Direction Fields

Radians

The transition from understanding angles in radians from degrees is a strange one for many students. Why is it that just after they spend all of this time learning how to measure angles in degrees do they now have to learn an entirely new form of measure?? Much of our folk understanding of angles is measured in degrees - there are 180 degrees in a triangle and straight line, and 360 degrees in a circle. But, as it turns out, we need a new unit of measurement to describe angles if we are to use that measure in some more advanced contexts.

The way we are originally taught to measure angles was a convention made up by the Babylonians since they had a base-60 numbering system. The number 360 was very important to the Babylonians, and so they decided to say that that was the number of degrees in a circle. If 100 was important to them, it would seem like they could have just as easily said that there were 100 degrees in a circle, and all of our subsequent calculations would by based on that standard.

Question: If there are 100 degrees in a circle, how many degrees are there in a triangle? How about a right angle? Did you need to make any assumptions in arriving at your answer?

This raises an interesting point: Do all numbers that define physical quantities have a certain arbitrariness to them? Is there something special about there being 12 inches in a foot, or 60 seconds in a minute? It turns out that most often these numbers are just conventions that were made up by some culture or person and then just continued onwards over time. I recently learned that Tibetans use a measure of one finger width to represent their fundamental unit of measure when drawing images, and that there are 12 such finger widths in a face.

Radians, however, represent a unique example of where there is an objectivity to a standard of measurement which most units don't have. This comes up when we start considering the trigonometric functions like sin(x) and cos(x), and particularly the use of infinite polynomial series to represent functions.

THE UNIT CIRCLE

The definition of a radian has to do with the definition of the circumference of a circle. To understand where a radian comes from, we consider a circle which has a radius of ONE and which is centered at the origin. This circle is called the unit circle, and its equation and graph are:








A radian is defined as the length of the unit circle's circumference that an angle will cut off. So, since the unit circle has a circumference of 2pi, an angle that scopes out the entire circle, which we would normally call 360 degrees, we now call 2pi. This is the actual length of the circle's arc that this angle will cut out if we took a tape measure and wrapped it around the circle. Similarly, if we were to consider an angle that scoped out half of our circle, which we would traditionally call 180 degrees, we now call that angle measure PI, since that is the length of the arc being scoped out.

For most purposes, being able to move back and forth between degrees and radians is as simple as using the conversion factor that 2pi radians equals 360 degrees. So, if I told you to find the number of radians contained in an angle of 40 degrees, we would set up the proportion below and solve for x:





Question 2: Find the number of radians in the following degree measures:

1. 30 degrees
2. 90 degrees
3. 150 degrees
   

Question 3: Find the number of degrees in the following radian measures:

1. pi/2
2. 4pi/6
3. 
   

INFINITE SERIES

The reason why we use radians to measure angles has to do with an extremely important yet difficult area of mathematics known as infinite series. In high school mathematics, there is a very special type of function we treat often known as a polynomial. It is basically the addition of a certain number of terms, where each term represents a different power of our variable. For instance, the following are polynomials:



To add to our confusion, we need to also remember that "degree" is a way that we classify different types of polynomials that has NOTHING to do with the word "degree" used to measure angles. A second-degree polynomial, for instance, is a polynomial with an x-squared as its highest power. A tenth-degree polynomial is a polynomial with an x^10 as its highest power.


In general, a polynomial is the addition of a whole bunch of individual powers of the variable, and we can write this in a compact form called sigma notation. In sigma notation, a general polynomial can be expressed as:


where M is the degree of the polynomial. A very important result in mathematics is that many types of functions that are NOT polynomials can be expressed as polynomials of an INFINITE degree, which can be written as:


What this means is that our polynomial keeps adding more terms with higher and higher powers of x, and this can be used to create any function that doesn't have any super sharp corners or breaks in it. This is true of the trigonometric functions sine and cosine, which can be approximated as the following infinite series:




But what value should we use to measure our angle "x"? To get the right answer, we must use radians in the above polynomial approximations. If you were to try to use degrees, your answers would start to get very very large, but we know that sine and cosine can never get bigger than 1 or less than -1. Radians, it turns out, are the objective measure of an angle that we can use in series approximations to get valid results for sines and cosines. In fact, there aren't any situations where we HAVE to use degrees, but here is a situation where we HAVE to use radians, making me feel that it would be best if we never learned degrees in the first place. Who knows anybody from Babylon, anyway?

Question 5: Based on the discussion above, calculate an approximate answer to the following expressions using the first 5 terms of the series approximations for sine and cosine.

Play Auditorium

This is a really cool game sent to me recently. It is called Auditorium, and is made by a company called Cipher Prime. The goal is to move around the circular pucks on the board and change their radius to control the flow that is coming into the screen. When the flow enters the circular puck, it changes direction depending on the circle's symbol direction, position and size. If you can move the flow to hit the things that look like meters, they will start playing music.

Your goal, or problem, for each level is to light all of the meters up at once by directing the flow through them all at the same time. It is really, really fun and visually beautiful. I have been using it with some of my students, and it definitely tests their logical reasoning, problem solving and visual reasoning skills, while engaging them visually, tactially and aurally. From a teaching vantage point, the fact that there are many ways to solve a problem is great for mimicking real world problem solving, and could be used as a way to identify different learning styles across student populations by analyzing a student's solution paths.

