## Math Excursions: Dimensions

This is a diversion from the usual education related posts to bring to light some concepts I personally find interesting in math. These are also not commonly understood by many people, so I hope my readers enjoy it. The first thing I want to talk about is the concept of Dimensions. It is common in science fiction, or even fantasy and new age babble to speak of dimensions as other “planes of existence”, or to describe something as coming from “another dimension”. Unfortunately, these types of descriptions we see in fiction are often ambiguous, or meaningless especially when we are referring to mathematical dimensions. In math, it all begins with a point:

Very Exciting

This point, has no length, no width, no height. In the simplest terms possible it is an object of zero dimensions. Next, when we look at two points:

We see that we can have these two zero dimensional points define for us a one dimensional object; the line. From this point its gets a little tricky, because in mathematics, to fully define two dimensions, we might take two lines and cross them in a perpendicular fashion, like so:

These two lines define two dimensions in what is called a Cartesian Plane, named after the Famous Rene Descartes, who initially came up with the idea. (I have a fun anecdote about how Descartes developed this, but I will post that another day). These two lines are what we call orthogonal to each other (these lines intersect each other at exactly 90 degrees). Orthogonality is how dimensions are described in mathematics. In order to define two dimensions, usually you need to show two orthogonal lines. Naturally this can be extended to the third dimension.

At this point, you might be asking yourself, can this keep going ad infinitum? Mathematically, yes it can and it is something that can be shown with an extension of the Pythagorean theorem. However, when we are talking about lines and orthogonality, we are really interested in seeing how many lines can we intersect with each other, where each line is orthogonal to all of the others? Look again at the 3 dimensional orthogonal lines. Where would you put a fourth orthogonal line?

This is incredibly hard to imagine spatially, simply because it is so foreign to our understanding of reality in 3D, that our brains really cannot visualize it. In math and on paper, the calculations make sense, but the intuition does not. It’s a fun crossroads in mathematics where what you would originally think is right actually becomes wrong, or weird. However, there is a way to visualize four dimensions simultaneously, but in order to do this we must visualize the fourth dimension as time.

See, you can be in a particular place on Earth, and we could give you latitude, longitude and elevation coordinates so that someone could find you. However, we could also say that you had these coordinates at approximately 6:30 PM, and that they changed slightly by 6:31 PM. I like to visualize this as an array of sheets where there’s a two dimensional me on the first piece of paper and another on the next, and so on in a nice straight line. Doesn’t this almost remind you of… a flip book?

That’s kinda what a flip book is; you are looking at two dimensions and a time dimension, where we our positions change relative to time, and it demonstrates movement. Your movement in space is an example of 4 dimensions.

Why is this important? Because it is the foundation of one of the most groundbreaking physics theories ever discovered; The Theory of General Relativity. In General Relativity, Einstein linked the concepts of space and time into a four-dimensional construct called (imaginatively) “spacetime”. The mathematics behind it is quite dizzying, so I will relieve my readers from it for the time being. In four dimensional space time, Einstein is able to define gravity as its “curvature”. Check out this video clip from my hero Carl Sagan, he describes more eloquently than anyone else can.

Once you put it all together, you see it clearly; In order to invent the universe, you first need tiny little points.

## How to Teach Mathematics in Secondary School

Relatively speaking, high school wasn’t too long ago, but sufficiently long enough for me to reflect on how this education affected me. After I had accrued a healthy amount of college tier mathematics, I see the big difference between how math is taught in college, and how to teach mathematics in secondary school. Before I delve deeper into this, there are some considerations that must be understood in order to see the difference between learning in college and learning in high school.The first and biggest piece of the puzzle in unsurprisingly maturity. For the most part (there are exceptions) once a student reaches college level learning, they are more or less prepared for it. If you are a United States college student with a smidgeon of fiscal awareness, you already know that you are paying a king’s ransom for the courses you are taking, and are likely using government aid to do it. It’s amazing how much pressure to succeed money puts on a student. I know it did for me, and I did not take it for granted. I worked hard because I knew if I threw caution to the wind or otherwise failed the courses I was taking that I would have to pay this money again. That’s not exactly what I had in mind, so I took advantage of any resources I could and knocked out the courses as best I could.

