Dr. Sarah's Math 3610 Class Highlights

### Dr. Sarah's Math 3610 Class Highlights Fall 2004 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.

• Mon Dec 6 Speedbump cartoon. Students go over the test questions. If time remains, then discuss geometry of the universe
• Mon Nov 29 Discuss the presentations. Discuss the end of the worksheet. Continue going over geometry of the earth problems. After problem 13, go back to Problem 11 (Pythagorean Thm). Review our Euclidean proofs and discuss what goes wrong in spherical geometry. Problem 14, and then 15.

• Wed Dec 1 Finish up problem 15. Speedbump cartoon Then discuss the final project and the last test. If time remains then search for "what is geometry."

• Fri Dec 3 Test 4
• Mon Nov 22 Sketchpad presentations.
• Mon Nov 15 Sketchpad test.

• Wed Nov 17 Continue going over geometry of the earth problems, including 11-12 using
• Dynamic Geometry activities on the sphere
2) Walter Fendt's Java Applet
and continuing with the other probelms as time allows.

• Fri Nov 19 Go back to problem 11 via beachball activity
• Mon Nov 8 Finish presentations and go over problems 1-4 on the geometry of the earth.

• Wed Nov 10 From the Sketchpad 4 folder, open up Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp and discuss the other folders and files as possibilities for short Sketchpad presentations. Continue going over problems on the geometry of the earth.
• Dynamic Geometry activities on the sphere
2) Walter Fendt's Java Applet

• Fri Nov 12 Review Problems 1-8, 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). Continue going over the geometry of the earth problems, including 9-12, and WHY SAS fails in spherical (compare to why it failed in taxicab geometry and why it was true in Euclidean).
• Mon Nov 1 Go over project 6 and paper folding activities. Discuss class readings. 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.
• 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.

• Wed Nov 3 Review Monday's activities. 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

• Fri Nov 5 Presentations on the geometry of our earth
• Mon Oct 25 Review hyperbolic geometry activities from class on Thursday. Go over the proof that equidistant is the same as parallel in Euclidean geometry, and discuss what goes wrong in hyperbolic geometry. Discuss Playfair's postulate in Euclidean and hyperbolic geometry. Use Prop 11, 12 and 27 of Euclid in order to prove the existence part of Playfair's axiom for Euclidean geometry. Discuss the fact that the existence part of Playfair's still works in hyperbolic geometry, but that we obtain infinitely many parallels (we'll see this in lab), because we do not need the full strength of Prop 27 (where the angles have to be equal) in order to have non-intersecting lines. Instead, there are many different combinations of angles that result in non-intersecting lines. If time remains then go over the test.

• Wed Oct 27 Discuss the confusion between Euclid's 5th and Playfair's. Prove that if we assume that Euclid's first 28 propositions hold, then Euclid's 5th postulate is equivalent to Playfair's axiom (ie prove iff). We'll see that this is not true in general later.

• Fri Oct 29 Presentations on folding:
-Folding an angle bisector
-Folding the perpendicular bisector of a line segment
-Folding the perpendicular from a given point to a given line
-Folding the perpendicular through a point on a line
-Folding a line parallel to a given line through a given point
-Folding to show that the sum of the angles in a triangle is 180 degrees
-Folding the intersection of the altitutes of a triangle
-Folding the intersection of the angle bisectors of a triangle
-Folding the intersection of the perpendicular bisectors of the sides of a triangle
-Folding the intersection of the medians of a triangle
-Folding to show that the square on the hypotenuse is equal to the sum of the squares on the two other legs of a right triangle
-Folding to show that (x+y)(x-y)=x2 -y2

Then go to the computer lab or library to work on finding book and web references for the Geometry of our Universe project.
• Mon Oct 18 Discuss last problem from project. Discuss taxicab activities. Discuss homework readings. What does parallel mean? (book definitions). Go over some pictures and discuss whether we think they should be parallel or not and whether they satisfy the definitions. Given parametric forms of two lines in three space, how can we tell whether they are parallel? Alternate ways to tell when objects are parallel. Definition versus theorems in geometry.

• Wed Oct 20 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. Show Dr. Sarah before you go on to the next file.
• What are the shortest distance paths in hyperbolic geometry?
Image of Shortest Distance Paths
• Is parallel the same as equidistant in hyperbolic geometry?
Image of Equidistant 1
Image of Equidistant 2
• Is Euclid's 5th postulate ever, always or never true in hyperbolic space?
Image of Euclid's 5th Postulate

• Fri Oct 22 Test 2
• Mon Oct 11 Presentations on the Sketchpad activities from last Wednesday. Go over the proof that the perpendicular bisectors are concurrent. Discuss taxicab circles and the relationship to the strategy for the Taxicab treasure hunt. Highlight the possible number of intersections of taxicab circles for different examples. Example 1   Example 2. Discuss the Relationship to the NCTM standards. Begin Taxicab activities in Sketchpad

• Wed Oct 13 Finish Taxicab activities in Sketchpad .
• Mon Oct 4 Collect the homework models. Sibley p. 55 number 6. Sketchpad via Sliding a Ribbon Wrapped around a Rectangle and Sliding a Ribbon Wrapped around a Box. Read the proof of trig identity and then fill in the details and reasons using similarity, trig and the pythagorean theorem.

