Unit Plan

Polyhedra Project

Strengths:  The Polyhedra project is a unique way to incorporate geometry, history, and art all into a mathematics classroom.  I think this could be a great introduction/hook project to start off a geometry unit because it has a non numerical and non formulaic approach.  Also, there is endless creativity options for the construction the origami models(Colours, materials, pictures etc).  For the typical math phobes, this could be an effective way to ease their way into geometry.  The artistic and historic type students will find themselves within their comfort zone and for that reason, they may find this project quite appealing and want to do further inquiry on Polyhedrons.  After finding the mathematical relations they might even be more motivated in the mathematics classroom.  For those who are comfortable with your standard homework, crunching out numbers,formula's, and problem sets, this type of project will certainly push them outside of their comfort zone which offers a beneficial challenge to them.  I think in particular, the writing portion of the project is extremely beneficial because we often neglect to do any writing in mathematics and it is such an important life skill.  Also, the presentation part of the project is great because it allows for all of the students in the class to have some knowledge on all of the topic areas.  You end up creating experts in specific fields, but at the end of the project most of the students will have the general knowledge of all the topics.

Weaknesses:  I thought some of the weaknesses of the project was that while there was certainly an artistic and written portion, there seemed to be very little in terms of the actual mathematics.  To a certain extent the historical component achieves this, for example, Euler's polyhedral theorem, but I don't think it offers enough in terms of PLO's and IRP's.  

Modifications, adaptations, extensions:  I think if I were to use this project in my class I would try and modify it to include less of a historical component while having a bigger emphasis on certain mathematical concepts, like surface area for example.  Personally, I think history is important because it offers interesting stories and it may provide a greater appreciation for mathematics, but at the same time, it can become quite boring and hard to find reliable information.  We also found that building five models is actually quite time consuming, especially if you are adding a lot of creativity, so depending on the group sizes you may want to cut down the number of models students have to build.   

For our project we decided to incorporate both surface area and an very basic introduction into probability by turning the polyhedra models into dice.  We also decided to cut out the historical component because we felt that geometry is already an interesting topic without it.  Possibly some of the "drier" topics will better be suited for history.  In addition, instead of having the students do all five of the origami models, we only required two for each pair.  We thought this might be better because on the presentation day, not everyone will have seen all the different types, providing some variety in the classroom.     

New math project outline  EDCP342A Names: Roz and Travis Project title: Polyhedra Dice
Grade level	Grade 9
Purpose	determine the surface area of composite 3-D objects to solve problems To investigate, create, and present findings on regular polyhedra. Introduction and inquiry into basic probability
Description of activities	Students will first be divided into pairs The pairs will then receive two of the four regular polyhedra (Not including tetrahedron) and be required to re-create a 3-D paper model.  Instructions will be provided on how to create the models. Each pair will then number each face/side of the polyhedra to create a set of dice. The pairs will then come up with a formula for calculating surface area for their polyhedra and then provide a numerical answer for their SA Students individually will then answer the following probability questions about their dice. How many sides are there for each of the five regular polyhedra? What can you say about the names of the polyhedra? What is the probability of rolling a 1 for each die? If both dice are rolled at the same time, what is the probability that the sum of the dice will be 10? * Extension:  What is the probability that when both dice are rolled, two 5's will land. * Students will then bring in their creations and surface area answers, and display them in class for all students to see.
Sources (bibliography, websites, etc.)
Length of time project will take (in and out of class)	3 classes for assigning of pairs, polyhedra, and creation of polyhedra dice, and surface area formulas . Students should have enough classtime to finish their models and proceed to inquiry on probability, otherwise it is to be on their own time. 20 mins at beginning of one class to present and display
What students are required to produce	In pairs, two 3-D polyhedra dice models. Surface area formula and numerical answer Individually, answers to probability questions
Marking Criteria:

Part A.(In pairs)  Two 3D polyhedra paper model dice.  /20
                          Proper, correct construction of model
                          Creativity of design (colour, patterns etc)

Part B.(In pairs)  Surface area formula and calculation /15
                         Clear understanding of what surface area is and correct answers

Part C (Individual).  Probability inquiry and answers /10
                             
                             

Poetic Mathematics

Is it where we begin

Is it where we end

Or maybe it's where we collide

Thought of as nothing and emptiness

Yet its existence is crucial and necessary

Once considered to be a devilish sin

Now considered to be at par

Alive as a numerator

Dead as a denominator

Falling in the dimension of purgatory

Waiting to embark on a journey to either side of good and evil

Positive and negative

0 or Zero

Teacher Bird Comments:

I think this is a great tool/way we can incorporate poetry and writing into mathematics.  Both the free writing and Poetry spark creativity that mathematics can often lack.  Also, it is a good way of challenging those who aren't comfortable with writing.  On the other hand, for the math phobes it can be more comfortable and provide an alternative outlet.  Having multi-modal and interdisciplinary approaches is good way to display the "funner" side of mathematics!

