Name: Rory Whitbread
Title of unit and grade / course: Shape & Space & Their Measurement / Mathematics 8
1) Rationale and connections:
a) It is important for students to learn the topic of space, shape and the measurement of each as they are broad reaching and provide a base for many future topics in mathematics and other subjects and even general everyday life. This subject area is also an excellent opportunity to include history with mathematics in the discussion of how varying shapes and measurements were used by other civilizations in other times. These subjects are included in the IRPs as many sections of higher grade mathematics depend deeply on their understanding.
b) Historical origins of this topic are vast. Many civilizations had broad working knowledge of geometrical shapes and their measurement. These were used for the buying and selling of commodities and for the division and taxation of land. Of course, mathematics was also performed by academics and for recreation and many of the ancient findings produced are still used widely today. The Pythagorean Theorem is one of the most famous in all of mathematics and is named for Pythagoras and his followers, but it seems it was being used by many more civilizations than just the early Greeks.
c) Shape, Space and Measurement relate closely to life outside of mathematics. First, they provide for a base of good spatial thought and reasoning. Second, they are topics that are easily found in every day life; from measuring the dimensions of a box that needs to made, to knowing how tall a ladder will be that will be able to reach the roof, to the age old measurement of land and structures. Having a good understand of these topics also helps with estimation - a valuable skill in a fast paced world.
2) Balanced teaching, assessment and evaluation plan
a) This is a very important unit of the Mathematics 8 curriculum and will therefore be worth 10% of the weight for grading. For this reason, multiple forms of assessment will also be used, with teacher assessment for a quiz and unit test and peer and student assessment utilized for a group project that will be completed throughout the 10 classes. The midway quiz and unit test will comprise 1% and 4% respectively. The remaining 5% will be assessed from the group project which will have 2.5% resulting from self assessment through journal work and 2.5% as evaluated by the group. With these forms of assessment in mind, there will be a good balance between summative and formative assessment as well as between instrumental and relational learning. All things considered, it is hoped that this will one of the favourite units of the students’ Math 8 year.
b) House Building Blueprints Project – The class project that will be worked on throughout this unit will be the basic design of a house or structure from the past, present or future. This project will be completed in groups of three or four with the group presenting their 2 or 3 dimensional final product to the class.
c)
Lesson topic Teaching strategies / approaches used
1) Surface Area Area and Surface Area will be reviewed and discussed. Specific applications for right rectangular and right triangular prisms and for right cylinders will be explored. The class will be placed into groups of three or four that will also be used for the unit project. Within these groups, tactile analysis of sample shapes will be used and nets will be constructed.
2) Surface Area Continued Students will be introduced to the volumes of the shapes explored last day. Relations of Surface Area to the dimensions and Perimeters of shapes will be explored. Students will then complete worksheets with questions relating to surface area.
3) Group Project Work Students will be introduced to the Unit Group Project (details given at end of unit plan). They will be given the full class to come up with a plan for what type of building they will create and which shapes will be included and how their projects will be presented to the class.
4) Pythagorean Theorem Students will be introduced to the Pythagorean Theorem with approximately 15 minutes of lecture involving concept and history. Then students will be split up into their groups to explore triangular shapes and perfect Pythagorean Triangles. They will be given knotted strings to explore these the way people would have thousands of years ago. Human triangles will be performed to conclude the class.
5) Group Work / Quiz Students will use the first half of the class to work on their group projects, with specific emphasis on the right triangular portions of their structures. The remaining half of the class will be used for a quiz on Surface Area and the Pythagorean Theorem.
6) Volumes of Shapes Early class will be devoted to lecture and discussion of the volumes of the shapes from lesson one. Relationships between Surface Area and Volume will be explored. Students will spend the last portion of class in groups discussing their group projects.
7) Group Project Work Students will have the entire class to work on their group projects. They will be asked to consider and discuss the cost of heating these structures as an application of volume.