• Wed Oct 6 Introduction to measurement. Discuss the reservoir problems. Then work on the handout in groups of 2.

• Fri Oct 8 Review Sketchpad reservoir problem and relate to measurement. Axiomatic versus metric perspectives of Euclidean geometry. Intro to taxi-cab geometry.
• Mon Sep 27 Introduction to "same shape" via pictures. Fig 8.4 Fig 8.21 Fig 8.32 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.

• Wed Sep 29 Go over regression in Excel and apply it to the the example on p. 212.
Part 1: Similar Triangles - AA Similarity activity sheet from Exploring Geometry with Sketchpad. Leave the Explore More part until later.
Part 2: 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. Leave the Explore More part until later.
Part 3: Then complete the Similar Polygons Sketchpad activity sheet.
Part 4: Go back to the Explore More parts of the worksheets.

• Fri Oct 1 Discuss Similarity Postulates based on Sketchpad Activities and the Sibley Reading. Discuss the Wile E assignment. Even/odd function proofs. If time remains then read the proof of the trig identity and then fill in the details and reasons using similarity, trig and the pythagorean theorem. work on the models.
• Mon Sep 20 Test 1

• Wed Sep 22 Discuss test 1 and begin Nova's "The Proof" video.

• Fri Sep 24 Finish "The Proof" video, share teacher activities from the Nova website, and go over The Burdon of Proof activity and the difference between legal system proof and mathematical proof.
• Mon Sep 13 Meet in 205. Go over the web links from the Worksheet on Archimedes and Cavalieri's Principle. Discuss web based Euclid's Elements (historical proof taken from it) which is a link from Problem set 2 solutions up on WebCT. Review the Pythagorean theorem - Euclid's historical proof and comparison with p. 8-9 in Sibley, which is a modern proof of p. 7 # 10 from Project 1. Discuss the benefits and difficulties of using the different methods, including original historical sources. Intro to algebraic extensions of the Pythagorean Theorem including Fermat's Last Theorem (next week) and Pappus. A geometric extension of the Pythagorean Theorem on Sketchpad.

• Wed Sep 15 Finish a geometric extension of the Pythagorean Theorem on Sketchpad. Consistency of axioms via minesweeper examples (and non-examples) and Euclidean geometry. Consistency does not imply uniqueness. Have the students create a minesweeper gameboard that is inconsistent and write up a proof that the gameboard is inconsistent - go around the room and examine each student's proof before presenting one version. Reading from Perry p. 50 on consistency. Godel's results. Intro to Euclid's 5th posulate - what it says and doesn't say and its negation. Historical overview of the 5th postulate.

• Fri Sep 20 University Cancelled Classes
• Mon Sep 6 Labor Day Holiday

• Wed Sep 8

Create a segment with the ruler tool.
Using the arrowhead tool, choose one of the endpoints and the segment too (by holding down the shift key as you select them)
Under Construct, use the Sketchpad feature to construct a perpendicular line through the endpoint.
Use the point tool to choose a new point on the perpendicular.
Use the ruler tool to construct the segment between the 2 points on the perpendicular line (ie before you do this, the entire line has been created, but the segment does not exist).
Use the arrowhead tool to select only the perpendicular line (but not the segment you just constructed)
Under Display, release on Hide Perpendicular Line.
Use the ruler tool to complete the third side of your right triangle.
Measure the right angle to verify that it is 90 degrees.
Measure the length of the three sides of the triangle.
Once you have all three lengths, under Calculate, click on the measurement of the base of the triangle in order to insert it into your calculation.
Continue in order to calculate the base*base + height * height - hypotenuse *hypotenuse
Move the points of your triangle around in order to try and verify (empirically) the Pythagorean Theorem.

Sketchpad has some built in explorations. 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(MacOS9)/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.

Read through Euclid's Proof http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html along with the appendix of Sibley to try and understand it.

We come back together and go through Euclid's Proof of the Pythagorean Theorem.

• Fri Sep 10 Review the calculus argument from project 1. Worksheet on Archimedes and Cavalieri's Principle. If time remains then in groups of two, evaluate five arguments showing that the derivative of an even function is an odd function. Decide which arguments are convincing to you, which arguments constitute a proof of the claim, what grades you think a teacher would assign to these arguments, and specific ways that each argument can be improved.
• Mon Aug 30 Take attendance and discuss homework readings. 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. 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. Handout Computer Lab directions and go to 205. Together (with a student up on the main computer), begin Euclid's Proposition 1.

• Wed Sept 1 Meet in the computer lab and continue activities. Complete Euclid's Proposition 1 To construct an equilateral triangle on a given finite straight line via the Sketchpad construction and script view, saving the file, and then the corresponding 2 column proof. 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. If time remains, then use a paper folding argument for Proposition 11.

• Fri Sept 3 Minesweeper Game 3. We then contrast with game 3 and learn that even if squares cannot be determined, knowing partial results can determine other squares. Minesweeper Proofs.
• Wed Aug 25 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. 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). Checkerboard challenge problem and the missing square. Handouts Main web page, Dr. Sarah's Office Hours,and syllabus.

• Fri Aug 27 Discuss homework readings. 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 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).