Mr. Edwards Math World

Simpson’s Paradox

Posted on May 19, 2012 by Brett Edwards

Probability theory is full of interesting perplexities that can interest almost any student. One thing making the study of probability so enjoyable is that so many results are counter-intuitive to what we would expect. A great example of this is called Simpson’s Paradox. There are different ways to illustrate the paradox but I will use boxes containing a mix of gold and lead balls (each ball weighing and feeling the same). We will place a mix of gold and lead balls into two different boxes, a brown box and a black box. Clearly we would rather select the more valuable gold ball therefore we want to determine which box has a better probability for such a selection. In the first scenario, the brown box contains 7 gold balls and 8 lead balls while the black box contains 4 gold balls and 5 lead balls. Which box has a better probability of selecting a gold ball at random?

The brown box has a superior probability of 7/15 (47%) compared to 4/9 (44%) in the black box. In the second scenario, there are 7 gold balls and 4 lead balls in the brown box with 11 gold balls and 7 lead balls in the black box. Again, which box has a better probability of selecting a gold ball?

Kind Trolls Riddle

Posted on May 8, 2012 by Brett Edwards

You are on your way to visit your Grandma, who lives at the end of the valley. It’s her birthday, and you want to give her the cakes you’ve made.

Between your house and her house, you have to cross 7 bridges, and as it goes in the land of make believe, there is a troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake.

How many cakes do you have to leave home with to make sure that you arrive at Grandma’s with exactly 2 cakes?

Posted in Fun, Riddles | Leave a comment

Finding a Sure Bet

Posted on May 4, 2012 by Brett Edwards

Pierre Simon Laplace said “probability theory is nothing but common sense reduced to calculation.” The intricacies of probability theory provide many opportunities for alert minds. The Virginia Lottery provided such an opportunity in 1992. One obvious rule that every lottery should know to follow is that the total prize money divided by the number of tickets sold (expected winnings) should always be less than the cost of a ticket. However, the Virginia Lottery broke this cardinal rule of lottery probability. This particular lottery chose 6 numbers from 1-44. Using the formula to determine the number of combinations, as seen below, we can find out the number of possible 6 number combinations.

n!/k!(n – k)!
44!/6!(44 – 6)!
44!/6!(38!)
(44 × 43 × 42 × 41 × 40 × 39)/(6 × 5 × 4 × 3 × 2 × 1)
7,059,052 combinations

Here’s the rub. The lottery jackpot was $27 million. Thus, the jackpot was higher than the sum total of potential combinations. Including the other potential prizes for 2^nd, 3^rd and 4^th prizes, the pot grew to $27.9 million. By taking $27.9 million and dividing it by the number of combinations we can find that the value of each ticket was worth about $3.95. A “mathematically minded” group of Australian investors noticed the opportunity and rounded up 2,500 other small investors to invest $3,000 each in an effort to purchase tickets for every possible combination ensuring a winning ticket. There remained the possibility of multiple winners and the logistical hurdle of buying 7 million tickets! After careful planning, the scheme was enacted and as expected the logistics of purchasing such a quantity of tickets proved too difficult. The group was able to purchase only 5 million of the 7,059,052 combinations. The winning numbers were announced and days passed before anyone claimed the winning ticket. The investors had in fact won but it took them days to find the winning ticket. Louis Pasteur said it well saying “chance favors the prepared mind.” Leonard Mlodinow tells this story and much more on probability theory in a great book called The Drunkard’s Walk.

Posted in Fun, Probability | Leave a comment

Amicable Numbers

Posted on April 30, 2012 by Brett Edwards

The Greek mathematicians had a special love for number that helped them discover a host of interesting numerical relationships. An interesting study to be done by any student of numbers is that of “amicable” or “friendly” numbers. Such numbers are a pair of numbers whose divisors sum up to the other. The Pythagoreans discovered that 220 and 284 are one such pair.

Divisors:

220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
284: 1, 2, 4, 71, 142

Sum:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
1 + 2 + 4 + 71 + 142 = 220

Some have even gone so far to connect this idea to the number of goats that Jacob gave to Esau as representing a special expression of Jacob’s love for Esau.

2,000 years after the Greeks, Pierre de Fermat continued these peculiar number explorations by finding the amicable pair of 17,296 and 18,416. Rene Descartes, never one to be outdone, discovered a third pair: 9,363,584 and 9,436,056. Leonhard Euler would put all of them to shame by finding a remarkable 62 amicable pairs. My favorite part of this story is that long after these brilliant mathematicians had seemingly exhausted the study of amicable numbers a little known mathematician named Nicoló Paganini entered the fray and discovered the second lowest pair of 1,184 and 1,210. Our students might be tempted to think there is little to be discovered within the realm of mathematics after such greats as Newton, Euler and Gauss. Paganini proved such a notion to be wrong and provides an entertaining and instructive lesson for our students.

Posted in Math History, Number Theory | Leave a comment

Finding God in Mathematics

Posted on April 30, 2012 by Brett Edwards

I had the pleasure of presenting a seminar at a conference for Atlanta Christian Math Educators hosted at Whitefield Academy on April 26. I want to make available my presentation and seminar notes for anyone interested.