In secondary school, this financial pressure is in almost all cases non existent. However, the pressure lies in constructing a future for oneself. Again, we are faced with maturity. How many high school students truly recognize the importance of the four years they spend in secondary school? You could answer this question by asking yourself this; Do you ever wish you could go back to your high school years and study harder? Do you know anyone who has said if they had it to do again, that they would put a higher priority to their studies? It is a double edged sword, because high school is really not simply meant for studies, but also as a breeding ground for healthy social interaction. People look back to high school and don’t remember taking their courses, or how well they learned, they look back at all the good times they had with their friends (or for some how hard it was for them socially).

Maturation – not only mathematically – plays a big role in the formative years that is high school. This insight might seem bleak to some educators because conveying enthusiasm for academics is not an easy thing to do. It’s an uphill battle, and there is much to accomplish in very little time. What can be done?

I want to limit the scope of this selection to dealing with how to garner interest, and effectively convey mathematical ideas to students who are very much busy with growing in other ways. There is a lot on controversy nowadays because of the “Common Core” debate that has sprouted up recently. Not only this, but the long standing debate of the effectiveness of teaching in relation to standardized tests heats up an already boiling over issue. I have some very radical opinions of this myself. I think in a phrase, it stands to obvious reason that high school – for both teachers and students – is not easy.

In any case, here is my two cents on how to teach mathematics in secondary school:

Temper Your Teacher Side With a “Cool” Side. This sounds a bit silly I imagine, but I found based on observation that the type of teachers that were the most effective in teaching were the ones who were happy to be at their job and were able to integrate their academic duties as a teacher with their sociability and likeability as a person. Make jokes! Be understanding, and don’t be overbearing in your professionalism. I had one teacher who was named Mr. Dick. That was actually his last name, but his likeability factor was off the scale. Students who have had him for more than one class or had gotten to know him would reverently refer to him as “The Dickman”. It was a strange irony, but it suitably demonstrates the point. He often integrated his personal interests in the material taught through anecdotes which were particularly useful. Sometimes the anecdotes had nothing to do with the material at all; he was just being a sociable kind of guy.

Don’t “Over-Temper” Your Teacher Side With a “Cool” Side. This little tidbit can do a disservice to your students in the very least, and get you in trouble at most. Don’t be too lenient, but be understanding. Don’t get too bogged down in long explanations about a concept, but don’t do away with them either. I think this is a rather hard part of math teaching, and why I chose not to go down that path. I really like to talk to the people I teach and get to know them, because it is easier to be a great teacher if you are not intimidating. But it’s hard to be a friendly, sociable person on one hand and be teacher or disciplinarian in the other.

Be Dynamic and Enthusiastic About What You Teach. You teach mathematics! You are teaching what Carl Friedrich Gauss called “the Queen of the Sciences” and what you dedicated many years of your life to understanding! Today’s lesson isn’t some boring useless drivel that you impose upon your students, and never treat it that way. I know through my own experiences that sometimes things can get tedious, and sometimes we don’t feel our best. But look deep inside you and pull out the best teacher you can on those days. Your students deserve it.

Don’t Be Unhappy if You See Disinterest. This is a big one for me, because I would love it if everyone was interested in math. However, the hard and fast truth is that people are interested in many things, and in reality there are very few people who choose mathematics as a career in some form or another. There are even fewer who develop more than a passing interest! It would seem that people tend to go one way or another: either you love it you you hate it. It’s the “cost of doing business” as it were, but at least foster as much interest as you can. People might be surprised someday when there’s a mathematical problem in their way and they remember what you taught.

Teaching mathematics in secondary school is obviously a gargantuan endeavor, and I respect the teachers who take it upon themselves to do it. I will stick with my tutoring, but I hope that any fledgling secondary school math teachers who read this take away something that helps them put their best foot forward.

For some quality information on how to teach mathematics to secondary school students from experienced teachers, get a copy of The Math Teacher’s Toolbox: How to Teach Math to Teenagers and Survive. A highly recommended read for both new and experienced teachers.

## Problem Solving Theory – Unlocking Mathematical Mysteries

The purpose and ultimate goal of mathematics is to find problems and solve them. This is done in two ways; (1) Pure mathematical problems are created and solved for their own sake, irrespective of their applications and (2) Problems in the real world are discovered, and using the pure mathematical concept, a solution is worked out. These are both equally important, and while this first way might not yield immediate results, it is important for giving those who try to find ways to mathematically solve problems in the real world. Almost always, the theory precedes the application. How does one go about solving a problem? Based on my mathematical learning and experience, this is my problem solving theory.