8) Review Work and Class Presentations The first portion of the class will be devoted to individual work on reviewing the material in the unit thus far. The second portion will allow students to complete final work on their group projects and get ready for presentations. The third portion will be given for the initial presentations.
9) Group Presentations All remaining groups will present to the class and concepts will be reviewed as appropriate throughout.
10) Unit Test The early class time will be devoted to a short review and class questions. The remainder of class time will be devoted to a Unit Test which will evaluate what was learned over the previous nine classes and will constitute 4% of the final grade.
3) In detail:
Surface Area
Rationale: To give students an understanding of surface area and its applications. The Surface Areas of various shapes will be explored in a hands-on fashion.
Instructional Objectives: Each student will understand the concept of Surface Area as it pertains to various shapes. A variety of different shapes will be looked at in two dimensional and three dimensional forms and students will work in groups to determine the Surface Area of each. The concept of nets will be explored.
Preparation: Students will spend the first brief portion of the class listening and discussing. They will then move into groups of three or four where group participation and exploration will occur. For this portion, various right rectangular and right triangular prisms and right cylinders will be handed out for tactile exploration and Surface Area Determination.
Introduction: A quick review and class discussion will be given of what various geometrical forms are. From here, areas will be reviewed and the concept of surface area will be developed and discussed.
Body: Splitting up the class into groups of three or four, the students will have some time to explore geometrical shapes (specifically right rectangular and right triangular prisms and right cylinders). Surface Areas for individual shapes will be discussed and determined. Nets will be created and discussed and their various uses will be determined.
Closure: Questions and discussion will be encouraged and the surface areas of the various shapes will be understood. Students will be asked to study shapes at home and to create a net for an object that they find at home that matches one of the shapes studied. These will be compared and discussed next day.
Hands-on Pythagorean Theorem
Rationale: To give students a review of the areas of various shapes from a different perspective, leading to a hands-on look at the geometry of the Pythagorean Theorem from its origins, without calculators or notebooks. It is hoped this will be an informative and enjoyable lesson.
Instructional Objectives: Each student will be able to understand various geometrical methods and specifically those of triangles through group work with knotted strings and ‘human triangles’.
Preparation: Students will spend the first portion of the class learning about the origins and the theorem itself. Afterward, students need only to clear space for themselves and be able to freely make use of the knotted strings in their groups of three or four.
Introduction: A quick review on the chalkboard with student involvement of previously performed geometry (and the origin of the word). The basics of the Pythagorean Formula will then be illustrated.
Body: Splitting up the class into groups of three, the students will have some time to explore geometrical shapes with their strings that include numerous knots which are each spaced 10 cm apart. It is hoped they will think of geometry as a Pythagorean might have with the same tools available. Questions will be asked and encouraged throughout the lesson amongst the groups.
Closure: The lesson will be concluded with human triangles, beginning with a right 3-4-5 triangle, built with people standing shoulder to shoulder. Approximations will be made and triangles will be balanced between those who are broader shouldered than others. Questions on the activity can conclude the lesson and summarize things that weren’t clear.
Volumes of Shapes
Rationale: Students will understand the concept of volume and will be able to make comparisons to Surface Area and contrast their differences. Students will also be able to use the concept with respect to their group projects.
Instructional Objectives: Each student will understand the concept of Volume as it pertains to various shapes. The same variety of different shapes from the pervious week will be looked at in two dimensional and three dimensional forms and students will work in groups to determine the Volumes of each. These volumes will be compared to the corresponding Surface Areas through discussion.
Preparation: Students will spend the first brief portion of the class listening and discussing. They will then move into groups of three or four where group participation and exploration will occur. For this portion, various right rectangular and right triangular prisms and right cylinders will be handed out for tactile exploration and Volume Determination (much the same as for the Surface Area lesson). Students will then move into their groups where they will continue to prepare their group projects. Materials for this will need to be provided.
Introduction: A quick review and class discussion will be given in regard to various geometrical forms. From here, the concept of volume will be developed and discussed. The students will then move into groups.
