Dr. Sarah's Math 3610 Class Highlights

Dr. Sarah's Math 3610 Class Highlights
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.

  • Mon July 28 Discuss parallel projects. Choose a short Euclidean geometry proof related to content in one of the two books [the proof does not need to be in the book]. Present the proof in your own words on the blackboard and in modern language in 5 minutes or less. This is an individual assignment.

  • Tues July 29 Discuss study guide for test 2. Choose a short Euclidean Sketchpad exploration, web applet, or computer exploration related to Euclidean geometry, and be prepared to present it. Be sure to put it in context by discussing why it is interesting or important.

    Amy: http://www.oswego.org/ocsd-web/games/bananahunt/bhunt.html
    Douglas: http://www.shodor.org/interactivate/activities/Angles/?version=1.5.0_13&browser=Mozilla&vendor=Apple_Inc.
    Jessica: http://www.frontiernet.net/~imaging/pythagorean.html
    Karen: http:nlvm.usu.edu/en/nav/vlibrary.html,
    http://www.hanssoft.com/html/games.html, and
    Katelin: http://www.saltire.com/gallery.html
    Kimberly: http://www.mathsnet.net/dynamic/pythagoras/index.html and
    Sasha: http://mathforum.org/alejandre/applet.polyhedra.html
    Spencer: http://www.cut-the-knot.org/pythagoras/FaultyPythPWW.shtml and

  • Wed July 30 Test 2

  • Thur July 31 Go over test 2. Discuss applications of hyperbolic geometry. How to Sew a 2-Holed Cloth Torus Oral abstract presentations. Course evaluations. Work on final project.

  • Fri Aug 1 Poster sessions. Peer and self-evaluation
  • Mon July 21 Timeline assignment poster sessions and peer and self-evaluation.

  • Tues July 22 Discuss taxicab circles and the relationship to the strategy of the game. Highlight the possible number of intersections of taxicab circles for different examples. US law is Euclidean. SAS in taxicab geometry. Sketchpad Activities

  • Wed July 23 Finish Sketchpad activities. Begin parallels in Euclidean geometery and review Playfair's postulate as well as Euclid's 5th. Compare with spherical geometry where Euclid's 5th holds, but Playfair's does not. Prove that parallel lines are equidistant.

  • Thur July 24 Review parallels. Prove Playfair's postulate and examine the relationship with Euclid's 5th in spherical geometry and Euclidean geometry. Begin hyperbolic geometry via the Escher worksheet. 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?
    Sketchpad Shortest Distance Paths
    Image of Shortest Distance Paths
  • Is parallel the same as equidistant in hyperbolic geometry?
    Sketchpad Equidistant 1
    Image of Equidistant 1
    Sketchpad Equidistant 2
    Image of Equidistant 2

  • Fri July 25 Discuss parallels in both books. Review hyperbolic geometry. Examine the crochet model.
  • Is Euclid's 5th postulate ever, always or never true in hyperbolic space?
    Sketchpad Euclid's 5th Postulate
    Image of Euclid's 5th Postulate
    From the Sketchpad 4 folder, open up Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp.
    Show that the existence part of Playfair's axiom works: Create a hyperbolic segment by constructing a parallel via perpendiculars. Measure the alternate interior angles on both sides to see that they are not congruent. Hence the uniqueness part of Playfair's does not hold, and in fact, we can create infinitely many parallels through a given point to a given line. In addition, our proof of the sum being 180 degrees will also fail because the alternate interior angles are not congruent on both sides.
    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.
    Investigate the Pythagorean Theorem in hyperbolic geometry.
    Discuss the model of hyperbolic geometry using hexagons and heptagons - a hyperbolic soccer ball.
  • Mon July 14 Continue Similarity. Read the proof of the trig identity and then fill in the details and reasons using similarity, trig and the Pythagorean theorem. Note that the Pythagorean theorem is a consequence of similarity as in Project 4. 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. Work on models for p. 216 number 4 (Loggers).

