Evelyn Lamb: Hello and welcome to My Favorite Theorem, a math podcast where we get mathematicians to tell us about theorems. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.
Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. It's Friday night.
EL: Yeah, yeah, I kind of did things in a weird order today. So there's this concert at the Utah Symphony that I wanted to go to, but I can't go tonight or tomorrow night, which are the only two nights they're doing it. But they had an open rehearsal today. So I went to a symphony concert this morning. And now I'm doing work tonight, so it's kind of a backwards day.
KK: Yeah. Well, I got up super early to meet with my financial advisor.
EL: Oh, aren’t you an adult?
KK: I do want to retire someday. I’ve got 20 years yet, but you know, now it's nighttime and my wife is watching Drag Race and I'm talking about math. So.
EL: Cool. Yeah.
KK: Life is good.
EL: Yes. Well, we're very happy today to have Fawn Nguyen on the show. Hi, Fawn. Could you tell us a little bit about yourself?
Fawn Nguyen: Hi, Evelyn. Hi, Kevin. I was was thinking, “How nerdy can we be? It’s Friday.”
So my name is Fawn Nguyen. I teach at Mesa Union School in Somis, California. And it's about 60 miles north of LA.
FN: 30 miles south of Santa Barbara. I teach—the school I'm at is a K-8, one-school district, of about 600 students. Of those 600, about 190 of them are in the junior high, 6-8, but it's a unique one-school district. So we're a family. It's nice. It's my 16th year there but 27th overall.
EL: Okay, and where did you teach before then?
FN: I was in Oregon. And I was actually a science teacher.
KK: Is your current place on the coast or a little more inland?
FN: Coast, yeah, about 10 miles from the coast. I think we have perfect weather, the best weather in the world. So.
KK: It’s beautiful there, it’s really—it’s hard to complain. Yeah.
FN: Yeah. It’s reflected in the mortgage, or rent.
FN: Big time.
EL: So yeah, what theorem have you decided to share with us today?
FN: You mean, not everyone else chose the Pythagorean Theorem?
EL: It is a very good theorem.
FN: Yeah. I chose the Pythagorean Theorem. I have some reasons, actually five reasons.
KK: Five, good.
FN: I was thinking, yeah, that's a lot of reasons considering I don't have, you know, five reasons to do anything else! So I don't know, should I just talk about my reasons?
EL: Yeah, what’s your first reason?
FN: Jump right in?
EL: Well, actually, we should probably actually at least say what the Pythagorean theorem is. It's a very familiar theorem, and everyone should have heard about it in a middle school math class from a teacher as great as Fawn. But yeah, so could you summarize the theorem for us?
FN: Well, that's just it. Yeah, I chose it for one, it’s the first and most obvious reason, because I am at the middle school. And so this is a big one for us, if not the only one. And it's within my pay grade. I can wrap my head around this one. Yeah, it's one of our eighth grade Common Core standards. And the theorem states that when you have a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
FN: How did I do, Kevin?
KK: Well, yeah, that's perfect. In fact, today I was—it just so happens that today, I was looking through the 1847 Oliver Byrne edition of Euclid’s Elements, this sort of very famous one with the pictures where the colors, shapes, and all of that, and I just I happened to look at that theorem, and the proof of it, which is really very nice.
FN: Yeah. So it being you know, the middle school one for us, and also, when I talk about my students doing this theorem—I just want to make sure that you understand that I no longer teach eighth grade, though. This is the first year actually at Mesa, and I've been there 16 years, that I do not get eighth graders. I'm teaching sixth graders. So when I refer to the lessons, I just want to make sure that you understand these are my former students.
FN: Yeah. And once upon a time, we tracked our eighth graders at Mesa. So we had a geometry class for the eighth graders. And so of course, we studied the Pythaogrean theorem then.
KK: So you have reasons.
FN: I have reasons. So that was the first reason, it’s a big one because, yeah. The second reason is there are so many proofs for this theorem, right? It's mainly algebraic or geometric proofs, but it's more than any other theorem. So it's very well known. And, you know, if you ever Google, you get plenty of different proofs. And I had to look this up, but there was a book published in 1940 that already had a 370 proofs in it.
