Assignment 4 - Unit Plan

EDCP342A  2010

Unit Planning Assignment (Assignment 4) Introduction

Due Fri. Dec. 10, 2010 (hand in electronically on your blog)

Donna Braaten

The overview of the unit topic (Sequences and Series - Math 12) is included below.  The links for the Unit Chart Overview, 3 Lessons (tabbed in an Excel spreadsheet) and Project Description are listed at the bottom of this post. 

Topic of the Unit

     The topic of the unit is Sequences and Series.  This topic is included in the curriculum as series are used throughout physics and in business calculations.  Understanding sequences is a prerequisite to understanding series.  Sequences and series are found throughout nature and help to predict what will happen in the future.  The learning I hope for my students is to understand and appreciate the inclusion of patterns and sequences in nature, music and business.  My hope is that they find it rewarding to recognize a pattern, express the pattern and then use it to predict what will happen next.  I also want for them to learn about several famous sequences and series, their discoverers, origins and uses such as Fibonacci Sequence, Harmonic and Taylor Series.  I think what makes this topic so interesting is its mystery, such as really considering what an infinite series really means or how recursion goes on and on.  It is mysterious but is also used to predict very concrete, practical numbers such as calculating the payments on a loan or the profit on an investment.  Discovering patterns is a way to make sense of the world.

Breakdown of Unit into Lessons and Projects

     I will start the Unit by introducing the students to patterns and rules.  They will learn and practice how to recognize a pattern in a list of items.  As it could be difficult to get lost in the topic, I will provide a roadmap to the unit in the first lesson.  This will provide navigation to the students to understand what they have learned and what is coming next.  The actual list of ten lessons will consist of:

1.     Getting Started with Patterns

2.     Sequences

3.     Arithmetic Sequences & Series

4.     Geometric Sequences

5.     Compound Interest

6.     Geometric Series

7.     Project Presentations Day

8.     Infinite Geometric Series

9.     Other Infinite Series

10.  Technology

     After understanding the importance and enjoying the satisfaction of recognizing patterns, the students will learn about famous sequences and series and their significance in many areas of life such as nature, business and music.

I will then focus on Arithmetic Sequences and Series followed by Geometric Sequence and Series.  Once they understand Geometric Sequences, I will have them apply this to calculating compound interest for loans and investments using current bank rates.

     We will then learn about more famous sequences and series though peer presentations.  I will follow this up with the introduction to Infinite Series and have them ponder such complex terms as zero and infinity.  I will end the unit with a focus on technology to show how spreadsheet applications can help detect, manage and manipulate series.

Pedagogy of the Unit

     In order to encourage students’ active participation, I will keep the momentum throughout the unit by employing interesting hooks as well as applying Daily Points.  Daily Points are a quick review exercise that not only reinforces the topic but also lets the students see the immediate benefit of doing the assigned homework.

     The introduction of a project will also help the students connect complex topics to actual people (the discoverers) and practical examples.  The created posters will remain on the wall as a constant reminder that there are people and applications behind patterns and formulae.  I will introduce financial terms as part of understanding compound interest to appeal to those who are more business oriented.  I will also express patterns found in nature and/or music for those who are more interested in the nature and the arts.  Lastly, I will utilize technology as many concepts can be reinforced through the use of a spreadsheet application.

Assessment and Evaluation

     In order to obtain a well-rounded assessment, many forms of review will be used which are both formative and summative. Formative assessment for this lesson will consist of the following:

· Teacher Observation during the introduction exercises to see if students are getting the main concepts and to understand where the trouble spots may be.

· Use Open-Response questions during the start of any new topic during the Teacher-directed instruction.

· Check student work during the in class activity to check student progress.

     Summative/End of Lesson Assessment will be based on:

·  Teacher Observation during the introduction exercises to see if students are getting the main concepts and to understand where the trouble spots may be.

·  Formal-Written evaluation - students will demonstrate their knowledge by successfully completing a unit test after all lessons in the unit are completed.

·  Authentic Assessment - students will demonstrate their connection to real world context by successfully completing a math project based on a prescribed rubric.

     Lastly as a mixture of both formative and summative, I will introduce the concept of Daily Points in which I will ask the students to complete 3-5 problems at the start of each class that is reflective of the homework.  One of the problems will be directly from the homework and they will be able to use their notes.  This will allow me to see how well they are grasping the new concepts while letting the students know that doing their homework is valued.