In addition, one of the most interesting and useful areas of applied mathematics is that of DYNAMICAL SYSTEMS, which often deals with the behavior of fields and flows like we see in this game. This game setting offers a setting to introduce the idea of Curl (in two-dimensions at least), which can be one of the most difficult in multivariable calculus and dynamical systems. The puck pieces in Level 2 (Spring) with the crescent on them are actually curl creators -- they cause the flow to curl into a vortex, depending on the radius of the puck.

Writing an equation for how to control the degree of curling in the flow is an important part of understanding a number of physical systems, like fluid dynamics (Wikipedia) and bird flocking:



Check it out here . . . It is really fun.

Play Auditorium

The SRMS Map

A map to traverse the different posts based on their relevant geographical or cultural location.


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Sangaku

In my exploration of the Internet in search of non-Western approaches to mathematics, I was delighted to discover a tradition in medieval Japan known as Sangaku, based on a form of traditional Japanese mathematics known as wa-san.

As I discovered from a number of sources, including here at Wikipedia , because Japan was cut off from Western societies during the period known as the Edo Period, the Japanese developed their own system to deal with the analysis of shapes and numbers. Citizens ranging from samurai, farmers and laborers would devise difficult mathematical problems and represent them in colorful wood block cuttings that they would hang from the temple walls. These served as both offerings to the gods, as well as challenges to other temple goers. An example is the picture above from 1847 appearing in the city of Uchiko.

There is apparently an excellent book, called the Sacred Geometry of Japanese Temples, which discusses the history of Sangaku and traditional Japanese mathematics in greater detail. I found this excellent set of geometry problems here (but note that the links to the answers no longer work!).

Let's start with the most straightforward one:

Q1) In the diagram below, three orange squares are drawn inside of a green triangle. If the radius of the blue circle in the upper left of the triangle is equal to 3, what are the radii of the other two blue circles?

Q2) In the diagram below, the three circles are tangent to each other and to the line beneath them. If the radius of the orange circle is 10, and the radius of the blue circle is 6, what is the radius of the red circle?

The Quadratic Formula

The quadratic formula is one of the most important expressions taught to math students, for reasons that extend much farther than solving quadratic equations. As it turns out, as also discussed in Rational Algebraic Expressions, second degree polynomials, or polynomials that have an x-squared in them, appear throughout science and Nature as a result of solving differential equations using a special technique called a transform. This fact is actually responsible for the appearance of oscillating systems in Nature, such as water waves, sound waves and light, as the quadratic formula is an excellent opportunity to introduce the concept of imaginary and complex numbers.

So let us revisit what the Quadratic Formula, shown above, actually is: it is a formula that helps us to factor a second-degree polynomial (a polynomial with an x-squared in it) when the factors are not readily apparent (which they very rarely are in real world situations). If you remember, when we are faced with an algebraic equation that looks like this,

x² + 5x + 6 = 0 (Equation 1)

we can sometimes solve this equation by factoring the second degree polynomial on the left hand side, and then setting each of these factors equal to zero. So, when we factor the polynomial above, we can rewrite Equation (1) as

(x+3)(x+2) = 0

And it turns out that this is only true when x+3 = 0, or x+2 = 0.

The reason this works is that if any product of two numbers is equal to zero, then either of those numbers, or both of those numbers must be zero, in order for the product to be zero. Any non-zero number times another non-zero number can never be zero, regardless of how small or negative either of them are.

And so if (x + 3) times (x+2) is equal to zero (which is the same thing as x + 5 x + 6 = 0), then either x +3 = 0 or x + 2 = 0. There can be no other way. And so, in this case, x = -3, or x = -2, which are called the roots of the polynomial. To see that this works, we can then plug in these values of x into Equation 1 and we will see that these values of x do indeed solve the original equation.


But how can we solve this problem if we cannot factor our polynomial into two factors? This is when the quadratic formula is used. We won't go through the derivation of the QF, which utilizes a principle called completing the square, but it is good to convince yourself that it works by looking at derivations like this, this and this. Once you convince yourself that the quadratic formula works, we can begin to understand how it works and what it can tell us about an equation we are trying to solve.

GRAPHING SECOND DEGREE POLYNOMIALS

Just as understanding the equation of a line, y = mx + b could enable us to graph a line on a coordinate axis using the slope 'm' and the y-intercept 'b' (more here), similarly, we can look at the general equation of a second degree polynomial, ax^2 + bx + c, and figure out what this equation would look like if we graphed it. By doing so, we can understand a lot about the particular equation we are solving.

If you recall from the posts regarding the equation of a line, it is helpful to understand functions by treating them in their most general form. In the case of a line, the function we considered was

y(x) = ax + b

and we learned that for different values of a and b, the graph of the function y(x) would look different. Because the highest power of x in the function y(x) is one, the equation of a line is an example of a FIRST DEGREE POLYNOMIAL. Because there are two parameters that we can control to make our line look different (a and b), we say that the equation of a line has two DEGREES OF FREEDOM. Any line in two dimensions can be graphed using the equation above by controlling the values of a and b.

Similarly, we can arrive at any parabola seen in Figure ## by changing the values of a, b and c in equation (2). Just as there was a rationale to how different values of a and b would affect the type of line we would get (the amount of slope, whether the slope was up or down, and where the line would intercept the y-axis), so too there is a systematic rationale to understanding how to make a particular parabola by changing the values of a, b and c.

