Math Projects Assignment

 “Math Projects” Assignment
Shannon Kennedy, Carly Orr, Marija O’Neill

EDCP 342 November 15, 2010

Part 1 & Part 2

Evaluation of Islamic Tiling Project: Tessellations

            We took a look at Susan’s Islamic Tiling project, and for the most part really enjoyed doing it! It is a beautiful mixture of history and culture with art and mathematics. When discussing this project we talked about many things that it does well, and did not have very many things that we would do to change it!

            One of the greatest strengths of this project is the fact that it forces students to discover symmetries on their own. While doing the project the need to identify reflections, rotations and translations comes about very naturally because it will help the student draw the pattern. An understanding of these concepts is also required in order to find the smallest repeating shape. This discovery based on need makes the concepts discovered far more meaningful for the students.

            After having discovered the symmetries within the pattern, students are then asked to describe the pattern in words. This is an interesting step because students are essentially being asked to describe a mathematical concept in English. Not only that, but in an English that they are comfortable with and makes sense to them. They aren’t being asked to talk about axes of symmetry or angles of rotation. They are simply being asked to describe a pattern, and they can do so however they choose. This is very powerful because it encourages the students to make sense of the mathematical concept of symmetries in their way, and they can express their understanding however they choose.

            That being said, we feel that there is one extra step here that is missing from this project, and that is that the students are never asked to translate this written expression of the pattern into a mathematical expression. We fear that without this extra step, students may not make a concrete connection between what they are doing and what they have learned in math class. For us as mathematicians it is very natural to think about a pattern in terms of reflections and rotations; however this may not be the case for all of our students. They might describe their pattern as “that same piece again only upside down” or “The head of the one lizard fits in between the head and left arm of another lizard”. Some students may need an extra push in order to convert their thoughts into mathematical ideas. It is also very important for students to be able to express themselves in the mathematical language, since that is how most information is conveyed in mathematics as well as many sciences.

            Other strengths of this project include the use of straight edge and compass. These skills are not often taught in the math classroom anymore, but can be very useful both for practical reasons (in design and construction) and as a mental exercise. Using these simple tools forces students to think about how shapes are formed and the relationships between lines and angles. It challenges them in their visual and tactile thinking abilities. As was mentioned in the assignment, this is the method that was used in ancient times, and so makes the historical aspect of this project very real for the students. They are doing something just as it was done hundreds, if not thousands of years ago. A slight warning should be mentioned here, in that some time will need to be taken in class to teach the students how to create equilateral triangles, draw perpendicular lines and bisect an angle with a straight edge and compass.

            The final part of this project brings in the creative and artistic aspects of this project, in that students are required to create their own tiling. This is fairly straightforward since it only requires a slight change to the previous tiling, however there is a wide variety in the possible patterns students could create. Adding creativity to the math class is always a bonus for a few reasons. It encourages the idea that there is room within this subject to create, discover and explore rather than simply following the rules. This is something that all mathematicians know, but which is rarely understood by high school students. Asking the students to create something also gives them ownership of their work and encourages them to be proud of what they have accomplished.

            This project relates almost directly to the section on symmetry in the grade 9 curriculum, so it could be quite easily implemented at that grade level. That being said, it could still be a very interesting project for older students as well! After all, we enjoyed doing it!

Part 3:  Devise our own Project

The following is the description of our new project.    Instead of asking students to try reproducing and figuring out symmetries on their own, we have asked the students to identify the symmetries and rotations from a given pattern.     We assume that the students have already completed a unit on symmetries (Math 9 IRP) and that this is a review, as well as an extension.  

In addition we have also added a step where the students write a reflection after doing the project.   In Part 1&2, we found that it was sometimes non-trivial to draw a tiling pattern with only straightedge and compass!   Perhaps the students might be surprised, start to wonder how amazing it must have been for artists long time ago (without the aid of computers or other tools) could have constructed such intricate and accurately repeating patterns.     It would be interesting to see their reflections.     We hope that students will gain a deeper appreciation for the rich historical and artistic and mathematical value behind these tessellations, as well as be able to notice interesting patterns that might appear in our modern life.   

MATH 9 PROJECT

MODERN WAY TO CREATE MEDIEVAL ISLAMIC TILING PATTERNS

Purpose

In these assignments, students will explore mathematical concepts behind some ethnic tiling patterns.   Students will examine patterns in terms of rotation and reflection.    Students will identify smallest repeating tile in a pattern, make variations, and reconstruct tiles with a compass and straightedge. 

Description of Activities

Step 1.  Choose one of the given Islamic tiling patterns (see attached handout).

Step 2.    Describe the pattern mathematically by answering the following questions.

  a).   Rotational Symmetry

i)              Find all the points of rotation in the pattern.  