Additional files:

Unit Plan Chart: 
https://docs.google.com/document/d/1gqYIaoiYCsL-w0_JiQzUNzCjZaosN165B6rMWRYgNSk/edit?hl=en&authkey=CIzj8vsF

3 Lesson Plans (sheets are tabbed at bottom):
https://spreadsheets.google.com/ccc?key=0AnL6aYoFt33fdDQ1NW9HZkxIQVg3V1FNQ2dKRTRRZ1E&hl=en&authkey=CPzinvYO

Project Plan:
https://docs.google.com/document/d/1yMI0siWsEL_H7cwHgw1fwg3QVqzN6DNw5Of5uG_RcA4/edit?hl=en&authkey=CKeMnfIK


Unit Plan Overview (posted here):
https://docs.google.com/document/d/1QFuF-E5oKIXo8jnbUwxb5xX9RTccFWrpL9OYmWHZVAI/edit?hl=en&authkey=CKTNjvIK

Math 11 project: Flatland comic strips

Team Members: Vincent, Mandeep & Donna

Part 1:  Trying out the project

Donna and Vincent were both very interested in trying out this project.  Both of them probably ended up spending more time than Susan suggested they do.  It was interesting to see how visually and conceptually different their end products were from each other. 

Vincent found that it was necessary to read the first half of Flatland in order to understand the general concepts behind the story (such as how Flatlanders “see”).  But he found this first half of the book very dry and not very entertaining to read.  He enjoyed the second part of the book, which told the story.  He found that watching the 30 minute Flatland movie partway through reading the first half of the book helped re-energize his interest in finishing the book.  Vincent thought mostly about how to draw the comic in a way that would convey the story of Flatland to the audience as efficiently as possible.  He concentrated on telling a story in a minimal way that would visually convey the concept of dimensions.  He had a few false starts before he starting drawing a comic that he found satisfying.  He settled on simple black and white line drawings with minimal words.

Donna thought that watching a math movie and reading a math story was in itself a very interesting task.  She was very curious and did not know what to expect.  The book, the film and the movie each had slight variations that became topics of discussion.  Creating a comic strip forced her to really think about the concepts, often going back to the book for reference.  She really enjoyed using colors and shapes, and if anything, spent too much time on the production since it was addicting.  Her comic was created on a computer and she spent a significant amount of time on the visual details of her comic. 

The fact that these two comics turned out so differently demonstrate that students will focus on the different parts that are of interest to them.  We both felt this was an educational experience that students will enjoy.


If you would like to see our comics, just cut and paste the following links into your browser. If there are any problems, just let me know and I will email them to you directly. 

Vincent's comic is located here: 

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B_WqVw36Np6_ZWUwYjYzNTgtNmY4OS00MGE2LWEyYWUtODQ2YzlkNmQ0OWE3&hl=en&authkey=CIfHqNkO

Donna's comic is located here: 

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B_WqVw36Np6_ZDJhZDJiNDktYzZlZi00NmIxLTg5ZjItYjk0ZmJjODM5Y2U1&hl=en&authkey=CMbM6dcD

Part 2: Look at the project with “teachers’ eyes”

The strengths of this project are that this is an enrichment assignment which allows students to develop a deeper understanding of geometric shapes and projections.  Having to create a comic telling the story of Flatland gives students the opportunity to apply their understanding of the book and the mathematical concepts discussed in it in a creative manner.  This project also enables students to take part in an inventive hands-on activity that the visual and artistically inclined students in the class will appreciate.   Reading the story of Flatland is also beneficial because it illustrates to students how fascinating and intriguing mathematics can be and encourages them to be imaginative and enquiring about the world and the math around us (ie, could there exist a fourth dimension and if it does, what would it look like?)  It also provides students with an alternative way of looking and comprehending geometric shapes (ie, a circle as a polygon with an infinite number of sides.)

The weaknesses of the project include having the possibility that students may spend too much time coming up with a comic strip and not concentrate enough on the mathematical aspect of this project.  Some constraints within the context of the classroom include there being is a lot of reading required for this assignment and some students may not have strong reading skills and thus may require a lot of time to finish the readings.  This may put pressure on the time constraints for this project.  The first half of Flatland is also very heavy with explanations which may make it difficult for some students (especially ESL and some LD students) to understand the novel and if they’re unable to comprehend the math topics discussed in the book, they will not gain anything from this activity.  Another issue with this project may be that some people (the girls in particular) may take offense at the sexism presented in the novel.  Getting a class set of the books may also be a problem in some schools.

