Category Archives: Math on the web

Geogebra has new skills

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A new version of Geogebra has been released, in beta. It’s called Geogebra 5.0, and you can see the news about it here. Or, here’s a direct link to launch it right away. Thanks to The Cheap Researcher for the lead on this. As readers of this blog may already know, I love Geogebra!

One of the main highlights is that Geogebra now supports 3D manipulations. Awesome! However, don’t get too excited–it doesn’t let you graph anything except planes. No surfaces. It will do geometric constructions, like spheres and prisms. Using parametric equations and the locus feature, you can coax it into rendering spirals or other space curves. [edit: I figured this was possible, but it actually wasn’t. Not sure why.]

Another highlight, which I find even more exciting, is that Geogebra now has a built in CAS. Here’s a screen shot of me playing around with a few of its features. It also has a ways to go, especially for those who are used to more robust systems like Mathematica/Maple/Derive/TI-89. But this is a great step in the right direction, and 10 points for the open-source camp!

Notice that it can work with polynomials in ways you would expect, it can symbolically integrate and derive (simple things), perform partial fraction decomposition, evaluate limits, and find roots. Here are a few more things it can do. Strangely, it had problems finding the complex roots of a quadratic (easy), but not a cubic (hard). Just take a look at my screen shot. Seeing that it did okay finding the complex roots, I wondered if it could also plot them for me. I started by entering (copying and pasting) the complex zeros as points in Geogebra, which worked. But then I discovered the new ComplexRoot[] function which approximates the roots and plots them on the coordinate plane all at once. Cool! Here’s the screenshot:

The seven complex roots of f(z)=z^7+5z^4-z^2+z-15

As you can see, I asked for the roots of a 7th degree polynomial. Since the polynomial had real coefficients, notice that every zero’s conjugate is also a zero, as we’d expect. And we also expect that at least one solution of an odd-degreed polynomial will be real (notice this one has only one real root, approximately 1.22).

That’s all I’ve discovered so far. I’ll let you know if I come across anything else exciting. Keep in mind that this is beta, so the final release will likely have all the bugs worked out and more features.

Vi Hart’s Blog

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It’s high time I gave a bit of press to Vi Hart’s Blog. If you haven’t checked it out, do so right away. It’s brilliant.  A number of people have pointed me to her blog, including one of my Calc students. Her little math videos are fresh, funny, and insightful. Denise, at Let’s Play Math, gave her some press too, which is what reminded me to finally make this post. Here’s the video Denise highlighted (the most recent of Vi’s creations):

This is particularly appropriate because there were a couple of us in our math department discussing this very question: In total, how many gifts are given during the 12 Days of Christmas song? It’s a nice problem, perfect for a Precalculus student. Or any student. Here’s a super nice explanation of how to calculate this total, posted at squareCircleZ. But before you go clicking that link, take out a piece of scrap paper and a pencil and figure it out yourself!

Here’s another nice video from Vi Hart:

You could spend a lot of time on her site. Here’s another awesome video. I’ll have to have my Precalculus class watch this one when we do our unit on sequences and series.

And you’ve got to love the regular polyhedra made with Smarties ,  right?

Plus, Vi Hart plays StarCraft, which is awesome too.  Back in the day, I really loved playing. I haven’t played in a while, and I certainly haven’t tried SC 2 yet, because then I’d never grade my students’ papers.

Bottom line is, you need to check out all the playful stuff Vi Hart is doing at her blog. Happy Wednesday everyone!

 

What if we Graded Toddlers?

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From techdirt.com

What If We Gave Toddlers An ‘F’ In Walking?

from the rethinking-education dept

theodp writes “To improve math and science education, Physics prof Dr. Yung Tae Kim thinks professors and teachers should take a page from skateboarding. ‘The persistence and the dedication needed in skateboarding — that’s what we need to be teaching,’ explains Kim. ‘No one says to a toddler, ‘You have ten weeks to walk, and if you can’t, you get an F and you’re not allowed to try to walk anymore.’ It’s absurd, right? But the same thing is true with math and science education. If you want to learn trig or calculus, it’s set at such a pace in schools that it guarantees that only the absolutely best students will learn it.’ Kim says it’s possible to ‘polish the turd’ of high school and college education, and lays out his plan for doing so in Building A New Culture Of Teaching And Learning (YouTube: parts 1–2–3), a video drawn from a farewell talk he gave to his Northwestern students. There’s more on The Way of Dr. Tae at DrTae.org and PhysicsOfSkateboarding.com.”

