Dr. Sarah's Math 3610 Class Highlights
Dr. Sarah's Math 3610 Class Highlights
Tues Apr 28
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.
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/anglebisection.html Katie Mullen
http://www.ies.co.jp/math/java/misc/oum/oum.html Kristen Johnson
two trees sketchpad exploration Kristen Eure
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.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
rainbow file from sketchpad Laura
Sketchpad Two Trees.gsp Chris
Sketchpad twotrees.gsp Rebecca
http://www.math.psu.edu/dlittle/java/geometry/euclidean/goldenratio.html Becca Horn
Take questions on test 2.
Continue applications of hyperbolic geometry.
How to Sew a 2-Holed
Thur Apr 16 Finish presentations.
image. hyperbolic activities.
Begin applications of hyperbolic geometry.
Tues Apr 7
Begin hyperbolic geometry via the
Save each Sketchpad file (control/click and then download it to the
documents folder) and then open it up from Sketchpad and follow the
What are the shortest distance paths in hyperbolic geometry?
Image of Shortest
Thur Apr 9 Review parallels in
hyperbolic geometry via
sometimes but not always holds, and
the existence of Playfair's via perpendiculars,
I-29 doesn't hold we obtain
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
crochet model, and
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
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
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.
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.
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 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
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.
A second example.
Begin similarity. Introduction to "same shape".
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
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.
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.
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
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
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?
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
Begin the Geometry of the Earth Project.
Groups choose their top four problems.
Induction versus deduction. An introduction to minesweeper games as an
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
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.