Name: Alex Shum Student Number: XXXXXXXX Attended Effective Lesson Planning workshop: workshop : Yes Teaching Lesson: Your favourite theory or theorist in your field of study. Lesson Title: Dido’s Problem: Problem: An introduction to the Calculus of Variations Learning objectives (list 1-2 specific objectives): At the end of the lesson, students should be able to:
TIME 15min 1 min
1) Determine the shape given a fixed number of sides and perimeter that maximizes enclosed area. 2) Recognize the shape that encloses the maximum area given a piece of string. string. (A Circle) Pre-assessment: (Ask questions, try to engage each member of the audience) What do perimeter and area mean? Who can tell me the formula of the area of a square? (Length x Width) What is the area of a triangle? (Base x Height)/2 The Pythagorean Pythagorean theorem? (In a right-angle triangle, the sum sum of the squares of the two shorter sides a,b is equal to the square of the third side, c ): ): a2+b2 = c 2. Opening your lesson: Tell the story of Dido: A young woman who fled from her homeland Phoenicia to escape her tyrant brother who had already killed h er husband. She escaped to Africa, where she struck a deal with the l ocal king, who in exchange for some treasure promised her the l and that an ox-hide could encompass. She cut up the ox-hide into long strips and sewed the strips together. What shape did she use? Learner engagement and participation
1.5 min
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Activity 1: Have students pair up and determine on graph paper, that if 1.5 min the perimeter of rectangle is fixed (at 24 units), what is the rectangle t hat maximizes the enclosed area? Goal: Have students recognize that the solution is a square (with a sidelength of 6 units). If students are having trouble, tell them to try rectangles with: a) Length: 2, Width: 10 b) Length 4, Width: 8 c) Length: 6, Width: 6 d) Length: 7, Width: 5 More importantly, explain that the maximizing shape has all sides that are the same length, and same angles: Square Activity 2: Have students to pair up and fill out a handout to determine 3.5 min the area of a few isosceles triangles with the same perimeter to determine the shape of the maximizing triangle. Goal: Determine that the shape is an equil ateral triangle. (All angles are the same, all side lengths are the same). If students are having trouble: Help them with using heuristic arguments.
Students do not need to rigorously prove the answer, simply have some intuition. I will provide rulers, so they can simply measure the height of these triangles. I will engage the students, and draw both isosceles triangles that give “zero” area, and show that area must increase from a certain point, then decrease after that. Question: What do you expect the 5-sided shape to be? 6-sided? Use pictures from slides. (Regular Pentagon, Regular Hexagon)
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Calculation with class (on Whiteboard/Blackboard): Ask students to 1.5 min assist in a calculation of comparing area of maximizing triangle with area of maximizing square (both with same perimeter). MATLAB Animation: Then show how as the number of sides i ncreases (with the same perimeter), the area increases. The resulting shape is a circle, having an infinite number of sides. Post-assessment:
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Provide a verbal quiz: asking if certain shapes are area-maximizing shapes. This will reinforce Objective #1.
1 min
Ask students to write down what they have learned: One-Minute-Paper, (to be collected by the instructor) Closure
1 min
Summarize that
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1) The shape with x sides that maximizes enclosed area given fixed perimeter has sides of the same length (and same angles). 2) The shape that maximizes enclosed area given fixed perimeter is a circle. If there is additional time: 3) Tell the rest of the story: That to maximize the land, Dido chose a semi-circle and used the shore as one of the sides. (To be covered in another lesson). 4) Explain that this is a popular problem i n a branch of math known as calculus of variations, and give two examples: Path Planning and Brachistochrone. Supporting materials/resources -
Powerpoint Slides, with MATLAB animations. Whiteboard/Blackboard Graph Paper Rulers/Calculator
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