One of the big leaps that happens when considering the parameters of a second-degree polynomial (a, b and c), compared to a first-degree polynomial (m and b), is that the shape of our graph -- in this case a parabola -- depends not just directly on the parameters themselves, but rather on the relationships of the parameters. In the case of a parabola, the value of 'a' does tell us the direction our parabola will open (up or down) and the value of 'c' will tell us the y-intercept of the parabola (the place where it will cross the y-axis). But these characteristics themselves do not provide enough information to know what the entire parabola will look like.

The first thing to notice about all of the parabolas in Figure 2 is that we can draw a vertical line through each of them about which the graph is symmetric -- if we fold the parabola along this line, it will fold back onto itself. This line is therefore called the axis of symmetry, and is arrived at by the equation:

x = -b/2a

It is important to notice that since this equation is x equalling a constant value, it represents the equation of a vertical line. Once we draw this line, we can begin to see where our parabola will be located on the coordinate axis, even though we still don't know exactly what it will look like and where it will be. What we can see, however, for each of the parabolas in Figure 2 is that the value that the parabola takes at the axis of symmetry is going to be the biggest or smallest value that the parabola ever takes (depending on if it is going up or down). As you will learn, a parabola can only change directions once, and does so at the axis of symmetry. Since the axis of symmetry only gives us an x-coordinate, we need to plug in that x-coordinate to our original quadratic equation to determine what the value of y is at the axis of symmetry. The x-y coordinates of the parabola at that point is called the VERTEX. Once we find the vertex, we can position our parabola vertically along the axis of symmetry.

THE ROOTS

Once we know where the vertex is, and if our parabola opens up or down, the next step is to figure out how wide or narrow our parabola is going to be. There are a few ways to do this, but the first way we will consider is to figure out the places where our parabola crosses the x-axis, which are referred to as the roots of our polynomial.

As mentioned above, in cases where we cannot factor our polynomial we will use the quadratic formula to determine the roots. If you notice from the QF written above, there are actually two solutions for the roots due to the +/- that appears before the square root. So in fact, the roots of a polynomial ax^2 + bx + c are given by


Though this may look rather complicated, we can break it down to make it much easier to grasp.

If you notice, the roots each have the term -b/2a in them, and then either add or subtract the same quantity from -b/2a: . So, the roots will be equally spaced around the value -b/2a, which they should be because we said that parabolas are symmetric around the line x = -b/2a.

Games that Teach: Algebra v. Cockroaches

THE EQUATION OF A LINE

If you ask most people who have been through high school mathematics what the equation of a line is, a surprising number can spit out "Y equals m x plus b", or y = mx + b. Many of them can even tell you that the "m" stands for the SLOPE, and the "b" stands for the Y-INTERCEPT, and that the number that we pick for each of these parameters controls the appearance of the graph of the line that we get.

But I also don't think many people understand why this equation is so important. Part of the answer is that this particular equation ISN'T that important. Though there are many examples of processes that behave like a line, most often we need much more complicated equations or functions to describe real world phenomena. But what IS important about the equation of the line are the techniques we will use to analyze equations of polynomials of which "mx + b" is an example. In fact, analyzing y = mx + b is one of the simplest cases of a function that we can analyze, which is why it is discussed so often.

To understand how to analyze our equation of a line, and how changing the values of m and b affect the shape of our line, I have been using the game below called Algebra v. Cockroaches, where students need to fill in the equation of the line that the cockroaches are traveling on to be able to exterminate them.

y = mx + b

Games that Teach: Gyroball

The following game, Gyroball, is an excellent exercise in spatial reasoning on a 3D coordinate plane, as well as a keen test of eye-hand coordination. In using it with different students and people, I am fascinated by how different people initially approach it. The key to Gyroball is patient control, which is something that is hard for many people.

I am also continually fascinated by how much people, especially young people, like to play games. And different students like different types of games -- understanding why and how would be interesting to understand learning and cognitive differences amongst students.

Your task for this post is to complete the first 5 levels of the game, and answer the questions in the form beneath the game. To do so, you might need a review of 3-d coordinate geometry.

Time Series

The Virtual Bead Loom

Although mathematics is often taught to be and thought to be solely the product of European culture, nothing could be farther from the truth. It IS true that Western European culture is responsible for much of the symbolic and logical structure of modern mathematics, and being fluent in this often arcane tongue is what is sometimes as thought of as being GOOD at mathematics. But the beauty of mathematics is that it tends to deal with concepts that are UNIVERSAL to the human experience, such as our understanding of shapes, patterns and quantities.

As mentioned briefly in the post Induction from Patterns, when I was in Guatemala recently I discovered that geometric patterns were an important part of Mayan civilization. I later learned that Guatemalan women have actually passed down their understanding of geometry to their daughters through the act of weaving. The patterns below are a picture I took in a street market in the town of Santiago de Atitlan along the shores of Lago Atitlan.




There is an excellent resource here called Culturally Situated Design Tools from the work of Ron Eglash at RPI which provide interactive environments to explore the connection between mathematics and art in indigenous cultures. There is a great overview of these tools in his paper here. The persistent theme across cultures is the presence of repeating geometrical patterns with different types of symmetry and regularity. There are tools to explore cornrow patterns in African American culture, beadwork in Native American cultures, and pyramid design and adornment in Mayan culture. There is little information on how to incorporate more advanced mathematics, particularly high school concepts, to the use of these tools, however.