Label these on the pattern.

ii)               For each point, what is the order of rotation, and what is the angle of rotational symmetry?   Label these on the pattern.

  b).   Lines of Reflection (Mirror Lines)

Do you see any lines of reflection?    

Label these on your pattern. 

          

You may wish to sketch or trace a larger simplified version of the pattern on a piece of paper, so you can clearly label on the page. 

Step 3.   Using the discoveries from Step 2, find the smallest (most minimum) repeating pattern, or the most basic tile shape which, if replicated will give the complete wallpaper pattern.

Step 4.  Using only straightedge and compass replicate this minimal repeating shape (by drawing) on a piece of paper.   Document your steps. 

Use the straightedge and compass constructions learned in class:

a)    the equilateral triangle,

b)    the perpendicular line,

c)     bisection of an angle and

d)    circumference of a circle.

Step  5.   Make a small change to your basic minimal tile shape.

Find a way to create your new minimal tile shape using compass and straightedge only.     Document your steps.   Make a pattern for your new tile shape from cardboard or heavy paper and cut it out.

Step  6.   Trace your new tile pattern repeatedly onto a large sheet of paper to find out what new tiling pattern you have now created.

Step 7.   Label the points of rotation, lines of rotational symmetry, and any lines of reflection clearly on the pattern.  

Step 8.   Write 100-150 word reflection on what you have learnt from this project (your learning process – for example, what surprised you? What did you find new or interesting? What part was hard, or easy, or fun for you?).    Give some examples of when you might come across tiling patterns in your daily life. 

Sources

You can see other more ethnic patterns from this website.

http://www.sfu.ca/~mdevos/303/wallpaper/

Some are quite intricate and complicated.   For this assignment, we will only be looking at patterns with rotations and reflections.

What Students Are Required to Produce and Marking Rubrics

Deliverable	Marking	Total
1.    One page display of original pattern showing rotations and reflection information.	Correctly identify all the points of rotation.  1 point Correctly identify all the angles of rotation.  1 point Correctly identify all the orders of rotation   1 point Correctly identify all the lines of symmetry 1 point Clear labeling 1 point	5 pts
2.    Step-by-Step account of how you made the minimum tile shape with compass and straightedge.	Clear, logical thought process.  2 points Correct minimal tile.   2 points Clean sketch of minimum tile.   1 point.	5 pts
3.    Pattern of your new tile,  cut-out on paper or cardboard.	Having a cut-out piece.   2 points.   New tile is slightly modified from old tile.  2 points. Cleanly sketched lines.   1 point.	5 pts
4.   Step-by-Step account of how you made your new file shape with compass and straightedge.	Clear, logical thought process.  2 points Correct minimal tile.   2 points Clean sketch of minimum tile.   1 point.	5 pts
5.   A picture of your new overall tiling pattern, made by repeatedly tracing your new tile pattern.	Correctly identify all the points of rotation.  1 point Correctly identify all the angles of rotation.  1 point Correctly identify all the orders of rotation.   1 point Correctly identify all the lines of symmetry.  1 point Overall pattern appeal.  1 point	5 pts
6.  Reflection write-up	Right number of words (100-150 words)  1 point Thoughtfulness of reflection.  1 point Learnt something new.    2 points Examples of tiling in daily life.   1 point	5 pts
TOTAL		30 points

Handouts

See attached handout with patterns to choose from.

Response to "Creativity, flexiblity, adaptivity, and strategy use in mathematics" by Christoph Selter

When my children were toddlers, my mom (a wise grandma) would always complement them whenever they are able to take one basic skill and apply it to another area of life.    She used the term "ban jia",  which means in Chinese "move to a different home".    Now that I am in teacher education,  I look back and recognize that this ability is a nice thing to complement a child for ... well, most of the time.

Once my 3-year old son saw a gorilla standing all alone in the corner of the zoo compound.   He said sympathetically,  "Look Mama,  that grandpa monkey is having a time-out!".     For my son,  he was learning how to  adapt  his knowledge of the "time-out" strategy (someone standing in the corner alone) to describe that gorilla in the zoo.  At the same time, he was trying to describe the gorilla based on his previous knowledge of what a monkey and a grandpa looks like.

Using "creativity",  "flexiblity",  and "adaptivity" to learn is a natural part of how human beings shape their understanding of the world.     We often see that in young children.     I guess you can call it a form of educated-trial-and-error.    When children make mistakes whereby the strategy they have chosen does not "solve the problem",   adults think it's cute and laugh about it.   However when adults make mistakes we are in danger of not being able to reverse out of a certain mental mode,  which could lead to frustrations of being unable to solve the problem.  

I was talking to a grade 9 math student recently who said he does not like substitute teachers.    So naturally,  I asked him "why?".      One reason is that substitute teachers might show them how to do a problem using another technique that is different from what his teacher has used.     Maybe he was so focussed on his old method,  that a new method disturbed him.   That's too bad.