It all begins with the problem. It’s intuitively obvious isn’t it? Without a problem, what would there be to solve? Problems arise in many different ways, but for the pure mathematicians, it manifests from the groundwork already laid down. For example, Since the time of the Ancient Greeks, prime numbers have been known to exist. Prime numbers are those whose factors (smaller numbers that can be evenly divided into the bigger number in question) are itself in one. To illustrate, consider the following:

4 is a number that has three factors. These factors are 4, 2, and 1. when we divide 4 by 4, we get 1, When we divide 4 by 2, we get 2, and when we divide 4 by 1, we get 4.

In the above picture, we are visualizing even division as the grouping of the chocolate chips where the number of chips in each circle are the same. This is only possible when the number of circles is what we are dividing the number of chips by! If you were to take 3 chocolate chips and divide them by two:

We see that the number of chocolate chips in each circle is not the same. In the second circle, we are missing one! This “missing chip” is what we would call the remainder. If you can evenly divide two numbers. There will be no remainder.

In the case of prime numbers, there are only two possible ways to evenly divide the prime. The numbers that divide primes are 1 and itself. Take 5 for example:

5 can only be evenly divided 5 and 1. Anything else, and we will end up having a remainder! Like I said, this fact has been known for a long time. But it was after the fact of knowing about prime numbers that we started to tinker with them, and come up with new problems that use prime numbers in some way. Arguably, the most famous example is the Goldbach Conjecture. Just as a side note, a conjecture is basically an unproven statement in math. Once you solve the conjecture (or prove it to be true) it becomes a theorem.

Say you were to take any even number bigger than 2 (4, 6, 8,…). The Goldbach conjecture says that any even number you can think of is actually two prime numbers added together. So, if you were to take 6 for example, we can see that it is the same as saying 3+3. 3 is a prime number, and the conjecture holds true. Also note that as of the time of this writing, nobody has been able to prove this true. This is an easy problem to state, but an immensely difficult one to solve. Play with it! Nobody said you can’t. Maybe you will find the key someday that solves it.

The whole point of this was to demonstrate the genesis of mathematical problems and the beginning of problem solving theory. Previous mathematical facts and a little thinking are what make up new problems both for people interested in pure math and those interested in real world applications. After the problem has been found, this is the process of how we might try to solve it.

Tinkering aroundIn this part, we basically break down the problem, find little relationships between other math facts we know, and try to see if there is some other relationship that we can find that is useful for solving the problem. When it comes to word problems (check out my article on word problems and their difficulty here) we are trying to bridge the gap between the math facts and the interpretation of the problem. All of this is tinkering. With the Goldbach conjecture, we might want to find out what else we know about even numbers, or what properties that prime numbers have, and put them in our mental repertoire to prepare us for the next step.

Experimentation – Now that we have some other facts we can mess with, we can now try and put it all together, and see if it works properly. This step is the proving ground for what we found in our tinkering. With the Goldbach conjecture, we might try to put together are facts of even numbers and primes in such a way that they make what the conjecture is suggesting is true all of the time. After we experiment, our results take us to the next step.

Success (or failure) This is where we finally find if our tinkering and experimentation gets us where we want to go. Through our efforts we see that we have succeeded or we find that we were following the wrong trail all along. More often than not, we find that our experiment failed, and that we must try again at the beginning. At this point, we might feel a bit defeated. We might take a break and think about what we did as a whole. What went wrong? How can we fix it? This is the beginning of the next step.

Take a Minute!I remember while taking calculus, I simply need to get up and get away. My brain needs some fresh air (or coffee) and I try to take my mind off of it for a bit. This does two things: (1) It keeps me from losing my mind, since people can only go for so long, and (2) It allows for gathering ammo for creative thinking. Surprisingly, the white walls of a dorm room do very little to get the creative juices flowing.

The SparkOkay we sat back down and we go through start at our tinkering process again. Hey, wait a minute, there was something I missed before! How did I miss that? We continue through our phases, and if we are lucky, we are met with success! The Goldbach Conjecture hasn’t had its “spark” yet, but when it does, even people not in the mathematical know will hear about it. I promise you that.

