Trig – The Oldest Practical Math

“If you ask someone what trigonometry is, they will probably mutter something about angles, sides, hypotenuses, and perhaps some strange acronyms they memorized long ago. This is pretty ignominious for a discipline that enabled navigation across oceans and eventually across interplanetary space.”

That’s an excerpt from the first chapter of Make Trigonometry, Build Your Way from Triangles to Analytic Geometry, by Joan Horvath and Rich Cameron, my guests today. Make Trigonometry is the third math book that Joan and Rich have written with us, the previous books being Make Geometry and Make Calculus. (Make: Trignometry is available in the Maker Shed as well as other booksellers.)

I want to welcome Joan Horvath and Rich Cameron. So Joan, tell us a little bit about what you’re doing in this book.

Joan: This book fits in between the other two pretty much. Trigonometry is an interesting beast in the curriculum in that it’s put different places by different people in different states and things like that, but it’s fundamentally the study of how do you measure things at some level. It comes out of surveying and some roots that are very, very old and it’s some of the oldest practical math or the oldest maker math, if you will.

Dale: Really?

Joan: It goes back 2000 years of trying to figure out, how far away is that object on the horizon and how tall is that tree? And is that guy that’s making my house actually doing it right? You can do a lot of trigonometry with basically nothing but a couple of sticks if you have thought it through. So it’s the ultimate practical math at its most basic level.

Dale: Joan calls herself a recovering rocket scientist. In that career, she worked on spacecraft headed to distant planets. Her co author, Rich, has a very different background as an open source developer who’s been involved in 3D printing since its emergence in the maker community. Together, they’ve come up with a way to teach math using 3D models that turns math into a hands on learning experience for students.

In this conversation, Joan and Rich talk about these 3D models, which you can’t see, obviously, but I’ll put a link below to a companion article on the Makezine. com website that will show off all of these models. 

Recently, Joan and Rich were in the Bay Area to participate in a workshop for coaches involved in FIRST Robotics. They have coached and judged FIRST competitions for quite a while.

Joan: FIRST Robotics for people who don’t know is a competitive robotics at a whole bunch of levels. But we work with some high school level teams. And so this was a, a first of its kind gathering of all the people who, uh, who work with teams. You know, trying to help them technically and, and managerially and all those things.

And so it was really something to, as somebody put it, get together when you’re not competing against everybody. And see some of the wonderful, some of the really wonderful teams up there in the Bay Area. There’s a couple of world class teams up there that it was fun to see their digs and see how much they do, in some cases, with very little.

Dale: Can you explain what the purpose of that workshop was, to bring people together?

Joan: Yeah. Teachers in general and robotics coaches in particular are kind of isolated, you know, they’re at a given school, there may be only one or two people who teach math, say, and you don’t get to collaborate with your colleagues very easily. And so for robot coaches, that goes triple because there’s a couple of dozen per city, maybe. It’s very hard to find each other. What these folks did just kind of word of mouth and said it’d be great if we all got together in person and put out a call for that and to their astonishment, they got people from South America and East Coast. They have over 300 people, apparently, just purely word of mouth.

 It was just interesting hearing how people, solve the challenges of doing competitive robotics and a lot of different environments. You never get to talk to these people, unless you see them at competitions and then you’re running around because well, it’s competition, right? Nice to just do that and be able to talk to people in relatively relaxed circumstances.

Einstein Tiles

Dale: Alright. Let’s transition back to the trigonometry book. Rich, I want to bring you in. Joan brought out a baggie full of monotiles. They demonstrate some interesting things. You said when you put them in front of kids, they go right at them like a real interesting kind of puzzle.

So how many sides are on that?

Rich: So, it’s I believe 13 sides. They call the shape a hat, sort of looks a little bit like a cowboy hat. I think if you turn it over, it looks a lot more like a shirt.

In various ways, they create these patterns that don’t repeat, so you can add these infinitely out in a pattern and the pattern never repeats. So most of them will be oriented the same, the same side up, you do have to flip a couple of them with this shape, in order to keep the pattern going. They came up with some newer ones, just very recently that don’t need any of them flipped.

Joan: It’s worth mentioning that this is brand new, a brand new discovery. Our math reviewer, Niles Ritter, when we did some simpler tiles said Oh, well, you have to talk about this new discovery. It’s been a holy grail for a long time to come up with a single tile that could be used non repeatably. Called an Einstein, which is, uh, German for one stone, an Einstein tile. It’s fun to incorporate in a reasonably basic math book, a very new discovery, which was made by an amateur, by the way. He went to some math departments to have it validated.

