Fermi Problem - Counting to 1,000,000!

Enrico Fermi chilling out with some computations.

Enrico Fermi was an Italian physicist and is considered one of the "fathers of the atomic bomb." He accomplished many things in his life, but today we're going to look at a style of problem named after him. He obviously did much more than this, and I highly suggest looking into him if you have the time. He's a pretty cool guy - and interestingly enough, I found out he died on my birthday. Well, years and years before it became my birthday, but I still think it's eerie. 

I've got the heebie-jeebies

Anyway, let's take a look at a Fermi problem. Some of you may be familiar with Fermi problems; for those of you who are not, and also are unmotivated to use Wikipedia, a Fermi question is defined as:  

"an estimation problem designed to teach dimensional analysis, approximation, and the importance of clearly identifying one's assumptions."

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In today's blog post, loyal readers, we will be considering this essential question:

How long would it take to count to one million?

Let's give it a go! 

First, let's assume that it takes exactly one second to say every single number. That means it would take one million seconds to count to one million. So, exactly how long is one million seconds? We'll do some simple arithmetic....

As always, click to enlarge my creations. 

It would be easy to leave this as our answer, but we all know that some numbers take much longer to say than others. The longest number I can think to say between 1 and 1,000,000 is seven hundred seventy seven thousand seven hundred seventy seven and the shortest number I can think to say between 1 and 1,000,000 is one. Let's time both of these: 

It took me about 3.26 seconds to say 777,777 and about 0.6 seconds to say 1. You are welcome to time yourself, but we're going to use my numbers. 

From this, we can conclude each number will take somewhere between 0.6 seconds and 3.26 seconds to say. We'll use the average of these two numbers for our calculations. We find the average is... 

We could take the easy route and round this number to 2 seconds. But, here on "Jennifer Writes About Math," we're about the most accurate way. Time for some more arithmetic! 

So, it would take about 22.338 days to count to one million! Good job finding that out, team!

Now, I would love to check our answer, but I don't have 22.338 days, let alone 1 day, to devote entirely to counting. This is where you lovely readers come in.... <3

Now, you may be asking yourself, "What the heck, why did we do this? This doesn't matter! There isn't a good way to check the answer!" Well, that's the thing. Math helps us explore the world around us. Maybe it doesn't matter in the grand scheme of things how long it takes to count to 1,000,000, but this process of discovery carries over into everything. How long does it take to get to the moon? How many gummy bears can fit into this tub? Fermi problems help us practice estimating and are a fun, valuable way to get people (specifically students!) involved in math. 

Because you're a pretty inquisitive person, you may also be asking yourself whether or not these kind of problems belong in the classroom. In my humble opinion, they do. Math is often approached as serious and intimidating, with one correct answer or procedure. As a result of this, some students understandably become disheartened and disinterested in the material. With Fermi problems, answers are less concrete and mathematical questions seem kind of goofy. Students don't need to have the correct answer, because the questions typically do not have a "correct" answer, nor is there a good way to prove or disprove an answer. Although the computations in the given example are relatively simple, the process of trying to answer the question can spark a new interest in mathematical exploration. Math is exploration, this is math, albeit rudimentary, and I think it belongs in the classroom. 

If you are interested in doing more Fermi problems, there are entire websites devoted to them. Here are a few...

This one lets you submit an answer, if you're into that.  

If you're more interested in how this could apply to a classroom, check out this resource.

Maybe you don't feel like solving any Fermi problems. That's cool, too. If you think of one you'd like to see me try out, leave a comment with your question! 

Note: additions to blog are marked in blue! 

"Trinomial Method"

From the looks of Mr. Felix’s classroom, you might think he was a narcissist. Through his genealogy, he was graced with the same name as the 1919 silent-film-star-turned-cartoon, Felix the Cat, and he was unashamed to fill the majority of his walls with posters of Felix.

In addition, he was affectionately known by the student body as “that guy with the flat-top hair cut that coaches the basketball team,” but even more than that, he taught the all-so-frightening Algebra II.