This might not be the most comprehensive or scientifically accurate problem solving theory, and it is completely possible that people might disagree. But my hope is that someone will understand the moral of this story; Do not, for any reason, Give up! I can’t express this enough. If you have trouble with math, try and follow these steps. In reality, we see that the spark in our problem solving theory is not some novel insight, but pure perseverance. I believe this completely, and I always encourage my students similarly.

For more on problem solving, check out Problem Solving & Comprehension: A Short Course in Analytical Reasoning. And for a personal favorite book from my high school years on prime numbers, read Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.

## Slow Learners: How To Find Mathematical Understanding

It’s a common fact that some of us learn faster than others. Most people fall around the average, but some of us have difficulties absorbing new concepts. Slow learners are very common in math, and I have personally tutored several who would self identify as slow learners. In one of my previous articles Math for Liberal Arts: Don’t Be Afraid!, I listed several suggestions I found would be effective in the process of learning math. Much of that article I also recommend to slow learners simply because I believe there to be some similar roots between those who chose to major in liberal arts, and those who show a sluggish tempo in picking up new mathematical concepts. Now, I believe that the very well debated “nature versus nurture” controversy is beyond the scope of this selection. However, in either case, I believe that there is the possibility of success in mathematics. Furthermore, If I were pushed, I would say that sheer productive effort is what tips the scale. Nuture would win me over every time.

When I was in college, I met an extremely bright young man who was attending college early. He was a part of a very prestigious high school math and science program. I shared a advanced math class with him, and one day I was having some difficulty with a concept. I asked him for his help, and his enlightenment was very useful. But what I remember the most was this concise little gem of advice he gave me: When it pushes against you, push back harder. At first I was confused, but with reflection I realized what he was saying. It gave me an insight on his intellectual character; although he was gifted with ability, even he had trouble understanding difficult concepts. I was not alone, and shared great company. He probably doesn’t even remember me, but I can tell you that this incident stoked my already burning fire for tutoring. Sometimes pushing back is beyond the strength of a student, but with someone stronger pushing with you, the impossible becomes possible.

The moral of this anecdote is obvious. Legitimate unrelenting effort is the key to learning not just mathematics, but learning anything. But don’t get me wrong; I certainly do not wish to imply that slow learners are not working hard. Quite the contrary; some of my slowest students were the ones who worked the hardest. What I am stating is that there is more to the puzzle. Simple muscling your way to understanding can sometimes be difficult, especially if the extent of current ability has been pushed to its limit. But as I have have stated in the above article, it would appear that math isn’t something reserved in an ivory tower only for those who are gifted. This is possible, and even if you have to work harder than your fellow student to get there, you can get there. View this as a simple encouragement to go forth and perform. Push back harder.

For an interesting perspective on how you too can overcome your wall of learning mathematics slowly, A Mind for Numbers How to Excel at Math and Science (Even if you Flunked Algebra) is a great book to read. I have personally recommended this to several of my students with really good results.

## How I See Math Word Problems – a Funny Pic, But a Big Issue

We have all seen this picture circulating the internet, and liked on our Facebook pages. This is for good reason. When it comes to solving these problems, people are often stumped. Why is that? It seems that almost every student I have tutored has had the same problem with word problems. It’s an amusing picture, but a rather interesting insight into one of the biggest roadblocks between students and the field of mathematics.

Do you know how I see math word problems? I see them as what should be; a huge motivation to learn math as well as the best way to engage students. You see, math word problems are often a way of seeing the real world practical value of the underlying “straight up” mathematics. This is the realm of applied math. No longer is it mere abstraction, or something that we learn by rote in our courses, but a tangible way to connect the concept with the question, “how do I use mathematics”? That is a really powerful question.

And… the question is somewhat difficult to answer. There are several cognitive models that attempt to explain why how I see math word problems (as an experienced tutor) might be different from how you see them. Some contend that it is related to how well people understand the English language; others believe that it is how say, algebra deals with generalizations versus how we tend to deal with them. Both positions are thoroughly researched by professionals and there is peer reviewed research and honestly, the verdict is still out on the specific cause. With my experience in tutoring, personally struggling, and finally learning math myself, I believe that becoming proficient with math word problems comes about in three steps: Concept learning, experience, and synthesis.