Rational Algebraic Expressions

Many people have heard that mathematics and music have a lot to do with each other, but often are unaware how. Although I am not a musician, I HAVE gravitated strongly to the areas of mathematics which deal most heavily with music, particularly the fields of SIGNAL PROCESSING and DIFFERENTIAL EQUATIONS. Though a thorough understanding both of these subjects involve mathematics much further along than that taught in high school, I have been finding a number of topics in high school mathematics where it is appropriate to introduce math's relationship with music.

A student of mine has recently been working on what are called RATIONAL ALGEBRAIC EXPRESSIONs, or when two polynomials are divided by each other. As you may remember, a RATIONAL NUMBER is a number that can be expressed as a FRACTION. So numbers like 3 (or 3/1), 3/4, 22/7 are all rational numbers. If you remember, the word RATIO and RATION are names for fraction, which is why these are called rational numbers.

You may remember that some fractions, when expressed as decimals, involve an endlessly repeating sequence. For instance 1/3 or one-third is equal to 0.33333... and so would need an infinite number of numbers to express exactly. Another example is above, 22/7 which can be represented as 3.1414141414... and is often used as an approximation to PI. But because these numbers can be expressed as a fraction, the decimal sequence will always a nice pattern to it. In the case of 1/3, the number "3" over and over again; in the case of 22/7, the numbers "14" over and over again.

Alternatively, there are numbers which, when expressed as a decimal, are also endless sequences but which CANNOT be represented in a repeatable pattern. The most prominent examples are the square roots of numbers that are not PERFECT SQUARES. So numbers like the square root of 25 can be represented by 5 or 5/1, but a number like the square root of 23 could only be represented in decimal form if we kept an infinite number of decimal places. And so these numbers ARE NOT expressible as a fraction, and are called IRRATIONAL NUMBERS.

An excellent exercise to test your ability to identify rational and irrational numbers, as well as perfect squares is Number Cop, by Hans Software, which I have embedded below . . . Use the drop-down menu to decide which types of numbers you would like to test your knowledge of.

From Rational Numbers to Rational Algebraic Expressions

Similarly, a RATIONAL ALGEBRAIC EXPRESSION is an algebraic formula that is represented as a fraction. So, for example, the following are rational algebraic expressions:


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The techniques that are stressed in teaching RAE's are finding what are called the ZEROS, or the places where the RAE is equal to ZERO, and the POLES, or the places where the denominator is equal to ZERO. Though it may seem like a completely random and useless subject to introduce, it turns out that understanding expressions like these, and the meanings of ZEROS and POLES, is essential to understanding many physical systems from bridges to car suspensions to musical amplifiers. We'll consider an example below from music and electronics, known as a BANDPASS FILTER, once we look at how to analyze expressions like the ones above.

So let's say that we have a typical RAE like:


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If we were able to factor the numerator and denominator, we would be able to determine the values of s that make this expression equal to zero (by finding the ROOTS of the numerator) or which make this expression infinitely large (by finding the places where the denominator is equal to zero. If we were to graph an expression like the one above, we would in general get a graph that looks like:

FILTERS

In mathematical and engineering lingo, a FILTER is an object that selects certain frequencies to pass, while blocking others. (To learn more about frequencies, click here). The most common implementation might be a radio tuner. In general, in any city every radio station is being broadcast all the time to every place in its range. So, the antenna in your car is actually picking up every station that is being broadcast in your area -- if you tried to listen to all of them at the same time, it would likely sound like a whole bunch of static.

When you set the dial of a radio to a particular station (106.1 on FM is my fave in the Bay Area), what you are doing is telling your radio antenna to only listen to a particular frequency, as the number of a radio station corresponds to the frequency of wave that is carrying that station's signal. So, when I set my radio dial to 106.1, I am telling a circuit inside of my radio to FILTER out every other frequency that it hears besides 106.1, and so my speakers only broadcast the music that is coming from that station.

The name of such a circuit is a BANDPASS FILTER, since I am able to set a particular range of frequencies, or FREQUENCY BAND, that are allowed to PASS. (BAND-PASS, get it?). Usually, this is implemented by a circuit as shown below; the mathematics of such a circuit turn out to be controlled by a RATIONAL ALGEBRAIC EXPRESSION.


The diagram above is an example of what is called an RLC circuit. The R, L and C are the names given to electronic components called a resistor, an inductor and a capacitor. When desigining a circuit like the one above, we are able to specify which frequencies are allowed to pass by changing the values of the resistor, inductor and capacitor. If we were to analyze this circuit using techniques specified HERE, we would find that the equation governing its behavior is given by the rational algebraic expression:



For the purposes of this lesson, the most important thing to notice here is that this expression has a polynomial on the top and the bottom of our fraction, and that the coefficients of our polynomials is goverened by the values of the resistor R, inductor L and capacitor C.

Algorithms

One of the most interesting and elegant fields of mathematics, particularly as it applies to modern computers and technology, is the study of algorithms. An algorithm is a systematic technique we can use to solve a particular problem, or even better, a whole class of problems. The study of algorithms is often concerned with finding the fastest or least expensive system that will solve our problem satisfactorily over and over again.

In some sense, an algorithm is a little bit like a recipe. Let's say that you wanted to makes some chocolate chip cookies from scratch -- the recipe is the step by step system that will take you from raw ingredients to a batch of cookies. In a sense, the recipe is your solution to the problem of how to make a cookie given a bunch of flour, butter, sugar, chocolate chips, pans and an oven. A recipe might be defined as "good" for a wide variety of reasons: if it uses the cheapest ingredients, it makes the best cookies, or it takes the least amount of time. And a good recipe is also one that we can use to make any amount of cookies we want, not just a pre-set number.

