Here I have finally figured out how to post a Excel file on my blog. I know, I am not as quick with the technology as I thought I was. When you open it on sheet one, you can change the turn, numerator, and denominator to see different patterns. There is a large image of the graph on another tab of the workbook.
Enjoy.
Spirolateral.xls
So this is an idea that has been bopping about in my mind for a while, and while last year was a crazy hectic year for so many reasons, I have worked to make this year less crazy. I want to do some reviews of articles that I read in Mathematics Teacher from NCTM. I could write letters to the editor I suppose, but the feedback loop on those is too short for this web 2.0 iphone instant message world we live in.
So I give you my thoughts on Chocolate Key Crytography by Dale J. Bachman, Ezra A. Brown and Anderson H. Norton in the September 2010 issue Page 100.
The title is intriguing especially in Mathematics Teacher because it promises something that I know very little about called the Diffie-Hellman Key Exchange, so initially I am very interested. Of course it comes with the requisite Mathematics Teacher cute graphics, but at least none of the painful pictures of students with faux engagement painted on their faces.
The article is broken down into several sections. The first covers what cryptography is, including a helpful breaking down of the word into cryptos and graphos for those of you new to the English language. Followed by an explanation that the internet is important and that concepts about cryptography can be taught to anyone, even high school students.
They then go on to describe a fundamental problem of key creation in cryptography. Essentially, you and the person that you want to communicate with have to pick a number together with out meeting. How can you do that? The authors have created a metaphor for this problem that uses M&M's and at that point one of their colleagues probably told them about how math teachers and Mathematics Teacher love M&M's, hence the article.
The article goes on from there to discuss some of the more interesting mathematics of the problem that includes group theory. The multiplicative group of integers . There is also a lot of discussion of how this is not hard and anyone can do it.
Why should we teach it? You may ask, and if you didn't you should ask yourself why you didn't. Well this shows that math has an application in the real world. Finally an answer to the internal question of when will ever use this. Of course, this doesn't answer when will ever use this. Because your students are not really "using" this. Someone else used to create the internet, and as cool as that is, it is something that is done, finished, kaput.
Overall I give it 3 out of 5. It is a nice application of math, but the authors don't give enough explanation or scaffolding to help a teacher present the ideas in the article to their classes. Their tell us that anyone can understand it, but their description is not detailed enough to help, and then they scare away a significant portion of the high school teachers of the world with the discussion of group theory. They also don't do a good job helping teachers to fit this into the curriculum. How could this connect with other topics and questions that high school students have to understand? Still I like cryptography, and I would like to see more computer science and discrete math covered in schools.
Great writing inspires. I have been reading over the last several months a small book that I have found very inspiring. Everytime I pick it up I get new ideas, and I challenge my assumptions about teaching. The most surprising thing about it is the book is not about math, not about teaching, not about students. It is about drama. It is called The Three Uses of the Knife by David Mamet. It speaks to the nature and use of drama.
I know that there is a general sense among many teachers that class does not need to be a show everyday, and clearly that attitude taken to extremes creates teachers that are more like Michael Scott of "The Office" than Dan Meyer.
But Mamet forcefully explains that humans have a dramatic urge. People want to see themselves as part of a giant play with themselves as the hero. As teachers we have a responsibility to engage students, and to ignore this paradigm seems short-sighted at best.
Mamet is very insistent that there are good forms of drama and bad forms of drama. He uses school, politics, the evening news as examples of bad drama carried out every day.
In the next few posts I will reflect on certain sections on Mamet's work and describe what I think it means for teaching. We will start the section he calls The Perfect Game.
Imagine the perfect game. Is it your favorite team thumping their opponent? Of course not, it is your team starting out well, looking dominant and then all of a sudden things fall apart. They change many things and nothing seems to work. Then all of a sudden when things look darkest, a player redeems himself and for some previous screw-up in the game, and scores the go-ahead score, but then the referee calls it back, and again the players must find a way to take the lead, and they do, but the other responds to retake the lead, only to have a more miraculous play occur for your team, and so on and so on.
This is the prototypical three act structure. Humans look for this structure in their lives all the time, and if you watch sports, or follow politics you will see that these activities are recast by commentators in this framework all the time. Mamet would tell us that this recasting is for our pleasure.
Mamet describes this as "Yes, No, But Wait...", and it repeats again and again.
You can imagine this in a classroom, students come in an see a problem something familiar that they know and can deal with. They deal with it easily and look to you for their deserved praise. Then you give them another problem similar, but with some unexpected twist. They work forward trying many things. Many of those ideas fail, and the way seems lost. You don't help them (much, if any) then at some moment through an idea spread through the class based solely off of students previously failed idea (or through some cryptic magic incantation you said under your breath) and the students are off and they solve the problem, or they don't stymied by some other detail they have left out, or you have given them a new problem with a new twist. In this way, we model "Yes, No, But Wait...". And basically we should believe that this is what students want, crave, desire. They do not desire the answer, the algorithm, the process. They desire to be heroes.
Part of what Mamet talks about is our never ending need to dramatize our lives:
For we rationalize, objectify, and personalize the process of the game exactly as we do that of a play or drama. For, finally, it is a drama, with meaning for our lives. Why else would we watch it?That idea about rationalizing, objectifying, and personalizing is so important. Students need to do that to really connect with the content that we are looking for them to master. If they don't do it this more complex and deeper level, then they will never remember it.
