Dr. Sarah's Math 3610 Class Highlights

Dr. Sarah's Math 3610 Class Highlights Fall 2002 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 9 Geometry of the Earth problem 13. Geometry of the Earth and Universe - From the Classroom to Current Research . Go over test 4. Search for "What is Geometry"? Come back together to discuss this.
  • Mon Dec 2 If you did not finish revising the Sketchpad test last week, then continue working where you left off. Use the Spherical Triangle Applet in order to create a right spherical triangle (ie drag points so that one of the angles is 90 degrees). Then use the given side lengths to determine whether the Pythagorean Theorem holds on a a sphere. Write out your calculation in depth (ie sketch the triangle and the angles and side lengths and the show your work from there) and show it to Dr. Sarah. Follow the directions at the bottom of the Escher worksheet. Read through David Henderson's Chapter 5 Introduction and then read through his Constructions of Hyperbolic Planes. Then skim through these readings a 2nd time as there is a lot to absorb. Begin working on the Hyperbolic Soccer Ball Construction which is due at the beginning of class on Wednesday.

  • Wed Dec 4 Review hyperbolic models, Finish sphere problems 10-13, if time remains then gluing spaces (torus, Klein bottle, 2 holed torus)

  • Fri Dec 6 Test 4
  • Mon Nov 25 Review Euclidean, spherical and hyperbolic geometry via Dr. Sarah's sheet. Redo anything that was graded as incorrect (see WebCT posting on this) on the Sketchpad test and show Dr. Sarah. Search for the answer to geometry of the earth Problem 12 (on the web). Read through the Final Project guidelines, the guidelines for oral presentaions and the guidelines for PowerPoint Review the answers to Geometry of the earth problems 1-9. If time remains, work on final project.

    Thanksgiving Break
  • Mon Nov 18 Sketchpad Test

  • Wed Nov 20 Discuss book and web references (library, Mathsci education center...) Beachball activity Continue with geometry of the earth problem 9...

  • Fri Nov 22Finish Finish the Beachball activity and continue with geometry of the earth problems via problem and then relate to hyperbolic geometry. Begin problem 10 and discuss for hyperbolic geometry.
  • Mon Nov 11 From Sketchpad 4, open up MacHD/Applications/Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp. 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...
    1. 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?
    2. The sum of the angles in a spherical triangle
    3. Theorems from Perry p. 247-248 and p. 255 in hyperbolic geometry - Do these in Sketchpad.
    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.
    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.
    4. Is the Pythagorean theorem ever, always or never true in hyperbolic geometry?
    Review material together as a class and if time remains, then work on final projects.
    Image of Sum of Angles
    Sketchpad of Sum of Angles
    Image of Hyperbolic Pythagorean Thm
    Sketchpad of Hyperbolic Pythagorean Thm

  • Wed Nov 13 Finish up proof that if we assume Euclid's first 28 propositions then Playfairs is logically equivalent to Euclid's 5th Postulate. Begin spherical geometry via globes. Problems 1 - 4 of Geometry of the Earth

  • Fri Nov 15 Review Problems 1-4 and continue working on the Geometry of the Earth (through problem 8 and compare to taxi-cab SAS and Euclid's proof)
  • Mon Nov 4 Review Friday's class. Why Study Hyperbolic Geometry? In Sketchpad, under File/Open, find the Sketchpad folder and then open up Sketchpad/Samples/Sketches/Investigations/Poincare Disk.gsp
  • 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
  • Given a hyperbolic line and a point off of the line, how many parallels can be formed? (Playfair's axiom)
  • Is Euclid's 5th postulate ever, always or never true in hyperbolic space?
    Sketchpad Euclid's 5th Postulate
    Image of Euclid's 5th Postulate


  • Wed Nov 6 Review Sketchpad files from Monday in the hyperbolic disk (redo each of them) and then complete Euclid's 5th (see Monday) exploration in the hyperbolic disk. If finished early then use Prop 11, 12 and 27 of Euclid in order to prove the existence part of Playfair's axiom for Euclidean geometry. Go over these together. Discuss the fact that the existence part of Playfair's still works in hyperbolic geometry, but that we obtain infinitely many parallels, 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.

  • Fri Nov 8 Collect project. Students present their work. Discuss final project. 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.
  • Mon Oct 28 Review taxi-cab geometry measurement and distances. Finish up the reason why ( Henderson's why as opposed to the proof we completed in class on Friday) SAS does not hold in taxi-cab geometry. 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 of 2 points at a time. Example 1   Example 2

  • Wed Oct 30 Taxicab Activities in Sketchpad

  • Fri Nov 1 Intro to Euclid's 5th Postulate. Review def of line as shortest distance path in taxicab geometry. What does parallel mean? Student responses, 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. Proof that parallel lines imply that they are equidistant all along their length in Euclidean geometry.
  • Mon Oct 21 Village Problem   Dynamic Solution on Sketchpad, Why are the perpendicular bisectors concurrent?   Mathematics Across Culture Solution of Project 3 number 5 on sketchpad.