The ways to adapt this project to try to overcome these weaknesses include having emphasis placed on the fact that this is a math (and not an art) project, so it will be mathematical content and understanding (not drawing skills) that will be assessed with this project.  Some strategies to deal with the time constraints of this project include having students read only Flatland for this assignment and, only if necessary, specific parts of Flatterland, Sphereland and/or the Scientific American article on the “planisphere,” with the teacher specifying which parts to read.  There is enough new mathematical material and concepts for students to learn and absorb just from reading Flatland; however, the projects that follow up with readings from these other books could be given bonus marks though.  To deal with the explanation-heavy first half of Flatland, there could be class/group discussions of the readings in class.  Also the project could be adapted to include the viewing of the Flatland movie and film (either during or outside of class) to help reinforce students’ understanding of the mathematical concepts (like dimensions) that are presented in the book.  There could also be a disclaimer before beginning the book and/or discussions while reading the book about the sexism presented in the novel so that students are prepared for it and not taken by surprise.  The teacher could also contact public libraries to try to obtain copies of Flatland if there is not a class set available.

Some ways to modify the project include:

•   Getting the class to write out and then perform a play and/or small skit about the story of Flatland
• Having the class divided into small groups and then having each group make their own movie about Flatland
• Allowing students to use computer graphics and/or photographs of common items and props to come up their comic strip about Flatland
• Having students use a simple, free programming environment of Scratch to create the 2D characters described in the story

Part 3: New project idea

Flatland Project – Personifying Shapes

Published by Edwin A. Abbott in 1884, Flatland:  A Romance of Many Dimensions is a satirical novella that tells the story of a world with only two dimensions. The inhabitants of Flatland are shapes, all with interesting characteristics and personalities.  Several films have been created from this original book, including a short film entitled Flatland: The Movie.

For this project, you will need to read the book and we will view the short movie in class.  We will then have a discussion about the concepts in the book.  Following this discussion, you will work in pairs to personify the shape of your choice with the result being a poster to be presented to the class.  You will have 1 full class period to work in pairs on this project.

1.     Read Flatland, available on line via Google Books (cut and paste link below).

http://books.google.ca/books?id=s2Tdsgf1KTwC&printsec=frontcover&dq=flatland&hl=en&ei=NADfTI_bH5OgsQOlmPnrCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCsQ6AEwAA#v=onepage&q&f=false

2.     On Friday, February 11, we will watch Flatland: The Movie in class.  Following the movie, each pair will be assigned to one of the two discussion groups that map to the two parts of the book: PART I: THIS WORLD and PART II: OTHER WORLDS.

3.     You and your partner will then chose a geometric shape (not restricted to those in the movie) and create a poster that expresses/advertises the shape, including but not limited to the following:

·      Accurately drawn shape

·      Origin of the name of the shape (eg pentagon => penta is 5)

·      Description of how to draw the shape and what tools were used (protractor, compass, ruler, etc)

·      3 mathematical facts about the shape (eg angles, lengths, area calculation)

·      3 places where the shape can be found in natural and/or man-made settings

·      2 thoughts/ideas about where the shape could be used

4.     You and your partner will then present your poster and ideas to the class on February 23.  Presentations should be approximately 5 minutes.

Marking Rubric

Criteria	Marks Allocated	Marks Assigned
Completeness of Assignment	Contains all the required parts: shape, name origin, 3 facts, 3 places and 2 thoughts/ideas where shape can be used.	/5
Mathematical Correctness	Angles, lengths and geometric relationships are depicted correctly and accurately.	/5
Creativity/Originality	Poster and presentation demonstrates unique and innovative ideas.	/5
Presentation	Presentation was well organized. Delivery was clear and effective.  Duties were shared amongst the pair.	/5
Bonus	Poster explains more than one method of drawing the shape.	/1
Total		/20

Other Project Information not included in worksheet above:

Grade Level: 10.

Essentials of Math 10: In the Section on Shape and Space, there is a Geometry Project with the following elements:

·       C3 measure lengths in both SI and Imperial units

·       C4 estimate measurements of objects in SI and Imperial systems, including length, area, volume, mass

·       C5 interpret drawings and use the information to solve problems

·       C6 draw top, front, and side views for both 3-D rod or block objects and their sketches

·       C7 sketch 3-D designs using isometric dot paper

·       C8 enlarge or reduce a dimensioned object according to a specified scale

·       C9 solve problems involving linear dimensions, area, and volume

·       C10 complete a project that includes a 2-D plan and a 3-D model of some physical structure

Applications of Math 10: Under Shape and Space, there is a Measurement Section:

·       C5 calculate the volume and surface area of a sphere, using formulas that are provided

·       C6 determine the relationships among linear scale factors, areas, the surface areas, and the volumes of similar figures and objects

·       C7 solve problems involving length, area, volume, time, mass, and rates derived from these

·       C8 interpret drawings, and use the information to solve problems

Purpose:

The purpose of this project is to raise the curiosity and understanding of geometric shapes.  The Flatland story gets them thinking about all the implications of living in a 0, 1, 2, and 3 dimensional lands.  It also raises their awareness of how to communicate such concepts based on what the audience already knows.

