Assignment 3

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.

Thinking Mathematically - P192 Sequence

I could not find a direct method in finding out the final digit. However, I found a shortcut. It is most easily explained by taking a closer look at page 2. So how does this shortcut work.

Take a look the example at the bottom of page 2. I am taking the 1st and 5th bits and putting it through an exclusive OR gate.

Exclusive OR gate:
inputs output
x y z
0 0 0
0 1 1
1 0 1
1 1 0

The output is 1 only if one of the two inputs are 1. So, I am taking the 1st and 5th bits and putting it through an exclusive OR gate. Then I take the 2nd and 6th, 3rd and 7th... Using this method I can skip 3 lines of binary strings.

I tried extending it and found that it works for every 9 bits, 13 bits and 17 bits. 13 is a special case though. If you are going to do every 13 bits you need an Exclusive NOR gate which looks like:

Exclusive NOR gate:
inputs output
x y z
0 0 1
0 1 0
1 0 0
1 1 1

I believe this also works with 25, 33, 45, 57, 73, 89... with 25, 45, 73 needing a Exclusive NOR gate.

You can see a pattern by subtracting each working number from the number prior. Starting at 9 you add 4 + 4 + 8 + 8 + 16 +16 + 32 + 32... Notice all of the numbers are 2 to the power of something.

----------------------------

I have something to add, a better explanation following my above method.

Given a bit string of any length, it can quickly be reduced by skipping bits! The problem is, which bits can you skip?

This is how it is done:

1. Take any bit in the string, n
2. Take a second bit, n + or - 2^m (Stay within your max string length)
3. Compare these 2 bits using an Exclusive OR gate, as written above
4. The output is your single resultant bit 2^m rows down

Example: A string of 20 bits

11101 00110 10100 00101

Take the first bit and another bit 2^m bits away. If m is 4, 2^4 = 16. If m is 5, 2^5 = 32. 32 is too large so take m = 4.

Looking at bits 1 and 17 (1+2^4) you compare them using an Exclusive Or gate. --> 1 + 0 = 1
Bit1 Bit2 Output
1(1) 17(0) 1
2(1) 18(1) 0
3(1) 19(0) 1
4(0) 20(1) 1

The new string is

1011

This is now easy to reduce to:

110
01
1

And we are done!

from Practicum

#1

MacNeill has what is known as an Advisory class right in the middle of every day before lunch. The interesting thing about Advisory is that in grade 8, a group of students are assigned to a single teacher. For the next 5 years that same group of students stay together with the same teacher for Advisory class. The goal is to create a comfortable environment and relationship between students and teacher, where students can feel safe and get help when needed involving school and possibly life situations.

Advisory runs for almost 30 minutes every day and what happens within depends on the teacher. Some teachers have the students do silent reading of the book of their choice. Most teachers however let the students to self study/homework.

#2

On the last friday, I dressed up as Geordi LaForge for Halloween. All the other student teachers said they'd dress up too but most bailed out. I was extremely dissapointed. However, my outfit was awesome and I was pleasantly surprised how many people knew the Star Trek Reference. One of my sponsor teachers even dressed up as Captain Kirk! We would have taken the prize if there was a staff costume contest.

Aside: I went to a conference earlier in the week about the adolecent brain and apparently adolecents do not read facial expressions the same way as adults...

On that day I got to supervise a math test with a TOC. One of the students, I got along well with but he completely bombed the test. During the test he asked me a question which I couldn't answer for him because of the test. Then he starting saying how poorly he was doing and how he really did study. In my mind I wanted to help but couldn't. On my face I had what I considered my pained want to help but can't expression.

What the student said to me was, "Mr. Thiessen, don't laugh."

Prime example of misreading of expressions.

On a different note, the class before I ran a review session. I prepared a practice test which covered all the material needed. If you could do this test you could do the test no problem. I gave them the numerical answers at the end of the hand out and even posted an answer key at the front of class with detailed solutions. What surprised me was even with an exam lingering 2 days away, students do not suddenly become more motivated to do math. Lesson learned.

Also, it doesn't matter if you are in grade 10, 11, or 12, there will be students who don't know how to graph y=mx+b

Division by Zero

Approaching zero,
Entering the dark.
To the blackest black,
We dive and divide.
Improbable,
But not impossible.
To exist at all,
Throughout space.

.
.
.

During the free write, the first word was 'divide'.