As teachers,  we should frequently emphasize to students that are are many ways to approach a problem.    If we can illustrate the value of being open to adapting different strategies,  our students will be able to extend their problem solving skills far beyond the classroom.     Our mathematical training should equip us to solve not just text-book, but real-life problems.

In our UBC math education class,  I have come to appreciate the way we examine different ways of solving the same puzzle on the board.    It has illustrated to me the incredible value of being creative, flexible, and adaptive when chosing a strategy to solve problems.

Looking at Word Problems

From Math 9 textbook:

"Alicia buys a $300 jacket on lay away.  She made a down payment of $30 and is paying $15 per week.  The total paid, P dollars, after n weeks can be represented by the equation P = 15n + 30.   a) create a table of values to show the total paid in each of the first 5 weeks  b) graph the data.  Should you join the points on the graph?  explain. c) how do the patterns in the graph relate to the patterns in the table?".


Is it practical?    $300 seems like an awful lot for a teenager to pay for jacket,  which made me wonder what "lay away" means?    I am not familiar with this term.   From the second sentence it is possible to guess what lay away means and it is actually possible to complete this problem without knowing what it means.     Some questions that come to mind are: why does Alicia want to buy this jacket, and why is she willing to pay for it over the next n years?    After I checked the dictionary, I learn that lay away means you don't actually get the jacket until you have finished paying it off.    That could be a long time!    What is so special about this jacket?    Just curious.

Creating a table and graphing the points is a good idea.  

when n = 1,   P = 45
when n = 2,   P = 60
when n = 3,   P = 75
when n = 4,   P = 90
when n = 5,   P = 105

Should we join the points on the graph?   This is a good question.  It makes the student think more about the application and quantity that we are plotting.   I'm not sure what is the purpose of asking part c).   The pattern is an even increment, a linear equation.  What is meant by "patterns in the graph"?   Perhaps the author want us to notice the linear relationship?

Extension:   I think an interesting question would be,  "how long will it take for Alicia to pay off the jacket?".    Perhaps,  we can also add the question, "do you think it is worth waiting so long for a jacket that you really like?"  

Puzzle Solving Exercise

Our group of 3 chose one puzzle from the book "Thinking Mathematically" (Fare is Fair,  page 168).   Individually we tackled the puzzle, documenting our mental conversations and our math work in two columns.

The Problem:  I wish to share 30 identical individual sausages equally amongst 18 people.  What is the minimum number of cuts I need to make?  What is the minimum number of pieces I need to create?

Practicum Stories - small life changing moments ...

First Story

I was observing in a computer class.  My SA teacher asked me to help one boy who was new to the  school.   I sat down beside him and discovered that he was quite behind in his assignment.   Even though I was not familiar with his assignment,  I went with him to look for the answer.    A few days later I saw that boy in the hallway,  and he smiled and waved to me.   That was a moment for me.  I knew that I had made a difference, however small, in the life of that particular student.    

Second Story

After hours of preparation and rehearsing and thinking through my lesson,  the time had finally arrived.  It was time for me to do my first math class.

I started with my activity to introduce the topic, and continued according to a lesson plan that I was quite excited about.    After class,  my SA came over to debrief with me.    We had done some microteaching at UBC,  studied some topics in class management,  did case studies,  and knew about Blooms Taxonomy and debated topics in social justice.    However,  at that moment,  none of the head knowledge could have prepared me for what I was about to hear from my SA.    The first sentence was:   "To be brutally honest with you,  if this is your class next semester,  then it would be a disaster."       She continued to tell me what she observed about my classroom management.    

It was the last block of the day.   I felt drained and overwhelmed from physical exhaustion and mixed-emotions.   I went to my car,  drove to a side street,  and had a good cry.     In the ensuing hours and days,  I continued to replay in my head the words she said to me,   pondered on what actually happened in the classroom,  and how I should respond.   I asked myself many questions.    I needed time to reflect, and remind myself why I chose this teaching path.    I learnt more than a few things:      

1.   Classroom management is easier said than done - it is a skill acquired by actually being in the classroom. 

2.   I must be prepared for negative feedback,  and know how to put that into perspective.

3.   I must work hard to develop a stronger teacher presence, and set clear expectations early in the course to avoid classroom management issues.
4.   I realised that techniques I use to discipline my own children in elementary school can still apply to these kids in highschool!
5.   Jumping in to teach in another teacher's class is like trying to cook in someone else's kitchen.

It was a valuable learning experience.   It has made me a stronger person.    The negative feedback was like a "wake up call".   I am thankful to my SA for being genuine and honest with me.   I know that she was well-intentioned, and did not want me to be setup for failure next semester.   She was doing me a favour by letting me know now,  rather than later.   I want to grow, and learn from my mistakes.   Having room to grow is a good state to be in.