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.
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.
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:
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!
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.
This is a classic problem solving activity that I first saw many years ago in Mathematics: A Human Endeavor. The NCTM has a nice applet to test cases one at a time.
Since I like playing with Excel and trying to extend what it can do, I wrote a spreadsheet to represent the problem. The end result is much better than expected — this representation uses a slider to show many cases in a short period of time, and with it I noticed patterns that I never had before (e.g. if you start at (1,0) on a rectangle whose sides have GCD=2, you get a loop that goes through all squares). Read more >>
Making a multiplication table is an excellent beginning Excel Activity. This video shows an efficient way to do it using absolute references, but there are many other approaches. I use an open-ended handout for this activity in class. This video is a good introduction to absolute references. Spreadsheet used in the video.
I was pretty excited when I figured out how to use conditional formatting in Excel to make a Sierpinski triangle. I used very tiny cells, and then used conditional formatting to color in the cells corresponding to odd numbers in Pascal’s triangle. The video also introduces Pascal’s triangle and explores some number patterns. Spreadsheet used in the video (Excel 2003).
To do the conditional formatting in later versions of Excel, go to the Styles Group on the Home Tab. Choose Conditional Formatting, and then choose New Rule. Choose the last option, “Use a formula to determine which cells to format.” Then enter the formula as in the video.
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