The study of algorithms, however, is not the study of how to FOLLOW the recipe. It is the study of how to WRITE the recipe given a type of problem, and the ingredients and constraints to solve the problem. To be able to solve the problem I posed in the post about Induction from Patterns, you would have needed to develop an algorithm to deal with the cases where the number of diamonds grew too large.

If you ever interview with a software company, they are likely to ask you questions that test your ability to develop algorithms. At a place like Google or Microsoft, they often ask less sophisticated algorithm questions to non-technical applicants to test their ability to think logically. These problems are often in the forms of puzzles or games.

One of my co-workers Mr. Ankamah recently showed me a Flash rendition of one such problem, known as the Missionaries and Cannibals. This is an excellent game since it requires somewhat non-linear and non-obvious techniques to accomplish the task. Also, it is analagous to techniques that are used in computer algorithms that perform very complicated tasks as efficiently as possible.

It is interesting that one of my students, of Mexican heritage, actually pointed out that a particular version of this game that I found (but can't anymore) is really racist, and he refused to play. In this rendition, the cannibals are depicted as dark black people, and the missionaries are white. This game, he remarked, made it seem like black people are savages and white people are helpless spiritual victims. I was very impressed with his insight, and didn't make him play that version.

But I found THIS version, which makes the cannibals look like alien monsters. He told me that this version was racist against monsters but since, I argued, there are no such things as monsters, I did not deem this argument as persuasive, and made him play anyway.

To begin, try the game at the link above, and answer these questions:

RIVER CROSSING PROBLEMS

We will consider in this section three examples of river crossing problems, including the Missionary and Cannibals problems mentioned above.

An often overlooked part of solving problems like these is developing a system to write down the various steps that you are trying to solve the problem, and to begin to find a pattern in the way the steps are executed. In the case of finding the triangles in the Latin American bracelets in Induction from Patterns, we called this process of extending our solution to a single case to the general case, induction. To see how this works, let's start with a different type of river crossing problem, namely the jumping frogs.


http://www.cut-the-knot.org/SimpleGames/FrogsAndToads.shtml

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QUICKSORT

...tbc

Rappeling Down Fissures

This past weekend I went to Yosemite and bought an excellent guidebook to hike the park. If you haven't been to Yosemite, you really must go. John Muir, the premier naturalist of American history, once said that Yosemite Valley was the most glorious temple he of Nature he had ever been lucky enough to see. That part of the Sierra Nevada mountain range is so different than almost any others in the world. As is mentioned in the guide book, the park itself ranges in elevation from 2000 ft to over 13000 ft, an amazing 11,000 ft vertical range over 1200 square miles.



This guidebook mentions a place that I would like to go to soon, called Taft Point and the Fissures, . Apparently there are fissures in some rocks in the high country where you can look down THOUSANDS of feet to the valley floor. That is amazing. I hope to have my own picture soon, but the one here I found on the web.

This morning I showed this to one of my students, and he thought that it would be a good idea to rappel down the inside of these fissures. I, deciding to crush his imaginative spirit in the name of Truth, told him that it might require a prohibitive amount of rope to rappel down that long of a distance.

And, much to his chagrin, that we could use math to figure out whether it would even be possible.


So HOW would we go about figuring out if we could carry enough rope to rappel clearly down the fissure from the top? We might begin by asking how much space that rope would take up, known as its volume, and see if we could fit it into a pack. We may also want to know how heavy that much rope would be, or its weight, to see if a normal human being could carry that much rope with us.

My student and I decided to do this calculation for rappeling down a 3000 ft rock, using a rope that is 1.5" thick. We further assumed that the rope could be modelled as a really long thin cylinder, allowing us to use the volume formula for a cylinder. And since most backpack sizes are given in cubic inches, and not cubic feet, we decided to report the total volume of the rope in cubic inches.

This problem was interesting as I was trying to explain how it is that we can derive the formula for the volume of a solid like a cylinder or a cube, but can't so easily with a more complicated shape like a sphere or a pyramid.

DIMENSIONS

In understanding geometric shapes, it is helpful to think about the idea of dimensions. Most people have a good intuitive sense of what is meant by one, two and three dimensions. A one dimensional object would be a line, a two dimensional object would be something that you could draw on a piece of paper, and a three dimensional object would pretty much describe most solid geometric objects that we encounter in our experiences.

The nice thing about certain types of three dimensional geometric objects, such as cubes, cylinders and other objects that have uniform cross sections (e.g. their cross section is the same everywhere along their height), is that we can extend our formula for the area of the cross section to a formula for the volume of the solid object almost immediately. Before we do so, it is helpful to refresh our memory for calculating the area of two dimensional objects.

An Introduction to Probability: The History of Probability

I was recently loaned an excellent book entitled Against the Gods, The Remarkable Story of Risk, and have posted the first two chapters below as a reading assignment. Though this book delves more deeply into mathematics as it progresses, the first few chapters detail the history of mankind's use of mathematics to understand risk and make predictions, and how this helped to separate humans from a time where we believed that things would be unilaterally controlled by the gods.

In understanding mathematics, it is very useful to understand how and why certain types of mathematics are invented. They weren't just invented to punish school kids or to make some people think they were smarter than others (though mathematics has proven effective for both of these purposes). Probability, for instance, was meant as a way to systematize information so that we could make repeatable and reliable predictions about events that have a component of randomness. This can extend from playing a game of Poker to deciding to take a road trip to Colombia and not having our caravan seized by rebels.

