User:Heather
From Prep08Wiki
Hello,
I am attending this course to develop a more interactive geometry course for my students.
I teach at McKendree University in Lebanon, IL.
I am a topologist and I study virtual knot theory.
Contents |
My goals for this seminar:
I teach a junior level geometry class for future math teachers - grade school and high school. I estimate that my average student is at Van Hiele level 2. Based on asking them about their geometry experience, they have no (or minimal) experience with geometry constructions from high school.
I hope to develop content for a block based course. I hope to select about 8 topics and set up a sequence of activities- interactive (intuitive), calculational, and then focus on proofs at the end of the block.
Tenative blocks: definitions, constructions, triangles, quadrilaterals, circles, transformations, similarity, area and volume
The course will have a portfolio requirement - the idea being at the end of the semester that the students will have a notebook with activities, definitions, etc that they can refer to in the future.
It would be good to develop a survey that I can administer through blackboard at the end of each block.
I would like to have the opportunity to discuss my course plans, goals, and materials during this seminar.
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Project Thoughts
This would be a good end of semeter project for my students. Most are teachers, so that drawing tessellations would likely be a student activity. I would like to set it up so that the students can use their knowledge.
1. Select a regular tessellation. 2. Modify the edges based using the following symmetries: a) reflection b) translation c) rotation d) glide
3. Select one of your tessellations and embellish it.
Paper:
3 pages Describe the 3 regular tessellation, write proof sketch of why only three.\
Identify and discuss the lines of symmetry in your selected tessellation. Show how you used these lines to create your embellished tessellation. Discuss which lines of symmetry are in your tessellation.
Portfolio Grading Rubric
I'm trying to create a list of grading criteria for my portfolio criteria: 1. Organization 2. Completeness (all items present) 3. Mathematical correctness
Any other suggestions are welcome.
Course Planning
Text: Michael Serra's Discovering Geometry and Patty Paper Geometry
Goal statement: This course is designed to prepare you to teach geometry and will include historical notes and in-class activities. We will approach geometry through the following methods: 1. Manipulative activities that will help you develop you geometry intuition. 2. Calculations that will give you experience problem solving with geometry. 3. Proof writing and the axiomatic method.
We will use a portfolio to track our development.
Portfolio Requirements: Block 1- Definitions
Reading: Chapter 1 and Chapter 13.1 Definitions: Define the following terms. Include labeled figures as needed. 1. Definition 2. Axiom 3. Congruence 4. Midpoint 5. Ray 6. Angle 7. Angle bisector 8. Complementary angles 9. Vertical angles 10. Supplementary angles 11. Vertical angles 12. Linear pair of angles 13. Regular polygons 14. Equilateral triangle 15. Trapezoid 16. Rhombus 17. Rectangle 18. Square 19. Chord 20. Tangent
Questions: 1. Why are the terms line, point, and plane usually left undefined? 2. What is wrong with the definition: A rectangle is a figure with parallel sides. 3. Write a brief biography about Euclid.
Patty Paper: Find a method to do the following using the patty paper. Write down each step. 1. Construct a line through two points. 2. Determine if two line segments (on the same piece of paper) are congruent. 3. Determine if two angles (on the same piece of paper) are congruent 4. Fold two intersecting lines on a piece of paper. Which of the angles form linear pairs? Which angles form vertical angles? 5. How do you construct congruent angles or line segments? Calculations Page 92: #8, 30, 42,44,48 Page 131 # 7 Page 142 # 23,24
Proof Samples: 1. Flow chart proof example 2. Two column proof example 3. Paragraph proof example
Portfolio Requirements: Block 2 – Constructions
Reading: Chapter 3, Chapter 13.2
Definitions: Define the following terms. Use labeled figures as needed. 1. Concurrence 2. Circumcenter 3. Incenter 4. Orthocenter 5. Medians 6. Centroid 7. Altitudes 8. Orthocenter
Questions: 1. Who is famous for this quote: “Instead of points, lines, and planes, one must be able to say at all times tables, chairs, and beer mugs.” What did the speaker mean? 2. What is the difference between draw, sketch, and construct? 3. What is the difference between absolute (or neutral) geometry, Euclidean, Hyperbolic and spherical geometries?
Patty Paper:
Find a method to do the following using the patty paper. Write down each step.
1. Construct an angle bisector.
2. Construct the perpendicular bisector of a line segment
3. Construct the perpendicular from a given point to a given line.
4. Construct the perpendicular through a given point on a line
5. Find a line parallel to a given line through a given point.
6. Construct the circumcenter of a triangle. What seems to be true about the circumcenter?
7. Construct the incenter of a triangle. What do you notice about the incenter?
8. Construct the medians of a triangle. What do you notice about the medians?
9. Contruct the altitudes of the triangle. Where is the orthocenter located? For an acute triangle, obtuse triangle, and a right triangle?
Calculations
Page 183: 12,13
Proofs: 1. What are the postulates/axioms? Compare and contrast from text vs. other text. 2. Flow Chart proofs page 700: 20-22 3. Kay – unique betweeness page 102 #14
Portfolio Requirements: Block 3 – Triangles
Reading: Chapter 4, Chapter 13.3
Definitions: Define the following terms. Use labeled figures as needed. 1. State the parallel postulate. 2. Triangle inequality conjecture 3. Side angle conjecture 4. Exterior angle conjecture 5. SAS axiom 6. SSS, ASA congruences 7. SAA
Questions:
1. Examine the explanation of why the triangle sum is always 180 degrees. Can you prove this in absolute geometry? Why or why not?
2. Why are SSA and AAA not triangle congruences?
Patty Paper: Find a method to do the following using the patty paper. Write down each step. 1. SSS investigation 2. AAA investigation 3. SAS 4. ASA 5. SAA 6. SSA 7. Isoceles Triangle conjecture
Calculations Page 218 # 2,4,5,9,12,14,17,20 Page 214 #3, 4,5,6,10,23 Page 223 #10,17 Page 233 # 2,5,7,9 Page 241 # 1,4 Proofs:
1. Prove SSS from the SAS axiom. Use flow chart proof methods. 2. Prove ASA from the SAS axiom. Use flow chart proof methods. 3. Prove: If M is the midpoint of segment AB and line PM is perpendicular to AB the PA is congruent to PB. 4. Prove: If PA is congruent to PB and M is the midpoint of segment AB the line PM is perpendicular to segment AB. 5. Select 1 of the above and write as a paragraph proof.
Portfolio Requirements: Block 4 – Quads
Reading: Chapter 5, Section 13.4, and section 2.6
Definitions: Define the following terms and state the theorems. Use labeled figures as needed. 1. Adjacent 2. Opposite 3. Diagonal 4. Kite diagonals theorem. 5. Isoceles Trapezoid theorem. 6. Triangle Midsegment Theorem 7. Corresponding angles 8. Alternate interior angles 9. Alternate exterior angles
Questions: 1. When are two quadrilateral congruent? Is SSSS a sufficient condition? Why or why not? 2. What are the congruence conditions for quadrilaterals? 3. What is the Sacheri Quadrilateral?
Patty Paper: Do the following explorations using the patty paper. Write down each step. 1. Parallelogram Properties 2. Rhombus properties 3. Rectangle properties 4. Kite properties 5. Polygon sum 6. Triangle midsegment 7. Trapezoid midsegment conjecture
Calculations
Page 304 # 10,13
Proofs:
1. Prove that the diagonals of a parallelogram bisect each other
2. Prove that SASAS is a congruence for quadrilaterals.
3. Argue that the parallel sides of a parallelogram are congruent.