Finding God in Mathematics Prezi

Finding God in Mathematics Notes

Posted in Seminars, Worldview | Leave a comment

Euclid – Proposition I.5

Posted on April 21, 2012 by Brett Edwards

Proposition I.5 – In isosceles triangles the angles are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.

Construction

Let AD and AE be joined (Post. 1). Let a point B be taken at random on AD. Describe a circle with radius AB to find point C (Post. 3). Let the straight line BC be joined (Post. 1). Let a point F be taken at random on AD between B and D. Describe a circle with radius AF to find point G (Post. 3). Let the straight lines FC and GB be joined (Post. 1).

Proof

AB and BC are equal to one another because they are radii of the same circle (Def. 15). In like manner, AF and AG are equal because they are radii of the same circle (Def. 15). ∠A is equal to ∠A because they coincide with one another (C. N. 4). Thus, ΔAFC is equal to ΔAGB (I. 4).

Because ΔAFC is equal to ΔAGB, the remaining angles AFC and AGF are equal and sides FC and GB are equal (I. 4). AF is equal to AG and AB is equal to AC, therefore BF is equal to CG because when equals are subtracted from equals, the remainders are equal (C.N. 3). Thus, ΔBFC is equal to ΔCGB (I. 4).

Because ΔAFC is equal to ΔAGB, ∠ABG is equal to ∠ACF (I. 4). Because ΔBFC is equal to ΔCGB, ∠CGB is equal to ∠BCF (I. 4). Thus, ∠ABC is equal to ∠ACB because when equals are subtracted from equals, the remainders are equal (C.N. 3). And because ΔBFC is equal to ΔCGB, ∠FBC is equal to ∠GCB (I. 4).

Therefore, it has been shown that in isosceles triangles, the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.

Q.E.D.

Posted in Euclid's Elements, Geometry | Leave a comment

Creative Carl

Posted on April 18, 2012 by Brett Edwards

Thinking well in the math class at times will require an ability of the student to think creatively and approach a problem from a counter-intuitive direction. There is an illustrative story of this principle. Once, a primary teacher asked an unruly but bright student to find the sum of the integers from 1 to 100. The student responded with the correct answer of 5,050 in a matter of seconds. This clever kid would go on to become known as “the Prince of Mathematicians” also known as Carl Gauss. Gauss’s path to finding the answer is very instructive to the math teacher and student. Most “normal” thinkers including myself would begin the laborious process of adding each integer to the next and 20 minutes later we would arrive at 5,050 or more likely not considering potential errors. Gauss, clearly gifted by God, thought of the problem from a different angle. Read more…

Posted in Algebra, Math History | Leave a comment

The Spider and the Fly Riddle

Posted on April 2, 2012 by Brett Edwards

This is a great “riddle” for application in a geometry class:

In a rectangular room, 30’ x 12’ x 12’, a spider is at the middle of an end wall, one foot from the ceiling. The fly is at the middle of the opposite end wall, one foot above the floor. The fly is so frightened it can’t move. What is the shortest distance the spider must crawl in order to capture the fly? (Hint: it is less than 42’)

Illustration courtesy of Brett’s impressive MS Paint skills.

Posted in Geometry, Riddles | 2 Comments

Asking the Right “Wrong” Questions

Posted on March 29, 2012 by Brett Edwards

Much of skillful teaching is in the details. This includes asking the right questions, placing such questions in the right order, and using a variety of techniques to arouse the intellectual interest of the student. This is an area where I feel there is much room for improvement in my own teaching. I want to consider some of these “details” in relationship to a specific problem. Obviously the application could apply to a variety of different problems. Let’s divide the polynomial 2x³ – 12x² + 19x – 12 by x – 4 and I will analyze a single step in this process to take a closer look at the details.

Any Algebra teacher can see the tempting error awaiting the careless Algebra I student. Before discussing the effective questions that can help students avoid such mistakes let’s review how this should not be taught. Read more…

Posted in Algebra, Pedagogy | Leave a comment

Is Infinity Equal to Infinity?

Posted on March 27, 2012 by Brett Edwards

Whenever mathematics approaches the concept of infinity, things begin to get a little fuzzy. Set theory seeks to study the nature and behavior of various sets of numbers. Georg Cantor would be the first to plumb the depths of this fascinating field. One attractive area within set theory is comparing sets of infinite numbers. There is much to say but I just want to share a great illustration given in David Foster Wallace’s book Everything and More: A Compact History of Infinity. It can be proven that the number of points between 0 and 1 is equal to the infinite points on the real number line. This seems so contrary to basic logic. However, the illustration leaves little argument against the truth. To start, our goal is to set up a one-to-one correspondence between the points on the real number line and the points between 0 and 1. For example, if we wanted to know if the number of people in a classroom was equal to the number of desks, we would simply ask all people to find a desk to sit in. If when everyone was seated, there were no empty desks nor any people without a desk we would consider the “set” of people in the room equal to the “set” of desks in the room. Using this analogy, if the points between 0 and 1 are seats, we are trying to find one seat for every single real number on the real number line. That is a lot of seats! Let’s see how the illustration proves this bizarre truth. We begin by taking the portion of the real number line between 0 and 1 and raising it above the number line as illustrated below.