So, to begin our study of probability, and how we can use mathematics to understand it, let's begin by taking a little history lesson . . .

Risk - Upload a Document to Scribd

How Loans Work

Understanding the mathematics behind how loans work is one of the most important ways in which mathematical literacy can help you. Loans enable us to buy things we cannot afford to pay for in cash, and are one of the most important tools in making business and industry possible.

Loans appear in many forms, from credit cards, to student loans, mortgages and when buying a car. Unless you become a gangster and use suitcases full of cash or watches as your means of transacting business, you will probably need a loan at some point in your life.

And keep in mind that just because you understand the mathematics of loans, you can still arrive at financial ruin by not having the discipline to follow what the math tells you. That is why there are so many depressed and depressing people in Las Vegas. They say that gambling was invented by God to punish people who are bad at math. Amen.

I. The Mathematics of Loans

When shopping for a loan, there are three numbers that you need to consider, all of which will change how hard it will be to repay your loan as well as how much money a bank will be making off of you.These numbers are (1) the interest rate, (2) the term, or length, of the loan, and (3) the monthly payment you will need to make to pay off the loan in the specified amount of time. Let's consider these in order, since they build off of each other.

(1) The Interest Rate

As the video in the Definition of a Bank post addressed, when a bank begins to loan you money, they charge you a fee to make sure that they continue to make money off of the money they lend you. This fee is called the interest, and controlling this rate is one of the most important concepts in the study of banking and money (at least, I think so, since I really have no idea).

Understanding interest mathematically is actually very subtle, and gives rise to some difficult concepts in calculus that we will not address here. Most simply, the interest rate represents a percentage of the amount of money being borrowed that you will have to pay off. So, if I were going to lend you $10,000 and charge you 15% interest, then one way to interpret this scenario is that you will need to repay me not only the $10,000 but also an additional 15% of $10,000 over the time you pay me back. In this case, that would mean that you would need to pay me an additional 1500$ beyond the original $10,000, making the total loan amount $11,500.

But unfortunately, loans are much more complicated than this. Instead of charging you a fixed percentage of your loan amount initially, most loans charge you a percentage of the amount you have borrowed over the amount of time you have borrowed it. So, let's say that you are charged 12% a year for a loan of $10,000. Then every year you are going to be charged 12% of the amount of money you borrowed. So, if the loan charges you interest once a year, then it might charge you 12% of $10,000 in the first year, or $1200.

But let's say that after one year you haven't paid back any of your loan. At this point, you owe the bank not just the initial $10,000, but $10,000 plus the extra $1200. So you owe $11,200 now, and the bank is going to charge you interest on it. So, now the interest isn't just on the initial $10,000, which would be an amount of $1200. It is actually 12% of the $11,200 you now owe the bank.

... to be continued . . .

Comparing Tax Plans using Averages and Percents

My coworker Mr. Roland recently started his own blog here, which is meant to be his compilation of mathematics teaching resources and miscellaneous commentary. Nice work Mr. Roland! Keep the posts coming.



The above picture is a graphic on this entry, entitled "Every American Should See This". It is particularly compelling in the context of teaching mathematics since it raises some very important concepts, namely how percentages relate to actual quantities and how the scale of a graph, (e.g. how the x-axis or y-axis is partitioned into values), affects how we can interpret a visual representation of information like this one.

Being able to quickly look at graphs and glean useful information and trends is an important skill across professions and people's interests. Graphs really are a case where a picture can be worth a thousand words, since within them is often contained meaning and insight that is difficult to convey in text. Being able to read, interpret and analyze a graph is important not only for professional reasons, but also to gain information and insight from magazines, newspapers and television.

Let's begin analyzing this graphic by looking at how the average tax cut of both of the candidates is calculated. Although most people understand the basic definition of an average from calculating their grades in school, averages can get more complicated when we consider averages using percents, as well as weighted averages.

Use the form at the end of the post to submit your answers to these questions.

As an example, let's say that your grade in a class was formed by averaging your scores on 5 exams, each graded on a scale of 0 to 100.

Q1) If you scored a 73, 89, 83, 100 and 92 on your 5 math exams, what would be your grade in this class?

Q2) If your scores on your 5 exams were x1, x2, x3, x4 and x5, what formula would you use to calculate your grade?

This is an example of calculating a simple arithmetic average by weighing each of our scores equally. Most of us have done it a million times without even noticing it, from splitting a check at a restaurant to calculating our grade in school.

In the graphic above, I was wondering how they calculated the average tax cut since there are a number of ways we can interpret an "average tax cut". And some of those ways will lead to more valid and effective insights than others.

Q3) Using the calculator in the side bar, what do you get for the average tax cut for each of the candidate's tax plans by simple averaging?

Note that this answer is slightly different than the average tax cut value displayed on the graphic. Why can't we just take the arithmetic average of the values we see above to determine the average tax cut of the population in a manner similar to averaging the grades in the example above?

The reason is that when averaging percentages in the example above, particularly in considering how to calculate a new average for the entire population, we must not only consider the actual percentages but also how many people are in each of those groups.

To see this more clearly, let's consider a simpler example.

Let's say that we had a population of 10 people and 7 of them made 35 thousand dollars a year, 2 made 100 thousand dollars a year, and 1 of them made 500 thousand dollars a year.

