MAED 314 – Assignment 3

MAED 314 – Assignment 3

• Vincent Collins
• Gregory Thiessen
• Rosemary Qi

November 23, 2009

Part 1 – (8) Develop a project of own choice based on the in-class library. Topics include Goldbach’s Conjecture (from novel), infinities, fractals, paradoxes, etc.

Grade: 11

Purpose:
This activity is both a history project as well as an activity to have the students think mathematically. These mathematical concepts can be very difficult to grasp and by doing this project, students will have to go beyond instrumental comprehension to fully understand the given topics and produce the desired results. Students will get the opportunity to work in small groups and present their findings to the rest of the class in a short presentation.

Description of Activity:
Project

In groups of 2 or 3 you will choose a topic from the provided list. Feel free to research and choose your own, however, please confirm your decision with me before proceeding.

• Goldbach’s Conjecture
• Infinities
• Fractals
• Paradoxes (Choose only one to work with)

A paradox is a statement that goes against our intuition but may be true, or a statement that is self-contradictory. Examples of some mathematical paradoxes are:

• Zeno’s paradoxes of motion (the dichotomy paradox, Achilles and the tortoise paradox, arrow paradox or the stadium paradox
• Russell paradox (Barbershop paradox)
• Greeling paradox
• Petersburg paradox
• Galileo’s paradox
• Etc.
Once you have made your decision, the following tasks must be completed:

1. Research your topic through the internet or by using books from the school library. I will provide a list of books that are available in the library.

2. Produce a visual representation of your topic. This can be in the form of a poster, tables, comic, performance, etc. Include a brief history of the topic including the person who developed the concept.

3. Present your work to the class in a short presentation of no more than 5 minutes.

Marking Criteria:
Your grade will be dependent on your understanding of the mathematical paradox which will be conveyed through your final product and presentation. Effort and contribution to the team will affect your final mark. There will be a peer evaluation at the end.

Due Date: 2 weeks from day assigned.
This project is expected to be completed primarily outside of class.

Sources:

http://www.stormloader.com/ajy/paradoxes.html
http://www.suitcaseofdreams.net/Content_Paradox.htm
Mazur, Joseph. Zeno’s Paradox: Unravelling the Ancient Mystery Behind the Science of Space and Time.

Part 2 – Evaluation of the project

This project is an enrichment project designed to make students think mathematically about concepts not ordinarily encountered throughout life. Benefits and weaknesses of this project are listed below.

Benefits:

Students will learn of various mathematicians who have contributed to mathematical knowledge over the centuries.

Students will have to critically think about concepts of math outside of what they usually experience.

Students will have to work in groups to accomplish tasks.

Students will have to present their findings in a clear manner such that peers will understand.

Weaknesses:

Not all of the topics are connected to the grade 11 IRP’s. Some of the mathematical concepts may be more suited for grade 12.

Students may have a difficult time comprehending some of the paradoxes.

Students may not comprehend the purpose or necessity of this project and may view is as another mundane task they must work through.

Trying out the project as a sketch
Our group has decided to tackle this project by focusing on Zeno’s paradoxes of motion.

History of Zeno of Elea

The Greek philosopher Zeno of Elea, born around 490 B.C., proposed four paradoxes that are still discussed 2500 years later. He wanted to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved.
Zeno's paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics. His four famous paradoxes include the dichotomy paradox, Achilles and the tortoise paradox, arrow paradox and the stadium paradox.
http://mathforum.org/isaac/problems/zeno1.html

Part 3 – New Math Project Design

Grade: 11

Purpose:
The goal of this math project is for students to examine cyclical patterns in an artistic, fun and unique way. The students are presented with a classic problem that has been posed in an abstract way, to see if they can discover the numerical relationship. The students will be challenged to recreate and experiment with the ‘celtic knot’ pattern and hopefully they will begin to see that math exists is science and in art too. This is an enrichment project that promote deeper thinking in the students.