Concept learning involves understanding the theoretical things that underlie all word problems. More specifically, I’m referring to the generalizations of math that can be applied to all situations (such as the famous Pythagorean theorem a^2+b^2=c^2, which doesn’t necessarily apply to one situation). In my experience, this is the foundation of learning applied math. If that foundation is shaky or ill constructed, the difficulty with word problems is significantly compounded. An easy example is counting. Toddlers are taught the basic counting numbers from one to ten. These numbers are often concurrently applied with a visual aid, such as taking ten pennies and counting them. They are introduced to the concept of number, and are shown a related application. In much the same way, I believe that all mathematical learning is similarly arrived at if it sticks to the minds of the students.

Experience is the sustained and guided application of the concepts learned. This can get a bit fuzzy because just as our toddler with his or her counting, the concept can be taught concurrently with the application, integrating concept learning and experience. However, as time goes on and more examples our shown, the student acquires a “mental repertoire” of concepts and their applications. This is an important and lengthy process, often taking up the most time. Essentially, this experience is building intuition. It is like taking a block of cold, hard clay, and working with it until it becomes warmer and softer, easier to use.

Synthesis is the final step in the process, where the building of intuition and past experience culminate in being able to apply concepts in novel ways. When a student reaches this level, it is likely that their aptitude for word problems is sufficiently advanced. This is where we want every student to be because they are approached by a unfamiliar obstacle and are able to see the patterns based on past experience and apply them similarly. It’s often a trial and error method, but one that brings forth results with perseverance. Before the student knows it, synthesis becomes easier and easier with additional each additional experience.

If I had to describe myself, I would say that I am a “mathematical egalitarian”, meaning that I think that everyone can learn math. Without this belief, being a tutor would be rather difficult. After learning the concepts, I always stress to my students the importance of the three P’s (practice practice practice). While a rather cliché approach, its usefulness for bringing about synthesis cannot be understated. I want my students to see math word problems how I see math word problems: an often hard mountain to climb, but not insurmountable.

For more information on word problems, a great beginning resource is math word problems for dummies. The approachability of this material is a great first step on the road to synthesis.

## Math for Liberal Arts: Don’t Be Afraid!

Courtesy of languageofmath.com

Among the college students I personally have tutored in the past, it is the liberal arts students that usually have the most trouble with the subject. Not that they are less important or otherwise unintelligent – this is certainly not the case! Many of my students had been very bright in their own subjects, and the reason why they chose liberal arts was due to a lack of interest or intimidation. In the past, it might have not come easy to them or was otherwise inaccessible due to their teachers or the pressuring of their parents. I respect great math teachers out there for their hard work, but it is no big secret that many math teachers are lousy! From my experience, these educators had either become disillusioned with the system that they are a part of or really didn’t have that passion within them not only for teaching math, but the passion for the subject itself. This unfortunately does a great injustice to the students. This is the origin of the math for liberal arts classes commonly seen in college; it is often watered down, and doesn’t reflect the beauty that resides within mathematical thought.

Before anyone gets offended at my rather seething assessment, please note that this is from my personal investigations talking to students and teachers as well as auditing these kinds of courses. Furthermore, if you are a liberal arts student, keep in mind that I believe that your disinterest is likely not due to a disability (despite the fact that legitimate conditions exist such as dyscalculia). I might even go as far as to say it is not your fault! You don’t have to be a genius or naturally gifted to be good at math. Terence Tao, a Fields Medalist (that’s the Nobel Prize for math) has a really fantastic blog article about the taboo of genius in math. Furthermore, this research study suggests a link between consistent interest and hard work to ability (keep in mind that the researchers noted that the debate between innate ability vs. hard work was beyond the scope of this paper. However, the paper does suggest that such “skills are not supported by neurobiological peculiarities and could be acquired by other people.”)

Because of these factors mentioned, I believe that all of math is math for liberal arts students. If you are in college, it goes without saying that you need mathematics courses to acquire your degree. It is just a matter of conveying the passion that those like me have for the subject to you. Here are a few suggestions:

• Relax. Take a few breaths. This just takes a bit of patience. As my friend Euclid said over 2000 years ago, “there is no royal road to geometry”. This is also true for all fields of mathematics. In order to learn it properly, math requires dedication to the subject. Not only does it require dedication, it also requires a fire in your heart to learn. Are you an artist? Believe it or not, there is beauty in mathematics, and that beauty has been expressed time and time again through the medium of art. Here is some of my favorite examples. This is called the Mandelbrot set, and though the use of functions and complex numbers, we get a beautiful fractal pattern that can be “zoomed” into forever. The patterns go on and on. This is possibly the best example I could think of, and perhaps this form of art-meets-math can inspire your own art in the future.

