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.

• Tues Dec 9 Go over test 2. Abstract presentations. Course evaluations.
• Tues Dec 2 Finish Applications Go over response to the assignment: Choose a short (new to our class) 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.
Austin: http://www.cut-the-knot.org/Curriculum/Geometry/TangentTriangleToEllipse.shtml
Brett: http://nlvm.usu.edu/en/nav/category_g_4_t_3.html
Casey: http://www.cut-the-knot.org/Curriculum/Geometry/TangentTriangleToEllipse.shtml
Cayce: http://www.analyzemath.com/Geometry/properties_triangles.html
Darrell: "Two Trees.gsp" under Investigations
Dewey: http://www.members.shaw.ca/ron.blond/SimilarTriangles.APPLET/index.html
Emily: http://aleph0.clarku.edu/~djoyce/java/elements/usingApplet.html
Edgar: Soccer Ball Application
Katy: http://www.cut-the-knot.org/Curriculum/Geometry/HingedPythagoras2.shtml
Kimberly: http://www.frontiernet.net/~imaging/pythagorean.html
Lianna: http://faculty.evansville.edu/ck6/GIAJSP/EulerLine.html and http://aleph0.clarku.edu/~djoyce/java/Geometry/eulerline.html
Mandi: http://www.saltire.com/applets/simtri1/simtri1.htm
Toni: http://www.cut-the-knot.org/Curriculum/Geometry/HingedPythagoras2.shtml

Take questions on Test 2. If time remains, then search for references for the final project.

• Thur Dec 4 Test 2
• Tues Nov 25 Finish the Euclidean proof presentations. Discuss parallel projects. If time remains, discuss applications of hyperbolic geometry. How to Sew a 2-Holed Cloth Torus.
• Tues Nov 18 Lab directions

• Thur Nov 20 Review lab work via the pictures listed in the Lab directions and the Hyperbolic parallel axiom image and the Pythagorean theorem image. Begin the Euclidean proof presentations.
• Tues Nov 11 Begin hyperbolic geometry via the Escher worksheet.
• 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
Review our Euclidean proof that parallel means equidistant and discuss what goes wrong in hyperbolic geometry.

• Thur Nov 13 Prove Playfair's postulate in Euclidean geometry and examine the relationship with Euclid's 5th in spherical geometry and Euclidean geometry.
• Tues Nov 4 Taxicab activities in Sketchpad.

• Thur Nov 6 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. Begin parallels in Euclidean geometery and review Playfair's postulate as well as Euclid's 5th. Prove that parallel lines are equidistant.
• Tues Oct 28 Meet in 205. Finish using the triangles to examine the area of regular polygons on the sphere and Colorado and Wyoming. 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. If time remains, then begin taxicab activities in Sketchpad

• Thur Oct 30 Discuss metric perspectives and coordinate geometry and do the missing square activity. Review Minesweeper and create an inconsistent game. Fill in a partial game to show that consistency does not imply uniqueness. Discuss Godel's 1930 theorem. Review taxicab 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.
• Tues Oct 21 Project 5 timeline presentation sessions and peer and self-evaluation.

• Thur Oct 23 Finish presentations. Use the triangles to examine the area of regular polygons on the sphere. Discuss Colorado and Wyoming.
• Tues Oct 14 Sphere activity 1. Sphere activity 2. 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. AAA on the sphere.
• Thur Oct 7 Test 1. Work on Project 5.

• Thur Oct 9 [Of the five Platonic solids] So their combinations with themselves and with each other give rise to endless complexities, which anyone who is to give a likely account of reality must survey. [Plato, The Timaeus] Euclidean angle defect. applet 1 and applet 2. Begin measurement. Quotations from Archimedes. 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. Orange Activity and Archimedes polygonal method. Worksheet on Archimedes and Cavalieri's Principle.
• Tues Sep 30 Ask students to share their ideas about Wile - how did they ensure the chase would always begin? How did they ensure Wile would catch the RR when the RR runs faster? Burden of Proof. Begin Euler's formula and platonic solids. Show there are 5 convex regular polyhedra in Euclidean geometry, but additional polyhedra in spherical geometry (infinitely many).

• Thur Oct 2 Take questions on test 1. Go over proof from project 4. Review the platonic solids - and how to remember the faces and vertices (and from there calculate the edges using Euler's formula). Continue platonic solids.
• Tues Sep 23 Finish presentations. Discuss similarity postulates. Similarity of quadrilaterals. 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. 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.

• Thur Sep 25 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 Sep 16 Take questions. Nova's "The Proof" video and notes.

• Thur Sep 18 Andrew Wiles worksheet. 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).
• Tues Sep 9 Meet in 205. Discuss the homework readings. Review the paper folding argument for Proposition 11.
Go over an application - a proof that the perpendicular bisectors are concurrent.
Build a right triangle in Sketchpad and investigate the Pythagorean Theorem.
Go through Behold Pythagoras!, Puzzled Pythagoras, and then Shear Pythagoras. Click on Contents to get to the other Sketches.
Go through Euclid's proof. Discuss Sibley Geometric Viewpoint p. 7 # 10 on Project 2. If time remains, then an introduction to extensions of the Pythagorean Theorem including a review of the Greenwaldian Theorem, as well as the Scarecrow's Theorem, Pappus on Sketchpad.

• Thur Sep 11 Finish Pappus' Theorem. Continue extensions of the Pythagorean Theorem. Review the Greenwaldian Theorem, and examine the Scarecrow's Theorem. 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.
• Tues Sept 2 Project 1 Presentations. Peer review.

• Thur Sept 4 Collect self-evaluation for project 1. Review ASULearn solutions. Review Euclid's Proposition 1 in Sketchpad. Hand out folding arguments. Use a paper folding argument for Proposition 11. 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.
• Tues Aug 26 Fill out information sheet. Form groups of 2 or 3 people and prepare to come up to the front of the room and present a partner's
1) Name
2) Something that will help us remember them
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? Also discuss 8/08 article Cows Tend To Face North-South
Begin the Geometry of the Earth Project. Groups choose their top three problems and turn these in to Dr. Sarah.
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. History of Euclid's elements and the societal context of philosophy and debate within Greek society. Intro to Geometric Constructions. Begin Euclid's Proposition 1 by hand and by a proof.

• Thur Aug 28 Take questions on the syllabus. Students are called on in random order to state and then prove something about a specific square in game 2 of minesweeper. Euclid's Proposition 1 in Sketchpad. At 4pm go to 205. Students complete Proposition 1 in Sketchpad and then work on project 1.