FN: Yeah. Even one of our presidents, I don't know if you know this, but yeah, this is some little nice trivia for the students.
FN: One of our presidents, Garfield, submitted a proof back in 18-something.
FN: He used trapezoids to do that.
KK: He was still in Congress at the time, I think the story is that, you know, that he was in the House of Representatives. And like, it was sort of slow on the floor that day, and he figured out this proof, right.
FN: Yeah. And then people continue to submit, and the latest one that I know was just over a year ago, back in November 2017, was submitted. That's the latest one I know. Maybe there was one just submitted two hours ago, who knows? And his was rearranging the a-squared and b-squared, the smaller squares, into a parallelogram. So I thought that was interesting. Yeah. And what's interesting is Pythagoras, even though it's the Pythagorean Theorem, he was given credit for it, it was a known long before him. And I guess there's evidence to suggest that it was by developed by a Hindu mathematician around 800 BC.
FN: And Pythagoras was what? 500-something.
KK: Something like that.
FN: Yeah, something like that. But he was the first, I guess he got credit, because he was the first to submit a proof. He wasn't just talking about it, I guess it was official, it was a formal proof. And his was rearrangement. And I think that's a diagram that a lot of us see. And the kids see it. It's the one where you got the big, the big c square in the middle with the four right triangles around it, four congruent right triangles. Yeah. And then just by rearranging that big c squared became two smaller squares, your a squared and b squared. Yeah.
EL: Yeah. And I think it was known by, or—you know, I'm not a math historian. And I don't want to make up too much history today. But I think it has been known by a lot of different people, even as far back as Egyptians and Babylonians and things, but maybe not presented as a mathematical theorem, in the same kind of way that we might think about theorems now. But yeah, I think this is one of these things that like pretty much every human culture kind of comes up with, figuring out that this is true, this relationship.
KK: Yeah, I think recently wasn’t it? Or last year? There's this Babylonian tablet. And I remember seeing on Twitter or something, there was some controversy about someone claimed that this proved that the the Babylonians knew, all kinds of stuff, but really—
EL: Well, they definitely knew Pythagorean triples.
KK: Yeah, they knew lots of triples. Maybe you wrote about this, Evelyn.
EL: I did write about it, but we won’t derail it this way. We can put a link to that. I’ll get too bothered.
FN: Now that you brought up Pythagorean triples, how many do you know? How many of those can you get the kids to figure out? Of course, including Einstein submitted a proof. And I thought it was funny that people consider Einstein’s proof to be the most elegant. And I'm thinking, “Well, duh, it’s Einstein.” Yeah. And I guess I would have to agree, because there were a lot of rearrangements in the proofs, but Einstein, you know, is like, “Yeah, I don't need no stinking rearrangement.” So he stayed with the right triangle. And what he did was draw in the altitude from the 90 degree angle to the hypotenuse, and used similar triangles. And so there was no rearrangement. He simply made the one triangle into—by drawing in that altitude, he got himself three similar triangles. And yeah, and then he drew squares off of the hypotenuse of each one of those triangles, and then wrote, you know, just wrote up an equation. Okay, now we're just going to divide everything by a factor. The one that was drawn, in you just divide out the triangle, then you just you end up with a squared plus b squared equals c squared. It's hard to do without the the image of it, but yeah.
EL: But yeah, it is really a lovely one.
FN: Yeah. And this is something I didn't know. And it was interesting. I didn't know this until I was teaching it to my eighth graders. And I learned that, I mean, normally, we just see those squares coming off of the right triangle. And then I guess one of the high school students—we were using Geometer’s Sketchpad at the time, and one of the high school students made an animated sketch of the Pythagorean theorem. And, you know, he was literally drawing Harry Potter coming off of the three sides. And you know, and I just, Oh, I said, yeah, yeah. You don't have to have squares as long as they’re similar figures, right, coming off the edges. That would be fine. So that was fun to do. Yeah. So I have my kids just draw circles so that they—just anything but a square coming off of the sides, you know, do other stuff.
EL: Well, now I’m trying to—my brain just went to Harry Potter-nuse.