Description of activities:
1. Follow the guide (see below) for producing a ‘celtic knot’ and produce a 4 by 3 knot. Decorate it as necessary to show how the pattern loops.
2. If the knot has m vertices along the left side and n vertices along the top side, determine the formula/algorithm for determining the number of loops present in an m by n knot.
3. What other patterns of loops can be made by combining loops of various sizes. What are the limitations of this pattern, if any?

Guide:
The (1) blocks are placed at the four corners of the pattern (see below), each rotated 90 degrees from each other. The (2) blocks are placed next to them, cascading along the edge of the knot pattern to form a perimeter. The (3) blocks are placed in the middle to fill the space and complete the knot design. The picture below illustrates how this works. Under ‘Graphic’, there is a drawing of a completed knot for reference.

Sources:
There is no source for the creation of this knot pattern. This is original artwork that shall now be examined for mathematical inquisition.
Length of time that project will take:
Students are given 1 week from the date assigned to complete this project. If time permits, a small section of class time may be devoted to beginning the project. Otherwise, it is to be work upon outside of class in groups of 2-3.
What students are required to produce:
1. The students must draw a 4 by 3 ‘celtic knot’ and decorate it to make it easy to see.
2. The students must derive a formula for determining the number of loops in an m by n knot.
3. The students must create a new knot pattern by opening & closing loops to create their own knot pattern.

Graphic:
m=4, n=4, # of loops: 4 (highlighted as purple, red, green & blue)

Marking Criteria:
1. /4 - Completed drawing of 4 by 3 ‘celtic knot’.
2. /4 - Formula for the # of loops.
3. /2 - Originality of knot created.

Celtic Knots

Celtic Knots have been a fascination of mine for the longest time. Over the years, I have played around with creating simple knots and have developed my own, reliable pattern for producing them. As the more clever of you will soon see, the template for creating knots follows a couple pretty simple rules. The knots I produce rely on a basic weave in the center and wrap around clockwise at the edges.

There is a lot of fantastic math that can be derived from 'knot' problems and they are a curious phenomenon of spatial geometry. Take a look at the pattern below. This is one of my more complex creations that shows just how boggling these knots can be.

How many loops are in this knot? How are they connected?

Here I have highlighted the loops in colour to make it a little easier to see what is happening. Of course, it's still not ideal. But you can begin so see why these knots are so interesting. In fact, you can create knots that consist of a single (highly complex) loop, but those introduce an entirely knew concept. So, to get your feet wet, I've given you a knot that has a lot more symmetry and the pattern is more easily distinguishable.

For those of you who'd like a real clear visual, I have deconstructed this knot into it's basic loops. Essentially 'spilling the beans' on the question posed above, there are 14 loops. They all fit perfectly and snugly together to form a 'somewhat' symmetrical square. How can this be? Certainly, I didn't just stumble across this pattern. This took a lot of thinking and working with these loops and patterns to make it work. But the point is that it totally does work and it opens up a world of possibility for mathematical and creative exploration. Can you make an even better knot than this?

SEQUENCE PROBLEM: SOLUTION

The geometry is some what subtle, but I argue it's entirely possible to formalize a solution for incredibly large sequences based on this relationship. As you can see by the tiers of the pyramid (discrete triangular fractal), this graphical solution works for all sequences involving numbers of adjacent 1's in powers of 2 (i.e. 1, 11, 1111, 11111111, etc.). Identifying sequences becomes more difficult for sequences that have other numbers of adjacent 1's, but the principle is the same and the geometry is not impossible to navagate. The full solution will involve similar pyramids for all powers of prime numbers. However, a majority of sequences can be solved with the geometry from the simplest of these (e.g. powers of 2 - above). I encourage you to try making a pyramid of your own. Start with the sequence of all 0's, with a prime number of adjacent 1's in the middle. Carry the numbers downward, look at the pattern, and see just how many random sequences of 1's and 0's can be accounted for.