  • Wed Oct 23 Test 2

  • Fri Oct 25 Axiomatic versus metric perspectives of Euclidean geometry. Intro to taxi-cab geometry.
  • Mon Oct 14 Finish up filling in the details of the trig identity. Read David Henderson's Proof as a Convincing Communication that Answers -- Why? and relate back to trig identity and to other proofs in the class. Read Measuring Volumes and discuss. Using Sketchpad as a discovery/exploration tool: 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?

  • Wed Oct 16 Collect and then go over Project 3

  • Mon Oct 7 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.
    Report back to the class when we come back together.
    If finished early, then work on the homework for Wed (see main web page) and/or p. 216 number 4 from Project 3.
    Similar Triangles activity from Exploring Geometry with Sketchpad.

  • Wed Oct 9 Review AA worksheet on Sketchpad. Discuss benefits and difficulties of this worksheet. Review geometric similarity modelling problem in Excel. Do Triangle_Similarity.gsp Sketchpad activity. Do Similar Polygons Sketchpad activity.

  • Fri Oct 11 Finish Sketchpad worksheets from Wed. 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.
  • Mon Sep 30 Finish The Proof - A Nova video about Princeton University Professor Andrew Wiles and Fermat's Last Theorem. Skim through Dr. Sarah's classroom activity sheet for Math 1010 - Introduction to Mathematics - on Andrew Wiles (What is a Mathematician) Examine The Proof website contents (especially Solving Fermat: Andrew Wiles, and the Teacher's Guide -- Click on Viewing Ideas | Printable Activity | Online Activity | Teachers' Ideas | and take notes on ideas from the Teacher's Guide for using this video in the classroom.

  • Wed Oct 2 Students discuss ideas from the Teacher's Guide from Mondays web reading. Work on the Math 1010 Andrew Wiles worksheet (mathematics). Reinforce ideas in The Proof in relationship to proof and critical reasoning. Highlight syllabus goals, what we have accomplished so far, and where we are going.

  • Fri Oct 4 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.



  • Mon Sep 23
    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
    Under File, release on open and open up MacHD/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 of I-47 from Book 1 Go through and take notes in modern language on the proof of Prop 47 so that you could present parts of it from your notes.

  • Wed Sept 25 Review Euclid's historical proof of Prop 47. Review the proof of the pythagorean theorem from Project 1 in Sibley. Discuss the benefits and difficulties in teaching with original historical sources. Relate our work on the pythagorean theorem to the proofs and reasoning standards assignment. Introduction to Fermat's Last Theorem.

  • Fri Sep 27 The Proof - A Nova video about Princeton University Professor Andrew Wiles and Fermat's Last Theorem.


  • Mon Sep 16 Finish project 1 presentations. Review philosophy of minesweeper axiomatic viewpoint, Euclidean axiomatic viewpoint, and constructions. Go over web based Euclid's Elements. Go through Postulates 1-5 and take notes. Compare the proof of your proposition from number 4 of project 2 with Euclid's proof. If you don't finish this in lab then finish this before Friday. If time remains then begin consistency (create a minesweeper game that is inconsistent and prove that it is inconsistent).

  • Wed Sep 18 Test 1

  • Fri Sep 20 Consistency of axioms via minesweeper examples (and non-examples) and Euclidean geometry. Consistency does not imply uniqueness. Proof that a 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. Discuss Wile E assignment. Begin the Pythagorean theorem via Euclid's proposition I-47 and comparison with Sibley p. 7 number 10 from Project 1.


  • Mon Sep 9 Meet in 203. Finish Euclid's Book 1 Proposition 1 via the Sketchpad construction and script view, saving the file, and then the corresponding 2 column proof. 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. Begin the construction of a square.

  • Wed Sep 11 Finish the java sketchpad of the construction the first side of a square and the corresponding 2-column proof. java sketchpad of completed construction. image of completed construction and script Burden of Proof. Construction of a parallelogram. Begin project 1 solutions.

  • Fri Sep 13 Call on students to present Project 1 problems.


  • Wed Sep 4 Paragraph proofs continued via an introduction to minesweeper games as an axiomatic system and resulting proofs. Groups of 2 work on paragraph proofs for Game 1 (after we've proven that B1 and B2 are numbers) and present them on the board.

  • Fri Sep 6 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. Intro to Geometric Constructions. Begin Euclid's Proposition 1 - To construct an equilateral triangle on a given finite straight line via the sketchpad construction.


  • Wed Aug 28 Fill out information sheet. What is geometry? Discuss with a partner and then report back to the class. 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. 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). Revisit the five proof exercise as a teaching methodology.

  • Fri Aug 30 Checkerboard challenge problem and the missing square. Intoduction to the history of proofs and the societal context within Greek society. Introduction to logic tables, two column proofs and paragraph proofs.