Total Length of Time:

The total time for this project is 3 class periods (1 for movie and discussion, 1 for group work time and 1 for presentations).  Outside the class would be for reading the book or further project time. The actual time students will have to work is 1.5 weeks.

Materials/Resources:

Materials needed for this project include poster board, compass, protractor, pencil crayons and markers.

Response to Selter's article on "Creativity, flexibility, adaptivity, and strategy use in mathematics"

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

Throughout this program, we have focused on the goal of teaching/facilitating students to think, rather than just instructing them to remember implementation methods.  This article further breaks down what is meant by getting students to think.  Specific takeaway points include understanding 3 main terms:

1.    Creativity:  ability to invent new or modify known strategies

2.    Flexibility: ability to switch between different strategies.

3.    Adaptivity: ability to apply appropriate strategies.

This led me to think about the relationship between the three, which is most important and which is easiest to learn.

In my effort to figure out which is most important, I further examined the article.  Which terms are discussed more, what are there definitions?  I used a basic approach – search for occurrence of the terms (or their roots).  It resulted in the following:

·      “strategy” – 104 occurrences

·      “adapt” – 75 occurrences

·      “creative” – 24 occurrences

·      “flexib” – 7 occurrences

My main ideas from this are:

1.    Strategy, although not defined in this article is referred to as “a plan of action”.  The article does indicate that a strategy can be deliberate or sub-conscious.  The most important part is that a student recognize that s/he is not blindly guessing at something and that that whether deliberate or sub-conscious, a student needs to solve today’s problems based on past information and learning.  One of the best things we can do for a student is to enable them to make the connections.

2.    The hierarchy of this terms for me would be flexibility -> adaptability -> creativity.  Although I think they are interrelated, I do believe that flexibility is the foundation for many of the other terms.

3.    Creativity does not necessarily mean to start with a blank sheet of paper.  I like that the definition includes NEW or MODIFY.  Many people are very creative but need to start with something existing or a problem to solve.  I believe that necessity is the mother of invention and also ensures that inventions have purpose.

In summary, I like the further breakdown of critical thinking as it makes it more approachable and less intimidating for new teachers to consider in their lessons.

Grade 10 Word Problems - Analysis

3 Grade 10 Word Problems from http://www.mystfx.ca/special/mathproblems/grade10.html

#1 - Brendan gambles while on a trip to see the Blue Jays
Brendan went to Toronto to see a Blue Jays game. He had $650.00 for the trip. It cost him $40.00 for his ticket to the game, $216.00 to stay 2 nights at a hotel, $329.00 for his flight and $29.00 for food. He met a man on Young Street who was running a small casino. Brendan bet $5.00 in Black Jack and won an amount of money that was triple what he had left after making the bet. He spent half of this at the ball game. How much money did Brendan take home?

Hint: He had 31 dollars left after making the bet.
Answer: [$750 - (216 + 100 + 40 + 329 - 5)] 3/2 = $90.00

I like that this problem provides a hint.  It also tries to be interesting by talking about baseball and gambling.  I find the gambling a bit risque given that they are grade 10.

#2 - Gary becomes teacher and earns half of Mary's salary
Mary and Gary graduated from university together. Gary became a teacher and earned half what Mary earned for 5 years. Mary spent 1/3 of her money; Gary spent 1/4 every year for those 5 years. Greg has $80,000 after 5 years. How much does Mary have?

Hint:
 Let x = Gary's salary
 Let 2x = Mary's salary
 Let 1/4x = Greg's spending
 Let 1/3(2x) = Mary's spending

Answer: Mary's savings = 5(Mary's salary) - Mary's spending = 5(2(21333.33)-1/3x(2(21333.33))) = $142222.22

I find any problem that provides salary data to run the risk of being very dated as salary data can change quite suddenly.  I also found that the example highlights that the person in the teaching profession is earning half of what the other "unnamed" profession is offering.  How much does Mary have is also unrealistic as it does not take in account any investments .. It implies that she just stuffed the money in a mattress or something.