Free write

zero, "nothing," location on a number line between -1 and 1, origin

 Divide, separating a group of people, numbers being divided by something, when there is a crack in something, division of sports teams like west versus east or north conference south conference west conference and then I don't know what to write about divide, two divided by one is two and eight divided by two is 4, it is the opposite of multiplication sort of, separating, parting, when you divide a group you are separating it into smaller groups.

zero, nothing, the origin, wasn't used/known in certain historical times, having no value, rating scale of 0 - 10 with 10 being the best and 0 being the worst.  We often associate zero with being a bad thing but in sports like golf it is a good thing :) lots of things have countdowns to zero, time in sports, liftoff, race's 3,2,1,GO! zero is a number inbetween -1 and 1.

Sponsor Teacher Response To Challenging Topics

In pre-calc 12, I think that permutations, combinations, and binomial theorem will be the most challenging and difficult for students to understand.  After emailing my sponsor teacher, she agrees completely.  Last year was her first year teaching this topic and she found that she had to go at a much slower pace than anticipated.  In addition, she had to explain the concepts with more detail and precision in comparison to other topics.  So some basic tips from her were to adjust the pace accordingly and to make sure students understand what information is key when reading word problems.

I think probability(perms, combs, binomial theorm) is a difficult topic in general because types of problems and the concepts are extremely different from what students are used to.  There are formulaic, logical, and common sense aspects to probability which make it difficult.  I think understanding what information to look for in word problems is a skill that needs to be developed so before introducing probability we should make sure to incorporate word problems throughout the year.  Also, the Khan academy had a couple of great videos on permutations and combinations that explain the concepts in a clear way.    

Lesson Plan for Microteaching in Partners - S.A. of Cylinders

Bridge: Each student is given a template of the cylinder and is asked to cut it out and build it.

Learning Objective: Students will learn to determine the surface area of a cylinder.

Teaching Objective: Students will build their own cylinders and in doing so, discover what contributes to the total surface area of a cylinder. Students will also learn the formula of a cylinder.

Pre-Test: Ask participants to work in groups of 4 and to find the surface area of the cylinder they have just cut out.

Activity:

Time	Teacher	Participants
2 min	Introduce topic and explain some of the practical applications of the S.A. of a cylinder.	Think about different applications of the surface area of a cylinder.
2 min	Ask participants to figure out the surface area of the cut out cylinder (pre-test)	Work out surface area of a cylinder
3 min	Explain the surface area of a cylinder and work out the formula with the class. Random fact: surface area and volume of a cylinder have been known since deep antiquity.	Listening to teacher.
5 min	Ask participants to work in groups of 4 on the cylinder worksheet.	In groups of 4, students will work on the worksheet.
3 min	Review and go over solutions for the cylinder worksheet.	Ask questions and check answers.

Post-Test: Participants will be able to check their understanding and knowledge when working on the worksheet and then checking it with the solutions provided at the end.

Summary: We will go through the formula of a cylinder again and ask if there are any questions from the group. 

Response to Boaler and Staples article

Although a bit lengthy, I enjoyed reading Boaler and Staples article as it gave some valuable data on the differences between the "standard" and "Railside" teaching styles.  The two main points of emphasis were on improving overall learning of mathematics, and creating/attaining an equitable environment.  I personally enjoyed reading about how Railside tried to reduce inequalities.  This is something that I previously thought was extremely hard to incorporate into a mathematics classroom but the teachers at Railside used some creative and effective pedagogy to attain this.  It basically boiled down to group work, collaborative discussion, multidimensional solutions, and justification for these solutions.  One idea that struck me was having groups work on a particular problem, then randomly picking one of the group members solution, if the solution was wrong then the rest of the group had to help out that individual.  This gave a sense of responsibility for the entire group.  In addition, the idea that

"high attaining students developed deeper understanding from the

act of explaining work to others,"

grabbed my attention because one of the math workshops I attended had a teacher from Oak Bay who was experiment with peer teaching and was having great success with it, despite it being somewhat controversial. I am quite intrigued by the idea of students teaching students because it potentially has great benefits for both students.  Overall the article presented some creative teaching tactics and they had solid evidence to back it up.

The actual study was done quite well in my opinion.  They had accurate numerical data, and even written feedback from students to back up their results.  They had a good sample size, duration, and methods.  In terms of weaknesses I couldn't see any major flaws in the study.  A minor weakness was the fact that Railside mathematics classes were half a year long where as the traditional schools were a year long.  I think the study could have been more accurate if they had another school that had half year math classes. Nonetheless I don't think the impact was that big as it just allowed Railside students the option of being able to take more math classes and get further ahead.  Also, the CAT 6 scores were significantly lower than the traditional schools but they justified this due to the language being unfamiliar to Railside students. A fairly legitimate argument in my opinion.

As a whole, I found their conclusions to be quite accurate and I found it difficult to come up with any arguments against them.