Problem Solved. (more or less) :o)

Two Column Solving: SEQUENCE PROBLEM

Gentlemen's Chess

Chess Invention: Gentlemen's (No Trading) Chess

Intro: The invention of Gentlemen's Chess spawned out of boredom of the tradition form of chess. The problem was that the regular rules for chess didn't inspire any creativity in game play and the games all seemed to go the same way. Bishop captures bishop, then pawn captures bishop back. Pieces of equal value were simply being 'traded', until the board was simplified and fewer (more valuable) pieces remained. It seemed too easy to lose your pieces and instead of fighting to protect a knight, we would just capture their knight in successive turns so that there was no net loss. Well clearly this wouldn't do. So we had to make a new rule. We said you couldn't trade pieces... what did that mean? It meant that a piece that was protected, couldn't be captured for whatever reason. It was permanently safe. A lovely notion. A piece that wasn't protected however, was fair game. We loved this idea and immediately started playing our new way. Almost immediately, we realized that this would create a stale-mate. Both players would move in such a way that all their pieces were protected all the time. This meant, that vast pawn-knight formations covered the entire board and it was impossible to move about. This lead to a second rule. Pawns can move in all directions. Boom! Suddenly, the board sprung to life and movement was possible, players started taking risks, pieces were lost and checkmates were performed. The game took on this whole new dimension. Strategies formed. Aggressive and defensive strategies were shown to each have their own pros and cons, and ultimately, Gentlemen's Chess was born.

Inventors:

Vincent Collins, Clarence Sng & Matt Hamm invented Gentlemen's Chess back in the 2003-2004 school year at UBC.

Rules:

1. If a piece, at the point of being captured by an opposing piece, is protected by a friendly piece, it cannot be captured.
   
2. Pawns are permitted to move in all 8 directions as long as they only capture in the normal forward-diagonal fashion.
   
3. Checkmate occurs when the king is threatened by a protected opposing piece.
   

Special circumstances:

3rd party protection: If a protected black rook is in between two white rooks, both white rooks are protecting each other. This is because if the black rook were to try to capture one of the white rooks, the other white rook, upon the black rook capturing, would be able to protect that space.

Protected mate: If a black queen is protected and on an adjacent square to a white king, checkmate is automatic, despite any aiding white pieces.

Removal of En Passant: Because pawns can move back and forth, En Passant no longer makes sense for pawn movement. Also, as inventors of the game, we've decided that En Passant doesn't add anything new or exciting to the game and is unnecessary.


Pawn promotion: Pawn promotion is a lot more common in this version of chess and therefore has the potential to sway strategy quite significantly. However, since pawns are so useful, some argue that when a pawn reaches the final row, it shouldn't have to promote (i.e. promote to a pawn). However, we maintain, that pawns must promote, even if that pawn only is sent to the final row to gain a better board position.


It is amazing that it has taken almost 6 years to finally start documenting the rules and history of this game. Please take the time to try playing chess this way. I guarantee you'll enjoy playing it and want to share it with others. :o)

Quick Write: Short-Practicum - Memorable Story

My sponsor teacher had dressed up as a navy officer for Hallowe'en and I had dressed up as a soldier. I swear it wasn't on purpose. The students all thought it was funny even though it was a little intimidating for them to have two 'officers' for teachers. Both my sponsor teacher and I volunteered to chaperone the high school dance and we looked like quite the pair. We were marching up and down the halls, guarding doors and looking tough. Well, then the fire alarm went off and we sprung into action like a pair of veterans. Hurrying toward the sound of distress! It turns out that the fog machine for the dance had been left on too long and the smoke alarm had been triggered. As the students began to file out into the sprinkling of rain upon the damp courtyard, the teachers stood in the doorway, huddled against the cold and watching the kids. All but my sponsor teacher and I who began marshalling students and warning them not to stray too far. Like sheep dogs we paced the perimeter, barking orders at stray students and keeping them safe. It was at this point that I realized my sponsor teacher was getting into character. He paced up and down the courtyard with his hands clasped behind his back. The street lamp casting him in shadow as he glared gloomily at the frolicking students. Finally, the alarm was turned off and we marched the students back into the building, stoically returning them to safety. It was quite an amusing sight. But we were a hit with the students and I think it made a great impression that will last with them until I see them in the long-practicum. :o)