#3 - The rotation on a turntable
If a turntable rotates through 720 degrees in one second, how many revolutions does it make in one minute.

Hint: there are 360 degrees in one revolution
Answer: 720 degrees/sec x 60s/min x 1 revolution/360 degrees = 120 rev/min.

I can see the benefit in the problem as wanting to highlight that one rotation is 360 degrees and ensure students pay attention to the units.  The turntable of today may be compared to the truncated grain pyramids of the Babylonians in that students may not have heard or seen a turntable.  Could bring one in and add some more interesting history to the lesson.

Fair is Fare - A math problem

Practicum Stories

Role of Gender

For 2 days of the first week, we were given a typical schedule of a student.  I was a Grade 10 girl.  It was eye-opening to really follow a schedule of an actual student.  I attended a Math, Science, Theatre, Art, English, French and Gym class.  The Aha moment came to me when I was sitting in the gym class.  It was the first class of the day where it was an all female class.  Walking through the locker room, I saw pictures of inspiring women sports figures and notices specifically targeted toward females.  As I was observing the class, I saw the same girls who were quiet in previous classes become more social as they played the team sport of volleyball.  From watching this, I decided it would be interesting to be part of a forum/club that has only girls.  As I played tennis in high school, I decided it may be a great opportunity to help out with the Girls' Tennis team.

Grade 8 Students

Everyone warns you before going into the classroom that it is classroom management that requires the most energy, focus, strategy.  Unconvinced, I spent many hours learning to multiply fraction 10 different ways.  Who'd thought you could use a number line for this?  Nonetheless, I learned the models, worked the homework, bought a tablet and put together my lesson plan and notes.  Armed with candy bar and pizza examples, and a game, I felt ready .... However, as predicted my many before me, it was the classroom management where I was not really mentally prepared to handle.  The grade 8's like to talk.  They listen, maybe try it, and then move on to their own conversations.  Although I tried circling the room many, many times, I found myself resorting to raising my voice.  This was slightly effective until they started getting louder ....  After two back to back classes of 30+ grade 8's, I almost had no voice left.  I was proud that I did not lose my temper and was no longer nervous, but I knew that when I come back in January to teach this class, I will need to find and use a more sustainable, sane-keeping strategy!

Feedback & Reflections on Team Microlesson

Overall, the lesson was okay but certainly had room for improvement.

• Most people felt that the use of the large prism shapes was a good idea especially when unfolding the shapes to flatten them.  However, some felt introducing all shapes at once was a little confusing and also 1 person felt they were distracting because I was fiddling with it.
• Most people like the hands on exercise but felt that the instruction of the exercise could have been much clearer.  We had decided not to use specific numbers, but almost all responses said we should have included the actual numbers.
• Most people felt that there was not enough connection to the real world.  Our real world examples were not convincing enough.
• Lastly, there were mixed views on how engaging the presentation actually was.  This seemed to be pretty split amongst the respondents.

My self reflection on the lesson was that I was disappointed and felt I could have done better.  We should have really had a more real world example up front.  The model would have been a good way to explain the surface area, but I did think it ended up being a distraction.  Also, the lack of concrete measurements for the hands on exercise was a mistake. In summary, it was a good learning experience.  We worked well as a team.

Team Lesson Plan

Surface Area of Prisms Hong Jiang, Edward Liao, Donna Braaten
Section	What	Time	Materials
Bridge	Show prisms.  Define what a prism is. Pose problem of how much paint, chocolate or gold it would take to cover.  What dimension do we need?  Intro to Surface Area and why you might need it.	2	- 3 large unfoldable shapes - prisms (rectangular, circular, triangular)
Leaning Objectives	- Each student will learn: -- how to calculate the surface area of a prism (rectangular, circular and triangular), building on past knowledge of area -- practical applications of measuring surface area -- to breakdown any shape to simplest known areas to be measured	1
Teaching Objectives	- Get all students involved. - Complete exercise with closing remarks	0
Pretest	- Review measurements of area including units	1
Participatory learning / Lecture	After opening (hook and learning objectives) - Open up each shape to break down shape to set of simple shapes for each - Split into groups to build a structure --- put wooden manipulative on the tables --- ask each group to build a village/structure (<= 10 shapes) --- have them calculate the service area - formula, then we provide dimensions	8	- wooden manipulatives - dimensions of the shapes
Post-test	Review their group work and hear their explanations	2
Summary and Wrap up	Review the key concepts: - break down the shape - add the areas	1