Dr. Sarah's Math 3610 Class Highlights
### Dr. Sarah's Math 3610 Class Highlights Spring 2006 Page

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 Jan 10** Fill out the
information sheet.
What is geometry? Since this course is aimed at future teachers,
why don't we work out of a high school geometry text?
Think about this, discuss with a partner, and
then report back to the class.
History of geometry including Egyptians, Babylonians, Chinese, and
Africans. Discuss Plato. Intoduction to the history of proofs and the
societal context within Greek society.
Introduction to inductive and deductive thinking
as methods for mathematical reasoning, teaching and learning.
Perry p. 5 number 1 (and its relationship to proof by induction).
Introduction to logic tables, two
column proofs and paragraph proofs.
Paragraph proofs continued via an introduction to minesweeper games as an
axiomatic system and resulting proofs.
Game 1
(Prove that B1 and B2 are numbers).
Handouts
Main web page,
Project Guidelines,
Checklist Points,
Sample Proofs.

**Thur Jan 12**
Students share something from the homework readings.
Review the concept of starting with axioms and givens and then proving things
with them (such as in the minesweeper games).
Finish
Game 1
proofs.
Minesweeper Game 2
For game 2, students are called on in random order to state and then
prove that a square is either a specific number or a bomb.
Minesweeper Game 3.
We contrast with game 2 and learn that even if squares cannot be
determined, knowing partial results can determine other squares.
Minesweeper Proofs.

**Tues Jan 17**
Fill in the index sheets.
Share something from the readings as we take attendance.
Review the concept of starting with axioms and givens and then proving things
with them (such as in the minesweeper games).
Intro to
Geometric Constructions. History of Euclid's elements.
Begin Euclid's Proposition 1.
Complete 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.

**Thur Jan 19**
Computer Lab Directions.
Go over Proposition 1 and 11 in Sketchpad. If you are done early, go over
the proof of Proposition 11 and be prepared to present it.
Use a paper folding argument for Proposition 11.
Built in Sketchpad feature for Proposition 11.
Build a right triangle in Sketchpad and investigate the Pythagorean Theorem.
Take out the Computer Directions Sheet and follow the directions to open the pre-made sketches that come with Sketchpad 4. Once you are in the Sketchpad folder, click on Samples, then on Sketches, then on Geometry and finally, open Pythagoras.gsp For future reference, I will write this as

Desktop/205Math(yourcomputersnumber)/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.

**Tues Jan 24**
Hand back the first project. Review the
Pythagorean Theorem - Euclid's historical proof and comparison with p. 8-9 in
Sibley which is a modern proof of Bhaskara's Proof p. 7 #10 from Project 1.
Discuss the benefits and difficulties of using the different methods,
including original historical sources. Begin
Worksheet on Archimedes and Cavalieri's Principle.

**Thur Jan 26** Go over the web links from the
Worksheet on Archimedes and Cavalieri's Principle.
Examine
SA
Review Euclid's Proof of the Pythagorean Theorem.
Intro to extensions of the Pythagorean Theorem including Pappus on Sketchpad,
the Scarecrow's Theorem, and
Fermat's Last Theorem.

**Tues Jan 31** Share from hw reading.
Discuss proofs, and consistency.
Consistency of axioms via minesweeper
examples (and non-examples) and Euclidean geometry and the historical
overview of Euclid's 5th postulate. Consistency does not imply uniqueness.
Hand out the Wile E assignment.
Go over the Scarecrow's Theorem, and
Fermat's Last Theorem. Begin
Nova's "The Proof" video

**Thur Feb 2** Finish "The Proof" video. Test 1.

**Tues Feb 7** Continue the notion of proof - Burden of Proof
activity, student even/odd function arguments.
Begin similarity. Read the proof of the
trig identity
and then fill in the details and reasons using similarity, trig and the
pythagorean theorem. If time remains, work on Sibley p. 55 number 6.

**Thur Feb 9**
Introduction to "same shape" via pictures.
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.
Work on Sibley p. 55 number 6. Come back together and discuss similarity
postulates. Applications of similarity:
Sliding a Ribbon Wrapped around a Rectangle
and Sliding a
Ribbon Wrapped around a Box.
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.

