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 Apr 28
    http://oneweb.utc.edu/~Christopher-Mawata/geom/geom1.htm Katie F
    Go over test 2
    Oral abstract presentations
  • Tues Apr 21 Chris' proof. Present Euclidean computer explorations:
    http://www.math.psu.edu/dlittle/java/geometry/euclidean/reflection.html Caitlin
    http://www.mit.edu/~ibaran/kseg.htmlJessica Placke
    http://www.math.psu.edu/dlittle/java/geometry/euclidean/anglebisection.html Katie Mullen
    http://www.ies.co.jp/math/java/misc/oum/oum.html Kristen Johnson
    two trees sketchpad exploration Kristen Eure
    http://aleph0.clarku.edu/~djoyce/java/Geometry/eulerline.html Olivia
    http://www.mathopenref.com/constcirclecenter.html Rose
    http://www.ies.co.jp/math/java/misc/oum/oum.html Sarah Gilliam
    http://www.saltire.com/applets/advanced_geometry/monthly_executable/monthly.htm Sarah Ploeger
    http://www.math.psu.edu/dlittle/java/geometry/euclidean/parallelline.html Tony
    http://www.cut-the-knot.org/Curriculum/Geometry/TangentTriangleToEllipse.shtml Lissa
    http://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/BOL13.shtml#explanation Melissa
    http://aleph0.clarku.edu/~djoyce/java/Geometry/Geometry.html Cati
    http://www.cut-the-knot.org/Curriculum/Geometry/TangentTriangleToEllipse.shtml Candace
    http://www.ies.co.jp/math/products/geo2/applets/pytree/pytree.html and http://www.math.psu.edu/dlittle/java/geometry/euclidean/goldenratio.html and http://jwilson.coe.uga.edu/emt669/student.folders/may.leanne/leanne's%20page/golden.ratio/golden.ratio.html Alana
    http://www.math.psu.edu/dlittle/java/geometry/euclidean/goldenratio.html Brice
    http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI45.html Megan
    rainbow file from sketchpad Laura
    http://www.mathopenref.com/constdividesegment.html Samantha
    Sketchpad Two Trees.gsp Chris
    http://home.gna.org/geoproof/ Erica
    Sketchpad twotrees.gsp Rebecca
    http://www.math.psu.edu/dlittle/java/geometry/euclidean/goldenratio.html Becca Horn
    http://www.cut-the-knot.org/Curriculum/Geometry/TangentTriangleToEllipse.shtml Angela
    Take questions on test 2. Continue applications of hyperbolic geometry. How to Sew a 2-Holed Cloth Torus.
  • Thur Apr 16 Finish presentations. Pythagorean theorem image. hyperbolic activities. Begin applications of hyperbolic geometry.
  • Tues Apr 7 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.
    Begin hyperbolic activities.

  • Thur Apr 9 Review parallels in hyperbolic geometry via Euclid's 5th sometimes but not always holds, and the existence of Playfair's via perpendiculars, but since I-29 doesn't hold we obtain infinitely many parallels to any line through a given point. The Hyperbolic Parallel Axiom states that if m is a hyperbolic line and A is a point not on m, then there exist exactly two noncollinear hyperbolic halflines AB and AC which do not intersect m and such that a third hyperbolic halfline AD intersects m if and only if AD is between AB and AC. How can we make sense of this axiom? Axiom pdf Then discuss models and pictures, including the hyperbolic internet, crochet model, and reef. Sketchpad file.
    Begin Euclidean proof presentations.
  • Tues Mar 31 Taxicab activities in Sketchpad. Playfair's Postulate in Euclidean geometry - constructing a parallel by using perpendiculars. Which Euclidean proposition are we using? Why are the lines parallel?

  • Thur Apr 2 Review activities from last Thursday. Discuss what goes wrong with the proof of I-32 on the sphere. Discuss what goes wrong with Playfair's on the sphere. Continue parallels in Euclidean geometry and review Playfair's postulate as well as Euclid's 5th. Prove that Euclid's 5th Postulate plus Euclid's other axioms before I-31 prove Playfair's. Euclid's 5th Postulate is vacuously true on the sphere so unlike what is listed on the web and in books, the statements are different. Review parallel ideas including same side interior angles being supplementary, alternate interior and corresponding angles being the same, equidistant lines, etc.
  • Tues Mar 24 Finish presentations. Discuss metric perspectives and coordinate geometry. Water Reservoir Problems. Review the proof that the perpendicular bisectors are concurrent. Play a few games of taxicab treasure hunt.

  • Thur Mar 26 Introduce taxicab geometry via moving in Tivo and relate to taxicab treasure hunt. Highlight the possible number of intersections of taxicab circles for different examples. US law is Euclidean. SAS in taxicab geometry. Euclid's proof of SAS and what goes wrong in taxicab geometry. 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. Discuss various ideas of parallel. Use a folding argument to show that parallel implies the sum of the angles in a triangle is 180 degrees, and then complete a Euclidean proof of I-32.
  • Tues Mar 17 Timeline presentations and peer and self-evaluations.

  • Thur Mar 19 Continue presentations.
  • Tues Mar 3 Finish Archimedes and Cavalieri's Principle. Sphere activity 1. Sphere activity 2. Sphere activity 1 and examine consequences, including AAA on the sphere implying congruence.

  • Thur Mar 5 Consequences for the formula for the area of a spherical triangle - 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. The missing square activity.
  • Tues Feb 24 Test 1

  • Thur Feb 26 [Of the five Platonic solids - the earth was associated with the cube, air with the octahedron, water with the icosahedron, fire with the tetrahedron, and dodecahedron as a model for the universe.] 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] Review the proof that there are only 5 regular Platonic Solids and discuss why there are infinitely many on the sphere. Euclidean angle defect. Nets 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. Archimedes and Cavalieri's Principle.
  • Tues Feb 17 Ask students to share their ideas about Wile - how did they ensure the chase would always begin? That he would continue to see him? How did they ensure Wile would catch the RR when the RR runs faster? Burden of Proof. Each group builds a model of a polyhedra and presents V, E, and F as well as a way to help remember the name. Euler's formula.

  • Thur Feb 19 Take questions on test 1. 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 Feb 10 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 Feb 12 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 Feb 3 Nova's "The Proof" video. Notes.

  • Thur Feb 5 Andrew Wiles and Proof. Henderson A second example. Begin similarity. Introduction to "same shape". Fig 8.4 Fig 8.21 Fig 8.32 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). 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.

  • Tues Jan 27 Discuss the homework readings. Go over Sketchpad's built in version of Proposition 11 as well as a ray versus a line in Sketchpad. Use a 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 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.
    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 Pappus on Sketchpad.

  • Thur Jan 29 Finish Pappus. A review of the Greenwaldian Theorem, as well as the Scarecrow's Theorem. 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 Jan 20 Project 1 Presentations and Peer Review.

  • Thur Jan 22 Review Project 1 solutions. Discuss suggestions from last semester. Review equilateral triangle construction. Replicate the construction in Spherical Easel and compare with the Euclidean proof. Go to 209b after 4pm and each student does the Euclidean construction of proposition 1. Work on proposition 11.
  • Tues Jan 12 Fill out information sheet. Form groups of 3 people and 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 four problems.
    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 Jan 15 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. Use a paper folding argument for Proposition 11.