What number should be associated with the white box?

So a lot has happened since my last post! I left my job as a "lowly" mathematics aide and finally got a position as a full-time math teacher! It's been a crazy few months to say the least, but I'm very happy to say that I love the school I'm in. It's really brought back the creativity as a teacher.

One really cool thing that the school has turned me onto is "Number Talks" or "Math Talks". Popularized by Jo Boaler, it is the act of taking a small time out of class to look at a problem and try to figure it out using only mental math. No pencils, no paper, no air writing. Only the brain. Not only am I impressed with the level of mental math students can possess, but the interesting patterns they use to get to certain answers are astounding!

For example, I used the following picture as a Number Talk. The question posed was:
What number should be associated with the white box?

I tried to make the question as vague as possible to see what the students could come up with. And the answers were very interesting! The nice thing about it is that the students can't be wrong! As long as they come up with a good logical argument, it's a perfectly viable answer (Do I smell a CCSS Math Practice Standard?).

If you haven't tried these yet, it's definitely worth a shot! This page is riddled with ideas!

Until next time, happy math!

Robo

James Tanton's Exploding Dots and the World in Base 1/2

Not long ago, I blogged about the conceptualization of polynomial long division and trying to find easier routes that would make division easier to understand, so students would find it easier to do.

Then I found James Tanton's Exploding Dots course.
This course is all about learning about different base systems, including "base x" where the base is unknown, which leads to how to perform all types of arithmetic in these bases with great conceptual understanding (though not necessarily with ease).

Then after completing the course, Tanton challenges the student to fuss around with different bases and understand what in the world is going on. So I did. In base 1/2.

Obviously, certain numbers have to be converted so that we can perform the operations. So here are number 1-30 in base 1/2.

Then I can show that the arithmetic works! Warning: It's a bit odd. Whatever "carrying" one must do in decimal arithmetic goes the opposite way in base 1/2.

And also, I can find out what decimal fractions look like in this base. It turns out some really cool results occur

Isn't that awesome?! Through a weird little course involving dots and boxes, I was able to show that an infinite geometric series with even powers of 1/2 adds up to 1/3. Insane!

Tanton does a great job here and I firmly believe that high school students would be able to understand these concepts and they can be moved through relatively quickly. Though it can't completely take over a polynomial chapter (where are the graphs?!), it can provide students with polynomial power that they own and shows students a whole new world of different bases! It's brilliant.

I encourage everyone to take a look and dare somebody to tell me this kind of math isn't fun!


Until next time, happy math!

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:

31,906,094,523,385,994,368,481,299,331,958,673,414,312,090,258,671,068,452,891,354,163,026,957,591,747,931,239,823,646,614,166,751,734,581,823,167,481,905,809,115,761,714,115,538,319,734,188,902,939,157,355,078,026,353,195,148,215,842,562,150,854,534,727,330,647,214,594,307,840,044,204,981,488,844,449,687,130,096,476,891,888,444,808,101,888

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!

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!

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!

Robo's Pi Day Problem 2016!

Well, it's just about that time of year! Pi day is quickly approaching. Though I don't find the exact merit of eating pie on Pi day, I do love good maths! So on my office window, I posted the following problem!

Obviously, I may not be able to get the pie to you readers, but still a fun little challenge!
Can two integers a and b actually be found that satisfy that equation?? Put your solutions below! If you do not want to know the answer because you are still nutting your way through the problem, STOP READING!

Happy math!

Robo


EDIT:
The following is a possible way to attack the problem using partial fractions!

Robo Versus the Subtraction Algorithm

Lately, I've seen many of postings on social media about using the following method to solve subtraction exercises:

AND EVERYONE IS OUTRAGED!
Well, parents are outraged. And rightly so, but I feel for the wrong reasons.

Many of parents feel this is Common Core's fault. That the new standards are forcing this obscure form of subtraction onto their students and it doesn't allow parents to help with their students' homework. On the contrary, it has very little to do with Common Core. The Common Core State Standards (CCSS) don't necessarily ever state the WAY something should be taught, rather it promotes conceptual understanding; which is exactly what this model is trying to do!

This model of subtraction clearly shows what subtraction is really about: the distance between two numbers. This is vastly different from what most American students learned back in their elementary school years:

The wonderful (or dreaded) subtraction algorithm!
And don't get CCSS wrong, it does actually promote it! In fact, it's a standard! (Check for yourself!)
The problem with the traditional algorithm is that many people don't know exactly how it works. They just know that it does work. And as a budding mathematician and high school teacher, this bothers me. But now it has become an epic battle!

Maths is about recognizing patterns and understanding how they work. All the way from arithmetic to category theory. We want UNDERSTANDING! But how can understanding be promoted if...well...we don't understand how things work!
AND AGAIN EVERYONE IS OUTRAGED!

I see the flaw in the "new way"s design. It seems to promote algebraic thinking before students even know what Algebra is (Is that such a bad thing?). It's not necessarily that the "new way" is hard to understand, it's just represented in the wrong fashion. There are definitely other ways to represent subtraction. For instance:

This diagram shows exactly what the "new way" is trying to get at except it uses a number line, which is always more mathematically viable than any algorithm...right? Also, I firmly believe any parent, despite the fact that they may not have seen it this way, would understand exactly what the problem is asking of them. It's a win-win! Theoretically, of course.

Also, there are plenty of other way to understand how subtraction works! We aren't just limited to algorithms. In fact, algorithms make math uninteresting and, in many cases which I've seen, deter the students' understanding of what is going on mathematically.

And not to be limited to only two ways to subtract, here are a few more representations of what subtraction could look like!

Subtract the same large quantity from each to get smaller quantities!

Think of subtraction as the inverse of addition. So 24 plus what number is 43? DANGER: still a promotion of algebraic thinking! But students do seem more comfortable with addition than subtraction.

Use base ten to your advantage! This is actually the more visual representation of how the standard algorithm works. And perhaps could be more effective than the algorithm! Given that your child knows how negative numbers work, of course.

Happy maths!

Robo