Q4) What would be the change in the amount of money paid in taxes by this population under each candidate's tax plans?

Now that we know the total change in money paid in taxes, we can determine the average tax cut by dividing this number by the number of people in our population.

Q5) Calculate the average tax cut for each candidate's tax plan in this population. Is this number greater than or equal to the average tax cut that would be calculated by simply taking the average of the percent tax cuts that apply to the categories represented in this problem? Why or why not?

What this population analysis illustrates is that the number of people in each income group affects the average tax cut. You can't calculate the average tax cut just by averaging the percentages.

So, returning to the graphic for the candidate's tax plans, to see what the actual tax cut is for the population we really need to know how many people are in each of these categories.

Q6) Using the Internet or another resource, make an educated estimate for the number of people in each of the income categories listed in the graphic above?

Q7) Use the information you gained in Question 6 to determine the change in the amount of the overall taxes paid, as well as the average tax cut for each of the candidate's tax plans. How does this compare to the values displayed in the graphic?

The Mathematics of Asthma

I found the article below in an issue of Scientific American in the context of teaching high school math, and I thought it was really interesting how many inferences somebody could make, or might want to make, if they applied themselves mathematically. It has made me start to see numbers in more of the things that I read.

This article is about something that affects me personally and, as it turns out, affects a number of my students as well. Many come from families with a high incidence of asthma.

Recently, the inhalers that I have been using for the past 20 years, known generically as albuterol inhalers, have undergone a major change. This is supposedly because the old inhalers emitted chlorofluorocarbons, or CFCs, which are damaging to the Earth's ozone layer.

But, in fact, there is likely a deeper issue at play.

One thing that pharmaceutical companies hate is when their patent expires on a drug, since then anybody can start manufacturing that chemical generically. In the United States, this period is around 17 years. Often what these companies do (as I learned on an episode of Law and Order) is that they change the drug slightly when their patent is about to expire so that they can get new patent protection. In the case of the albuterol inhalers, the shift in government regulations allowed them to re-apply legal protection to albuterol inhalers so that they can again apply premium pricing on it. It is a total racket, if you ask me.

So, I've been having some students read the article below and answer some questions about it. It demands an understanding of percents, fractions, stepwise problem solving, and mathematical reading literacy.

To view the document, hit the full screen icon at the upper right of the document viewer, and use the +/- buttons to zoom in and out.

"Change Is In the Air" - Upload a Document to Scribd

Based on the article above, answer the following questions . . .

Simultaneous Equations

In the video below, the Portuguese soccer star Cristiano Ronaldo is racing a car. I am left to wonder just how fixed this whole thing is, but let's pretend for a second that it isn't. In either event, it makes for some good math problems.

In particular, I would like to see if we could use math to determine whether it would ever be possible for Cristiano Ronaldo to actually win a race like this one against a Bugati. And, if so, how we should set the race up to make this happen. There are a number of important concepts from high school mathematics that are involved in solving such a problem, including understanding the behavior of polynomials as well as how to model a real world phenomena using systems of equations and the techniques to solve such equations.

To make things easier, let's start with a simpler case than the scenario we see in the video above. In particular, let us consider what happens if the equations that we are trying to solve are linear equations, and can be expressed as a line. An overview of the equation of a line, as well as a test to check your understanding, are given in the post Algebra v. Cockroaches.

THE EQUATION OF A LINE

One of the most repeated concepts in teaching high school algebra is the concept of linear equations. It is almost a little bit annoying how much they stress this concept without telling you why it is important. For instance, you may remember or already know that the equation of a line is given by the form . . .

y = mx +b

.. where we say that m is the slope, and b is the y-intercept. To understand what these terms mean, it is helpful to look at the graphs of some lines. The slope of the line is a measure of how steeply the line increases or decreases. In the picture below, line (1) has a positive slope because it is increasing as we move to the right. Line (2) has a negative slope because it is decreasing as we move to the right.

In a scenario like a race, or an object moving with some speed, a linear equation arises when that object is traveling at a constant speed. So let's say that we were driving a car at 55 miles per hour, and we wanted to know how far we drove in 3 hours. Or 2.5 hours, or 10 hours?

To arrive at an answer, we would multiply 55 mph by the amount of time spent traveling in hours, and that would tell us how far we have gone. In this case, we could write a linear equation where y represents the distance traveled in miles, and x represents the time traveled in hours. In this case, our equation would be

y = 55x

In this case, if we were to graph this equation, we would see that it was essentially plotting the position of the car at different times. And if we were to draw this graph, it would appear to be a line with a slope of 55. In this equation, the y-intercept, or b, is equal to zero.

Now let's say that instead of measuring how far we had driven, we wanted to know how far away we were from our house. If we started driving when we were 7 miles away from our house, and we were traveling at 30 mph, what is the equation we could use to determine how far away we are from our house?

In this case, our equation would be

y = 30x + 7

and this graph appears as well in

... TO BE CONTINUED ...

Drum Making

AN INTRODUCTION TO CIRCLES

The number of interesting shapes that there are in Nature is too great to treat all of them, but if there are a few that every person should know a thing or two about , they are: circles, triangles and rectangles, and in 3-d, spheres, cylinders and cubes.

In this lesson, we are going to stick with circles, which could very well be the most important of all of them. We see circles everywhere in Nature. We see them as wheels, in the middle of eyes, and as the cross section of trees. They are important components of early artistic traditions, including Tibetan mandalas and Native American symbols and forms.