Body: Splitting up the class into groups of three or four, the students will have the bulk of class time to explore geometrical shapes (specifically right rectangular and right triangular prisms and right cylinders). The volumes of individual shapes will be discussed and determined. Questions will be asked of each group as necessary to encourage further exploration.
Closure: Questions and discussion will be encouraged and the volumes of the various shapes will be fully understood. The students will then continue to work on their group building projects and specifically discuss how the volume of their rooms will be determined by their shapes and what result this will have on the heating of their buildings. This will be included in their group presentations.
b) House Building Blueprints Project The class project that will be worked on throughout this unit will be the basic design of a house or structure from the past, present or future (creativity will be encouraged throughout the project). This project will be completed in groups of three or four with the group presenting their 2 or 3 dimensional final product to the class. The project must include varying shaped rooms and structures with measurements, with at least one right angle triangle being featured. Floor areas of rooms, surface areas of walls and roofs and internal volumes for heating considerations must be determined and presented.
Individual Expectations and Grading
Three journal entries will be given (one for each class dedicated to the project work). Other entries from additional individual and group work will be encouraged. Each entry should include successes and drawbacks that were encountered throughout the project and how each related to the concepts that were learned in class. The role each student accepted and what was accomplished as part of the larger group should also be discussed.
These entries will be given a mark out of 6.
One final journal entry will be given after the student’s group presentation. This entry will include summary information on what was discovered as part of the group’s presentation. Again the successes and drawbacks of what was found during this presentation will be encouraged to be listed and described, specifically regarding unit concepts. The relative success or failure of the rest of the group presentation will not be considered here – only the observations of the individual student.
This summary entry will be given a mark out of 4
The Total Mark for all journal entries will be out of 10 and will comprise 2.5% of the final grade.
Group Expectations and Grading
Students will be expected to perform well in cohesive groups. Within each group, the roles of each student will be chosen and agreed upon with regard to the final presentation. Students will work together to explore concepts and how they relate to their specific project. Again, creativity and participation by all will be highly encouraged.
This group work will result in a mark out of 5
Students will present their blueprints and/or three dimensional projects. Number and size of rooms will be discussed. Considerations for surface area and volume of a variety of different shaped and sized rooms will be necessary for full marks. Also, as specified, there must be at least one right angle triangle featured with use of the Pythagorean Theorem in determining its size. Presenting skills will be encouraged but not evaluated.
This group presentation and final product will result in a mark out of 5
The Total Mark for each member of the group (each members mark will most likely be the same) will be out of 10 and will comprise the other 2.5% of the final grade.
Friday, December 11, 2009
Group Project - Recreational Mathematics
Martin Gardner
Martin Gardner was born on October 21, 1914. A prolific writer, Gardner has published 70 books to date on topics including: recreational mathematics (our interest here), pseudoscience, magic, philosophy, literature, and even a couple works of fiction. His most recent work of fiction is a story about Oz involving Klein bottles. Gardner is most fondly remembered for his column Mathematical Games, which ran in Scientific American for 25 years.
Gardner has no formal mathematics training outside of high school and even had a hard time passing his high school calculus course. His background is in philosophy, which he studied at The University of Chicago. When asked what he likes about mathematics Gardner responded: “There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known.”
After serving as a yeoman in World War II he became a contributor to Esquire magazine. His first submission was a short story entitled “The Horse on the Escalator.” From Esquire he became assistant editor of the children’s magazine Humpty Dumpty (Humpty was the head editor). After submitting a piece on flexagons to Scientific American in 1956 he was asked to head a monthly column focusing on recreational mathematics.
Gardner claims that the number of puzzles that he has invented could be counted on one hand. He could often be found scouring Manhattan bookstores for books of recreational mathematics that he would use for inspiration. Gardner believes that his column was so successful because of his lack of experience. While writing the column he would also be solving the problem for the first time.
In 1981 Gardner retired from Scientific American to focus on his other passions, mainly debunking pseudoscience. Gardner inspired many a future mathematician with his column which lives on in various collections that have been published since his retirement.
