Author Archives: Mr. Chase

Constructing a trapezoid using the side lengths

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Lloyd left a comment on a post of mine yesterday, asking:

how do you draw a Irregular quadrilateral trapezoid with fixed dimensions for the two parallel bases and the two legs with no angles given using geometry tools?
top base= 328
bottom base= 223
left leg =220
right leg= 215

How would you answer this question? It’s not trivial. You’ll quickly find that if you do a straight-edge and compass construction, you’ll need the height of the trapezoid.

If we let a, b, c, and d be the side lengths of a trapezoid with a and c as the bases, can we express the height h as

h=\sqrt{b^2-\left(\frac{d^2-b^2-(c-a)^2}{2(c-a)}\right)^2}

This number is constructable, but would take some work to actually construct it on paper. Perhaps we can return to that particular question later. For now, we can let GeoGebra show us the general idea. I’ve made this applet in which you can change the side lengths and the trapezoid will be constructed. I used the height formula above to calculate the height, and the applet shows this value.

Footnote:

Just to hit home my usual point one more time, the figure above is ALWAYS a trapezoid, even when sides b and d just happen to be parallel. Just remember that a parallelogram is a special case of a trapezoid!!

Nine important equations

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Certain Uncertainty: Schrödinger Equation

From Wired.com…

9 Equations True Geeks Should (at Least Pretend to) Know

By Brandon Keim

Even for those of us who finished high school algebra on a wing and a prayer, there’s something compelling about equations. The world’s complexities and uncertainties are distilled and set in orderly figures, with a handful of characters sufficing to capture the universe itself.

For your enjoyment, the Wired Science team has gathered nine of our favorite equations. Some represent the universe; others, the nature of life. One represents the limit of equations.

We do advise, however, against getting any of these equations tattooed on your body, much less branded. An equation t-shirt would do just fine.

The Beautiful Equation: Euler’s Identity

Also called Euler’s relation, or the Euler equation of complex analysis, this bit of mathematics enjoys accolades across geeky disciplines.

Swiss mathematician Leonhard Euler first wrote the equality, which links together geometry, algebra, and five of the most essential symbols in math — 0, 1, i, pi and e — that are essential tools in scientific work.

Theoretical physicist Richard Feynman was a huge fan and called it a “jewel” and a “remarkable” formula. Fans today refer to it as “the most beautiful equation.”

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I was glad to see that the first “must have” equation was Euler’s Identity (note that “Euler’s Identity” is the accepted name for this, not to be confused with Euler’s Formula or Euler’s Polyhedron Formula or any of the other amazing facts named for Euler). I think there’s large consensus in the math community that this is, indeed, a breathtaking equation. It may not be the most fundamentally important, but it definitely showcases why mathematicians delight in math.

I’m ashamed to say it, but I hardly knew any of the other equations. I knew Boltzman’s equation; Maxwell’s equations and Schrödinger’s equation have come up in some of my graduate coursework, but the others I hadn’t ever seen. One might argue that the other equations are not so important. (If you like arguing about such things, join those commenting on the article).  You should still look through the list yourself; how many of these equations do you know?

Granted, this was a general article that encompased all “true geeks” not just math geeks. But still, don’t we all want to be a true geek?

(Oh, and happy birthday to Johan (III) Bernoulli, who had no notable equations named for him :-))

7 billion

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Looks like by all accounts, there are now 7 billion people in the world today. At least that’s what wikipedia says. Here are two blog posts on the subject from the math blogging community. I have to say, I was surprised to see that wolframalpha doesn’t know anything about this important event (I’m sure it won’t be the last time alpha disappoints me). Anyway, I hope everyone feels humbled to be just one of the crowd.

And, happy birthday to Karl Weierstrass!

9999

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Happy 9999th day of life to Mr. Chase! 🙂

I’d like to see 9999 candles on a cake!

Actually, I guess tomorrow is the big day. Maybe I’ll post again then :-).

Bonus points for:

  1. calculating my age (easy)
  2. calculating my birthday (a bit harder)

Why are infinite series so hard to grasp?

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I’ve posted on infinite series a few times before. But I was inspired to touch on the topic again because I saw this post, yesterday, over at the Math Less Traveled. Actually, the post isn’t really about infinite series as much as it is about p-adic numbers and zero divisors. I’m excited to read more from Brent on this subject. But I digress.

The point I want to make with this post is that students struggle with wrapping their minds around convergent infinite series, and yet they live with them all the time. Students have inconsistently held beliefs about infinite sums.