• Find a tutor. Not just any tutor will do, but one that truly has the spark of math inside him or her. How could you possibly find this in someone? Ask them what they think of mathematics. Do you see their eyes light up? Do they delve into a description of its beauty? Is he or she able to explain concepts well and give you a reason for thinking its important? This is the passion that you see. Seek it out in a tutor and you have won half the battle.
• Forget about your past. I have heard the story many times; students who struggle with mathematics sometimes are met with the wrath of their parents or teachers in their younger years. Sometimes, this frustration led to abuse. This negative conditioning can carry over into a person’s adult years, and can scar them so much that they have an almost pathological aversion to mathematics. It is those types of people who have done such a disservice to education that fuels my passion to teach. Sometimes a hurdle is hard to jump, but understanding, encouragement, and patience go a long way.
• Find an application that is relevant to your field of study. This falls in lockstep with the first point; if you find a way to apply math to what you love, then you can find a reason to sustain your interest. Do you have a passion for Psychology? Statistics plays a vital role in understanding the results of well constructed experiments. Without it, correlations and causes could not be rigorously determined. This is why quantitative methods are taught to budding psychologists to this day.

I can’t say it enough; all math is math for liberal arts. It will take time and effort, but it can be done. Believe in yourself and take action.

When it comes to math for liberal arts, I believe that the visual arts are the most representative. Check out Viewpoints: Mathematical Perspective and Fractal Geometry in Art. This comes highly recommended from a personal professor of mine, and I found it enlightening.

## 100 Digits of Pi – How Do they Calculate it?

Pi is a number found throughout mathematics. Just like a thief in the night, Pi can pop up somewhere when you least expect it. Fundamentally speaking, Pi is simply the circumference (the length of the perimeter) of the circle divided by its diameter (length of circle straight through the middle).

There is a simple experiment you could do at home: Take a piece of string and, a ruler, and a few circular things in your house (peanut butter caps, wheels, etc).

1.) Wrap the string around the circular object and measure it.

2.) Then measure the diameter by using to placing the ruler right in the middle of the circle.

If you did everything correctly, you might get some measurements very close to pi or even accurate to a decimal place or two. Sometimes your measurements might look like 3.3 or 3.2 and this is due to the fact that (1) no matter how precise with you are with measuring, you are likely to be off a bit and (2) the circular object you use is likely not a perfect circle. We know that, if you were perfect with your measurements, and a had the most perfect of circles, your measurement could be very very good. You could even get the measurement out to 100 digits of Pi, and this is what it would look like:

3.141592653589793238462643383279502884197169399375105820974944592307816406286208996280348253421170679…

See those 3 dots? That is called an ellipsis and is used in math to denote that something keeps going on. Pi is an interesting number in that if you could calculate it out far enough, there would never be a time where it would end or have a forever repeating decimal (like 1/3 = .33333333333…). Pi will keep going on and on, and never end.

No matter what circle you find in the universe, if you were to do the above experiment on it, it would always approximate or equal Pi. How interesting is that? It is where the world of math and the physical world collide, which helps us to understand that math can be used to understand the universe around us.

Nowadays, the first 100 digits of pi is small change; computer scientists have calculated the number out to trillions of digits. Most applications of Pi in the real world do not require more than ten to fifteen digits, so why do computer scientists waste their time with calculating Pi? As it were, calculating Pi is a very good way of testing the capabilities of your computer. It is a popular “stress test” for computers.

Warning, Don’t Try MSpaint at home

In any case, back to the question at hand. How do mathematicians find the value of Pi so accurately. Obviously mathematicians don’t sit around with strings and rulers all day, despite the amusement that the English department might derive from such a scene. In fact, there are many equations that can be used to calculate pi. This particular equation is called an Infinite Series:

Courtesy of Wikipedia.org

Don’t get too intimidated! If you look closely, you see that the top part of the fractions are always 4, and the bottom is always an odd number. As the series continues, the bottom odd number increases 1, 3, 5, 7, 9, and so on consecutively. Since we are adding and subtracting, we are always going above the value of pi, then below the value of pi. If you were to continue adding these terms, your answer would converge on pi, meaning that it would get closer and closer to the actual value.