The problem will be presented in class.
Thoughts on the project:
This project is a good way to show students that math problems exist in popular culture and can be done just for the fun of it. It’s also a chance for students to pick a problem of their own choosing that will hopefully be a fun challenge for them and not just another problem where they aren’t interested in the answer (except that it’s correct). One of the great things about the problems in Scientific American is that they are phrased in a way where you are actually interested in finding the answer. During class it’s hard to make questions that are interesting for students, and showing them that they can actually find math problems that are “fun” will hopefully improve their opinion of math and willingness to engage in the learning of it. While I was finding the solution to the problem, I wondered how easily I would be able to find the solution if I were still in high school and didn’t have a degree in mathematics. If I assigned this project to students I would worry that the problem they pick might be so hard that they get the impression that math is just something that they will never be able to understand and use fully. On the other hand, picking problems for the students doesn’t allow them to find problems that are interesting to them and takes away a lot of the benefit of the project.
I would most likely use this project as an enrichment assignment to be completed between units, since there is no specific topic that is being used or learned here, it’s simply a general interest project. It would be fun to have everyone present their project, but I don’t know if I would have the time to spend doing in-class presentations. Instead, maybe I would have them create a poster and put the posters up in the classroom. Another thing I could do if I had one class to spare is make this the basis of an in-class math fair where everyone would create a booth and go from group to group trying to solve the other problems.
To put together a math fair I could modify this project so that the end result is a booth rather than a report and poster. If the students were not in grade 12, I could adapt the project so that I choose a number of problems that I know are solvable for their grade level and allow them to choose only from my selection of problems. Another idea would be if I had a grade 12 class and a lower grade class (grade 8-9) I could assign this project to the grade 12’s, have them make a booth with tools to allow anyone to solve the problem and bring the finished projects to my grade 8-9 class for the students to work on. I could make part of my evaluation of the grade 12’s that they present the problem in a way that is understandable and solvable for a grade 8 class.
Martin Gardner was born on October 21, 1914. A prolific writer, Gardner has published 70 books to date on topics including: recreational mathematics (our interest here), pseudoscience, magic, philosophy, literature, and even a couple works of fiction. His most recent work of fiction is a story about Oz involving Klein bottles. Gardner is most fondly remembered for his column Mathematical Games, which ran in Scientific American for 25 years.
Gardner has no formal mathematics training outside of high school and even had a hard time passing his high school calculus course. His background is in philosophy, which he studied at The University of Chicago. When asked what he likes about mathematics Gardner responded: “There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known.”
After serving as a yeoman in World War II he became a contributor to Esquire magazine. His first submission was a short story entitled “The Horse on the Escalator.” From Esquire he became assistant editor of the children’s magazine Humpty Dumpty (Humpty was the head editor). After submitting a piece on flexagons to Scientific American in 1956 he was asked to head a monthly column focusing on recreational mathematics.
Gardner claims that the number of puzzles that he has invented could be counted on one hand. He could often be found scouring Manhattan bookstores for books of recreational mathematics that he would use for inspiration. Gardner believes that his column was so successful because of his lack of experience. While writing the column he would also be solving the problem for the first time.
In 1981 Gardner retired from Scientific American to focus on his other passions, mainly debunking pseudoscience. Gardner inspired many a future mathematician with his column which lives on in various collections that have been published since his retirement.
The problem will be presented in class.
Thoughts on the project:
This project is a good way to show students that math problems exist in popular culture and can be done just for the fun of it. It’s also a chance for students to pick a problem of their own choosing that will hopefully be a fun challenge for them and not just another problem where they aren’t interested in the answer (except that it’s correct). One of the great things about the problems in Scientific American is that they are phrased in a way where you are actually interested in finding the answer. During class it’s hard to make questions that are interesting for students, and showing them that they can actually find math problems that are “fun” will hopefully improve their opinion of math and willingness to engage in the learning of it. While I was finding the solution to the problem, I wondered how easily I would be able to find the solution if I were still in high school and didn’t have a degree in mathematics. If I assigned this project to students I would worry that the problem they pick might be so hard that they get the impression that math is just something that they will never be able to understand and use fully. On the other hand, picking problems for the students doesn’t allow them to find problems that are interesting to them and takes away a lot of the benefit of the project.