EL: I’m sorry, I’m sorry.
FN: That’s a good one.
KK: So I have forgotten when I actually learned this in life. You know, it's one of those things that you internalize so much that you can't remember what stage of your education you actually learned it in. So this is this is an actual Common Core eighth grade standard?
FN: Yes, yes. In the eighth grade, yeah.
KK: I grew up before the Common Core, so I don't really remember when we learned this.
FN: I don't know. Yeah, prior to Common Core, I was teaching it in geometry. And I don't think it was—it wasn't in algebra, you know, prior to these things we had algebra and then geometry. So yeah.
My third reason—I’m actually keeping track, so that was the second, lots of proofs. So the third reason I love the Pythagorean theorem is one fine day it led me to ask one of the best questions I'd asked of my geometry students. I said to them, “I wonder if you know how to graph and irrational number on the number line.” I mean, the current eighth grade math standard is for the kids to approximate where an irrational numbers is on the number line. That's the extent of the standard. So I went further and just asked my kids to locate it exactly. You know, what the heck?
EL: Nice. Yeah.
FN: And I actually wrote a blog post about it, because it was one of those magical lessons where you didn't want the class to end. And so I titled the post with “The question was mine, but the answer was all his.” And so I just threw it out to the class, I began with just, “Hey, where can we find—how do you construct the square root of seven on the number line?” And so, you know, they did the usual struggle and just playing around with it, but one of the kids towards the end of class, he got it, he came up with a solution. And I think when I saw it, and heard him explain it, it made me tear up, because it's like, so beautiful. And I'm so glad I did, because it was not, you know, a standard at all. And it was just something at the spur of the moment. I wanted to know, because we'd been working a lot with the Pythagorean theorem. And, yeah, so what he did was he drew two concentric circles, one with radius three and one with radius four on the coordinate plane, and the center is at (0,0). If you can imagine two concentric circles. And then he drew in y=-3, a line y=-3. And then you drew a line perpendicular to that line, that horizontal line, so that it intersects the—right, perpendicular to the horizontal line at negative three. And it intersects the larger circle, the one with radius four.
KK: Yeah, okay.
FN: So eventually what he did was he created, yeah, so you would have a right triangle created with one of the corners at (0,0). And the triangle would have legs of—the hypotenuse would be, what, four, the hypotenuse is four. One of the legs is three, and the other leg must be √7.
KK: Oh, yeah, okay.
FN: Yeah, yeah. So it's just so beautiful.
EL: That is very clever!
FN: Yeah, it really was. So every time I think about the Pythagorean theorem, I think back on that lesson. The kids really tried. And then from √7, we tried other routes. And we had a great time and continued to the next day.
EL: Oh, nice. I really liked that. That brought a big smile to my face.
FN: The fourth reason I love the Pythagorean theorem is it always makes me think of Fermat’s Last Theorem. You know, it looks familiar, similar enough, where it states that no three positive integers a, b, and c can satisfy the equation of an+bn=cn. So for any integer value of n greater than 2. So it works for the Pythagorean Theorem, but any integer, any exponent greater than two would work. So I love—whenever I can, I love the history of mathematics, and whatever, I try to bring that in with the kids. So I read the book on the Fermat’s Last Theorem, and I kind of bring it up into the students for them to realize, Oh, my gosh, this man, Andrew Wiles, who solved it—and it's, you know, it's an over 300 year old theorem. And yeah, for him to first learn about the theorem when he was 10, and then to spend his life devoted to it. I mean, I can't think of a more beautiful love story than that. And yeah, so bring that to the kids. And I actually showed them the first 10 minutes of the documentary by BBC on Andrew Wiles. And just right, when he tears up, and, you know, I cannot stop tearing up at the same time because it—I don't know, it's just, it's that kind of dedication and perseverance. It's magical, and it's what mathematicians do. And so, you know, hopefully that supports all this productive struggle, and just for the love for mathematics. So, kind of get all geeky on the kids.
EL: Yeah, that is a lovely documentary.
FN: Yeah. Yeah. It's beautiful. All right. My fifth and final reason for loving this theorem is Pythagoras himself. What a nut!
EL: Yeah, I was about to say, he was one weird dude.
FN: Yeah, yeah. So, I mean, he was a mathematician and philosopher, astronomer and who knows what else. And the whole mystery wrapped up in the Pythagorean school, right? He has all these students, devotees. I don't know, it's like a cult! It really is like a cult because they had a strict diet, their clothing, their behaviors are a certain way. They couldn't eat meat or beans, I heard.
FN: Yeah. And something about farting. And they believed that each time you pass gas that part of your soul is gone.
EL: That’s pretty dire for a lot of us, I think.
FN: Yeah. And what's remarkable also was that, the very theorem that he's named, you know, that's where I guess one of his students—I don't remember his name—but apparently he discovered, you know, the hypotenuse of √2 on the simple 1x1 isosceles triangle, and √2 and what did that do to him. The story goes he was thrown overboard for speaking up. He said, hey, there might be this possibility. So that's always fun, right? Death and mathematics, right?
EL: Dire consequences. Give your students a good gory story to go with it.
FN: I always like that. Yeah. But it's the start of irrational numbers. And of course, the Greek geometry—that mathematics is continuous and not as discrete as they had thought.
EL: Well, and it is an interesting irony, then that the Pythagoras theorem is one really easy way to generate examples of irrational numbers, where you find rational sides and a whole lot of them give you irrational hypotenuses.
EL: And then, you know, this theorem is the downfall of this idea that all numbers must be rational.
FN: Right. And I mean, the whole cult, I mean, that revelation just completely, you know, turned their their belief upside down, turned the mathematical world at that time upside down. It jeopardized and just humiliated their thinking and their entire belief system. So I can just imagine at that time what that did. So I don't know if any modern story that has that kind of equivalent.
EL: Yeah, no one really based their religion on Fermat’s Last Theorem being untrue. Or something like this.
FN: Right, right. Exactly.
EL: Yeah. I like all of your reasons. And you've touched on some really great—like, I will definitely share some links to some of those proofs of the Pythagorean theorem you mentioned.
So another part of this podcast is that we ask our guests to pair their theorem with something. So what have you chosen for your pairing?
FN: I chose football.
KK; Okay, all right.
FN: I chose football. It's my love. I love all things football. And the reason I chose football is simply because of this one video. And I don't know if you've seen it. I don't know if anyone's mentioned it. But I think a lot of geometry teachers may have shown it. It's by Benjamin Watson doing a touchdown-saving tackle. So again, his name is Benjamin Watson. I don't know how many years ago this was, but he’s a tight end for the New England Patriots. So what happened was, he came out of nowhere. Well, there was an interception. So he came out of nowhere to stop a potential pick-six at the one-yard line.
EL: Oh, wow.
FN: I mean, it's the most beautiful thing! So yeah, so if you look at that clip, even the coach say something to remember for the rest—anybody who sees it, for the rest of their life just because he never gave up, obviously. But you know, the whole point is he ran the diagonal of the field is what happened.
EL: Yeah, so you’ve got the hypotenuse.
FN: You’ve got the hypotenuse going. The shortest distance is still that straight line, and he never gave up. Oh, I mean, this guy ran the whole way, 90 yards, whatever he needed from from the very one end to the other. No one saw Ben Watson coming, just because, we say literally out of nowhere. Didn’t expect it. And the camera, what's cool is, you know, the camera is just watching the runner, right, just following the runner. And so the camera didn’t see it until later. Later when they did film, yeah, they zoomed out and said, Oh, my God, that's where he was coming from, the other hypotenuse, I mean, the other end of the hypotenuse. Yeah. But I pair everything, every mathematics activity I do, I try to pair it with a nice Cabernet. How's that?
KK: Not during school, I hope.
FN: Absolutely not.
EL: Don’t share it with your students.
FN: I’m a one glass drinker anyway, I'm a very, very lightweight. I talk about drinking, but I'm a wuss, Asian flush. Yeah.
EL: Yeah. Well, so, I’m not really a football person. But my husband is a Patriots fan. And I must admit, I'm a little disappointed that you picked an example with the Patriots because he already has a big enough head about how good the Patriots are, and I take a lot of joy in them not doing well, which unfortunately doesn’t happen very much these days.
KK: Never happens.
EL: There are certain recent Super Bowls that I am not allowed to talk about.
FN: Oh, okay.
KK: I can think of one in particular.
EL: There are few. But I'll say no more. And now I'm just going to say it on this podcast that will be publicly available, and I'll instruct him not to listen to this episode.
FN: Yeah, now my new favorite team actually, pro—well, college is Ducks, of course, but pro would be Dallas Cowboys. Just because that’s the favorite team from my fiance. So we actually, yeah, for Christmas, this past Christmas, I gave him that gift. We flew to Dallas to watch a Cowboys game.
EL: Oh, wow. We might have been in Dallas around the same time. So I grew up in Dallas.
EL: And so if I were a football fan, that would be my team because I definitely have a strong association with Dallas Cowboys and my dad being in a good mood.
FN: There we go.
EL: And I grew up in in the Troy Aikman era, so luckily the Cowboys did well a lot.
FN: Well, they’re doing well this year, too. So this Saturday, big game, right? Is it? Yeah.
KK: I feel old. So when I was growing up, I used to, I loved pro football growing up, and I've sort of lost interest now. But growing up in the ‘70s, it was either you're a Cowboys fan or a Steelers fan. That was the big rivalry.
KK: I was not a Cowboys fan, I’m sorry to say.
FN: I never was either until recently.
KK: I was born in Wisconsin, and my mother grew up there, so I’m contractually obligated to be a Green Bay fan. I mean, I’m not allowed to do anything else.
EL: Well, it's very good hearted, big hearted of you, Fawn, to support your fiance's team. I admire that. I, unfortunately, I'm not that good a person.
FN: I definitely benefit because yeah, the stadium. What an experience at AT&T Stadium. Amazing.
EL: Yeah, it is quite something. We went to a game for my late grandfather's birthday a few years before he passed away. My cousins, my husband and I, my dad and uncle a ton of people went to a game there. And that was our first game at that stadium. And yeah, that is quite an experience. I just, I don't even understand—like the screen that they've got so you can watch the game bigger than the game is like the biggest screen I've ever seen in my life. I don't even understand how it works.
FN: Same here, it’s huge. And yet somehow the camera, when you watch the game on television, that screen’s not there, and then you realize that it's really high up. Yeah.
KK: Cool. Well, we learned some stuff, right?
KK: And this has been great fun.
EL: Yeah, we want to make sure to plug your stuff. So Fawn is active on Twitter. You can find her at—what is your handle?
FN: fawnpnguyen. So my middle initial, Fawn P Nguyen.
EL: And Nguyen is spelled N-G-U-Y-E-N?
FN: Very good. Yes.
EL: Okay. And you also have a blog? What's the title of that?
FN: fawnnguyen.com. It’s very original.
EL: But it's just lovely. You're writing on there is so lovely. And it, yeah, it's just such a human picture. Like you really, when you read that you really see the feeling you have for your students and everything, and it's really beautiful.
FN: Thank you. They are my love. And I just want to say, Evelyn, when you asked me to do this, I was freaking out, like oh my god, Evelyn the math queen. I mean, I was thinking God, can you ask me do something else like washing the windows? Make you some pho?
KK: Wait, we could have had pho?
FN: We could have had pho. Because this was terrifying. But you know, it's a joy. Pythagorean theorem, I can take on this one. Because it's just so much fun. I mean, I've been in the classroom for a long time, but I don't see myself leaving it anytime soon because yeah, I don't know what else I would be doing because this is my love. My love is to be with the kids.
KK: Well, bless you. It's hard work. My sister in law teaches eighth grade math in suburban Atlanta, and I know how hard she works, too. It's really—
FN: We’re really saints, I mean—
KK: You are. It’s a real challenge. And middle school especially, because, you know, the material is difficult enough, and then you're dealing with all these raging hormones. And it's really, it's a challenge.
EL: Well, thanks so much for joining us. I really enjoyed it.
KK: Thanks, Fawn.
FN: Thank you so much for asking me. It was a pleasure. Thank you so much.