“Math Projects” Assignment

(#6) Math 10 Project: Geometry of Design

group: Michelle, Meghan, Paul

This is certainly an interesting project from the point of view of math appreciation.  There are many students who are turned off by algebra and numbers, and for them this playful visual exercise can do a great deal to draw them in to the world of math.  The practicality of this project may be called in to question, as there is notoriously short time in class.  However, this may be a criticism of the school’s scheduling more than the assignment itself, which is wonderfully enriching and deserves more time than it will inevitably be allotted.



Strengths:

• Requires very little base knowledge.  This makes it accessible to everyone in the class.  It may inspire the struggling students to see a project in which their previous weaknesses in mathematics are not compounded.
• Combines math, history, nature and art.  Let’s face it, most people find art and history far more interesting than mathematics.  Showing how math can relate to the these “fun” subjects will go a long way towards showing students how enjoyable math can be.
• This project enables students to choose their own piece of art or object from nature, which allows for broad ranges of interest.  
• After students complete this project they will see how mathematics is in so many different aspects of everyday life that they probably did not realize before.  One of the main outcomes of this project is math appreciation.   

Weaknesses or Constraints:

• Students may be able to find geometric shapes in well known art on the Internet.  This will prevent the student from playing around with shapes and tracing paper to find geometry in the item themselves.
• Perhaps the topic might be too narrow and not really interest all students.
• Project will be hard to cram into a busy curriculum as it is time-intensive and does not cover many PLOs.

Possible Modifications, Adaptations or Extensions:

• One possible modification of the project is having the students orally present the steps they took to finding geometry as opposed to the step by step write up. This might benefit some students, that  are more comfortable verbalizing their process than writing out the steps. Another advantage is students will have a chance to share their findings with their peers.
• Perhaps students could then make their own piece of art work which is based on geometric shapes or mathematical ratios.  

Our Project:

Math12 Project: Mathematics in Art Design

Grade Level: We chose to do this project in Math 12 because geometry is no longer in the curriculum and this is a great enrichment project for these math students who have knowledge about geometric and trigonometric and shapes.
Our project will span two classes over a period of one week.

Purpose:  The purpose of this project is for students to understand and see how mathematics is within many works of art, even though it is not obvious.  The planning of many pieces of art includes mathematical shapes, calculations and ratios.  Students will play around with and sketch geometric shapes, the golden rectangle, logarithmic spirals etc. and see how they can be manipulated to form interesting compositions.  Students will develop their own work of art which is based on their chosen mathematical elements. 
    

Description of Activities:

Day 1:   

• Show students examples of famous art which contains and is based on geometric shapes or ratios.  There will be a PowerPoint presentation or overheads with examples and also a hand-out for students which includes: the Mona Lisa, the Parthenon, Egyptian art, Mondrian, daVinci etc.  
• After students see how these basic shapes and ratios are behind the planning of many works of art students will learn how to draw, using a compass and ruler, these basic geometric shapes.  Some shapes they will practice drawing are the golden rectangle, logarithmic spirals, polygons, stars and possibly trigonometric functions.
• Students will decide by the end of this class which geometric shape(s) or ratio(s) they want to use to construct their artwork drawing.  

Homework:  Students plan and sketch their artwork drawing.  


Day 2 (One Week Later):  

• Students make their drawing on large poster board.  The drawing can be done in markers, charcoal or pastels.  By the end of the class students should be able to have their work finished.  If the project is being done in the art room then students could use other materials such as paint, collage, etc.  

Sources:
Kimberly Elam (2001). Geometry of Design: Studies in Proportion and Composition. (NY: Princeton Architectural Press.)
Miranda Lundy (1998).  Sacred Geometry. (NY: Walker & Co.)


Timeline of Project:  Two classes separated by a week.  Students are required to plan their artwork between the two classes and come to the second class with a complete plan so that they have the maximum amount of time to complete the artwork in the second class.