I would most likely use this project as an enrichment assignment to be completed between units, since there is no specific topic that is being used or learned here, it’s simply a general interest project. It would be fun to have everyone present their project, but I don’t know if I would have the time to spend doing in-class presentations. Instead, maybe I would have them create a poster and put the posters up in the classroom. Another thing I could do if I had one class to spare is make this the basis of an in-class math fair where everyone would create a booth and go from group to group trying to solve the other problems.
To put together a math fair I could modify this project so that the end result is a booth rather than a report and poster. If the students were not in grade 12, I could adapt the project so that I choose a number of problems that I know are solvable for their grade level and allow them to choose only from my selection of problems. Another idea would be if I had a grade 12 class and a lower grade class (grade 8-9) I could assign this project to the grade 12’s, have them make a booth with tools to allow anyone to solve the problem and bring the finished projects to my grade 8-9 class for the students to work on. I could make part of my evaluation of the grade 12’s that they present the problem in a way that is understandable and solvable for a grade 8 class.
Wednesday, November 18, 2009
Reviewing my group's work
It was great to see the solutions (and attempts) provided for the Jacobian Locks problem by my team member's - Jill and Caleb. Jill's was very neat and gave great visuals of stick people in the village along with great charts illustrating the various scenarios they went through in solving the problem. She made the mathematics behind her solution very easy to follow and came up with, what appears to be, the correct solution. Well done Jill!
Caleb's solution was great to read as well from the pictures on his whiteboard. He used set theory which also provided an easy way to follow his thinking. His many notes along the way made it enjoyable to read. The "oh poo!" comment when he realized it was not working, reminded me of something I have exclaimed at that point myself.
Both of these solutions were great to read and it made me wish that my own weekend had not been turned upsidedown and that I had personally had more time to work on this problem.
Caleb's solution was great to read as well from the pictures on his whiteboard. He used set theory which also provided an easy way to follow his thinking. His many notes along the way made it enjoyable to read. The "oh poo!" comment when he realized it was not working, reminded me of something I have exclaimed at that point myself.
Both of these solutions were great to read and it made me wish that my own weekend had not been turned upsidedown and that I had personally had more time to work on this problem.
Sunday, November 15, 2009
Jacobian Locks Problem Solving
The problem from page 176 in Thinking Mathematically goes like this:
A certain village in Jacobean times had all the valuables locked in a chest in the church. The chest had a number of locks on it, each with its own individual and distinct key. The aim of the village was to ensure that any three people in the village would amongst them have enough keys to open the chest, but no two people would be able to . How many locks are required, and how many keys?
I initially figured that there had to be some kind of trick to this. I knew that if every villager had a key to each lock, then there would be no way to prevent two villagers from opening any number of locks. So I began to think of tricks. First, I thought that there is only one lock and each villager has one third of a key that together form a full key. But this would mean not every lock has its own individual key. Then, I thought maybe the chest is up in the air and it requires a human ladder to reach it, but two people could not. This would require only one lock and each villager to have one key. This might be the solution, but I doubt it. Then I thought maybe the chest is very big (it does hold all the valuables for the village) and has three locks spread far around it. Each villager has one key that opens a lock which closes again right away. Three villagers could do this but not two. The same would work for three locks with three keys that break upon their use. Then every villager would need only one key, but the keys would not be distinct. Then I got to thinking that there must be something in the order or way the locks are open.
A certain village in Jacobean times had all the valuables locked in a chest in the church. The chest had a number of locks on it, each with its own individual and distinct key. The aim of the village was to ensure that any three people in the village would amongst them have enough keys to open the chest, but no two people would be able to . How many locks are required, and how many keys?
