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MATH307 Calendar (click on a date to view the day's activity, PowerPoints, assignments, etc.)
Today we determined how much high school geometry we remembered and examined a few high school geometry textbooks in order to discuss the sequence of the topics covered, paying specific attention to the importance of the topic of parallel lines.
Today was a shortened day due to convocation. We basically just played around with verifying a few theorems using GSP.
Assignment Supplemental Problems #1
We continued playing around with GSP, examining the nine-point circle and Morley's triangle, along with other relationships.
We began our discussion of axiomatic systems of geometry. We examined Euclid's elements, looking at his definitions, axioms, common notions, and a few propositions.
Today we began discussing some Finite Geometries
Assignment Supplemental Problems #1
We continued our discussion of Finite Geometries, discussing Properties of an Axiomatic System and then proving a theorem from four-line geometry.
Assignment Supplemental Problems #1 Supplemental Problems #2
After submitting Assessment #1, we completed our discussion of Finite Geometries by proving a theorem in four-point geometry and creating a model for Fano's geometry. We ended the day by reviewing the textbook homework.
Assignment Assessment #2 Supplemental Problems #2
We began to discuss the Incidence Axioms today. We also submitted Assessment #2 and #3.
We completed our study of the Incidence Axioms by proving a couple of theorems as results. We then introduced the Distance Axioms and what they tell us.
After examining some HW problems from 2.3, we developed the concept of betweenness and began a proof using it.
We completed our proof from Wednesday and then defined segments, rays, lines, and angles from a set theoretic perspective and proved a theorem concerning two rays being equal. We then motivated the copying of a segment by a hand construction on the board.
Assessment #4 was returned and reviewed. We then discussed the Segment Construction Theorem and began discussing angles.
We discussed the Angle Axioms today.
We completed our discussion on the Angle Axioms today. We then began to discuss our final axiom for points, segments, lines, and angles; although we only set the stage for it with some questions, concepts, and definitions.
Reminder of overall course goal Notes to finish from previous day Assignment
We completed our study of points, segments, lines, and angles by motivating the Plane Separation Axiom and using it to prove Pasch's Postulate. Students then began their construction assessment in front of their peers.
A couple of "pushes" on the homework from 2.5 and 2.6 were given, Assessment #6 was handed out, and more students completed their construction assessment in front of their peers.
We began to discuss the concept of congruence and what it can do for us with respect to proving triangles congruent.
We continued discussing the concept of congruence of triangles today, examining the SAS "hypothesis" and taxicab geometry.
We continued discussing the concept of congruence of triangles today, examining the SAS Postulate and some of the theorems that can be proven by using it.
We worked in groups today to discuss the proof of a couple of theorems stemming from the SAS Postulate and ASA Theorem.
We completed our proof of the SSS Triangle Congruence Theorem and then completed Assessment #7.
Assignment GSP Slides to Review for Next Class (be sure to maximize to be able to click through all slides; and be sure to begin on page 1)
We carefully went over Assessment #7. We then walked through the Moment for Discovery on p.161 in our text in order to develop a model that satisfies all of the axioms accepted thus far but still not the familiar Parallel Postulate.
Class was cancelled for today due to Steve being away at a conference. However, GSP slide-notes were made so students could download the lecture notes for the day.
GSP slide-notes (be sure to maximize so you can see the pages, and begin on page 1) Assignment
Today we cleaned up the posted notes from last Wednesday and Friday. We played around with the Poincare Model for Hyperbolic Geometry and the Spherical Geometry Model.
Poincare Model for Absolute Geometry Spherical Geometry Model Assignment
We discussed the AAS, SSA, HL, HA, and LA triangle congruence theorems.
We began discussing quadrilaterals in absolute geometry.
We completed our discussion on quadrilaterals in absolute geometry.
Sacch_Quad_Poincare_11-2-09 Sacch_Quad_Spherical_11-2-09 Assignment
We discussed circles today.
Class notes to finish Assignment
There was no class meeting today. Students should examine the following notes leftover from Wednesday and complete the assignment.
Class notes to finish Assignment
We finished up our discussion on circles in absolute geometry. We then began to discuss what the acceptance of the Parallel Postulate can now do for us. We did this by looking at Euclid's 27th and 28th propositions (which did not need the P.P.), and then his 29th proposition (which did need the P.P.). http://aleph0.clarku.edu/~djoyce/java/elements/elements.html (A website version of Euclid's Elements) Assignment
We continued our discussion of the importance of the Parallel Postulate in proving "what we know" about geometry. Assessment #8 was then given.
We discussed special types of quadrilaterals (parallelogram, rhombus, square, trapezoid) and proved some theorems about them.
We examined some consequences of the midpoint connector theorem. We discussed parallel projections and the side-splitting theorem. We constructed a parallelogram inside of a quadrilateral, divided a segment into n congruent segments, and found the centroid of a triangle.
We discussed similar polygons today. We showed that similar triangles do in fact exist, theorems used to prove them, and then how to prove the Pythagorean Theorem by using similar triangles. Assessment #10 was given.
There was no class meeting today. Students should be working on Assessment #10.
Assessment #10 was collected. We then proved the Pythagorean Theorem from similar triangles, used similar triangles to develop trigonometry, and derived the Law of Sines and the Law of Cosines.
Assessment #10 was returned and discussed. We then constructed the geometric mean of two segments, derived the Law of Cosines, and derived the Cevian Formula.
We started the day discussing the ambiguous case with the Law of Sines, along with the time in which the ambiguity is amplified. We then discussed regular polygons, their existence, the sum of the interior angles, the sum of each interior angle, and which ones can be constructed using only a compass and straightedge.
We took Assessment #10 (which is really #11) in class and then discussed tessellations and proved which regular polygons will tessellate the plane. We then discussed semi-regular and irregular tessellations, examining a few websites. Assessment #11 (which is really #12) was then given to be submitted on Monday.
GSP Notes Assignment Assessment #11 (really #12)
Assessment #11 was returned and reviewed. A reminder of Assessment #12 due this afternoon was given. We then spent the rest of the day stating and verifying via GSP our circle theorems.
We proved several circle theorems, examining the role that the inscribed angle theorem plays in each. We then proved the inscribed angle theorem.
We summarized the following three topics: Area, Coordinate Geometry, and Transformational Geometry. The take-home part of the final exam was given.
Assignment Take-Home Final Exam
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