It can be hard to wrap ones head around this; how do you get the actual value of pi if the digits never end? In reality, we never do get the actual value. Our calculations only become more and more accurate but it will never reach the exact value. With this in mind, 100 digits of pi in the perfect world of math is awfully inaccurate. Good thing we live in the real world.

## Quadratic Equation Problems and You (With Practice Problems)

In high school and early college, I used to tutor several students who were taking basic algebra and had several issues with quadratic equation problems. The biggest issue was understanding how to factor quadratic equations, which they would then use to find the “roots” or the the solutions that equal zero. A good example would be something like:

x2+2x+1 = 0

This is one of the most basic quadratic equation problems, and like most quadratic equations it follows what is called a general form:

Ax2+Bx+C

A, B, and are what we call coefficients. Coefficients are some number that we multiply the by. In algebra, it is common to put a variable next to a number to say that we are multiplying them together. In our example above we have the term 2x. The number 2 is the coefficient, and our variable is x. This is very important to remember this when we solve this example.

In order to solve this problem, we would commonly write out this:

(x + ?)(x + ?)

Now, the pluses in this expression could also be minuses, in any particular combination you can think of. I always told students that while considering the general form of quadratic equations, ask yourself:

What two factors of coefficient add or subtract together to make coefficient B? Now, look again at our example:

x2+2x+1 = 0

Our “C” is and our “B” is 2. Now that we have them identified, ask yourself the question: What two factors of coefficient add or subtract together to make coefficient 2?

we know that -1 times -1 is 1. but, if you add those factors together, it gives us -2. That’s not what we need. We also know that 1 times 1 is one, and that this adds up to 2. This appears to work. So, we have our two factors, and we can plug them in to our question marks above:

(x + 1)(x + 1)

Now, we must use our FOIL method (First, Outer, Inner, Last) to multiply them together:

Our result is:

x2 + x + x + 1

Which we can simplify to:

x2+2x+1

So we know that the factors are correct! Now that we know our factors, we simply have to make the factors equal to zero:

(x + 1)(x + 1) = 0

Now, we have two equations that we can look at:

(x + 1) = 0

(x + 1) = 0

In this case they are the same, but they can be different. Why does it work out this way? Because if you solve for one factor, you get 0(x+1) = 0. this is obviously zero, but it also works the other way around!

In this particular example, x = -1. This is because the factors are the same.

And there you have it! This is the solution to the example. In order to master this concept, you must follow the three P’s (Practice Practice Practice!). I will include several problems for you to explore on your own.

Assignment: Solve x for y = 0.

1. y= x2+10x+21
2. y= x2-7×-18
3. y= x2-13x+30
4. y= x2-13×-30
5. y= x2+5x

## Teaching Geometry – How to Teach Geometry Effectively

Geometry is possibly the oldest and most intuitive form of mathematics. From the ancient Greeks to the present day, geometry has played a role in forming the world’s architecture, construction techniques, and even physics from Isaac Newton to Albert Einstein. For those of us who remember geometry in high school, teaching geometry consisted of learning different rules of angles and lines, and regurgitating them in the form of rudimentary proofs. While the importance of understanding proof cannot be understated (since it is the foundation of mathematical thought), reflecting on my personal geometric education made me realize that this method could leave students wanting.

The real challenge reveals itself in how to teach geometry in a way that fosters the intuitive nature of the subject. It’s such a powerful and enduring subject that surprisingly enough is much more than just a collection of theories regarding lines and angles, it’s a field of study whose implications has led the zealous to murder, and established the tradition of rigorous formalism that we see today.

I could imagine my readers faces at the mention of murder in a selection about mathematics. Murder you say? On its face, it is quite a stretch to go from teaching geometry to a capital crime, but strangely enough (at least according to legend) it has happened. This takes us again back to ancient Greece during the time of the Pythagoreans. The Pythagoreans were a religious group founded by Pythagoras (of the famous Pythagorean Theorem). Pythagoreanism was a melding of mysticism and mathematics; it was an esoteric brotherhood that appreciated the beauty of mathematics, their religion profoundly based on its perfection. One of their well known beliefs was that all numbers were commensurable or able to be expressed as ratios or fractions. However, there was a Pythagorean named Hippasus of Metapontum who is credited with the discovery of incommensurables or numbers that cannot be expressed as ratios or fractions. According to the story, his discovery was made while riding on a ship with some of his fellow Pythagoreans. Allegedly, they were so shocked by his claims that they sentenced him to death by drowning and threw him overboard. Alas, incommensurables exist and are now known as irrational numbers. The square root of two to Pi are just two well known examples.

