Secrets to measuring a piece of paper – Numberphile


Ten years ago I taught some eighth graders science. The principal said “Oh Cliff, you can teach exactly as you wish.” So my idea in teaching is, every day have the kids go home and do an experiment. And, first thing I did was say, okay, hey peoples, here’s a homework assignment. Measure the area of this sheet of paper. Use metric units and show your work. And, by the way, here’s a ruler for you. So, I don’t know, what’s the area of this sheet of paper? Just go do it. Bring it in tomorrow.

So the obvious thing to do is you take your ruler, you measure across here, 20 centimeters this way. You go across this way, you measure up and down, 26 centimeters this way. 20 centimeters times 26 centimeters should be around 520 square centimeters. Okay, that much you would do, I’d do. What my students didn’t realize is, of course, this reflects the type of science problem, or math problem, that has lots of wrinkles and difficulties. For example, oh, what do you mean by the area of this sheet of paper? The front side, or the front plus the back? Or maybe just the back side? Oh! Did you include the circles that have been knocked out? Do they count as part of the area? Worse than that. When you measured this, did you realize that its height over here is different from its height over here? Let’s put this piece of paper onto a square sheet of paper. Oh, this isn’t a rectangular piece of paper. In other words, multiplying height times width doesn’t work. You’re gonna have to do at least some kind of outer product or an integral perhaps. The rulers that I gave to my students 9 10 11 12 13 14 15 16 17 17 18 19 centimeters. Uh, there’s something wrong with Theophilus Measure Company. It’s an Imperfecto. In other words, this is a comment about doing science Always examine the assumptions that you’re making whenever you make a measurement. Whenever you write something. Hey! Look for the places where you have ambiguous statements.

What’s the area of this sheet of paper? Should I count the edges? Don’t trust your measurement tools. Good science does not mean getting the perfect answer. Good science means figuring out what could go wrong. BRADY: “How did your students do? How many of these embedded tricks did your students pick up on?”Oh, so, I had two classes, fifteen sixteen people in each class. This is 10, 12 years ago One kid right away picked up. Hey, there’s these holes in it. I’ll have to subtract the holes, if, you know, pi r squared, he figured, and he subtracted from it. Another kid said, wait a second, I don’t know what you mean by area. In fact, two or three kids said, oh, there’s a front side, or front plus back, or was just the back side? Nobody picked up on The Imperfecto ruler. Brady: “And the shape?”And the shape? Nobody really saw that.

Nobody saw this coming, and no one recognized that, oh, they’d been handed a bill of goods. Brady, people write to me and say I’m thinking of going to college. How do I write an essay? People write to me and say I’m trying to get into grad school, what should I say on my application? My idea is, what I want to see is somebody who’s got to figure things out, who says “I don’t know now, but I got to know!”and somebody, somebody who’s got enthusiasm. Not somebody who’s boring or puts me to sleep.

Brady: “Cliff, is that natural? Do you learn that or does it come naturally?”Does it come naturally? Who knows? I mean, to me, to me, I can’t stand … not … I want to know things. I gotta know things. What’s the what’s the – Wait wait wait wait wait wait what’s the mathematician, the German mathematician? “We must know, we will know”, What’s his epitaph? What was his name? Uh hell, come on.

You must know. Oh. Hold on. We must know we must know. I’m gonna look him up real fast. I’ll take a second. Well, he’s curious. He’s gone off to find out who the mathematician is. Brady: “Is that an iPhone?”Yeah. Brady: “I didn’t think you’d be an iPhone guy.”Hilbert! It was David Hilbert! On his grave it says, “We must know, we will know.”I like that feeling. …1894 1895 in Switzerland, came up with an absolutely wonderful device. A calculator called the millionaire. This device – check this out – this will not only … hey! Go over here, over here…


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