I initially figured that there had to be some kind of trick to this. I knew that if every villager had a key to each lock, then there would be no way to prevent two villagers from opening any number of locks. So I began to think of tricks. First, I thought that there is only one lock and each villager has one third of a key that together form a full key. But this would mean not every lock has its own individual key. Then, I thought maybe the chest is up in the air and it requires a human ladder to reach it, but two people could not. This would require only one lock and each villager to have one key. This might be the solution, but I doubt it. Then I thought maybe the chest is very big (it does hold all the valuables for the village) and has three locks spread far around it. Each villager has one key that opens a lock which closes again right away. Three villagers could do this but not two. The same would work for three locks with three keys that break upon their use. Then every villager would need only one key, but the keys would not be distinct. Then I got to thinking that there must be something in the order or way the locks are open.
Tuesday, November 10, 2009
Two Memorable Moments from Practicum (Quick Write)
1) Teaching Completing the Square - It was my second lesson. The first went well, but was a learning experience. But, with Completing the Square, I got the AHA! moment from the class. I invoved humour and previous review and it felt good to look out at every face, most smiling, some nodding, and many getting the beginnings of a difficult topic.
2) Halloween Costume - Dress up day was the last day of the two week stretch. I had developed a rapport with many of the students and was starting to feel I was well liked. One student that had me wondering on day one, "am I going to have problems with this one" sat in the third seat from the enterance. I walked in in my calculator costume and he said, "you made my day by dressing up as a calculator". It brought an instant smile to my face. As I walked away, I heard him say to a friend, " I want to dress up as a calculator".
2) Halloween Costume - Dress up day was the last day of the two week stretch. I had developed a rapport with many of the students and was starting to feel I was well liked. One student that had me wondering on day one, "am I going to have problems with this one" sat in the third seat from the enterance. I walked in in my calculator costume and he said, "you made my day by dressing up as a calculator". It brought an instant smile to my face. As I walked away, I heard him say to a friend, " I want to dress up as a calculator".
Sunday, November 1, 2009
Quick Writes
Divide: A word with many meanings - some destructive and othersconstructive. A noun or a verb. A divide could be an area across a valley where one might want to build a bridge. A divide could be a stretch between cultures - a social divide. Of course, to divide is the cutting of another object or number into smaller pieces. Divide is the root word of division, which is the noun that is used to claim where a divide ha been made. Divide begins with the letter D and contains two i's. No other basic mathematics skill begins with the letter D
Zero: Zero is a unique word - a word to describe nothing - truly nothing. Zero is a concept that seems obvious today, but in the early days of mathematical study, zero did not exist. it was completely left off the number line of integers. It is an amazing concept when you think of it - something that's sole purpose is to describe nothing. Zero makes adding, subtracting and multiplying very easy. But something very interesting happens with division. Zero can divide as many times as one wishes. I can give you zero things many, many times over. However, when I try to divide by zero, it becomes mathematically impossible. Why, you ask? Well to dived by zero means that I am not dividing at all, which makes it impossible.
Zero: Zero is a unique word - a word to describe nothing - truly nothing. Zero is a concept that seems obvious today, but in the early days of mathematical study, zero did not exist. it was completely left off the number line of integers. It is an amazing concept when you think of it - something that's sole purpose is to describe nothing. Zero makes adding, subtracting and multiplying very easy. But something very interesting happens with division. Zero can divide as many times as one wishes. I can give you zero things many, many times over. However, when I try to divide by zero, it becomes mathematically impossible. Why, you ask? Well to dived by zero means that I am not dividing at all, which makes it impossible.
Division by Zero
Some feel a division at the root is all bad
To seprate into groups
Breaking apart makes things lesser
And limits their worth
When seen through clear lenses
It is in all that we do
Thus division by zero should be liked most of all
For it, and it only, does not yield to divergence
To seprate into groups
Breaking apart makes things lesser
And limits their worth
When seen through clear lenses
It is in all that we do
Thus division by zero should be liked most of all
For it, and it only, does not yield to divergence
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