I was just discussing the same point with my father in law this past week. Our education system needs to change in fundamental ways if we want students to truly learn at their own pace. We do a bit of a disservice to students who need to take the material a little slower. There’s nothing wrong with taking things slowly. Likewise, we do disservice to students who could complete the coursework in half the time.

[Hat tip: Tim Chase, as usual :-)]

 

Pythagorean Theorem

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Just when I said the other day that we care less about the ancient Greek mathematicians than we did 200 years ago, I turn the attention of my readership to the Pythagorean Theorem. It’s a simple repost: I liked the following video that Denise @ Let’s Play Math posted yesterday and wanted to share it. It also connects nicely to the post my dad made the other day about beautiful proofs.

Give the Babylonian’s some credit

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So says this CNN article from Friday.

Over 1,000 years before Pythagoras was calculating the length of a hypotenuse, sophisticated scribes in Mesopotamia were working with the same theory to calculate the area of their farmland.

Working on clay tablets, students would “write” out their math problems in cuneiform script, a method that involved making wedge-shaped impressions in the clay with a blunt reed.

These tablets bear evidence of practical as well as more advanced theoretical math and show just how sophisticated the ancient Babylonians were with numbers — more than a millennium before Pythagoras and Euclid were doing the same in ancient Greece.

“They are the most sophisticated mathematics from anywhere in the world at that time,” said Alexander Jones, a Professor of the History of the Exact Sciences in Antiquity at New York University.

He is co-curator of “Before Pythagoras: The Culture of Old Babylonian Mathematics,” an exhibition at the Institute for the Study of the Ancient World in New York.

more…

[Hat tip: Mr. Gherman]

Google Ngram

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This is super fun. Google has just released this tool for playing with word frequency data from a huge amount of scanned literature (5 million books dating as far back as 500 years). You can read more about it here, including some nice research that’s already being done with the full data set that’s also been released. (also here)

For example, here’s a graph of the appearance of the word “homeschool” in the collective Google corpus.

You can also compare the appearance of words. For example, here’s informal evidence that we care less about ancient Greek mathematicians (BC) and more about European mathematicians (17th and 18th century) than we did 100 years ago.

Not very rigorous, I’ll admit. But it’s an example of what kind of interesting trends can be instantly teased out. As this article quotes Erez Lieberman-Aiden of Harvard University, “It’s not just an answer machine. It’s a question machine.” I think that’s a nice way to put it.

 

The Important Theorems Are the Beautiful Ones

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Dr. Gene Chase guest blog author here again.

What makes a math theorem important?

The usual answer is that it is either beautiful or useful. If like me you think that being useful is a beautiful thing, then important theorems are the beautiful ones.

But what makes a theorem beautiful? For example, why is the Theorem of Pythagoras widely regarded as beautiful: and a, b, and c are not 0 if and only if a, b, and c are the sides of a right triangle? (OK, break into small groups and discuss this among yourselves! An answer appears at the bottom of this post.)

But the theorem 1223334444 = 1223334443 + 1 is not beautiful, won’t you agree?

If the theorem is geometric, we can appeal to visual beauty. For example, three circles pairwise tangent have a beautiful property that is animated here.

But beautiful theorems do not have to be geometric. Numbers are beautiful. For example, Euclid’s theorem that there are an infinity of primes is beautiful. No one has been able to draw a beautiful picture about that, although people have tried from astronomer and mathematician Eratosthenes in 200 BC to science fiction writer and mathematician Stanislaw Ulam in 1963.

For $15 you can have a mathematical theorem named after you. But I can guarantee that it won’t be beautiful. So if you want a theorem named after you, give Mr. Chase the $15 instead and he’ll find one for you. Don’t use 1223334444 = 1223334443 + 1. I claim that as “Dr. Gene Chase’s theorem.”

Answer to discussion question above: Most folks say that a beautiful theorem has to be “deep,” which is just a metaphor for “having many connections to many other things.” For example, the Theorem of Pythagoras has to do with areas, not squares specifically. The semicircle on the hypotenuse of a right triangle has an area equal to the sum of the areas of the semicircles on the adjacent sides. And so for any three similar figures.

Do you remember the joy that you feel when you first learned that two of your friends are also friends of each other? That’s the joy that a mathematician feels when she discovers that the Theorem of Pythagoras and the Theorem of Euclid are intimate with each other. But I’ll leave that connection to another post.

Math is about surprising connections. Which is to say, it’s about beauty.