Problems with multiple entry points can lead to interesting conncections. The precalculus and calculus class started off the year with an investigation of infinite nested radicals. Specifically the following:
               
                   Let a_(n+1)=sqrt(2+a_n) with a_1=sqrt(2). Find a_n as n goes to infinty.

Using some nimble algebraic manipulations, the Calculus class showed the limit to be 2. The Precalc class was able to arrive at the same conclusion through investigating the sequence using a spreadsheet.

The next obvious question was:

What are the requirements on positive integer values of p so that the limit converges to a whole number? That is, when will a_(n+1)=sqrt(p+a_n) with a_1=sqrt(p) converge to a whole number?

The precalc students showed that this happens precisely when p is of the form m(m-1) for positive integer values of m. Further, if p is of the form m(m-1), then a_n will equal converge to m as n grows large.

This intersects nicely with the work of the calc students who were able to show that a_(n+1)=sqrt(p+a_n) with a_1=sqrt(p) converges to [1+sqrt(1+4p)]/2. Outstanding work, though they were left wondering the requirements on p for (1+4p) to be a perfect square. The precalc class was able to supply this answer. If the sequence converges to a whole number m for p of the form m(m-1), then [1+4m(m-1)] must be a perfect square!

A simple investigation into infinite nested radicals led to some number theory discoveries. As usual when attempting to answer a question more questions have to be asked.

(1+4p) being a square root, doesn't ensure the sequence converges to a whole number. Or does it?

Certainly you can find a whole number p to so that the sequence will converge to any whole number m. Or can you?

Does their exist a whole number p so that the sequence will converge to 1?

Lots of loose ends here, I fear they may have lost interest...

Mistakes are a necessary component of learning. The history of mathematics as a discipline is no different. It is rife with mistakes, wrong conclusions, guesses, controversies, conspiracies, secrets, and murder. It is a messy, uncertain business. We will allow it to be so as well; without, of course, the conspiracy, secrecy, and murder! Often, we will have an end in sight, but the journey is perhaps more important than the destination. The great mathematician Karl Friedrich Gauss said, “I have had my results for a long time: but I do not yet know how I am to arrive at them.” Many times in life, we have a desired solution or outcome but are at pains with just how to realize it. GK Chesterton says of people in such a place, “It isn’t that they cannot see the solution.  It is that they cannot see the problem.” If this is true, then strategies, methods, and ways of thinking that allow us to make sense of the problem while giving direction towards resolution are the keys to mathematical proficiency. This is what we are about here. “It is the story that matters not just the ending.” Paul Lockhart.

Pafnuty Chebyshev 1821-1894

Chebyshev is perhaps best known for the inequality that bears his name. He proved that for any population with a finite mean average, μ, and finite standard deviation, σ, the probability that X lies inside the interval (μ-rσ,μ+rσ) is greater than or equal to 1-1/r**2.

When we study the interactions of events, we sometimes run into places where our intuition fails us. This is especially fertile ground, albeit sometimes uncomfortable. When we have mathematical models in place, we are asked to trust the math. To the extent that we are able to take this leap, our understanding grows and the previously counterintuitive becomes obvious.

Dr. Jones and the physics class recently ran into such a situation while studying the mechanics of dynamic motion. The context was an object being pushed across a flat surface in such a way that the object moves with a constant velocity. Noting that constant velocity implies an acceleration of zero and applying Newton's Second Law of Motion, we reach the conclusion that the net force on the object in this direction of constant velocity is zero. For many of us, however, this is counterintuitive. How, our rational minds insist, can the net force be zero when the object is moving? Once we take the aforementioned leap of trusting the math, this fact becomes a more palatable. Indeed, after sitting with the idea for a while, it becomes obvious. The frictional force between the object and the surface is exactly counterbalanced by the force pulling or pushing the object. The net force is zero and the object is not slowing down or speeding up. 

Another example that lies somewhat comfortably in the zone of proximal development of high school mathematics students is the Monty Hall Problem. In his game show, Let's Make a Deal, Monty had three doors. Behind one of the doors was a desirable prize, behind each of the two others, a goat. The contestant would choose a door. Monty would then have one of the doors with a goat behind it opened. The contestant was then asked whether they wanted to stick with their original choice or switch to the other unopened door. The door of the contestant's choosing was opened revealing their choice. The question is what is the best strategy. Should one stay with the original choice or switch? Does it even matter? The answer is counterintuitive to many. Indeed, the mathematician Paul Erdos refused to accept the answer as did thousands of viewers. Erdos was ultimately convinced when showed a computer simulation of the game. Gardiner's Python Programmers are investigating this question. We started with writing programs to simulate the game. Then, we will write additional programs to investigate the strategies of sticking with the original choice versus choosing. Once we have the answer, we have the opportunity to align our intuition with our findings. 

In the fast paced world of Secondary Education we need to take advantage of the opportunities to wrestle for prolonged periods with questions that are not easily answered. When the counterintuitive becomes the intuitive, we have truly shown how flexible and inventive our minds can be. We are taught to really examine and consider carefully the evidence and the argument. There is a healthy place for doubt and speculation, even in the confines of our own certainty. It is not so bad, we discover, to admit that we are wrong about something. To learn to marvel at your own growth is a worthwhile endeavor.

Python Programming

The Python Programmers will start meeting on either Monday, December 11 or Tuesday December 12. Stay tuned for details. We welcome people of all ages and experience!

The Algebra I class will attempt to offer a answer to the following question:

Is it possible to predict  a person's height given their shoe size?

Stay tuned for them to submit their findings.

The Python Programmers have recently investigated code that generates the first n terms of the Fibonacci Sequence. The following discovery was also made:

If n is the nth term in the Fibonacci Sequence and n+1 is the nth plus one term, Then as n goes to infinity, the ratio of n+1/n converges to the Golden Ratio. We also found that this is the case with the Lucas Sequence.

Under current investigation is the question of what types of numbers must be our first and second term in the sequence for this surprising property to hold. That is, does always hold when the first terms are positive integers? integers in general? rationals? irrationals? transcendentals?

Ask a programmer about these questions, they are exploring some good stuff!