Students are Required to Produce:  

• a large composition drawing which incorporates at least one mathematical element in the planning.  
• all their planning materials such as sketches of shapes, rough drafts and rough work of their final composition.
• A written or spoken explanation of the mathematics involved.

Marking Criteria: Rubric (20 marks total)

Use of Class Time (3 marks)

• 0 - Absent
• 1 - Come to second class without plan, make poor use of class time
• 2 - Come to second class without plan, make good use of class time
• 3 - Come to second class with plan, make good use of class time

Presentation (2 marks)

• 0 - Unfinished or missing
• 1 - Poor effort, obviously done at the last minute
• 2 - Decent to good effort.  Most students will receive this mark.

Mathematical Content (10 marks)

• 0 - Missing
• 2 - No apparent mathematical content
• 4 - Mathematical concept attempted but incorrectly applied (such as an unmeasured freehand spiral)
• 6 - Mathematical concept properly applied, but with errors in explanation.
• 8 - One mathematical concept properly applied and explained
• 10 - More than one mathematical concept properly applied and explained

Completeness (5 marks)

• 1 mark for each of the following completed on time: Planning materials, rough draft, explanatory piece
• 2 marks for final draft submitted on time

Justification for Rubric: Since this is a math class and not an art class, the artistic component is negligible.  The largest component is the mathematical content.  It follows a linear progression where 0 marks are earned for doing nothing, and 8 marks are earned for doing the minimum.  An extra 2 can be gained by going a little bit beyond.  The students would, of course, be made aware of this beforehand.  It is expected that most students will attempt to incorporate 2 mathematical concepts, and even if they completely mess one of them up, they’ll still achieve 8 out of 10.  We feel that marks should not be difficult to obtain on enrichment projects such as this.

Response to Creativity, Flexibility, Adaptivity and Strategy use in Mathematics

The story at the beginning of the article really emphasizes the importance of adaptability.  It was great that this student who normally struggles with math was proud of his innovative method to solve the problem, however, I feel he has missed a very important concept of mathematics.  Ferit failed to see, or was not lead to see, the many methods and strategies that must be employed to similar problems.  He believed that since this strategy worked for a particular problem, that it must work for all the other problems like it.  
A key element to problem solving is recognizing which strategies work best and employing those first. The four words in the title of this article, creativity, flexibility, adaptivity and strategy, are very important in terms of problem solving.  Students must not be afraid to get creative.  Even though they may very well chose a method that leads to no solution, they always learn something in the process.  These students who use creative problem solving methods will not only find unconventional ways to solve problems but develop their creative aspect which will benefit them in all areas of their future.  Students must be flexible when attempting mathematics.  The teacher can be a great source of inspiration for being flexible. 
I feel that it is quite true about students who have more previous knowledge are less able to be flexible and adaptable within mathematics.  I feel that rather than constantly building on concepts, problem solving requires students to forget what they know and start from scratch.  I feel that students who will be more successful are the ones who try something different and are not stuck within the constraints of previous concepts and methods.  But how do we as teachers get students to do this? How do we help students tap into this creative side and loose their fear of getting stuck.  I feel that we must lead by example.  When evaluating problem solving we must place value on the thought process behind it and also place value on all the methods they tried which did not lead to a solution.  I think that the 'Two Column Problem Solving' type assignments are a great way to emphasize the importance of creativity in mathematics.    

Two Column Problem Solving

Word Problem Analysis

You have a four-wheel drive all-terrain truck, and you need to make a 27 000-km trip. Each tire can be used for 12 000-km. The four tires of the truck are new, and you have five new tires in the back of the truck. How can you use the nine tires to complete the trip?

(MathPower11 Ch5-Rational Functions, Equations and Inequalities)

Practical?
This problem is not very practical. Although, it deals with tires on a truck and making a long trip, it has this really weird aspect of having to change the tires during the trip. The reason this problem is odd is because no one usually drives long enough in a single trip for new tires to wear out. Say you drive for eight hours a day at 100km/hour it would be over 33 days of driving. I feel that this problem would have a more practical aspect if it was a semi-truck because they may drive this many kilometers. But changing the problem to a semi-truck brings the problem to eighteen tires and not four being used at one time, so maybe this makes the problem more complicated than intended.