Dale: Interesting. So is that part of trigonometry?

Joan: It is in the sense that it’s in the book but, you know, trigonometry is also — I like to think of it –as a continuous arc between geometry and calculus because trigonometry brings you into some spaces that you really need as a foundation for calculus that aren’t really geometry.

The boundary between geometry and trig is pretty soft and pretty squashy. We overlapped a little bit in the books that books stand alone. 

I guess if you’re going to be technical, it’s probably topology, but fundamentally it’s really pretty simple looking at angles and thinking about how do how do shapes go together? I’m technically trained as an engineer, but I find playing with these and playing with their cousins that are simpler. You can play around with what kind of tilings can you make to do interesting patterns that do repeat.

Rich: And which ones don’t because of the angles involved.

Dale: One of the key aspects of all three books really is that you are making use of 3D printing and design tools to allow people to explore these subjects, so that the students to not just memorize something, but it makes it tangible, both in the physical sense, but also in a manipulative sense in your mind. These are objects with properties and you can do things with them. 

Joan: If people can play with something… so this is one of the fundamental models, it’s in both the geometry book and the Trig book. And so what this is is a triangle with a 90 degree angle in it, and a slider, and it’s scaled so that the long side of the triangle, the hypotenuse to its friends, is 1 in some units.

And so the ratio of this side of the triangle to this hypotenuse is called the sine of this angle. And so you can play around with, as you change this, how does that angle, how does the sign of that angle change? Because you can read it off on the side here. Math teachers have really liked this because it’s dynamic.

You can draw a whole bunch of triangles, but you can change this and reason about let’s see, as this angle gets bigger, this ratio gets bigger until it eventually becomes one when I’m all the way over here. As this angle gets smaller, it’s going to come down to zero eventually, as this gets smaller.

The fact that you can play with this dynamically, and look at it and reason about it really helps. Originally we had some grant support to do geometry for blind students. And so this one comes out of that– teaching them about this, being able to feel it and think about it.

We try to have classic universal design that if you make something that works really well for people who maybe learn differently or have to learn differently, then often it’s better for everybody– to learn hands on and lead with inquiry, lead with exploration. And say, okay, now you did all that. Now, the way you can do that symbolically is like this. And you have something to really hang on to.

Dale: To be able to test it out in your head is very useful.

Joan: Rich is phenomenally good at holding things in his head. Our joke is that we’ll start working on one of these things and I’ll say we should have a model of this and we’ll argue about it for a while. He’ll go off and do it and 20 minutes later, come back with something in OpenSCAD. Then while it’s printing for three or four hours, I’ll go off and try to prove it analytically to myself. And he’s always right. It’s really annoying. 

Dale: Rich, are you the model builder for most of the models we have in the book.

Rich: Yeah. I’ve been using OpenSCAD, since 2010 designing 3D printers in it, all kinds of stuff. Robots. 

Dale: Explain what that is for, for someone who doesn’t know OpenSCAD.

Rich: Well, it’s a programming interface, so you…

Dale: Programming. 

Rich: So you write a little program that describes the object you want to make.

Joan: And, it’s an open source language as well. So it’s free and you can use it anywhere.

Dale: And to some degree, you’re taking equations and adapting them as programs in OpenSCAD and getting an object out of that you can do something with.

Joan: Math incarnate. This guy is kind of math incarnate.

Rich: It’s not as maybe not as user friendly as something like Tinkercad, but there’s a lot of stuff you can make in it that you couldn’t make in Tinkercad. These are models of various types of waves. This is a planar wave, and this is a radial wave. We just have the equations in our OpenSCAD file that describe these different waves. 

Dale: In Tinkercad, they’re kind of difficult to draw those kind of more complex patterns.

Joan: It’s hard to be precise in Tinkercad. So if you’re trying to do something, where you have real structure that you’re trying to represent, it’s… I don’t know how you would do this Tinkercad, which is a wonderful place to get started, so don’t get me wrong, Tinkercad is really interesting for making lots of basic stuff.

But if you really want to show relationships and then write equations behind that and say here is what it is in math, and that’s powerful.

Dale: How does a teacher use your book? Or to that degree, a parent? 

Joan: The intent is that it’s project first. So there are projects and you can certainly do the projects. And we try to keep the complicated projects in each book for last chapter, although interestingly, the stuff at the end is often what is really popular if we, for instance, are at a Maker Faire and have it out on a table. The tiles are, for instance, are in the last chapter of it. Here’s a whole bunch of concepts you’ve learned, now here’s some fun projects that take them all together.