The history of geometry is not completely bleak, and we see this in another Greek mathematician named Euclid of Alexandria. Known as the “Father of Geometry”, Euclid wrote a set of thirteen books known as The Elements. Euclid did something revolutionary; his work began with some fundamental facts or unquestioned assumptions called axioms. From these axioms, Euclid constructed several proofs that not only dealt with geometric figures, but also with prime numbers. Yes, even numbers themselves were not viewed like we see them today (1,2,3…) but as the lengths of lines. Until this very day, Euclid’s Elements remains a relevant text, and was the introductory textbook in geometry for over 2000 years. Its fun to think that our very mathematics education is still affected by a man who lived millenia ago.

Okay, so these are some interesting facts, but what does this have to do with teaching geometry? One of the best things for teaching a new concept is context. What is geometry? Why does geometry matter? Where did it come from? It’s the seed that is planted that, if nurtured, will lead to a growing interest not only in geometry, but in mathematics as well. And you can see that this visualization of numbers started with lines and angles. When we consider it this way, we see that geometry is the door to mathematics. Plant the seed.

## Geometry for Dummies – How to Do Geometry and Understand It

I suppose I am in a real geometric mood, because this is my second consecutive post on the subject. However, I personally find geometry to be one of my favorite topics in math, and its subtle relationship to the rest of mathematics is overwhelmingly important. Some might say that titling the article Geometry for Dummies is a way of implying that perhaps there is a way to get around the hard work involved with sufficiently learning the subject. For those who say that, I must once again invoke Euclid.

Euclid lived during the time of the Ptolemy I Soter, a general who was under the command of Alexander the Great. Later, he ruled over Egypt and was an admirer and personal benefactor of Euclid. According to legend, Ptolemy found Euclid’s Elements to be too difficult of a text, and asked Euclid for an easier way to learn geometry, to which he replied, “There is no Royal Road to geometry”. I should assure my readers that although geometry may be a daunting subject to the uninitiated, it is not inaccessible.

In this article I want to go over some of the basics of how to do geometry, which covers some of the fundamental ideas that we see in the Elements of Euclid. Remember that Euclid’s work is a seminal masterpiece that has survived as a mathematics textbook for over 2000 years. If you can understand some of his basic axioms, then you are on the fast track to learning geometry:

1. If you draw two points, a straight line (or line segment) can be made with them. Draw any two points on a sheet of paper. No matter how you draw those two points on the paper, you can always draw a straight line between them.
2. A straight line segment that we were able to make out of those two points in axiom one can be extended forever either way. You could make that straight line go on forever, and it would still be a straight line.
3. Take any line segment that ends in two points, like the one that we had in axiom one. Pretend this was a string, where one point was stuck into a piece of paper and the other had a pencil attached. If you were to move the pencil around, the drawing would look like a circle and the distance between the center of that circle is the edge of it would be the length of the string! That would be anywhere on the edge of that circle! That length of string would be called the radius of the circle.
4. There is a special type of angle called a right angle. A right angle is where two lines intersect at exactly 90 degrees. This kind of looks like someone just took an edge off of a square. The 4 axiom says that, it does not matter what two right angles you take, they will be equal to each other.
5. Now this is the big one. This one has been debated by mathematicians of the past, and created new fields of geometry that we use to this day! Say you took two straight lines. If you drew them straight enough with respect to each other, could you make it to where they would never touch? Perhaps you could, but how would you know? All you would have to do is draw a line through them! Are any of the inside angles on one side make an angle less than two right angles? If so, those lines will touch eventually. Are all the inside angles right angles? They will never touch. This one may not be as intuitive as the other four, but this illustration should help you to understand.

The Fifth axiom is known by mathematicians as the parallel postulate, and many attempts were made over the years to ensure its truth. Mathematicians found that in some cases (when you are not drawing on a flat piece of paper) that this axiom does not work. This led to the field of Non-Euclidean Geometry.

These concepts I explained aren’t exactly Geometry for Dummies in the sense that they make it shorter to learn geometry. This is just the first step. These are intuitive concepts that you will see throughout your geometric journey. The path is rougher than a Royal Road, but the satisfaction of the journey is profound.

Happy Learning!