  • Tues July 15 Burden of Proof. Platonic Solids - showing there are 5 convex regular polyhedra in Euclidean geometry, but additional polyhedra in spherical geometry (infinitely many). Euclidean angle defect. applet 1 and applet 2

  • Wed July 16 Measurements with and without metric perspectives. How were circumference, area and volume formulas obtained via axiomatic perspectives and before coordinate geometry and calculus II? Orange Activity and Archimedes polygonal method. Worksheet on Archimedes and Cavalieri's Principle. Sphere activity 1. Sphere activity 2.

  • Thur July 17 Take questions on the first test. Discuss polygons and polyhedra in both texts. Review Sphere activity 1 and examine consequences, including whether the difference between the angle sum and pi is detectable for a 1 mile square area triangle in Kansas. Use the triangles to examine the area of regular polygons on the sphere. Discuss Colorado and Wyoming. AAA on the sphere. Discuss folding activities of the sum of the angles in a triangle is 180 degrees. Discuss a proof using Euclidean axioms. Discuss what goes wrong on the sphere. Discuss metric perspectives and coordinate geometry and do the missing square activity. Begin Reservoir problems.

  • Fri July 18 Test 1. Reservoir problems. Go over the proof that the perpendicular bisectors are concurrent. Begin taxicab geometry via moving in Tivo, and play a few games of taxicab treasure hunt.
  • Mon July 7 Fill out information sheet. Form groups of 2 or 3 people and prepare to present a partner's
    1) Name
    2) Something that will help us remember them
    3) Something that you both have in common other than this course or your interest in mathematics
    Next discuss how can we tell the earth is round without technology?
    Mention the related problem on Project 2 for Friday [Wallace and West Roads to Geometry 1.1 8].
    Where is North?
    Begin the Geometry of the Earth Project. Groups choose their top three problems and turn these in to Dr. Sarah. Next, read through the handout.
    Induction versus deduction. An introduction to minesweeper games as an axiomatic system.
    Axiom 1) Each square is a number or a mine.
    Axiom 2) A numbered square represents the number of neighboring mines in the blocks immediately above, below, left, right, or diagonally touching.
    Examine game 1. Students are called on in random order to state and then prove something about a specific square in game 2. History of Euclid's elements and the societal context of philosophy and debate within Greek society. Go to the computer lab in 205. Intro to Geometric Constructions. Begin Euclid's Proposition 1. Time at the end of class to work on Project 1.

  • Tues July 8 Project 1 Presentations. Review Euclid's Proposition 1. Hand out folding arguments. Use a paper folding argument for Proposition 11. Begin Euclid's Book 1 Proposition 11.

  • Wed July 9 Collect homework and discuss project 1 solutions. Finish 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.
    Go over an application - a proof that the perpendicular bisectors are concurrent.
    Build a right triangle in Sketchpad and investigate the Pythagorean Theorem.
    Go to 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.
    Return to the classroom.
    Go through Euclid's proof. Discuss Sibley Geometric Viewpoint p. 7 # 10 on Project 2. Introduction to extensions of the Pythagorean Theorem including Pappus in Sketchpad, a review of the Greenwaldian Theorem, as well as the Scarecrow's Theorem.

  • Thur July 10 Discuss the homework readings. Go over images and quotations. Highlight that the Yale tablet is Sibley The Geometric Viewpoint 1.1 3 and The 'hsuan-thu' [Zhou Bi Suan Jing] is similar to Bhaskara's diagram in Sibley The Geometric Viewpoint 1.1 10, and the connection of Eratosthenes to Wallace and West Roads to Geometry 1.1 8. Fermat's Last Theorem. Nova's "The Proof" video notes. Begin worksheet on Andrew Wiles and Proof.

  • Fri July 11 Finish worksheet on Andrew Wiles and Proof. A second example. Begin similarity. Introduction to "same shape". 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. Groups prepare short presentations on SSS, SAS, AA, SSA, AAS, ASA, HL (Hypotenuse and leg of a right triangle - ie SSA in a right triangle). Discuss similarity postulates. Look at a proof of SAS and discuss what goes wrong on the sphere for large triangles. Applications of similarity: Sibley The Geometric Viewpoint p. 55 number 6. Sliding a Ribbon Wrapped around a Rectangle and Sliding a Ribbon Wrapped around a Box. Similarity of quadrilaterals.