Math Topics

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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Random Cool Math Thing #2

Pythagorean Card Table
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Cube Intersected by a Moving Plane

This is my first Sketch Up video.  It is surprising how many shapes the cross-section of a cube can take on, and how hard they are to visualize (for most of us).   The Sketch Up file I used is in the 3D Warehouse.

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.

Random Cool Math Thing #4

Visual Quadratic Reciprocity

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Random Cool Math Thing #3

Odd Perfect Squares
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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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Linear Transformations with Graphics

I made this video when I was thinking about ways to teach the basics of linear transformations and matrices by illustrating their connections to computer graphics.  I wanted to show the actual matrix calculations that lead to the graphics, as well as a slider to show the graphics dynamically.  Since I made the video, GeoGebra added a spreadsheet, and it’s actually much easier and more versatile to do the same thing in GeoGebra (I’ll post that link soon).  However, it was a fun challenge to figure out how to push Excel to do graphics in this dynamic way.  To date, this is my most popular video on youtube.   Spreadsheet used in the video.

Clock Buddies — A Round Robin Tournament Activity

This is a great first day of class activity, which works well with math phobic students (I use it in a math for elementary teachers class). Students make a list 1:00, 2:00, etc. on a piece of paper — however many times I assign — and then they have to make an appointment with a different student in each slot. Students move around the room making appointments and learning each other’s names. If there are an odd number of students, I passively participate, accepting appointments whenever students come and ask me.

At some point a student or a few students will announce that they are done, and I tell them they aren’t done until everyone’s schedule is filled in. It’s an interesting question to figure out a good number of appointments to give them so that they will get stuck, but not overwhelmed; I usually go for a few more than half the number of students (12 for a class of 20). Read more >>

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