One into another. Over another. Fraction. Dum doodle dee. There is a line with a dot above and bellow. Or you could see a backslash. A divider! Two sides to something, ............ Cut a cucumber or an apple into 2, 3, 4 pieces. Oh it's a miracle. Big number over smaller number could be bigger or smaller. Divide those people into groups.

The second word was 'zero'.

The name of a character in Code Geass... What is 0? Is it nothing? Is it very very small, approaching nothing? What do you get when you divide by 0? Calculators say undefined. But what is a number divided by something so incredibly small? Something so big... Infinity. That's right, Infinity. Yeah, capitalize that! Where does 0 exist? Where in the Universe, Galaxy, Solar System, Star, Planet, Continent, Household, Pocket?

Would I use this in a 9, 10, 11 classroom? Depends on the class and how I feel they learn. Writing a poem about zero is interesting, but is it useful? When doing the free write it wasn't as if I had a revelation about zero. I didn't come to any new conclusions. I just wrote what I already knew with the knowledge of limits as x approaches 0. This is what would probably happen if my high school students were given the same assignment. They would write what they knew already without learning anything new or coming to an understanding about the topic.

Math lesson reflections

Our lesson on logarithms did not go as planned. Our original plan was as follows,

1) Ralph do a short history as a hook
2) Write up some of the rules on logarithms
3) Derive multiplication and division laws for logarithms
4) Have the students do two quick questions to review the logarithm rule blog(x) = log(x^b)
5) Have an engaging question for the class to do

For 15 minutes this was way too ambitious of a time-line. The history went a bit longer than expected taking up ~50% of our time, not giving much time to work on the lecture portion and student centered activity.

The big issue from what I gleamed from the comments was that there was not enough activities for the students. So how would I change it for the future?

I would cut the history to 2 minutes max and quickly write up the basic rules of logarithms, such as for log(0), log(1), when base is the same as what is being logged. For this type of presentation, we can assume that we are just doing a review of the rules and the students have already learned it before.

I would cut out the derivations for the multiplication and division laws. In fact, I don't think I'd include those laws at all for this presentation. For 15minutes we need to keep it extremely short. What I would do is write a function on the board, starting with y = x and ask the class, 'without using a calculator how would I graph y = log(x). The purpose of this activity would be for the students to recognize that when x <= 0, there would exist no y. When x = 1, y =0. When x = 10, y = 1 etc. I would then get them to do the same graph but with a different base instead of 10 and see how the different graphs compare.

I could then give them an equation such as y = x^2 or x^2 - 3 and we could repeat the above process. They can check their results using a graphing calculator after wards. The purpose of this activity would be for them to recognize that a log of a negative number or 0 is always undefined and a log of 1 is always 0. Also, to graph it without a calculator, the students would have to recognize how to choose easy number to work with when dealing with logarithms. For example, if you were working with a base 2 logarithm, you would want to solve for when the inside is 2^n.

Example, for y = log(x^2 - 2) [base 2]
If y = 1, then x^2 - 2 = 2. Solving for x you get 2. Next you do for y = 2, then x^2 - 2 = 2^2 = 4. Solving x you get root(6). It's even easier to graph the logarithm if you were t first graph y1 = x^2 - 2 first, and then look for key locations on y1 such as when y1 = 0,1,2,4,8,16, 1/2, 1/4, etc.

That's all I have to say.

Booppps Lesson Plan - Exponentials and Logarithms

Bridge (The Hook) - The history of logarithms, who invented it and why was it invented.

Teaching Objectives - To cover/review some of the laws of exponentials and logarithms before going over a more challenging problem together as a class. If there is additional time we have an extra problem that we can challenge the class with.

Learning Objectives - Have the students apply various log rules to solve a question, displaying their understanding.

Pretest - Go over the rules of exponentials and logarithms and then have the students answer a couple simple one or two step questions.

Participatory Activity - Work through together as a class a challenging logarithmic problem. Graph, without a calculator, y = log(x+2) - log(x-1) where the log is base 3.

Post Test - The participatory activity can be considered as the post test since there is only 15 minutes for the lecture. If there happens to be time we have an additional question to give.

Summary - Tell the class there will be an exam next week!

Flatlander

Just finished the book 'Flatland'. I'll talk about it more later, for now I'd like to just put up this link which someone sent to me a long while back: http://www.tenthdimension.com/medialinks.php.

I remember reading a Sci-Novel a while back which involved a 3-D world and a 2-D parallel world. This was a long time ago and could possibly just be part of my imagination... I'm going to see if I can find.