Memorable?
The only thing memorable about this problem is wasting time on a trip to constantly rotate and change tires on the shoulder of the highway.

Solvable?
Yes this problem is solvable in more than one way. There could be the issue of which solution is most efficient, meaning the least number of stops to change tires.

Can there different positions or interpretations?
As being someone who has car knowledge, I have an issue about driving for extended periods of time and at high speeds when the tires have different levels of wear. This is not only bad for the car but also not so safe. If the car was only front wheel drive or only rear wheel drive then we could have the two front tires at the same level of wear and the two back tires at the same level of wear. But with this problem, the truck is all-wheel drive so the tires should all be the same so the tires are balanced.

How would kids interpret it?
I think that kids would interpret it correctly. I don't think that there would be any major misunderstanding about what the problem is asking.

Is this problem strange?
Yes! This problem is quite strange, although there I'm sure exists stranger problems. This problem does relate to driving and lifespan of a tire in kilometers. I think this problem should not be worded as "you" have a truck... but rather that someone wants to make this trip and they have these tires etc.

Rewrite or Extend?
Since I have knowledge about cars and changing tires I would add to the question that the tires need to be as balanced as possible. Also, students may not be familiar with any of this if they do not drive or are not in automotive class.

Practicum Stories

On my last day of my practicum I taught a Math9 class where we did a review of exponents and unit quiz. I spent approximately half the class going over questions they had from a quiz which they got back. Then the students were given 20 minutes to review and independently study for the quiz. I was not sure how this would go because they are a younger grade and I wasn't sure if they would stay on task studying for the quiz. However, I wondered around the room and answered questions, and the students took advantage of this time for the most part. My sponsor teacher had said that the students needed to do the quiz that day because they were already behind from the other Math9 classes, but they also may not have been completely ready for it. He had suggested that we do a group quiz or something along those lines where they could collaborate with other students.
I came up with a type of group quiz, but one that everyone handed in their own papers and not one from the group. Students were given 20 minutes to work on the quiz by themselves, but I told them that they would have a chance to get help from other students after. They were quite excited about this, but I held off telling them the details. After 20 minutes they were each given 3 slips of paper with their own names on each. These were their "lifelines" where they could get help from 3 different students. When students asked another for help, they would give the helper their slip of paper. So as each students handed in their quiz they would also hand in the slips of paper of the students they helped. However, each student could only help up to 5 people, which prevented the top students getting bombarded with everyones questions.
This group quiz went really well. My sponsor teacher was nervous and excited about doing something like this with the class because he had never done anything like this before. He is a more traditional lecture style teacher and he wants to start employing more creative activities in his classes in the future. He said that he would most certainly do this type of group quiz with his future classes.

I had another great experience with a Math11 class in which I introduced polynomial functions and properties of their graphs. I took the class to the computer lab and had them use the graphing program graphmatica to explore different graphs of polynomial functions. I chose to use the computers rather than graphing calculators because I felt they were more user friendly. I personally had a hard time figuring out the calculator in high school and I didn't want the lesson to be focused around how to use the calculator. I made a worksheet with linear, quadratic and higher order polynomial functions and a chart had to be filled in by the students. The different headings were: shape of graph, degree, leading coefficient, domain, range, minimums, maximums, x and y intercepts.
I was thrilled with how well the lesson went and how involved the students were graphing the functions. My sponsor teacher was very happy with the outcome and will be using the computer lab for graphing in the future. He had never used anything other than the graphing calculators for lessons like this.

