(Alternate Title: What is a Radian?)

**What I wanted them to grok:**

Discussed before they started: what is pi? Could you cut a string exactly pi inches long? Most of them said "no". They thought it was theoretically possible, but couldn't be measured precisely enough. I promised them that they'd walk out of class with a string of length pi in their pocket.

**The Activity:**

Materials: Circular object (collected from around my room: a CD, a water bottle, paper plates, cardboard mailing tube, etc etc)At the appropriate time we discussed this, and came around to "pi radii of the circle". Many of them still didn't like this and insisted it "wasn't really pi", so we had to have the discussion about operating in the perfectly abstract realm...some of them noticed that if their circle had a radius of exactly one centimeter, then the string would be pi centimeters...that seemed to help. Then I had them convert a bunch of degree measures to radians and back, and come up with formulas.

Metric ruler

Marker

String

Strip of Paper (I purchased adding machine paper at big box office supply store)

Tape

Scissors

1. Measure the diameter of

your circular object. Calculate the radius to the nearest 10th of a centimeter and record: r = _______ cm

2. Cut a strip of paper at least 8 times longer than your radius, and tape it to a flat surface.

3. With a marker, make a mark on the edge of your circular object. Make a corresponding mark near the end of your strip of paper. Line up the two marks, then roll your circular object one complete revolution. Make a second mark on the paper at exactly one complete revolution. What does the distance between the two

marks on the paper represent? Verify this value by using the radius to calculate

it.

4. Cut a length of string the same length as the distance between the marks. Set it aside.5. Using one of the marks on your strip of paper as "zero", use your ruler and a pencil to mark off intervals along the paper of the same length as your radius. About how many radii before you get to the one-revolution mark? This value should be the same for all of the various circles. Why?

6. Take your length of string, fold it in half, and cut it exactly in half. This string is precisely pi long. You and your partner can walk around all day with pi in your pocket! The question is….

drumroll.....Pi whats?

The scary thing was, most of them didn't recognize it from last year until they started writing down a formula. But I'm hoping that more of them actually know what the heck a radian is, and this makes the unit circle a slightly easier mountain to climb. We shall see.

Oh my gosh, radians elude students so much! I posted about it last year (http://samjshah.com/2008/03/13/83/). But I think I'll try your activity this year; I like that your activity had students use all different size circles (kind of crucial for it)!

ReplyDeleteShow themEuclid's Book V. Everything in it is studying numbers, and all the numbers are geometrically viewed in reference to some arbitrary unit. In fact, they didn't even call them "numbers". All the ancient Greeks ever talked about were geometric proportions.

ReplyDeleteI posted the document to scribd in word format - the formatting is a little messed up (page breaks, and the drawing of the circle didn't translate) - but here it is for download.

ReplyDeleteAck, this should be better.

ReplyDeleteCool! I wish I had the time (and the attention of my students) to show this to some of the students I tutor. They're in calculus, and still don't understand radians!

ReplyDeleteIt's really a nice lesson!

ReplyDeleteBut you hit it - kids are not retaining enough year to year, and even unit to unit. Would more practice help? Longer units? More depth, less breadth? (my theory...)

It's certainly not just your students; I think we all see it.

Jonathan

Yes I'm firmly in the "mile wide inch deep" camp of being critical of our math curriculum.

ReplyDeleteLove your blog, but the string is not pi long. Your students were correct, there can be no physical example of pi, only a rational approximation.

ReplyDeleteI make a point of talking about this to my students as an eventual segue to complex numbers. Irrational numbers have no physical counterpart anymore than i and its friends. It helps them get over the title: "imaginary"

If the string can't be pi long, then no string can be exactly anything long. It can't be 1 inch long. It can't be 1 foot long. Not _exactly_, anyway. But if you're saying that you can have a circle whose radius is exactly 1 long, then I disagree with you, I say the string _can_ be exactly pi long.