The simplest convergent series is a geometric series \sum_{n=1}^\infty a_n=a_1r^{n-1} which converges to \frac{a_1}{1-r}. The easy proof of this fact goes like this: we look at the sum formula for a finite geometric series, s_n=\frac{a_1(1-r^n)}{1-r} and we notice that

\lim_{n\to\infty}\frac{a_1(1-r^n)}{1-r}=\frac{a_1}{1-r}

for |r|<1.

But this proof isn’t very satisfying for the student encountering infinite series for the first time ever. Evaluating the limit feels like ‘magic.’ The idea of adding up an infinite amount of things and getting a finite value is unsettling. I admit, it sounds like quite a lot to swallow. That being said, however, students have no problem declaring the infinite series

0.3 + 0.03 + 0.003 + 0.0003 + \cdots

to be 1/3. It’s not “close to” 1/3, it’s not “approaching” 1/3, it IS EQUAL TO 1/3. And my Precalculus students already accept this as fact. So without even thinking about it, they’ve been living with convergent infinite series all along. Hah!

Once they finally shake their denial, they can more easily accept the convergence of other infinite series like \sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}. At first when students encounter a series like this, they think, “surely we can’t say the sum is EQUAL to \frac{\pi^2}{6}. It must be close to \frac{\pi^2}{6} or approach it, but equal to?” But the same students make no such distinction with 0.3+0.03+0.003+\cdots = \frac{1}{3}.

So there it is. An inconsistently held belief about infinite sums. To the student: You cannot have it both ways. Either you must agree with, or deny, both of the following equations:

0.3+0.03+0.003+\cdots = \sum_{n=1}^\infty 0.3(0.1)^{n-1}=\frac{1}{3}

1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}

But to believe one equation is true and the other is only ‘kind of’ true is inconsistent. I rest my case. 🙂

Teaching math with applications

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Wow, if the title of this post didn’t grab you, I don’t know what will. Pretty riveting, right? Has anyone ever thought of teaching math with applications? </end sarcasm>

This is the basic thesis of a recent article from the NY Times, “How to Fix Our Math Education” by Sol Garfunkel and David Mumford:

THERE is widespread alarm in the United States about the state of our math education. The anxiety can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. Bush’s No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not.

All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.

Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a “reform” version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.

For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a “group of transformations” or a “complex number”? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.

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from toothpastefordinner.com

To be fair, the real thesis–if you read further in the article–is that we should primarily teach applications and math can swoop in and rescue us if and when it’s needed:

Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.

If you haven’t gathered already, I don’t agree at all with this thesis. It’s my opinion that math should be taught as math, respected as its own field of study, and a valuable part of a high school liberal arts curriculum. Students should value math for its inherent, abstract beauty. Applications are of course a must in any course. But I find that in the text resources I’ve used, the applications are often contrived. Extremely contrived. Doing math should feel like playing a game, like working on a puzzle, or like arguing.

In fact, when high school students ask why are we learning this?, I FIRST respond with the things I just said: It’s part of a liberal education; it will make you a well-rounded, intelligent person who can hold conversations with other smart people in other fields; and it’s fun. I mention SECOND what applications exist for the math we’re learning. For high school students, if we’re honest, most of them will never need any of the math we’re teaching. Seriously. If you’re not working in a math or science field, when was the last time you had to factor a polynomial?

The authors go on to say “Science and math were originally discovered together, and they are best learned together now.” But this is not universally true. In many cases, the ‘useless’ math was developed first (think of number theory for instance) and then only later were applications discovered (think of the RSA or El Gamal public key encryption schemes).

So why learn math? There was a nice post about this yesterday on one of my new favorite blogs, dy/dan, titled “Cornered By The Real World.” He highlights this great article by Samuel Otten in this August’s Math Teacher magazine. I highly recommend reading the whole article. As a taste, I’ll include the same snippet that Dan shared:

I believe that thinking and acting as if the justification for teaching and learning mathematics is found solely in everyday applications can be dangerous. Mathematics does not exist only to serve other professions, nor is it merely a collection of algorithms and procedures for dealing with real-world situations. Such a mind-set essentially paints our discipline into a weak and lonely corner and leaves undefended many of its greatest aspects.

I could say more about this, but I’ll spare you. I’m passionate about making math a subject worth learning all on its own. If I say more, I’ll start to sound like Paul Lockhart.