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
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.
Darrell: "Two Trees.gsp" under Investigations
Edgar: Soccer Ball Application
Lianna: http://faculty.evansville.edu/ck6/GIAJSP/EulerLine.html and
Robby: Applications/Sketchpad/samples/sketches/geometry/Fractal Gallery.gsp
Take questions on Test 2. If time remains, then search for
references for the final project.
Thur Dec 4
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
Tues Nov 18 Lab directions
Thur Nov 20 Review lab work via the pictures listed
in the Lab directions
Hyperbolic parallel axiom image and
the Pythagorean theorem
image. Begin the
Euclidean proof presentations.
Tues Nov 11
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
Is parallel the same as equidistant in hyperbolic geometry?
Sketchpad Equidistant 1
Image of Equidistant 1
Sketchpad Equidistant 2
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
Go over the proof that the perpendicular bisectors are
Begin taxicab geometry via moving in Tivo, and play a few games of
taxicab treasure hunt. If time remains, then begin
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
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.
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.
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.
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
Continue platonic solids.
Tues Sep 23
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
Thur Sep 18
Andrew Wiles worksheet. A second example.
Introduction to "same shape".
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
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
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.
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 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
2) Something that will help us remember them
Next 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 three problems and turn these in to Dr. Sarah.
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 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.