I can remember his words clearly: “If you miss tomorrow’s class on factoring, you are going to fail my class.” My fear drove me to class the next day, and I sat through what was, in retrospect, one of the strangest and most unique lectures of my life. I did not realize how strange this lecture was until, well, today when a few of my classmates looked at a paper I had on factoring and asked why I had T-charts all over it. 

You might be thinking to yourself, “T-charts… factoring… what do these even have to do with each other?” Well, sit back, enjoy the show, and let me share with you Mr. Felix’s “Trinomial Method,” because somewhere behind the flat-top and basketballs is a pretty smart guy.

Polynomials may look like big scary monsters, but they are really easy to tackle with the right method.

TRINOMIAL METHOD

Let’s say we’re given the snazzy polynomial

3x² + 12x + 9,

and asked to factor it. First, we’re going to draw a cute little T-chart

that will also double as abstract art. Now, we’re going to take the first and last coefficient of the polynomial and multiply them together, and place this number in the upper left of the T-chart. Then we will take the middle coefficient and place it in the upper right of the T-chart.

Next, we will list all of the ways the leftmost number can be factored. We’ll get something like this:

Don't forget all possible ways to factor! 

So many factors...

Now, we will take each of these numbers and add them together (and also realize it’s easier to type it than to write it by hand), to get something like…

You might notice that 3 + 9 = 12 is in the 12 column… hmmm… I wonder if that means anything.

Let’s use these numbers to split up the middle term in the polynomial. We’ll get something like

3x² + 12x + 9 = 3x² + 3x + 9x + 9.

Now we’ll factor this, and we get

3x² + 3x + 9x + 9

3x (x + 1) + 9 (x + 1)

(x + 1) (3x + 9),

thus factoring our polynomial.

“How does this work, exactly?” you might ask. 

Let's flush this out.

Well, the general thought process behind it is that you need to find two numbers that when multiplied together equal the first and last coefficient multiplied together, but when added together equal the middle coefficient. Obviously, this does not work easily for all cases. You can try a few on your own if you’d like. But, in essence, this is it:

There are a few cool things about the “Trinomial Method.” If your leading coefficient is 1, s and t are how you factor your polynomial. As an example:

Math is trippy sometimes.

Obviously, the more time you spend using this method, the better you will get at it. Eventually you won't need to list out all of the factors and it will come more naturally to you.

Destroy all the polynomials with factoring! 

This can greatly reduce time spent factoring. Although it's important to learn how to factor the traditional way, Mr. Felix's "Trinomial Method" can be very helpful and is definitely worth looking at. 

So, thank you Mr. Felix! You keep doin' you. 

Why I Choose to do the Maths.

I was riding the bus home from a class downtown. Unofficially, most of my class decided to start sitting together. We started discussing classroom topics, our professor, and how cold the weather was getting. 

The all-important portion of the conversation hit - when we all determine those who are like us and those who aren't - our majors. We went in a circle. A few people wanted to teach history, and others elementary education, English, biology.... and then the question came around to me. 

"What is your major?" 

From their reactions, you might have guessed I started describing a hernia, or what it is like to coat yourself in seaweed and pretend to be a seaweed monster. My answer was "Math," but all of the super sleuths out there who read the title of the post already knew that. 

I have noticed overwhelmingly negative reactions to even the mention of majoring in math. If I were to graph the responses I get from people, it would look something like this:

(Go ahead, click on the picture. Something good will happen. I spent time on it so you should at least admire it a little larger.) 

Apart from the shock of being immediately chastised, I started to wonder why so many people did not like math. Did they dislike it in the same way I dislike cantaloupe or strenuous physical activity? Did they dislike it because they had a teacher that did a less than stellar job? 

As someone who thoroughly enjoys math at this point in her life (middle school was rough for all of us), it greatly confuses me as to why so many people severely dislike math. How can someone dislike something so expansive and large as math? Math is so many different different things. Math is its dictionary definition. Math is the study of patterns. It is what I do for homework. It is what I think about when I see roller coasters. It is what I use to calculate how much that shirt I need is on sale. 

So, why do I choose to do the maths that so many people dislike? 

I choose to do math because I was born too late to explore the world and too early to explore space, but math allows me to explore both in ways that explorers haven't before. Math can be used in an infinite number of ways that I couldn't possibly begin to list.

