Spring 2013 Math 311 Class Announcements and Resources
Test IV grades and Course Letter Grades (for those who took Test IV)
Here are your Test IV grades (and your course letter grade if you took Test IV). Grades for all will be up on Bronco Web soon.
Test IV Study Guide
Here it is! Test IV below from Spring 2012 is a resource, and the relevance of problems from that test is discussed.
Test III grades posted
Here are the Test III grades posted by the ID number on your test paper. Here are solutions with some indication of things I was looking for when grading. Please tell me if you find errors or typos in the solution set!
Materials for Test III...
are the fourth test from Spring 2012 and Test III which you already have below. You should be able to tell what is relevant by considering what we have covered...but I'll write a little more later about what you should study specifically. Some remarks now appear below, and I may post more study information about theorems to emphasize.
Relevance of problems on the sample test: Problem 1 is relevant. Problem 2 is not relevant. Problem 3 is relevant. The posted document does not have pictures in it for reasons that I have discussed, but you should be able to draw the picture. Problem 4 is relevant, though it is motivated by things ahead of us in the text; it doesn't require anything that we havent covered. Problem 5 is relevant, and was a homework problem! Problem 6 is not relevant. Problem 7 is highly relevant! Problem 8 is relevant. I am more likely to ask you to prove the Scalene Inequality than the Triangle Inequality given where the lecture stopped, but you should be working on your own proofs of the Triangle Inequality. Sample Test III problems 1,5,6,7 are also relevant to your exam.
There will be no sample test available for our Test IV: this is the exam the last class took during the final exam period.
Triangle inequality hint: Let ABC be a triangle. We want to show that d(A,B) + d(B,C) is less than d(A,C). If d(A,B) is greater than or equal to d(A,C) we are done, so assume d(A,B) is strictly less than d(A,C). Use the PCP to construct D on ray AC such that segment AB is congruent to segment AD. B*D*C (why?). It is now enough to show that the length of segment BC is greater than the length of segment DC. Show this by showing that the measure of angle DBC is less than the measure of angle BDC (use the exterior angle theorem) then use the Scalene Inequality.
AAS proof note: I'm convinced that there is no way to prove AAS without using the exterior angle theorem, which makes it less attractive as a test proof (because of the need for cases – but see that I actually handle the cases quite compactly below). By the way, the ASA proof does not need cases, because the application of the Angle Construction Postulate in it does not depend on the position of the new point in the same way the application of the Exterior Angle theorem in the AAS proof does.
Here is my version of the proof. Suppose ABC and DEF determine triangles and segment AC is congruent to segment DF, angle BAC is congruent to angle EDF and angle ABC is congruent to angle DEF. Construct point B' on ray AB such that segment AB' is congruent to segment DE. Observe that triangle AB'C is congruent to triangle DEF by SAS: if we can show B=B' we are done. A consequence of this congruence is that angle AB'C is congruent to angle DEF which we know already is congruent to angle ABC. Suppose for the sake of a contradiction that B is not B'. Then either A*B*B' or A*B'*B. If A*B'*B, notice that angle AB'C is external to triangle BB'C and greater than the remote interior angle B'BC = ABC by the Exterior Angle Theorem – but these angles are equal! If A*B*B' the argument is exactly the same with B and B' interchanged (cases finessed by symmetry!). In either case we have a contradiction, so B=B' and the two triangles ABC and DEF are congruent. This presentation makes the problem of cases look not so serious.
This does mean that I marked some of the early homework 7 papers incorrectly (in your favor) because I thought there was a clever way to do it with just the Angle Construction Postulate...make sure you understand the correct proof. Needless to say, if you identify a problem with my proof here, tell me about it so I can correct it (or correct your reading of it).
Test III Date is Not April 4th
I already said this in class. There will be two more exams and no cumulative final; Test III will be later, I have not decided when; Test IV will be held in the final exam period. Relax and have fun with Dr. Teitler. I will post some homework tomorrow (the second).
Test II Grades; Solutions will Appear Here
Here are the grades on Test II posted by the ID on your test paper. Most of you have your papers, but this also allows you to see the class distribution of grades. Solutions will appear in this space when I have typeset them, sometime this week. Here are the solutions.
The Purpose of Homework (and Academic Standards)
Please be aware of the reasons why I am so disturbed about seeing a proof apparently copied from a solution manual or web resources on so many papers in the recent assignment. The purpose of homework is not primarily to provide things for me to mark; homework is a relatively small part of the grading base in this class, and I give credit for it it primarily to encourage you to do it. The reason that I assign and grade homework in this class (or any class) is so that you have a chance to get feedback on your work and hopefully have a chance to improve. You do not learn much by copying from a solution manual or other remote source onto your paper. I waste a not inconsiderable amount of time if I evaluate work actually done by professionals at my level which appears on your paper. If you do look at a proof taken from an outside source in the course of doing your homework (which I don't especially recommend), I suggest a different approach. Read it. If you think you understand it...wait an hour and then write it out in your own words without looking at the original. You will learn more that way.
