question. It is still under heavy debate by many mathematicians.

If you define that a side is a straight line then a circle does not have a "side"

If you concur that a side can be a curved line then a circle will have only one "side"

Then again if you have a polygon with N sides... and you let N go to infinity then the polygon will become a circle with an infinite number of "sides"

So which one do you agree with?
 
Well, Mr. West How would you explain this concept to a kindergartener who is learning about shapes. Last year I told them one however, I do not want to give the wrong information, but could you imagine explaining infinite sides to a 5 year old?
 
I would defiantly go with the polygon with an infinite number of sides as the explanation. You might want to explain the second law of thermodynamics as well, or maybe Einstein's Theory of Relativity.

Kids are really smart.

A way to illustrate the polygon idea would be to draw a polygon on a paper on the wall. Then have the class slowly walk backwards until the polygon is small... It should then look like a circle.

What do you think?
 
RSD there is still one more way to view how many sides there are to a circle.

Ans: Two

There is an "In"side and "out"side to a circle, :D
 
Although, I agree students are very smart, I would hope that you are joking with teaching the students about the different theories. With that being said I will try the polygon on the paper technique, My boyfriend is drawing one now. Thank You for your assistance Mr. West.
 
The infinite sided polygon sounds like a good approach, but an infinite sided polygon is not a circle. The point at which the polygon becomes an ACTUAL circle is when the infinite sides become points. Because a circle is made up of an infinite number of points, not sides. As long as the shape has any length to the sides, it is still classified as a polygon. It might make a good visual, but make sure they understand that a circle is points. At some future time (if they don’t already know) they will learn about tangents to a circle and you can’t have a line perpendicular to a line on a circle, they would be parallel.

So since we know that a circle has tangents and a tangent can’t be tangent to a line, then a circle has no sides.
 
The way I would explain it to kindergarten kids is like this....

Before the kids arrive I would draw a line segment on the board and with your finger, I would then erase a few points

When they come in, ask them...
Teacher : is this a line?
Students : no
Teacher : why not, what’s missing?
Students : some points.

So you fill in the missing points

Teacher : is it a line now?
Students : yes
Teacher : Actually we call it a line segment because it doesn’t go on forever, but lets call it a line.

Teacher : so can you see that a line is just a bunch of points?
Students ; yes (hopefully)
If not, start with one point and draw another further down the board and keep going until you complete a line (if anyone had said no)

Teacher : So if we pull the ends around of this line what do we make?
Students : a circle (hopefully)
If not, be ready with a piece of rope and a marker. With them watching, cover the rope with dots and then stretch it into a circle. (If anyone had said no)

Teacher : so when you think about a circle, you can imagine it as an infinite number of points.
Teacher : since a point has no side, a circle can have no sides.
Hopefully they will accept this, but you might have a really bright student that realizes a square is made up of the same line segment.

Hopefully you won’t have to go into this detail, but always be ready for the out of the box thinker

Should this happen, be ready to explain that we use certain words in math to express different things. Remind them that you called the line a line segment, because they mean different things. Then explain that it is no longer a line segment when you connected the ends to make the circle. Math is like a foreign language, different words mean different things. Then draw the xy plane and draw a rectangle in the plane. Ask them if it has sides. They will say yes. Then continue the sides of the rectangle across the plane (so you have 4 lines with a box in the middle). Tell them that we no longer call these sides. Math has it’s own word, we call them regions. Once a line segment goes on forever(continuous) , it no longer has a side, it has a region (above and below the line). Once you connected the ends on the rope, it is continuous, so it has no sides. It does have regions (inside and outside). The two sides of a circle that I joked about in my first post.
 
alohacharlie said:
The way I would explain it to kindergarten kids is like this....

Before the kids arrive I would draw a line segment on the board and with your finger, I would then erase a few points

When they come in, ask them...
Teacher : is this a line?
Students : no
Teacher : why not, what’s missing?
Students : some points.

So you fill in the missing points

Teacher : is it a line now?
Students : yes
Teacher : Actually we call it a line segment because it doesn’t go on forever, but lets call it a line.

Teacher : so can you see that a line is just a bunch of points?
Students ; yes (hopefully)
If not, start with one point and draw another further down the board and keep going until you complete a line (if anyone had said no)

Teacher : So if we pull the ends around of this line what do we make?
Students : a circle (hopefully)
If not, be ready with a piece of rope and a marker. With them watching, cover the rope with dots and then stretch it into a circle. (If anyone had said no)

Teacher : so when you think about a circle, you can imagine it as an infinite number of points.
Teacher : since a point has no side, a circle can have no sides.
Hopefully they will accept this, but you might have a really bright student that realizes a square is made up of the same line segment.