I want to teach math because I would like to inspire students to overcome the societal stigma against math, and perhaps use it to pioneer the future. Now, don't get me wrong, I do not wish to, nor could I, get every student to bask in the warm, fulfilling sunlight that is math. Students are interested in different things and it would be unrealistic to expect every student to get butterflies at the thought of completing a difficult proof. In fact, I do not get butterflies when I think about chemical compounds. My goal, however, is to give students new value and appreciation for math, and maybe even a deeper understanding of the world while I'm at it. 

One day, it would be pretty sweet if people didn't uncontrollably vomit at the thought of math or someone choosing to major in it. Until then, I'm just going to start answering "What is your major?" with "Explorer." 

Post 1 - Counting Circles

Up until MTH229, I had no idea what a counting circle was. 

To those unfamiliar like I was, a counting circle can best be described as a game to teach math students arithmetic and critical thinking, among other things. Before the game starts, students form a circle. The circle aspect is important, but we'll get to that in a bit. A random student is picked in the circle and given a number; the number depends on the level of the students as a whole, and should be non-threatening for the first few times the game is played. Next, an operation is defined to the class. Typically, the operation is addition; however, subtraction can be obtained by adding a negative number. The first student applies the operation to the initial number given. Regardless of whether or not the student applies the operation correctly, the number the student states is recorded on the board. The next student in the sequence applies the operation to the number the student before them presented. Rinse. Repeat. This continues for at least a full circle until the teacher pauses the class. At this point, the teacher prompts the students to solve for a number further in the sequence. After each student signals to the teacher they have solved it, the teacher then prompts the students to share ways in which they found the answer. This, in whole, is a counting circle. 

There are a number of different things I enjoy about counting circles, both as someone who participates and someone who hopes to be a math teacher one day. One of the more important aspects, I believe, to a counting circle is well, the fact that it is a circle! It can be symbolic for how math builds upon itself or even demonstrate how patterns can be 

e n  d   l    e     s      s       , 

such as with the operation the students preform. Working within in the circle also builds a sense of a team within the student community and familiarizes the students with one another. Because there is no pressure on whether or not the student preforms the operation correctly, students feel less fearful of mistakes. When students feel less frightened, they are more likely to participate in class and put forth genuine effort because there is nothing at stake. When the teacher prompts the students to solve for a number later in the sequence, it allows the students to practice pattern-seeking skills. It helps students transition to higher thinking in math, and gives them a greater understanding of both patterns and seeking out patterns. 

What I believe to be the most important part, though, is when the students share the ways in which they solved for the later number in the sequence. This allows students to practice two very important skills - communication and critical thinking. As a student explains their method, they are practicing communicating their thoughts in a way that is accessible to others. I personally find explaining a concept to someone else helps me grasp the concept better as well. It also may help students develop a more advanced math vocabulary ("subtract" as opposed to "minus", as an example). 

Through this sharing process, students are exposed to the many different ways a solution can be obtained, and even the individuality of each student. Many teachers, intentionally or unintentionally, convey the idea that there is one "right way" to solve a problem. To this day, I recall my 7th grade math teacher who would not allow me to solve a problem any other way than her own. Somehow, she developed the idea in me that math was not discovery, but resuscitation. In the process of her teaching, critical thinking was completely lost. My mother, who taught middle school math, and I would argue because I wanted her to "show me how to do the problem!", whereas she wanted to help me form new ways to solve it. Try as she might, I did not rework the way I viewed math until my senior year of high school. I believe counting circles are a good way to combat this idea in students and develop their critical thinking skills. 

In all, I believe counting circles are a very engaging way to involve students in class and build important lifelong skills. They are useful in many ways - from building a sense of a team, to practicing communication, to creating an accepting classroom environment. As someone who was initially taught to avoid thinking critically, I think counting circles are a great way to both introduce critical thinking to math, and develop it further. Counting circles are quite amazing things because they develop many important concepts and ideas within students without the negative connotation of the word "math". Because it is viewed as a game, students are likely to be excited to participate! For many students, math is something they find difficult to be excited about, and this is a cleverly disguised math activity. Although it may not be practical to use counting circles daily, I believe semi-regular use of it would have a positive impact on any classroom.