Dr. Sarah's Math 3610 Class Highlights

### Dr. Sarah's Math 3610 Class Highlights Spring 2006 Page The following is NOT HOMEWORK unless you miss part or all of the class. See the Main Class Web Page for ALL homework and due dates.

• Tues Jan 10 Fill out the information sheet. What is geometry? Since this course is aimed at future teachers, why don't we work out of a high school geometry text? Think about this, discuss with a partner, and then report back to the class. History of geometry including Egyptians, Babylonians, Chinese, and Africans. Discuss Plato. Intoduction to the history of proofs and the societal context within Greek society. Introduction to inductive and deductive thinking as methods for mathematical reasoning, teaching and learning. Perry p. 5 number 1 (and its relationship to proof by induction). Introduction to logic tables, two column proofs and paragraph proofs. Paragraph proofs continued via an introduction to minesweeper games as an axiomatic system and resulting proofs. Game 1 (Prove that B1 and B2 are numbers). Handouts Main web page, Project Guidelines, Checklist Points, Sample Proofs.

• Thur Jan 12 Students share something from the homework readings. Review the concept of starting with axioms and givens and then proving things with them (such as in the minesweeper games). Finish Game 1 proofs. Minesweeper Game 2 For game 2, students are called on in random order to state and then prove that a square is either a specific number or a bomb. Minesweeper Game 3. We contrast with game 2 and learn that even if squares cannot be determined, knowing partial results can determine other squares. Minesweeper Proofs.
• Tues Jan 17 Fill in the index sheets. Share something from the readings as we take attendance. Review the concept of starting with axioms and givens and then proving things with them (such as in the minesweeper games). Intro to Geometric Constructions. History of Euclid's elements. Begin Euclid's Proposition 1. Complete Euclid's Book 1 Proposition 11. Go over Sketchpad's built in version of Proposition 11 as well as a ray versus a line in Sketchpad.

• Thur Jan 19 Computer Lab Directions. Go over Proposition 1 and 11 in Sketchpad. If you are done early, go over the proof of Proposition 11 and be prepared to present it. Use a paper folding argument for Proposition 11. Built in Sketchpad feature for Proposition 11. Build a right triangle in Sketchpad and investigate the Pythagorean Theorem. Take out the Computer Directions Sheet and follow the directions to open the pre-made sketches that come with Sketchpad 4. Once you are in the Sketchpad folder, click on Samples, then on Sketches, then on Geometry and finally, open Pythagoras.gsp For future reference, I will write this as
Desktop/205Math(yourcomputersnumber)/Applications/Sketchpad/ Samples/Sketches/Geometry/Pythagoras.gsp Go through Behold Pythagoras!, Puzzled Pythagoras, and then Shear Pythagoras. Click on Contents to get to the other Sketches.
• Tues Jan 24 Hand back the first project. Review the Pythagorean Theorem - Euclid's historical proof and comparison with p. 8-9 in Sibley which is a modern proof of Bhaskara's Proof p. 7 #10 from Project 1. Discuss the benefits and difficulties of using the different methods, including original historical sources. Begin Worksheet on Archimedes and Cavalieri's Principle.

• Thur Jan 26 Go over the web links from the Worksheet on Archimedes and Cavalieri's Principle. Examine SA Review Euclid's Proof of the Pythagorean Theorem. Intro to extensions of the Pythagorean Theorem including Pappus on Sketchpad, the Scarecrow's Theorem, and Fermat's Last Theorem.
• Tues Jan 31 Share from hw reading. Discuss proofs, and consistency. Consistency of axioms via minesweeper examples (and non-examples) and Euclidean geometry and the historical overview of Euclid's 5th postulate. Consistency does not imply uniqueness. Hand out the Wile E assignment. Go over the Scarecrow's Theorem, and Fermat's Last Theorem. Begin Nova's "The Proof" video

• Thur Feb 2 Finish "The Proof" video. Test 1.
• Tues Feb 7 Continue the notion of proof - Burden of Proof activity, student even/odd function arguments. Begin similarity. Read the proof of the trig identity and then fill in the details and reasons using similarity, trig and the pythagorean theorem. If time remains, work on Sibley p. 55 number 6.