The intent is that if a kid’s a little bit older, they might be able to go ahead and use our models to make things. You don’t have to be able to write OpenSCAD from cold. You can download our files, our files are all open source, they’re on GitHub. We talk in the book about how to download them, so a kid can download them, a parent can download them, and if they have a 3D printer, they can print them.

 Some of the models will work on paper. In Trig, we have a fair number of things that just need a ruler and a piece of string, like figuring out how tall a tree near your house is, by just measuring the distance from where you are to the tree and then backing up enough so that a ruler exactly covers the tree and you can get a pretty good estimate of how tall a tree is and things like that. We have some calculations in this one that are fundamentally very simple surveying.

So you can get a sense about what are people doing when they have these far more sophisticated surveying objects, you can do all that with a stick and a piece of string if you have to and measure it as best you can. The hardest part is measuring out the distance from yourself to the wall. We had some funny looks from the neighbors when we were testing, testing it out on my roof, which is what we do out front. It’s like what are those people doing now? 

Dale: Give me an example. 

Joan: One of the last projects in the book is this little robot arm which requires that you learn a bit of inverse kinematics.

Rich: It’s a serial robot arm with three servo joints. In order to get the tip of that to a certain coordinate in space, you have to give it the coordinates, and then you have to convert those coordinates to spherical coordinates, which involves a lot of trig.

And then you have to do some more trig to– you create a triangle from the base of the arm to the tip, and use Law of Sines and Law of Cosines to figure out what the angles are involved changing the elbow joint angle, so that your tip is at the right distance from the center joint.

Joan: These are 3D printed and then these are just little hobby servos. So we talked through how to make this and how to play with it and, you know, how you would expand it for robots in general. This is a special case that, that the trig is tractable, at the level of this book but it’s fun to do. It’s pretty simple, pretty small set of 3D prints that you can fit on all the pieces on your average small 3D printer. So we tried to keep everything so it would work on a beat up school 3D printer. 

Dale: What Rich is describing is something like a program would do to move that 

Rich: So the arm is driven by an Arduino sketch that uses all these different trigonometric functions, and that’s included in the book. And we walk through how it works.

Dale: That’s cool. What’s the content piece of what you’re trying to cover in Trig?

Joan: The subtitle is something like triangles to waves or something like that. The most basic trigonometry is I have some kind of triangle and I’m going to figure out things from it, like for instance, how tall a tree is because here I have, you can imagine I have a one foot ruler here, and for people listening, I’m holding up my finger inside a triangle. Imagine I have a one foot ruler inside a triangle, and I know I can figure out from that, because the ratios of various things are the same, how tall my tree over here is.

We start out with the basics of single triangle, single triangle trig, as I like to call it. It turns out that you can, in ways that are reasonably straightforward, and we have some models that work from Play-doh, that we roll out on Play-doh. I don’t know if we want to go into that here because it’s sort of visual. We have models that would go from a single triangle to a continuous function, and when you do that, you get a wave. One of the nice things you can do… with what we’re doing is you can start talking about, and when you go into 3D, what happens? We talk about waves in a plane, and I’m holding up a ripples in a pond. If you imagine I’m taking my finger and poking it in the pond, I would get a pattern of concentric circles that would go out. 

Rich: We start with the triangles, and you’ll notice as the angles change in here, you can get a sine and cosine of 1 or of 0, but when you go in between, you know, this angle is 45 degrees, halfway between, 0 and 90, the end of this, you see it’s not 50 percent of the way there. It’s actually about 71%. The path it takes describes this wave. These arcs that are sort of circular at the top and, you know, straight lines going through the, the center as they cross from positive to negative, and so sine wave looks like this, and rotate this model, you get a cosine wave. The ways that those interact and, you know, why they look like that, and we can talk about the unit circle model, which we can, as Joan was showing, we made a version that we can roll along a surface, along a piece of Play Doh, we can make an impression of this that draws a sine wave, that’s on the cover image of the book. Other things that we’ve got, so we talk about the angles involved in ellipses, for example, so angles of reflection.

Dale: So let me just try to describe that. You have an ellipse

Rich: So these two holes are the foci of the ellipse. You can draw a line from any point, from one of the focus to any point on the edge of the ellipse, and going to the other focus, and the the total length of those two lines will always be the same, for a given ellipse. So as one gets shorter, the other gets longer. And the angle inside… Is always, it’s the angle of reflection, so if you reflected light starting at one of these points, bouncing off the surface, it would go to the other point. So the outside angles here are equal at all times. 

