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?