Monday, November 15, 2010

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.   


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. 

You can see other more ethnic patterns from this website.
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


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


30 points

See attached handout with patterns to choose from.


  1. Well done, group! Your mark on the written part of the project is "Very Good". You have done an excellent job on all aspects of analyzing the given project in terms of student learning and engagement, and you have thought critically about how to improve it. I very much like your idea of bringing more mathematical language to the study of symmetries in these tilings.

    Your own project is well-presented, and you've put some thought into your assessment rubric. My only question there is the number of marks you've given in various places for "clean sketches" and "right number of words" (4 out of 30 points in all). That's not a huge percentage, but neatness and exact word count seem so much less significant than other things you are evaluating (like logical thought processes, thoughtfulness of reflections, learning something new, correctly identifying patterns, etc.)

    Your new project is essentially the old project with some rethinking, and that is the only thing that separates your Very Good mark from an Excellent. If possible, I would have liked to see a new project that was a little farther away from the original one, though you've done very well with adapting and improving this one.

    Looking forward to seeing your presentation today (which I will also send you comments on.)

  2. Susan's assessment of your project presentation:

    Shannon, Marija, Carly: Islamic tilings

    Good “sketch” of the original project, and very good insights into what kids can learn from this. I like your analysis of the discovery aspects of the project. Good critical thinking!

    New project: I really like the idea of bringing more mathematical naming and description to the project. Great idea to bring a reflective element to this project too! I like the idea of bring other cultural patterns (beyond Islamic tilings) as well. On seeing your presentation, I think that your new project really is different from the original one – so your mark is upgraded!

    Marija: Fantastic analysis of the 17 wallpaper patterns and ways to code them! Lovely!

    Nice poster!

    Excellent powerpoint presentation!

    Overall mark: Excellent!