Micro-Teaching Feedback Reflection

We were fortunate because we were able to do our lesson with the entire class. One advantage is that we were able to get more feedback. Another advantage is that I experienced the difference in classroom management from a smaller group compared to the entire class. I had a bit of difficulty getting everyones attention at the end. This showed me that I must be more assertive and speak louder, and simply waiting does not work.
We decided to do a jigsaw activity and one major advantage I realized is that it takes the pressure off the teacher. Rather than the teacher having to plan a lesson covering all the material, the students take over the teacher role and teach each other. The teacher still should walk around and engage with students but it was nice to have a break. Also, the students are more actively listening and learning at each group. They knew they had to go back to their triad and teach the others so this caused them to pay attention.
We should have allowed more time at the end for them to teach each other. We received a few comments on the feedback sheet stating there was not enough time here.
One area on the feedback sheet we received a low mark was connecting our topic to other areas of math. We perhaps could have brought this topic more broad and tied in other ideas.
We had many positive comments about using history (Babylonian, Greek), visuals (ancient tablet, cuneiform script) and relating to real like (base 60 numbering system). I really enjoy history and I worry with a lesson like this because some people find it dry, but I was pleased to see that some comments said that we made a dry topic interesting.
An area I could improve with my group was to involve everyone more since my structure was more lecture type than the other two. I asked for volunteer and when people hesitated I hesitated too! I should have had thought of a different method of engaging them if no one volunteered.
Overall, I am really pleased with out micro-teaching lesson. I have only seen this jigsaw activity in my other classes with humanities related topics or reading. I see now that this can work in a math class. But I must remember that there will not be 3 teachers.

Group Micro Teaching – Apprenticeship and Workplace 10

Bridge: 2 min

Catch interest by having students form groups of three and numbering themselves 1,2,3 for jigsaw activity in which they will become the expert on some aspect of the history or modern use of the Pythagorean Theorem.

Learning Objective:

Students will be able to demonstrate an understanding of the Pythagorean Theorem by describing historical and contemporary applications of it.
> To learn a historical use for Pythagorean Theorem and to understand where Pythagoras' inspiration came from.
> To understand that this theorem came about from a practical application, not pen and paper theory.

Teaching Objective:

> Learn and improve our methods in conducting an engaging and educational lesson that will reach everyone.
> To interactively show the motivation for the Pythagoras Theorem, introduced to Pythagoras by the ancient Egyptians.
> To try out the Jigsaw strategy.

Pre-test: 1 min

> Who can tell me what we already know about the Pythagorean Theorem?
o Looking for formula, triples, and to see if anyone might already know where it comes from.

Participatory Learning: 7 min

> Split groups into 1’s, 2’s and 3’s.
o 1’s at Egyptian Station (Deb)
o 2’s at Babylonian Station (Michelle)
o 3’s at Modern Application Station (Nadine)

Station 1 – Egyptians
> Explain how ancient Egyptians used a right-triangle to redistribute fields after yearly flooding.
> Explain how they made a right triangle using rope and knots
> Explain how Pythagoras became involved/interested
> Make a right-triangle with a piece of string and ruler. Make a 3, 4, 5 triangle and see if they can find a right angle in the classroom with it.

Station 2 – Babylonians
> Explain Babylonian Mathematics and history of Pythagorean Theorem.
> Teaching Babylonian numbering system and ancient tablet.
> Briefly cover other ancient cultures in which Pythagorean Theorem was known and used

Station 3 – Modern Uses
> Talk about how surveyors use their equipment to set up right hand triangles in order to calculate distances etc…
> Have students using laptops research other possible modern uses for Pythagoras.
> If they cannot find any, have them brainstorm a list of possible fields that they think may use the Pythagorean Theorem.

Materials Needed:
o String and rulers
o White board, diagrams of tablet and Babylonian numbering system
o Laptops, paper to write ideas down on, pens

Post-Test: 3 min

Groups of three re-form and students teach other members about their area of expertise. Only have 1 minute each to share.

Summary: 2 min

Students complete the statement: The one thing I learned today that I didn’t know before was _______________________________________________________________.
(Orally if time allows.)


Group: Deb, Nadine, Michelle