But to be able to use circles in doing calculations via equations, we must get a much more rigorous definition of them.

Q1) What is your definition of a circle?

Everyone knows a circle when they see one, but to understand the algebraic definition, it is helpful to consider this more deeply.

A circle may best be defined as all of the points that are at the same distance from a given point, called the CENTER.

So if I were to take a point C as my center, as in Figure 1, and wanted to know all of the points that were 2 units away from the center, those points would make a circle. The distance that I set as the distance away from the center I am interested in is known as the RADIUS.

So now let's say that we wanted to draw a circle on a piece of graph paper like in FIgure 2. Let's further say that we are going to draw a circle whose CENTER is at the ORIGIN, or the point (0,0). We can determine the equation that defines all of the points that are a given distance away by refreshing our understanding of the distance formula.

The distance formula, you may recall, is given as d = ((x1-x2)^2 + (y1-y2)^2). We will consider where this formula comes from HERE, but for now, let's ask the question "What are all of the x-values and y-values that are a distance 2 away from the point (0,0)?" In this case, we could say that one of our points in the distance formula is (0,0), so x2 and y2 are both zero. Now, if I gave you another point (x1,y1), you could tell me the distance between it and (0,0) by plugging in that value and solving for d.

But in this case, I am asking the question: Given a distance, say 2, what are all of the x's and y's that are that distance away from (x2,y2) = (0,0)? So in this case, I would plug in 2 for the distance, and the points that satisfy this are the points (x,y) for which

2 = (x^2 + y^2)^(1/2)

The Definition of a Bank

I was recently tasked with giving a 20 minute presentation to high school kids about the current financial crisis and I found the incredible video above which explained the concept of a bank.

A few weeks ago I attended a conference at Mission High School called Teachers For Social Justice, or T4SJ. During one of the workshops, entitled "Using Mathematics as a Weapon", I was partnered with a woman who has apparently been organizing conferences on Teaching for Social Justice for a while. I have been struggling with how to convince students that learning math is important to them, and she pointed out the importance of teaching them how to manage money and understand loans and finance to make sure that rich bankers don't take all of their money. So, I entitled my presentation: "The 2008 Financial Crisis, Or How to keep Rich People's Hands Off of Your Money". I didn't end up giving the presentation, but I did make some slides and will post them with some explanation when I get a chance.

Induction from Patterns

FIGURE 1.

I found the diagram above on a site called Puzzler's Paradise, and decided to adapt it to teach some spatial and inductive reasoning.I thought this problem was interesting because during my recent trip to Guatemala I bought the bracelets below that had this and similar patterns on them. Use the form at the end of this post to submit your answers to the Questions.



On Puzzler's Paradise, the question for the red white and black pattern in Figure 1 is:

Q1) How many outlines of triangles can be drawn in the above pattern?? (HINT -- triangles can be of different sizes)



This is a good question since it requires spatial reasoning on multiple spatial scales. As you can no doubt tell, there are multiple ways to form triangles in the above pattern, and each of those "ways" has many triangles of that type that can be formed.


The power of mathematics and inductive reasoning is to be able to take a pattern like the one above that is amenable to simply counting up all of the solutions, and make a broader conclusion about how the number of triangle outlines relates to the size and nature of the pattern itself. Such analysis represents an important tool in problem solving, namely using a smaller, more manageable circumstance to infer or induce the properties off larger problems of this type.

Before proceeding to our induction though, let's take a second to see how well you remember the definitions of different kinds of triangles.

Q2) Of the triangle outlines you identified above, how many are equilateral triangles? How about right triangles?

Now, what if we had a pattern like the one above which, instead of having 4 black diamonds had 25 black diamonds running across the center? Or 50 diamonds? In these cases, you wouldn't want to have to draw or make the pattern, and then count up all the triangles like you have just done. Instead, you might want to abstract your answer to the 4 diamond problem, and see if you can use that answer to solve the problem for ANY number of diamonds. This, mathematically, could be referred to as the N-diamond problem.

The process of preceding from a simple instance, like the diagram above, to the more general case of a pattern where we may not want to have to manually count all of the triangle outlines is an example of reasoning by induction. This type of reasoning is an important tool in the problem solver's arsenal. By seeing how the question can be solved for a simpler case (in the pattern above, for the 4 black diamond case), we might be able to arrive at a system that allows us to make conclusions for much more complicated cases (like for 25 black diamonds across).

To begin, we may want to either look at another simpler case, and determine if there is a pattern from the case of 4 diamonds to 5 diamonds. This is particularly important to make sure that the pattern is the same for every number.

Q3) Draw the pattern above for the case where there are 5 black diamonds across the center. Repeat Q2 for the case of 5 diamonds across.

So what did that lead you to believe? Is the pattern the same for every number? What do you think the answer to Q2 would be if there were 6 diamonds?


Q4) Determine the relationship between the number of black diamonds across the center of the pattern above, and

1. the number of triangle outlines,
2. the number of right triangle outlines, and
3. the number of equilateral triangle outlines.

Your answer can be in the form of a table that includes the number of each of these quantities for all multiples of 5 from 10 to 100.

Q5) Is this relationship linear in any of these cases? For the cases where it is linear, if x represents the number of black diamonds across, what is the equation of the line that represents the quantity to be calculated?

EXTENSION QUESTION

Can you use inductive reasoning to answer the question below?

Q6) Everyone at a party shook hands with everybody else exactly once. In all, 66 handshakes took place. How many people were at the party?