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.
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.
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.
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?
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?
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.
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.
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.
Are you a math teacher? If you are then you probably have built up a tolerance to the funny looks; a tolerance to the subtle comments that are meant to infer that you are VERY different from everyone else. You must have this tolerance or how else could you continue in your profession. I understand. I have it too.
But here is the thing. It dulls us to the messages around us. Something is wrong in in the state of math education. Most of us know this, though some still fight it. Most of us know that it is the charge of math teachers to make it better. We must raise an entire generation of children that meet math teachers and say, "Yeah, math was always too easy for me so I decided to do something difficult like teach grammar instead." That is our charge, and to make it happen we need to assiduously evaluate everything that we do. We must find the mistakes and fix them. I don't know if what I am about to propose a solution to is the most pressing problem, but it definitely is one. We may be able to learn something from the problem, and I hope something from my proposed solution. Here goes...
Complex Numbers
Let's take what a child is supposed to know about complex numbers after Algebra 2. Algebra 2 is or was a terminating course in high school math. So this may be all that any one every learns about these crazy things. Here are the objectives from the Algebra 2 book I am teaching out of during summer school:
Our latest post on determinants made the carnival of mathematics at catsynth.com. Which reminds me of one of the things that makes me happy in the world... Cats falling compilation videos on youtube. One of my AP Calc BC kids showed me this one. Those kids are so smart.
I really love the music. I always wondered what the purpose "Man in motion" was.
So far we have found that we can use the determinant to find area and volume, and that they can also help us reform the concept of proportion. Determinants can provide a way to calculate this quantity in some cases where other methods could be much more difficult. We have seen that the determinant is a generalization of proportion. What more?
One of the most important properties of determinants is that if two rows (or columns) are the same then the determinant is zero. This can be easily seen from the area interpretation of the determinant, because if two columns are the same then the area defined is degenerate for that dimension. The box has been flattened.
This is very similar to another way in which we discriminate things in math. In algebra we often see the quantity (x-a). This comes up in factoring or even in creating polynomials that have specific quantities, such as Lagrange Interpolation.
Using determinants to discriminate
Just like the expression (x-a) has the ability to test if the value of the variable x is equal to a, Discriminant will allow us to test if many variables have desired quantities simultaneously.
Example: Finding the equation of a line
In this example uses the multi-linearity of the determinant to find the equation of a line through two points. For example let’s try to find an equation of a line through (3, 5) and (-2, 3). To do this with determinants we set up the following 3x3 determinant equation.
Clearly, the points (3, 5) and (-2, 3) satisfy this equation, so that solution set does go through those two points. Is it a line? Well from any method that you choose to evaluate the determinant you can see that the coefficient of x and y will be numbers. Therefore it is a line.
Why are the in the third column? Well we need to make sure that we have a square matrix to take the determinant of, and we need to make sure that the three rows have the same number. So that we can get the determinant to evaluate to zero when we insert the points we know.
Example: Equations for other curves
One problem that we often ask students in Algebra 2 is to find the equation through three points. Let’s do one now, for instance find an equation through the points (3, 5), (-2, 3), and (6, 9). Then we set up the following determinant equation.
The numbers in the first column are the squares of the numbers in the second column. In fact, we can find an equation for a circle through those three points by the following equation.
Conclusion
Determinants may have more use and meaning that we have given them credit for in the standard high school curriculum. Can we bring them to the students in a meaningful way? Can we show them their usefulness and their beauty? I encourage you to explore this with students, friends
Mathematics is the art of giving the same name to different things. - Henri PoincareSo in this case it is not the quite the same name, but I have sort of "invented" some terminology. The idea of "schlope". Schlope is quite similar to slope. The slope of a line tells you how much to go up for every 1 that you move to the right. Schlope tells you what to multiply by for every one step to the right. Schlope is very important for exponential functions. Generally, in exponential situations we talk about the "rate of growth". If the rate of growth is 8% then the function that models that growth has "1.08 to the t" in it. That 1.08 is what I call "schlope". So, schlope is the number that you actually use in the formula.
My wife and I wrote an article in Highlights magazine about factorial. You probably know the magazine from the doctors office. You can read the article here Big numbers at a banana-split bar but you miss all the pictures. Guess you will have to find it at the dentists office.
If I was a student who had just finished a course in Algebra I in most schools across America I think that I would think that the following were the most important things that we had done. I would think this because we had spent so much of our efforts on it.
This is a continuation of a discussion of determinants started here. And there is more here.
Proportions
So if determinants are important how important are they and when can they be introduced reasonably. One place that they could possibly come up is in reference to proportions. As a reminder, a proportion is equation where two ratios are equal. One method to solve equations such as this is the Means-Extremes Property. This is more commonly known as “Cross-Multiplying”. I think I speak for many teachers that cross-multiplying is a “bane of existence”. Cross-multiplying is a rule that is often overused. It seems to quickly rise to the top of all students list of favorite methods so that whenever in doubt about how to proceed in a problem with fractions teachers often here the idea put forth that the correct method might include cross-multiplying. This is probably because few students really understand what this method does or why it works
Connections between determinants and proportions
after cross-multiplying we know that