There are similar functions that we talk about with parabolas, so parabola follows a similar function, so in this case, the distance from the one focus to the total of the distance from the one focus to this line here, which is parallel to a line called the directrix is a constant value. So the distance from the focus to this point and down to a straight line down to this directrix is always the same. 

Joan: And that has an application that it makes parabolic reflectors. So if there’s a light wave coming straight down, and bouncing off the side of it anywhere, it’s all going to go to the focus. Or alternatively, if I have at light at a focus, like a parabolic reflector for a headlight or something, they have a light in the middle, all of it’s going to come out in nice, plain waves because of the the properties of parabola.

So you can use this to talk about, how do parabolic reflectors work. There are hyperbolic reflectors as well. So you can talk about this. So these topics are the analytic geometry part of the book. So trigonometry and analytic geometry are often taught together and sometimes have the horrible name “pre-calculus”, which I find kind of weird because they’re a discipline in and of themselves. I always like to think that you’re learning math that’s inherently interesting at a moment and not because you’ll need it later. I hate it when people say you’ll need this math later.

Dale: Talk about how the standards in math are evolving. I think they are different in different parts of the country.

Joan: There were standards across the country called Common Core, which I forget what fraction of states adopted them, but it was a large fraction. It wasn’t a hundred percent. It was a lot of them and those were updated in the late 2010 someplace and they’re updated about every ten years or so. And so the most recent updating should have been 2021 or so.

But of course that was mid pandemic when everything was nuts. The new standards are still kind of in flux. So some people are still using the old ones. Some are using the new ones. Standards in education are a bit of a political football everywhere at the moment, which doesn’t help.

When we’ve been writing our books, we decided to, instead of necessarily saying we line up with these standards here, we’ve been listing topics because we figure that teachers can work from a list of topics and whatever standards they’ve been asked to follow, they can translate pretty easily.

Trigonometry has a little table in the back of what topics are covered where. People can work off that pretty easily, I think. 

Calculus is a little bit different because there isn’t really, there aren’t really formal standards for calculus. Most people think of that as a college subject that you’re teaching, you know, maybe you’re teaching a little differently in high school. So, there aren’t really standards for that per se.

Dale: You and Rich have used these materials and models in, and taught summer programs, haven’t you?

Joan: This summer, we taught calculus again at IEA. It’s a gifted program, gifted enrichment program, and we had kids from age 9 to 11 taking calculus.

We went at it very conceptually, using our models, having them make models, having them play with it and backtracked into trigonometry a little bit because those kids were a little younger and hadn’t seen some trig things. Fortunately, we had models from this book to sort of fill in some of trig, so that was convenient. We had taught kids 12 to 14 before, so we went a little bit younger with this one, just because that, happened to be what was in the program this year.

But kids love to build things, as you know. One of the gratifying things about having these is we do a lot of events and we’re really looking, we’re looking forward to Bay Area Maker Faire. We’ll be there the first weekend of that. One of my favorite moments is I guess it was LA Maker Faire, or maybe it was Orange County, I don’t remember. Anyway, one of them, we had all their models out, and they’re very colorful, and they always draw a huge gaggle of kids, and disproportionately girls, which I think is pretty great. I don’t know why that is, but it is. And so there’s a little girl in a princess outfit, and she must have been like six, comes up, and she’s playing with all the models, and she’s trying out all the things, and having us talk them through, you know. So finally, her mom says, you have to let some other children play with these.

And she’s towed away, shrieking at the top of her lungs: “No, more math! Oh no, more math, please!” You hear this descending please as she’s dragged away. And it’s like, oh, I high fived Rich and said, “our work here is done.” That’s what we want, right? Because you want math to be fun, you want kids to shriek when math is over.

And maybe not so loud.

Dale: Some of the advanced topics are really accessible to even younger people. Given the right context, they’re able to grasp a lot of these concepts.

Joan: Well, I hate the phrase you’ll need it later. There’s a marvelous, marvelous little book called The Mathematician’s Lament, by Paul Lockhart, who’s a math teacher in New York someplace. He says, if we taught art, the way that we teach math, you’d say, okay, in kindergarten, you’re going to learn to cut straight lines because you’ll need it later. And in third grade, you’ll, something else, you’ll learn to, to pencil in a box or, and you do all these things and then by the time you get to high school, you say, okay, now you can draw something.