ReplyDeleteI do have a nice discussion during this lesson about how in math we are often dealing in abstracted perfection that can really only exist in our minds, not in the messy real world.

And also, Kevin, thanks for commenting. I hadn't seen your blog before. Interesting reading.

ReplyDeleteThanks, I wouldn't call it interesting, barely contained rants. You seem to have your stuff together, though. Inspiring to say the least. I like your argument about 'exactly 1 unit long' I'll think about that. If you don't mind I'd like to reply to that thought once I have a retort. Just some fun mathematical banter, again if you don't mind.

ReplyDeletekeep up the good work!

Kevin

Of course I don't mind! That's what I'm here for.

ReplyDelete

ReplyDeleteCould you cut a string exactly pi inches long? Most of them said "no".They were right.

I say the string _can_ be exactly pi long.That's meaningless. You can just as well say the same string is exactly 2 long. Does that mean pi = 2?

Uh...that is the desired response, Jack. Pi Whats? It's the title of the post. I kind of wonder if you read for meaning, or are just looking for a way to start an argument.

ReplyDeleteYou could ask it in terms of whatever units you like, if it makes more sense to you that way.

The correct response to "I say the string _can_ be exactly pi long" is not "Pi Whats?" but "You're wrong." Pi is a dimensionless number, not a length. Pi inches is a length, but no actual string can be exactly that long.

ReplyDeleteI've been pondering this ever since I posted and I admit I'm not entirely won over by Kate's argument (though she's a teaching Hero of mine!).

ReplyDeleteBut she did say that there is class discussion on this topic so maybe we're not seeing everything.

I've been thinking that perhaps if we assume the string is exactly 1 unit long then, assuming such a thing were possible then perhaps its not too bad to suppose you could have a string pi units long.

My objections border on the basic uncertainty inherent in measurement. Now, for the sake of argument I've been thinking about measurement. Things like time are defined by resonsance in atoms and those are definitions now. So in some sense a second can be perfectly measured. Also since the speed of light comes out of Maxwell's equations it, combined with a second, can determine a distance. So the question sort of boils down to, can light travel pi units? But here we run into Planck length and the fact that a photon is really more of a spread out blob.

Mind you (Kate and other readers) I view this more as an intellectual exercise. But I rather enjoy exercise, intellectual and otherwise.

Cheers,

Kevin

Kevin, if you're going to bring modern physics into play, then I can throw out all sorts of arguments that the question "what is the length of this string" has

ReplyDeletenodefinite answer, even with infinitely precise rulers and assuming that spacetime is a continuum.Within any reasonable sense of high school mathematics, (and filling in common English colloquialisms, such as omitting units when they're clear from context) the answer is clearly "yes". The point is no more to argue the fine points of modern physics (Kevin) than it is to show how clever you are about grammar and usage (Jack). It's to start introducing the concept of radian measure.

Go here for my rather lengthy rejoinder.

ReplyDeleteOne of the ways I try to bring some life to radians, is tell them that of all the units that we use, radians are one the most likely to be universal. By which I mean aliens are most likely to use them too. I point to the way that we mark an angle. That mark is often meant to reflect the size, but that would not be a good way to measure because it would matter how far away from the vertex you made the mark. But if you look at the ratio of arclength to radius then you get a constant.

ReplyDeleteAlso I think that it is important to say that "radians are the number of radiuses".

I also take this opportunity to impugn the degrees system that they love so well. I ask them why there are 360 degrees in the circle, then after some good guesses let them know that it is because there are about 360 days in the year. This goes back to the Babylonians. The Babylonians actually knew that there were 365 days in a year, but assumed that god would want the number to be easily divisible and that the remainder were celebration days for his creation.

Anyway, love the lesson. You are one of my teaching heroes too.

Matt

lovely! Thanks for sharing!

ReplyDeleteThis is very cool. Have definitely stolen it for use with my classes.

ReplyDeleteWill blog about it when both groups have both done it.

Thanks Kate!