• Thur Feb 9 Introduction to "same shape" via pictures. Fig 8.4 Fig 8.21 Fig 8.32 Use the Triangle_Similarity.gsp file (control click and save the file. Then open it from Sketchpad) to complete the Similar Triangles - SSS, SAS, SSA worksheet. Work on Sibley p. 55 number 6. Come back together and discuss similarity postulates. Applications of similarity: Sliding a Ribbon Wrapped around a Rectangle and Sliding a Ribbon Wrapped around a Box. Introduction to geometric similarity and its application to geometric modeling via Mathematics Methods and Modeling for Today's Mathematics Classroom 6.3. Go over p. 214 Project 1, and the example on p. 212.
• Tues Feb 14 Students present from Andrew Wiles worksheet and the hw readings. Finish p. 214 Project 1, and the example on p. 212. Axiomatic versus metric perspectives of Euclidean geometry and intro to measurement. Work on reservoir problems. Work on models for p. 216 number 4 (Loggers). Go over project 5 hints on WebCT.

• Thur Feb 16 Students work on the reservoir handout in groups of 2. Go over the proof that the perpendicular bisectors are concurrent. Introduction to taxicab geometry via moving in Tivo, play a few games of Taxicab treasure hunt.
• Tues Feb 21 Coordinate geometry and measurement versus axiomatic geometry. US law is Euclidean. SAS in taxicab geometry. Discuss taxicab circles and the relationship to the strategy for the game. Highlight the possible number of intersections of taxicab circles for different examples. Example 1   Example 2. Discuss the Relationship to the NCTM standards.

• Thur Feb 23 Taxicab Activities.
• Tues Feb 28 Parallelism

• Thur Mar 2 Begin hyperbolic geometry. Save each Sketchpad file (control/click and then download it to the documents folder) and then open it up from Sketchpad and follow the directions.
• What are the shortest distance paths in hyperbolic geometry?
Image of Shortest Distance Paths
• Is parallel the same as equidistant in hyperbolic geometry?
Image of Equidistant 1
Image of Equidistant 2
• Is Euclid's 5th postulate ever, always or never true in hyperbolic space?
Image of Euclid's 5th Postulate
From the Sketchpad 4 folder, open up Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp. We begin with hyperbolic geometry theorems.
Show that the existence part of Playfair's axiom works by constructing a parallel via perpendiculars. Measure alternate interior angles to see that they are approximately congruent. Then drag the parallel, changing the angle to show the uniqueness portion fails. Measure the alternate interior angles to see that they are not congruent.
• Tues Mar 7 Take questions. Review Thursday's activities and look at the crochet model of hyperbolic geometry. Playfair's and Euclid's 5th in Euclidean and hyperbolic geometry.

• Thur Mar 9 Test 2. Assign earth project and search for web references.
• Tues Mar 21 Folding Presentations

• Thur Mar 23 shortest distance Review Playfair's and Euclid's 5th postulate, and then discuss the Hyperbolic Parallel Axiom: If m is a line and A is a point not on m, then there exist exactly two noncollinear halflines AB and AC which do not intersect m and such that a third halfline AD intersects m if and only if AD is between AB and AC.
• What is the sum of the angles in a hyperbolic triangle? How large can the sum of the angles get? How small can the sum of the angles get?
Image of Sum of Angles
• Escher worksheet
• Is the Pythagorean theorem ever, always or never true in hyperbolic geometry?
Image of Hyperbolic Pythagorean Thm   Image
• Tues Mar 28 Earth presentations.

• Thur Mar 30 Continue going over geometry of the earth problems, using the child's balls and dynamic geometry activities on the sphere: