Dr. Sarah's Math 3610 Class Highlights

### Dr. Sarah's Math 3610 Class Highlights Fall 2003 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 Dec 2 Test 4

• Thur Dec 4 Go over test 4. Discuss final project topic and presentation. Take questions. Search for "What is Geometry"? Come back together to discuss this. Discuss the geometry of the universe.
• Tues Nov 25 Continue Geometry of our Earth Go back to Problem 11 (Pythagorean Thm). Review our Euclidean proofs and discuss what goes wrong in spherical geometry. Problem 14. Review Euclidean, Spherical and Hyperbolic Geometries and discuss test 4.
• Tues Nov 18 Go back to Problem 10 (Sum of Angles). Look at the Euclidean proof and discuss what goes wrong in spherical geometry. Begin the Beachball activity

• Thur Nov 20 Sketchpad test.
• Tues Nov 11 Discuss the hyperbolic models readings from lab on Thursday. Begin presentations.

• Thur Nov 13 Continue presentations.
• Tues Nov 4 Review hyperbolic geometry activities from class on Thursday. 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 on Thursday), 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. Discuss the confusion between Euclid's 5th and Playfair's. Begin proof 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.

• Thur Nov 6 Finish the proof that if we assume that Euclid's first 28 propositions hold, then Euclid's 5th postulate is equivalent to Playfair's axiom. From the Sketchpad 4 folder, open up Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp. We begin with hyperbolic geometry theorems from Perry p. 247-248 and p. 255..
• 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.
• Create a transversal and measure angles to see that alternate interior angles are not congruent.
• Theorem 5: If AB is a parallel halfline for a line m from point A and AK is the opposite halfline for AB, then AK is not parallel to m.
•   Theorem 6: If AB is a parallel halfline for a line m from point A and AK is the opposite halfline for AB, then AK does not intersect m.
We discussed the fact that many of Euclid's propositions (beginning with Prop 29) depend on Euclid's 5th postulate. Since this postulate is sometimes but not always true in the hyperbolic disk, then we must examine the validity of the propositions...
• 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
• Read David Henderson's Chapter 5 Introduction and then Constructions of Hyperbolic Planes Come back together and discuss the readings and examine some of the models. If time remains then do web searches for Tuesday presentations.
• Tues Oct 28 Divide up geometry of our earth problems. Taxicab Activities in Sketchpad Begin Intro to Euclid's 5th Postulate. Review def of line as shortest distance path in taxicab geometry. What does parallel mean? Student responses

• Thur Oct 30 Finish Intro to Euclid's 5th postulate. 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.
Why Study Hyperbolic Geometry? 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?
Sketchpad Shortest Distance Paths
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?
Sketchpad Euclid's 5th Postulate
Image of Euclid's 5th Postulate
• Look at Interactive Euclid's Elements and go through the links for the 5 postulates.
• Tues Oct 21 Test 2
• Tues Oct 14 Review Sketchpad reservoir problem and relate to measurement Axiomatic versus metric perspectives of Euclidean geometry. Intro to taxi-cab geometry.

• Thur Oct 21 Solution of Project 3 Number 5 on Sketchpad. Why are the perpendicular bisectors concurrent? Review taxi-cab geometry measurement and distances. Play a couple of games of Taxicab Treasure Hunt What is an effective strategy for this game? Relationship to taxicab circles and NCTM Standards. Using Graph/Show Grid in Sketchpad, draw taxi-cab circles about different points and find the possible number of intersections for different examples. Example 1   Example 2 If time remains then study for test 2 by reviewing activities and links from the class highlights page from Thursday Sept 11 up through Tuesday October 7. If additional time remains then look at solutions on WebCT.
• Tues Oct 7 Go over Sketchpad worksheets from Thursday. Discuss Similarity Postulates based on Sketchpad Activities and the Sibley Reading. Read proof of trig identity and then fill in the details and reasons using similarity, trig and the pythagorean theorem. Finish up similarity by Sibley p 55 number 6. Look at homework models. Discuss upcoming due dates. Hand out and discuss Project 5, including Sibley. p. 38 number 5 part a. Introduction to measurement. Axiomatic versus metric perspectives of Euclidean geometry.

Thur Oct 9 Go over Sibley p. from Tuesday's class on Sketchpad via Sliding a Ribbon Wrapped around a Rectangle and Sliding a Ribbon Wrapped around a Box. Intro to measurement via Measurement Standards for Grades 9-12 and Measuring Volumes. In groups of 2, work on this problem in Sketchpad: Suppose the government of a small developing country wants to construct a water reservoir for three villages. Where should the water reservoir be placed so that it is the same distance from all three villages? Is the same distance point always the best place for the reservoir? Explain. Then work on handout.
• Tues Sep 30 Questions from Thursday. Highlight syllabus goals, what we have accomplished so far, and where we are going. 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. Go over regression in Excel and apply it to the the example on p. 212. Hand back Wile E assignments and discuss revisions. If time remains, then begin homework readings.

• Thur Oct 2 Questions on readings. Geometer's Sketchpad game. Choose a partner. Draw a triangle and measure its parts (side lengths and angles) using the program, but do not show your triangle to your partner.
Part 1: Choose roles so that one person will hand measurements to the other: Try to give your partner 2 or 3 measurements and instructions that will always ensure that they will be able to draw a triangle with the same shape as yours. Note: Since we only require similar and not congruent triangles, pairs of corresponding sides you give to your partner must only be proportional (via the same proportionality constant) to your measurements, while angles should be equal. Try to come up with a complete list of all of the possible sets of 2 or 3 measurements and instructions that will always result in similar triangles.
Part 2: Switch roles so that the person who received measurements in part 1 will now hand measurements to the other: Try to give your partner 2 or 3 measurements and instructions so that they will not necessarily be able to draw a triangle with the same shape as yours. The person who is given the measurements should try and draw 2 differently shaped triangles from the measurements and instructions. Try to come up with a complete list of all of the possible sets of 2 or 3 measurements and instructions that will NOT ensure that they have the same shape.
Part 3: Similar Triangles - AA Similarity activity sheet from Exploring Geometry with Sketchpad. Leave the Explore More part until later.
Part 4: 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 5: Then complete the Similar Polygons Sketchpad activity sheet.
Part 6: Go back to the Explore More parts of the worksheets.
Part 7: If time remains then look at the main web page for upcoming homework and work on the models for Tuesday and/or read over Project 5.
• Tues Sep 23 An algebraic extension of the Pythagorean Theorem, paying special attention to the concept of proof, the way mathematicians do research, and what algebraic geometry is. The Proof - A Nova video about Princeton University Professor Andrew Wiles and Fermat's Last Theorem. When finished, revisit the concept of proof.

• Thur Sep 25 Discuss ideas from the The Proof website contents and call on students to share their thoughts on these activities with the rest of the class. Read David Henderson's Proof as a Convincing Communication that Answers Why Give some examples of proofs from class that did and did not satisfy his idea of a proof. The Burdon of Proof activity and the difference between legal system proof and mathematical proof. A geometric extension of the Pythagorean Theorem on Sketchpad. If finished early before we come back together, work on Reasoning and Proof Standards Assignment by going through the web readings.
• Tues Sep 16 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. Intro to extensions of the pythagorean theorem, which we'll look at in depth next week. 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 consistent 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. Discuss web based Euclid's Elements (historical proof taken from it) which is a link from Problem set 2 solutions up on WebCT. Intro to Euclid's 5th posulate - what it says and doesn't say and its negation. Historical overview of the 5th postulate.

• Thur Sep 18 Test 1
• Tues Sept 9 Checkerboard challenge problem and the missing square. Call on students to present Project 1 problems. Taking the last problem further -- begin the Worksheet on Archimedes and Cavalieri's Principle

• Thur Sept 11 Meet in 205. Finish Worksheet on Archimedes and Cavalieri's Principle

Algebraic Pythagorean Theorem in Sketchpad
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.

Geometric Pythagorean Theorem in Sketchpad
Sketchpad has some built in explorations. Under File, release on open and click on Desktop, then on ASU-FS4.Apps, then on Mac Software, then on Math Dept, and then on Sketchpad, then on Samples, then on Sketches, then on Geometry and finally, open Pythagoras.gsp For future reference, I will write this as Desktop/ASU-FS4.Apps/Mac Software/Math Dept/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. Discuss the benefits and difficulties of using the different methods, including original historical sources.

• Tues Sep 2 Review first week including the equilateral triangle construction and take questions. Highlight the difference between a construction and a proof. Minesweeper Game 2 and Game 3. 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. We then contrast with game 3 and learn that even if squares cannot be determined, knowing partial results can determine other squares. Minesweeper Proofs. Compare with project 2. 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. Be prepared to share your thoughts with the rest of the class. If time remains, Checkerboard challenge problem and the missing square.

• Tues Aug 26 Fill out information sheet. What is geometry? Discuss with a partner and then report back to the class. History of geometry including Egyptians, Babylonians, Chinese, and Africans. 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). 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. Groups of two work on paragraph proofs for Game 1 (after we've proven that B1 and B2 are numbers) and present them on the board. Handouts Project Guidelines, Main web page, Dr. Sarah's Office Hours,and syllabus.

• Thur Aug 28 Meet in 205. Questions on readings for homework? 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. Together (with a student up on the main computer), begin 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. Also, do 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. Read p. 10 Example 1 of Sibley. Model the construction of a square: java sketchpad of the construction the second side of a square and the corresponding 2-column proof java sketchpad of completed construction. image of completed construction and script. If time remains, students contruct a parallelogram. If time remains, work on Checkerboard challenge problem and the missing square.