Wednesday, April 27, 2016

The Graph of Robo

Since I've finally gotten my feet sunken into the realms of math education, I've been enthralled by the power and elegance of polynomials. They're predictable, yet so powerful when it comes to working in several bases at once.

One beautiful use of them in within the "Everything Formula", which is a beautiful inequality that is able to graph absolutely any picture you want it to, given you can fit it in certain parameters. I will explain how it works in this post, but Matt Parker will always give a better mathematical explanation than I can.

The formula looks something like this:

1/2 < floor(mod(floor(y/17)* 2^(-17*floor(x) - mod(floor(y),17)), 2))

Ok, the formula is not so beautiful, but the outcome is so cool! For instance, I can graph my little nickname "Robo".

So let's find out how polynomials happen to be involved here and how I feel this could be a fun project in the classroom.
The main key of finding anything in the "everything formula"s graph is finding the elusive "k" value you see on y-axis. Every k-value will be different for each picture. For instance, the k-value in this graph is:


Which is incredibly large! How in the world can we expect to find this number?
Well, we use binary! Or a polynomial of base 2. The binary number is easy enough to find, though very laborious to come up with. All you do is start at the bottom left of where your "picture" begins and if you want it to be a black square, you put a 1 and a white square is a zero and you go up the picture. Then start at the bottom again and go up again; putting ones and zeros wherever applicable. 
For example, the binary number that works for my ROBO graph is the following:

Convert that to base 10 then multiply by 17 and VOILA! You have your very own ridiculous k-value for your graph.

It's an interesting use of polynomials and definitely can be a fun thing to do in the classroom. I wouldn't go completely in depth as I've done here, but you mathy people would probably would like to know how it works. 

For instance, rather than writing out the entire binary number, maybe it's better to do only a portion of the binary. But whatever binary numbers they do happen to come up with, have them write out the polynomial in base 2! It's definitely worth seeing what crazy math is going on in the background. Besides, most of the squares will probably be white anyway, so the polynomial should still be relatively short. Keep it to may the first 20 squares? But then at the end of the day, they will have a number that is literally theirs!

To create a graph and find corresponding k-value, I used this website. There is no way I expect me, you, or any student to figure it out long-hand.

Enjoy and happy math!

Friday, April 22, 2016

Is There an Alternative to Polynomial Long Division?

For whatever reason, I've always been very familiar with the division algorithm. But much like my peers, I never really understood how it worked (until recently), I just knew it did work and I was able to use it in my arsenal of mathematical weapons. Polynomial long division is really no different either, but these algorithms don't really get to what is happening.

Division has a very simple definition: How many ways can I partition or group objects into equal amounts? Admittedly, with large quantity of numbers and abstract ideas such as polynomials, this definition is really hard to wrap your head around.

So how can we answer these abstract division problems in another fashion? Is there another way that promotes conceptual understanding? Well of course there is! There's actually quite a few ways to do so.

The easy answer is synthetic division but any maths educator knows that it only work with a linear divisor. The following are examples of many different ways we can look at division, including the algorithm.

Without further ado, ladies and gentlemen: The Standard Algorithm

Oh! And the synthetic division, of course!

Now the next one is a similar take as the standard algorithm. But rather than multiply and subtract, it's just a bunch of factoring. But I do believe it does grasp at the concept of how many times (x+2) can go into this quadratic:

Essentially the goal is to start with the quadratic term and make it look something like the denominator. And then once that is simplified, just go down to the linear term and do the same thing until there is no reducing left to do. This is also how I divide fractions in my head:

Just split up the numerator into something the denominator can divide into and work from there. It's the same idea!

Last but not least is the box method! If you know the box method for multiplying, it's the same concept except working backwards (well duh, Robo). I'll just leave my solution for the example above below:

If you would like to see how this is done in detail, my man James Tanton has a brilliant video that explains it in depth.
What I love about the box method is that, not only does it do well in explaining that multiplication is just the area of a rectangle, but it can also be reverse engineered to do the inverse and does it in such an elegant and easy way. Hence, this is my go-to option when it comes to division. Sure the algorithms are good and dandy, but they just do not have the same power and grace that this box method has.

Don't get me wrong, division is a very hard concept, especially when dealing with the abstract. There really is no out-and-out best way, but know that there are always other representations of it, and use those to your advantage.

Until next time, happy math!

Wednesday, April 13, 2016

Robo's Long Term Goals

As of recently, I've tried to be the model of the "growth mindset" with my students. Not so much with the material that they may be studying, but rather as a teacher. I try to show them how a positive mental attitude mixed with the idea that I am always doing my best to learn new perspectives and outlooks when it comes to approaching a problem is beneficial to them as a life-long learner, rather than saying "Oh, probability?! I've always been bad at probability, so here are a few things to memorize so that you can at least take the test!" (I'm still not the best at probability though...but I'm getting there!)

So over the next few years, I'm setting a few goals that I would like to meet in order to maintain my own growth mindset as a teacher and also install the idea of a growth mindset into my students.

Goal 1: Nix the Tricks
A couple of months ago, I read Tina Cardone's book Nix the Tricks, It confirmed my belief that I am not alone in the fight of attempting to achieve conceptual understanding for my students. Yes, there are a few "tricks" I may still use (I'm comfortable with students using exponentials and logarithms as inverse operators), but overall, nixing the tricks from my teaching only benefits everybody. Conceptual understanding is hard to achieve when everything that is known is based off a shortcut. And if a concept isn't understood, the harder it is to do well in a class; and the harder the class is, the more likely a fixed mindset of "I will never get this material" will start to sink.
So nix the tricks! Teach the underbelly of what is going on! I've done my best to get students to wrap their collective heads around concepts and it's all worked out well so far.

Goal 2: Omit the word "should" from my vocabulary
I have to admit, I am a huge James Tanton fanboy. His ideas, creativity, and sheer honesty motivate me to become a better teacher. I must've read most of his essays at least twice, which is why I was surprised that I never caught onto this idea until yesterday. In his essay 12 Points About My Teaching, Tanton describes why he believes he has been successful in his teaching endeavors. His eighth point was that he knows there are mathematical ideas that he does not know. There are ideas that students just don't know. And that's okay! It is okay not to know things. But it is not okay to not want to know them. It is also not okay to tell students that a certain topic is something they should know, especially when reviewing.
Every year, I, along with every high school Algebra teacher, struggle teaching rational functions because many students struggle with fractions in general. Students don't remember that common denominators are necessary when adding or subtracting fractions, or that when dividing fractions, all you must do is multiply by the reciprocal of the second fraction. And the struggle is justified.

While working with a student last week on this exact topic, I told them that adding and subtracting fractions is something that they SHOULD know how to do. But really...should they? Many students don't work with these ideas for YEARS until working with rational functions. They rely on their calculator to do the dirty fraction calculations for them. Should we expect them to remember?

I had to catch myself and I rephrased. I told that students that "adding and subtracting fractions is something we can 'expect' you to be able to do." With literal handmade quotation marks. I don't blame students for not being able to remember that. Hell, I haven't seen calculus in years, so I've had to re-learn everything so that I can help students. So I understand the struggle completely. So rather than saying they SHOULD know something, focus more on review! Before exploring how to use operations on rational expressions, take a day to go over addition and multiplication of fractions. And don't just review how to do it, but explain why they work. Younger students are good at following blindly. Refresh their older selves as to why common denominators are needed to add fractions.

I do urge anyone who reads this, especially math educators, to look at both Cardone's book as well as Tanton's essay(s). They are really eye-opening,

Until next time, happy math!