Dr. Sarah's Math 3610 Class Highlights
Dr. Sarah's Math 3610 Class Highlights
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
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 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
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
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
US law is Euclidean. SAS in taxicab geometry.
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
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
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?
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
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
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
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.
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
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.
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
Fri July 18
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.
Mon July 7
Fill out information sheet.
Form groups of 2 or 3 people and prepare to present a partner's
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?
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
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
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
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
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
Sibley The Geometric Viewpoint 1.1 10, and
the connection of Eratosthenes to Wallace and West Roads to Geometry
Last Theorem. Nova's "The Proof" video
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".
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.