**Tues Feb 14**
Students present from Andrew Wiles worksheet and the hw readings.
Finish p. 214 Project 1, and the example on p. 212.
Axiomatic versus metric perspectives of Euclidean geometry and intro to
measurement. Work on
reservoir problems.
Work on models for p. 216 number 4 (Loggers).
Go over project 5 hints on WebCT.

**Thur Feb 16** Students work on the
reservoir
handout in groups of 2.
Go over the
proof that the perpendicular
bisectors are concurrent.
Introduction to taxicab geometry via moving in
Tivo,
play a few games of
Taxicab treasure hunt.

**Tues Feb 21**
Coordinate geometry and measurement versus axiomatic geometry.
US law is Euclidean. SAS in taxicab geometry.
Discuss taxicab circles and the relationship to the strategy for the game.
Highlight the possible number of intersections of taxicab circles
for different examples.
Example 1
Example 2.
Discuss the
Relationship to the NCTM standards.

**Thur Feb 23**
Taxicab Activities.

**Tues Feb 28** Parallelism

**Thur Mar 2**
Begin hyperbolic geometry.
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
Is parallel the same as equidistant in hyperbolic geometry?

Sketchpad Equidistant 1

Image of Equidistant 1

Sketchpad Equidistant 2

Image of Equidistant 2
Is Euclid's 5th postulate ever, always
or never true in hyperbolic space?

Sketchpad
Euclid's 5th Postulate

Image of Euclid's 5th
Postulate

From the Sketchpad 4 folder, open up Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp. We begin with hyperbolic geometry theorems.

Show that the existence part of Playfair's axiom works by constructing a parallel via perpendiculars. Measure alternate interior angles to see that they are approximately congruent. Then drag the parallel, changing the angle to show the uniqueness portion fails. Measure the alternate interior angles to see
that they are not congruent.

**Tues Mar 7**
Take questions.
Review Thursday's activities and look at the crochet model of hyperbolic
geometry.
Playfair's and Euclid's 5th in Euclidean and hyperbolic geometry.

**Thur Mar 9**
Test 2. Assign earth project and search for web references.

**Tues Mar 21**
Folding Presentations

**Thur Mar 23**
shortest distance
Review Playfair's and Euclid's 5th postulate, and
then discuss the
Hyperbolic Parallel Axiom: If m is a line and A is a point 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.

From the Sketchpad 4 folder, open up Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp
What is the sum of the angles in a hyperbolic triangle? How large
can the sum of the angles get? How small can the sum of the angles get?

Image of Sum of Angles
Escher worksheet
Is the Pythagorean theorem ever, always or never true in hyperbolic
geometry?

Image of
Hyperbolic
Pythagorean Thm Image

**Tues Mar 28** Earth presentations.

**Thur Mar 30** Continue going over
geometry of the earth problems,
using the child's balls and
dynamic geometry activities on the sphere:

1) Brad Findell's Elliptic/Spherical Toolkit for Sketchpad

2)
Walter Fendt's Java Applet

including why Playfair's is not the same as Euclid's 5th in spherical geometry (by relating this to the Euclidean proof that these are equivalent statements if we assume the first 28 propositions of Euclid), and
WHY SAS fails in spherical (compare to why it failed in taxicab geometry
and why it was true in Euclidean).

**Tues Apr 4**
Review Thursday's activities related to the
geometry of the earth problems and hand out
the review sheet. Continue with the sum of the
angles. Go over two Euclidean proofs and discuss what goes wrong in
spherical geometry. Then begin the
Beachball activity.

**Thur Apr 6** Presentations.

**Tues Apr 11**
Finish the
Beachball activity and then
discuss AAA on the sphere.
Discuss some of the problems from
pcmi
the Pythagorean Theorem and Problem 13 of
the geometry of the earth problems.
Go back to our Euclidean proofs and discuss what goes wrong in
spherical geometry.

**Thur Apr 13** Finish
geometry of the earth problems.
Take questions on the study guide for test 3.
Review axiomatic versus metric perspectives. Discuss "what is geometry."
Discuss modern geometry perspectives such as computational geometry,
algebraic geometry, and differential geometry, and related applications such
as the "shape of space." If time remains,
conduct web and library searches for final project.

**Thur Apr 20** Test 3

**Tues Apr 25** Students go over the test questions.
Take questions. Learning evaluations.