You’d hate art because it’s boring. I think people treat math that way because I think calculus is the point where things are interesting because it underlies our modern knowledge of physics. And that’s why Newton invented it, or Leibniz, depending on your religion, but we won’t go there. But anyway, people could teach these calculus concepts as we have, in age 9 or 10, if you have a physical model, because it’s so much physicality to it. Then you have this underlying place, and you can see where you’re going, and you can start to daisy chain on these other things that you learn along the way.

I know Rich has opinions 

Rich: A lot of, math, our calculus book, in particular, covers some stuff that I think has been arranged the way it is and is taught in the order it is and to make it easier for the teachers to grade than actually for the students to understand.

So there’s a lot of memorization involved and just repeating stuff back to make sure that you’ve memorized it because that’s what’s easy to assess as a teacher. But we meet a lot of people who took calculus or took trigonometry and didn’t retain any of it because they just memorized enough stuff to pass a test and didn’t really understand it.

So we hear every time we show this stuff, we hear people saying, oh, I wish I had this stuff when I was learning this concept.

Dale: What’s some of the feedback you’ve gotten from teachers who have used your books?

Joan: Well, we haven’t really heard that much from teachers, per se. I mean, we have heard from students.

Somebody came up to us at Maker Faire and said my teacher used that that same Lego bricks for getting started with Calculus. And somebody came up and said, My teacher used that in class and it really made it so much clearer.

So the book hasn’t been out that long and so, by nature of things we haven’t heard that much. But clearly with students, they really enjoy getting into this and some of them want to build houses with the models instead and stuff like that because they’re kids.

We also try to talk historically in our books. So here’s how the person who designed this and who developed this in the first place, this is the problem they were trying to solve. And here’s what they were thinking about.

And here’s the problem. How would you solve it? Here’s a hint and here’s an object. And then, let’s walk through with some simple object or something. In Trig, we have them make a very simple slide rule, either with paper or with 3D prints. I was right on the cusp of not really using a slide rule. I used a little bit. 

But it’s a really good way to learn logarithms and to get a lot of number sense, because you really learn how to what does it mean to have a logarithm and stuff when you’re moving around physical things. We had our nine to 11 year olds playing around with these; we just printed up a bunch of slide rule scales to move around.

Dale: Who invented the slide rule?

Joan: It’s about this about the time of the Mayflower. There’s three or four people that did variations on it, but it’s contemporaneous with the Mayflower. It’s old. Napier rationalized it. The first one was with some weird base that escapes me at the moment for reasons that made sense to them at the time, but base 10 is Napier, but earlier than that, there were a couple of others.

Navigation drove things for quite a while because they had to do a lot of analysis at sea or analysis in a trench with their Duke asking them, what angle do I point the cannon?

 Ballistics and navigation drove math for a lot of the period where trigonometry is being developed. And that’s a lot of its roots are there 

Dale: I would suspect that a lot of it is also used in the space program 

Rich: The ellipses I was showing earlier, so the foci are also the points ,where a gravity body would be. So if I’m going around the ellipse, I’m orbiting something that’s at one of these two foci.

I’ll go faster when I’m closer to the focus and further when I’m further away. The parabola and the hyperbola are shapes of orbits I can take. If I don’t get close enough to a body at a low enough speed to get into a stable orbit, I’m going to take a parabolic or hyperbolic path that passes it. These are, yes, very, very relevant to orbits and rocketry as well.

Dale: Anything else we need to touch on that you want to kind of emphasize in your book?

Joan: I think that the overall thing that we have tried to emphasize is that you can start with a physical model and or a project or an exploration and learn all the notation. That’s a little bit backwards from how people tend to do it. 

 It’s so valuable to do the model up first, and I think we’re there, there are a few other people who do this but I, I think we’ve covered a lot of the core subjects with physical models.

Dale: Well thank you for writing this book, and it’s really just coming out now. 

Joan: I did want to mention, by the way, that both Trigonometry and Calculus, have exclusively on the Makershed, have an EPUB version, which if you happen to be visually impaired or be teaching somebody who’s visually impaired, it has screen readable equations.

And as far as I know, we are, the only ones I know of that have trig and calculus that way. And then Geometry is a PDF with alt text on the illustration. So if you’re visually impaired, this is three books that might be very, very helpful

Dale: Well, thank you, uh, Joan and Rich. 

Joan: All right, thank you

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