Hopefully you won’t have to go into this detail, but always be ready for the out of the box thinker

Should this happen, be ready to explain that we use certain words in math to express different things. Remind them that you called the line a line segment, because they mean different things. Then explain that it is no longer a line segment when you connected the ends to make the circle. Math is like a foreign language, different words mean different things. Then draw the xy plane and draw a rectangle in the plane. Ask them if it has sides. They will say yes. Then continue the sides of the rectangle across the plane (so you have 4 lines with a box in the middle). Tell them that we no longer call these sides. Math has it’s own word, we call them regions. Once a line segment goes on forever(continuous) , it no longer has a side, it has a region (above and below the line). Once you connected the ends on the rope, it is continuous, so it has no sides. It does have regions (inside and outside). The two sides of a circle that I joked about in my first post.
Click to expand...

Ouch! This gives me a headache!

If a "side" is defined as a line segment, then a circle doesn't have any sides...NO part of a circle is a line segment. At the kindergarten level, I think that would be sufficient. Most kindergarten kids are not ready to deal with the kind of abstractions involved with this question.

If you feel the need to do so, you might show examples of regular polygons with lots of sides so that the students could see that the more sides there are, the shorter the sides are compared to the radius of the polygon, and the more the polygon resembles a circle. But when it IS a circle, there are no longer any "sides" (line segments) involved.

No wonder my kindergarten granddaughter is confused by some of the things she's learning!!!
 
How are 5 year olds not going to understand points drawn on a piece of string? The leap from multiple sides of a polygon to no sides of a polygon is a difficult concept for middleschool kids to graps. But thanks for your input.

RSD, I would change one thing though, I would give the students a bunch of markers and let them each draw a point until it is completely covered.
 
And just a personal thought MRSPI,
I am the one with the headache because my highschool students don’t understand most of the terms used in math. Consecutive, integer, region.... are words they seem to never have been taught. The best time to teach kids about the terms in math is when they are young. Kids are like sponges, if you expose them to it, they will pick it up. If you dumb it down for them, then you have a bunch of confused dumb kids
 
I'll add my personal thought, ALOHACHARLIE (wow...isn't it impressive when you type someone's name in all caps???)

I'm a very big advocate of introducing students to proper math vocabulary and expecting them to use it all of the time. If you have EVER told your students to "plug and chug" or something similar...you are violating that principal. Of course, I don't know you, so I don't know if you've done that. But, "bending a line segment around until its ends meet" sets my teeth on edge....unless you are going to get into non-Euclidian geometry. I sure wouldn't go "there" with kindergarten kids.

There is, however, a limit to what you can expect students to grasp at age 5 or 6. Circles are NOT polygons, plain and simple. For kindergarten kids, a circle has no sides (unless you want to go with "inside" and "outside," which I don't object to...but that doesn't deal with sides as line segments.
 
Boy oh boy oh girl... :wink:

Here's joker RSD's original question:
So do circles have one or zero sides, because it is just one continual line so is that one side or is it refered to as something else?

Why bring 5 years old kids into this?

So a math-shattering decision that a circle has one side is made:
by how much will this bring down the price of groceries?
 
Ok mrspi,
Your comment makes no sense. I have never heard the phrase plug and chug, but it has no barring on this problem. It is not a math term. It is a phrase. And I am not advocating the proper use of the King’s English. I am advocating the introduction of math terms. So what I want from you mrspi is what I sometimes ask the students when they lose sight of the matter at hand. “Pay attention, no really, pay attention to what I am saying”. Not what you think I said. I did not define a circle as wrapping the ends of a line segment. It is a VISUAL for cognitive learning. So a 5 year old can get it. I am not sure why you think you needed to attack my idea in the first place, but if you do, then make sure you are not totally off base. And if you think you can suggest a better idea, then do it, don’t piggyback on mine.

And thanks Denis for trying to defuse the situation, but this has to be said.
 
To RSD,
My apologies to you for the way your post turned out. It was my intention to give a helpful suggestion without it turning into a pissing contest. I understand if you chose not to use it, but know that it was sent with the good intentions. I hope that the ugly direction that this post took does not discourage you from coming here to get fresh ideas in the future. I haven’t been posting long, but this is the first time I’ve seen it turn like this. Denis had the right idea, I should have ended it, but I will not respond to any more of her rants so think of this as a fluke. It is a good site to get ideas (whether you like them or not). Good luck in your career.
 
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