Grading a complete homework assignment for a class of 30 students takes a lot of time; many hours. If I find myself spending 40 minutes or an hour going through a problem on a homework assignment and the work on more than half the papers looks identical and closely resembles (down to details of notation) a document found on the web, this is demoralizing for me, and it does not serve the purpose.
Now, about academic standards. This is a writing course. Copying proofs from a source on the web or a solution manual is plagiarism in exactly the same sense as copying a paper or large parts of a paper from some public or private source in an English or History class. I do not want to spend my time trying to detect plagiarism; I'm not in the least interested in “getting” any individual on these grounds. What I want is to grade your work, and not for the supposed delight of taking off points but for the purpose of giving you feedback and trying to help you understand the material.
Sample Test II Papers and Study Materials
Here is some study material for Test II including guidance as to what problems on the test papers to study and solutions to Homework 5.
Both Test II and Test III from last year are relevant: we are ahead of where they were.
Here is the Test II paper. Here is the Test III paper. Everything on the Test II paper is relevant, and some of the Test III.
Test I Grades
Here are the Test I grades posted by the ID number on your test. Here is my solution set that I made before the exam. Of course, some of the questions can have different, creative answers. This file contains some but not all of the comments that occurred to me while I was grading the exam.
Test I rescheduled
to Thursday, Feb 21st.
Here is the Spring 2012 Test I paper. Pictures have been added.
A universal remark about old tests is that I draw pictures by hand, and these are not present in the PDF versions of old tests that I have readily available. In a geometry class this is more of a problem than in some other classes, though I do think that you can work out from what you now have the kinds of questions that I was asking. I am thinking that I might post a version of this test with diagrams on Thursday or Friday; if I did it now I would answer one of your homework problems!
Logic material (Feb 7)
Here is the manual of logical style. This is a work in progress; there will be changes and additions.
Euclid's Elements (for Jan 22)
Take a look at this online presentation of Euclid's Elements. I'm also planning to put one or more paper copies on reserve in the library. The online version should be adequate for any class purposes, though.
Welcome to the class. This is the class announcements page, where I will make class announcements and post documents for the class, such as homework assignments (when they are not just problems from the book), sample proofs, lecture notes, and so forth.
New material will be added at the top. Material from the legacy section which becomes officially Useful will be copied in at the top when it becomes Useful.
Legacy material from the Spring 2012 document which might possibly be useful
Here find a document with more proofs from the chapters we are currently working on. Update April 20th at noon; I am not planning to make any major additions (unless someone requests something that I think is reasonable). Make sure that you study the definition of congruent triangles and the two proofs that base angles of isosceles triangles are equal; this is also content relevant to the test. The coverage of the exam is chapter 3, except for the brief last section which talks about parallel postulates. Stuff in the notes is obviously fair game (I have added new comments and study questions throughout the notes, make sure you read them through again!). Anything you did in homework is fair game. Of course, anything in the book is, too, but we do have to take time limitations and level of difficulty into account. Happy studying! Tentatively, I am planning to be generous about supplying axioms and less generous about supplying definitions. Make sure you are familiar with important definitions. It would not hurt at all to review the axioms too.
Notes on Neutral Geometry
Here find the notes on neutral geometry. These will be updated frequently.
ADDITIONAL EXERCISE for assignment 2
Find a model of incidence geometry in which none of the three parallel postulates hold. The idea is that more than one of the three situations can hold: for example, you could build a model in which there is a line L1 and a point P1 with just one line through P1 parallel to L1 and a line L2 and a point P2 with two distinct lines through P2 parallel to L2. (The third situation is to have no parallels to a given line through a given point).
Postulates and definitions for section 2.3
The undefined notions of incidence geometry are point, line, and lies on (as in “point P lies on line L”).
postulate 1: for each pair of distinct points P and Q there is exactly one line L such that P and Q both lie on L.
postulate 2: for any line L, there are at least two distinct points P and Q such that P and Q both lie on L (there may be more!)
postulate 3: there are three distinct points P, Q, R such that there is no line L such that P, Q, and R all lie on L.
Those are the axioms. We also give a definition of parallel lines and three possible parallel postulates.
Definition: lines L and M are parallel iff there is no point P such that P lies on L and P lies on M.
Postulate 4a (Playfair's Postulate): for any point P and line L such that P does not lie on L, there is exactly one line M such that P lies on M and M is parallel to L.
Postulate 4b (Hyperbolic Parallel Postulate): for any point P and line L such that P does not lie on L, there are at least two lines M such that P lies on M and M is parallel to L.
Postulate 4c (Elliptic Parallel Postulate): for any point P and line L such that P does not lie on L, there is no line M such that P lies on M and M is parallel to L. (this is equivalent to saying that there are no parallel lines at all).
Geometry with three-point lines
Here is the description of the geometry with three-point lines that we developed in class:
line 1 ABC
line 2 ADE
line 3 BDF
line 4 FEC
line 5 GDC
line 6 AFG
line 7 GBE
Questions to think about before the next class: is this the same as Fano's geometry given in the book? Which if any parallel postulate does it satisfy?