Math Topics

Intersection of Three Cylinders

Puzzle books I had as a kid often asked what the intersection of three cylinders looked like.  Even looking at the answer in the back, I still had trouble seeing it.  Making this video helped a lot.   The Sketch Up file that I used is available in the 3D Warehouse.

Trigonometry Yoga

Which is bigger, the sine of 40\(^{\circ}\) or the sine of 50\(^{\circ}\)?

This is a great trigonometry assessment question.   Unfortunately, virtually none of my college students who haven’t used trig for a while can answer it (without a calculator).   For a lot of students, trig is one of those subjects that just didn’t stick very well.

So, we start trig functions all over, but this time with circle definitions and some “trigonometry yoga” (no trademark; I made up the name when I was thinking of a title for this post).

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\(k^x-x^k\) Slider and \(x^y=y^x\) graph

Use the slider to change the value of k, and to see dynamically how the graph of \(h(x)=k^x-x^k\) changes. Negative values have \(x^k>k^x\). Which value of k gives a graph that is never negative? Why?

Here is a very pretty graph of \(x^y=y^x\) with areas where \(x^y < y^x\) shaded in green and areas where \(x^y > y^x\) shaded in purple.

I am working on an article about some problems related to these equations, including The Biggest Product Problem. There’s a lot of interesting stuff here!

The Tangent is a Tangent!

In my post Trigonometry Yoga, I discussed how defining sine and cosine as lengths of segments in a unit circle helps develop intuition for these functions.

I learned the circle definitions of sine and cosine in my junior year of high school, in the class that would now be called pre-calculus (it was called “Trig Senior Math”). Two years earlier, I’d learned the triangle definitions of sine, cosine, and tangent in geometry class. I don’t remember any of my teachers ever mentioning a circle definition of the tangent function.

The geometric definition of the tangent function, which predates the triangle definition, is the length of a segment tangent to the unit circle. The tangent really is a tangent! Just as for sine and cosine, this one-variable definition helps develop intuition. Here is the definition, followed by an applet to help you get a feel for it:

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Random Cool Math Thing #2

Pythagorean Card Table
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Pick a Number Between 0 and 1 — with Fractions!

This applet is a pick a number game, where the secret number is a fraction between 0 and 1. Enter a numerator and denominator and the computer will tell you if your fraction is too high or too low, until you guess the Secret Fraction and win.

Your previous high and low guesses are listed on the page (unless you hide the list), and there is a number line that graphs the guesses. You can change the number of divisions in the number line; the labels update automatically.

The game has several levels. Level 1 uses only fractions with denominators 2, 4, and 8, whereas Level 6 uses denominators from 2 to 16.

The applet doesn’t fit well on the page with my website theme, so you can open a new window and try it here. (Note: if you don’t see a fraction in the pink box, go to the bottom of the page and choose, “View as Java Applet,” which is what I wrote this applet as.  It does not seem to have converted correctly to HTML 5).

This applet was the most challenging to write of all the GeoGebra applets I’ve written to date — I wasn’t sure it was possible to create this game using GeoGebra, so it was quite satisfying when it came together. Here were some of the tricky parts (you can download my original file here):

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Random Cool Math Thing #3

Odd Perfect Squares
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Real Life Fraction Problems

It’s hard to make sense of multiplication and division of fractions.   Before fractions, multiplication can almost always be seen as repeated addition, and the answer is never smaller than both the numbers you started with.   Division is usually sharing, and the number of cookies each child gets is always smaller than the total number of cookies (unless someone is selfishly dividing by 1 and not sharing with any of her friends).

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Farey Fraction Visual Patterns

The Farey Sequence, \(F_{n}\) is the list of all simplified fractions between \(\dfrac{0}{1}\) and \(\dfrac{1}{1}\) with denominator less than or equal to \(n\). For example,

  \( \large F_{5}=\dfrac{0}{1},\dfrac{1}{5},\dfrac{1}{4},\dfrac{1}{3},\dfrac{2}{5},\dfrac{1}{2},\dfrac{3}{5},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5},\dfrac{1}{1}\)
 
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Constructive Criticism of Khan Academy’s 1st Video on Fractions

The video below is my entry into the MTT2 contest, where people are making commentary videos on videos from the Khan Academy (KA). This contest began when John Golden and David Coffey posted their “Mystery Teacher Theater 2000” video, where they provided running commentary on a KA video on arithmetic with integers. Their video was quite sarcastic, and many people criticized it for not being constructive. I aimed for constructive in my video, but I think the effect of John and David’s video was also quite constructive, as it’s opened up more space for dialogue (some productive, some less so) than a less provocative video (like mine) is likely to do. Read more >>