Spirolateral Break

Spirolaterals are an idea that I don't remember where I read about them, but the idea is relatively simple. You take the the decimal expansion of some fraction and using the digits create segments of those lengths then rotate some fixed number of degrees. I have an Excel spreadsheet that creates these pictures. If I figure out a way to post the file I will. Let me know if you know of a way to do that.

Here is 8/147 with a 90 degree turn at each step.

Math Elective

I have been reading a number of the posts on David Bresoud's blog at maa.org. They are slowly bringing me to the conclusion that my profession is in danger. More and more I see math education mirroring the american automotive industry. We are certain that there will always be a demand for what we provide. We are certain that people will desire a version of that product that is nearly identical to version we produced 20 years ago. This may not be true, and we have gone a long way to "reform", but there is still a long way to go. Bresoud articles seem to point out consistently our ridiculous adoration of calculus as the bridge between high school and college mathematics. He points out also how this leads to all kinds of systems and policies that lead people to take math they don't want and they don't need. The math they do want and do need is often under-supported, under-credited, and under-appreciated.

Those are my interpretations of his articles. You may have your own. One of his more recent posts, here, talks about the indicators of success in college. And of course points out what many know, the SAT and ACT are not particularly good predictors of college success. Which is what I understood their importance to be in the first place.

So what if SAT and ACT went away?

At first, I was excited by the prospect. So many pieces of ridiculous mathematics could be jettisoned from the curriculum. We would have freedom to create programs that make sense for today. It would be easier to rest control of the curriculum from calculus and refocus on statistics and discrete math. More and more students are going in biological sciences (need statistics) and fewer and fewer are becoming engineers (need calculus).

Then the dark side of it hit me. If math teachers could say you need this for SAT, then what is the likelihood we could keep our requirement status. Would you really need four years. How many students and parents would like their child to not HAVE to take math. Based on how many people regularly tell me that they hate math or were never any good at it, A LOT.

And what if happened quickly. Where would we be?

This is why I feel we have to teach math as if it were an elective. Every class. Every class needs to have a clear purpose. Students should not leave high school without knowing how to use excel, because if you are going to do real math in the real world you are going to use excel at some point. So screw the calculator and get the computer.

How much data is being produced today? Way more than can be analyzed currently, but they are looking for people to do it. Why has AP Statistics grown and grown. When will it plateau?

I guess I have a lot of questions, but I don't know how to rattle the colleagues I see around me to the coming danger.

Visualizing Numbers Real and Imaginary

In the video below I try to give some of the "patter" I use when I introduce the idea of "i" for the first time. I think that there is a great benefit in doing it this way because it helps strengthen their understanding of what real numbers do as well. It also emphasizes that numbers often get paired with operations. I actually could make that clearer. Anyway, feel free to steal this introduction to use in your own classes.

Visualizing Complex Numbers

In my continued quest to spread an understanding of complex numbers I put together this little dance sequence with my Summer School Algebra 2 students. I used the function f(x)=(x-1)^2+1 determine the dance sequence. This function has two complex roots (1+i) and (1-i). I had students stand at 1+i, i, -1+i, 1, -1, 1-i, -i, and -1-i, then we went through the three steps of the function. These were "minus 1", "squared", and "plus 1". The most important visual here is to get a sense of what squaring does to the complex plane. This includes some expansion of the numbers with distance greater than 1 and a wrapping of plane on top of itself. Watch the video and let me know what you think?

How is a function like a recipe?

I was talking about function notation with my students, and trying hard to differentiate between f, f(x), and f(x)=x+3. The metaphor that I tried was recipe. Does anyone else have a good metaphor that helps to distinguish these concepts?

Sneetches = Inverse Functions

Teaching summer school today. I had a student not understand what f(f^(-1)(x))=x was trying to say. I was searching for a process that would help her understand doing and undoing. And then it hit me. Sylvester McMonkey McBean.

Conic Movie

I have been reading the book Conics by Kevin Kendig. This movie was made with Grapher on my Mac, and displays the viewpoint of taking conics and projecting them down on the sphere.

Make Complex Numbers Real

This post is continuation of this rant.

We do such a bad job with complex numbers that students often assume that the word Imaginary in front of Number is an adjective that i is not really a number. It is some sort of imaginary thing.

One important reason that students find this easy to believe is the fact that it is not possible to order the complex numbers. Ordering is a basic and fundamental concept for numbers. Ordering is first thing that students are to learn about numbers. We stress the idea of ordering of the numbers, we use the ordering of numbers to help explain addition and subtraction. So if we can neither tell if i is neither bigger or smaller than 0, then it might not really exist.

So makes imaginary and complex numbers real. What makes them real is that can represent things that happen in the real world. Real numbers can represent length and direction (direction in one dimension, left or right, up or down). Complex numbers can represent length and direction in two dimensions. So 3+4i can represent 3 steps to the right and 4 steps up. You can represent movement in two dimensions with complex numbers. But that is not different than vectors. What makes complex numbers so different is the existence of a coherent multiplication rule, one that even allows a coherent definition of division.

So it is multiplication that makes complex numbers so special. What does multiplication in complex numbers do. It does two things, it is capable of doing what multiplication of real numbers does, which is dilation. Multiplication by a positive real number performs a dilation on the points of the complex plane. Multiplication by a negative real number performs a dilation and a 180° rotation. In particular, multiplication by -1 is just a 180° rotation. Now how about multiplication by i. If you take any complex number and multiply it by i the result is a 90° rotation about 0. And from all that we know about i, this makes tremendous sense. We know that i is the square root of negative one, so that multiplying by i does half of what multiplying by -1 does. We know that i^4=1 so that multiplying by i four times does nothing to a number, just as rotating by 90° four times returns things to where they started. It turns out then that multiplying by complex numbers models rotation! This is what is important. Turns are everywhere. Everywhere in life there are rotations of all sorts and these numbers allow you to do computations with coordinates that affect the desired rotations. Want to turn 90°, multiply by i. Want to turn 45°, multiply by sqrt(i). This is why polar form for a complex number exists, and it is the importance of DeMoivre's Theorem.

As a final note tonight. Let's bring it back to algebra. Let's say we have to solve x^6=1. Then we are looking for 6 solutions. Well then these are complex numbers that multiplication is the same as rotating 0°, 60°, 120°, 180°, 240°, and 300°. In other words 1, 1/2 +sqrt(3)/2 i, -1/2 +sqrt(3)/2 i, -1, -1/2 -sqrt(3)/2 i, 1/2 -sqrt(3)/2 i. But what is better is seeing it on the complex plane.



That is right the solutions to this equation are vertices of regular polygon, and more this generalizes. So that the regular polygons can be associated to the polynomials x^n-1.