SECOND SEMESTER:
4/10: Work on the handout from class. To get started, find F_5 by subtracting areas. Then draw corresponding figures for F_6, F_7, etc.
3/29: p.170 #33, 36, 37, 39, 41, 43, 45, 56-58. Quiz Monday.
3/28: Finish the problem I wrote on the board.
3/26: In the HARDCOVER, p.168-169 #5, 6, 8, 11, 13, 16, 17, 19, 21, 23
2/29: Don't forget to bring the mone
y for the field trip b Frida, March 2nd. It costs $16.50.
2/14: Try to find the flaw in the attempted proof I handed out in class. There will be a proof-based quiz on Thursday.
2/8: Work on the handout. If you can't prove Ceva's Theorem, try to prove property (1) , the one about fractions. You can also try to prove the corollaries without proving the theorem itself.
2/7: Finish the proof I started in class. For those of you who were absent, read through Euclid's proof of Proposition VI.3.
1/31: For the quiz on Thursday, know how to find the length of a line segment (equivalently, the distance between two points) given coordinates. Also know the equation for a circle with center (h,k) and radius r. That's it.
1/30: In the hardcover, p.638 #3-7, 9,10,12,19,22,27,31-35,40.
1/24: In the hardcover, do p.22 #34, 35, 38, 40, 41. Bring the hardcover to class on Friday (we won't need it Thursday).
1/12: Finish the handout. Remember, this is Euclid III.35, so if you get stuck, see how the proof is done there.
1/9: Finish finding the errors in the false proof sheet I handed out on Friday. The error in the second will likely be different depending on whether you know complex numbers or not. Even if you don't, there is still a subtle error that you probably never thought of before.
1/6: Work on proving the theorem I wrote on the board. Also, try to find the errors in the "proofs" I handed out in class. I'll collect these on Monday.
1/5: Read through the proof of Proposition 3.21. This is an easy consequence of Proposition 3.20, so it makes sense that the proof is short and relatively easy. Those with the Dover edition may wish to read the commentary following the proof.
FIRST SEMESTER:
9/8: Think about the problem from class, which was to find 1+2+3+...+98+99+100 (the sum of all integers from 1 to 100). Remember that there's a very short solution. Don't add them all the long way- the whole point is to avoid that. Don't worry if you don't get it. Just give it a shot and think about it. I'll give you one (small) hint: That triangle of dots I drew would look better if it was a rectangle...
9/9: Work on getting the Green Lion edition of Elements if you haven't already. We'll begin using it around the middle of next week.
9/12: Represent the identity (a-b)^2=a^2-2ab+b^2 geometrically. ^ means raise to the following power, so a^2 just means "a squared". As a bonus, try to explain why 1+1/2+1/3+1/4+... can be made larger than any number if enough terms are added. Hint: 1/3 +1/4 is greater than 1/4+1/4 and 1/5+1/6+1/7+1/8 is greater than 1/8+1/8+1/8+1/8.
9/14: Continuing what we started in class, construct an isosceles triangle which has its two equal sides three times as long as the remaining side and do your best to prove that your construction works. Remember that this is a straightedge and compass construction and we're limited to basically drawing circles and straight lines. I don't care too much if your proof is flawless, just write down the steps in your construction (there may be more than 4).
9/15: Read the proof of proposition 1.5 that I handed out. Make sure you understand what the theorem is saying (sometimes reading the first few lines of the proof will clarify this) and come to class with any questions you have. IMPORTANT: You'll see the reasoning behind each statement in the proof. C.N 3 means Common Notion 3, I.3 refers to propostition 1.3, Def. 12 is Definition 12, Post. 4 is Postulate 4, and so on. Common Notions, Postulates, and Definitions can all be found before Book 1.
9/16: Study the proof of proposition 1.6. Don't just give up and say you don't understand it (it's easier than the first one I did on the board today). Help me out and come with specific questions like why does Euclid do this? why are these two sides of equal length?, etc.
9/17: Study the proof of I.7. We'll skip this in class and prove I.8 a different way, but the proof of I.7 gives good practice with the theorems (propositions) we've proved so far. I'll likely include a skeleton version of this proof on a quiz and have you fill in the details.
9/20: Go through the proof of I.9. There will be a quiz on Friday consisting of one proof. I'll type up a skeleton proof and you'll have to fill in some details. You'll also be given a handout of the propositions we've proved so far so don't try to memorize the proposition numbers.
9/21: Review the proofs we've done so far (review from Euclid, as my proofs on the board are a little informal at times). Make sure you understand the proofs of I.1, I.5, I.6, I.8 and I.9 in particular. Practice quiz tomorrow.
9/27: Look through the proofs of I.11 and I.12. Also, you should have the Larson book by now- we'll be doing some problems from it starting tomorrow.
9/28: p.47 (in hardcover) #14-19(all), 28-36 even.
9/29: Reread the proof of I.16 in Elements, and read the proof of I.17.
10/3: Read through Euclid's proof of I.18. This is an easy proof and you should be able to follow it by now. Remember, quizzes on Wednesday and Friday.
10/5: Read through the proof of I.19. Note how easy it is because of the previous theorems we've proved.
10/6: Quiz tomorrow. It will be open book, open notes. To study for it, review the proof of I.16 and remember that in class we didn't quite finish the proof.
10/11: Read through Euclid's proof of I.24. We'll briefly discuss I.22 and I.23 in class. They're constructions, so they're crucial to Euclid's development of the theory but not so much for us.
10/13: Read Euclid's proof of I.26. This is an extremely important theorem, adding ASA and AAS to our two existing triangle congruence theorems, SAS and SSS. Even if you get lost in the proof, make sure you understand the content of the theorem.
10/14: Read a little bit about the parallel postulate. This is a good start. If you have the Dover edition of Elements, start reading on p.202. Also read this article about proposition I.16.
10/19: Read through the proofs of Propositions I.27 and I.28.
10/20: Read through the proofs of I.29 and I.30. We already started 1.29 in class.
10/21: In the hardcover textbook, p.146 #13-23 odd, p.153 #3-17 odd. Bring this book to class on Monday.
10/24: Study for the quiz tomorrow. It's open book (Euclid only) and open notes.
10/26: Read through the proofs of Proposition I.30 and I.32. I.31 is Euclid's construction of a parallel line, which we won't particularly emphasize. If you were in class today and you understood the proofs, just skim over the theorems quickly.
10/27: Read the proofs of propositions I.33 and I.34.
11/1: Try to find the area of a trapezoid using techniques similar to the ones we used in class.
11/2: Read Euclid's proof of the Pythagorean Theorem (Proposition I.47).
11/3: Read section 9.2 (starts on p.535) in the hardcover. Make sure you understand the examples and do #1, 2, 7, 10, 12, 18, 32 on p.538-539.
11/14: In the hardcover, p.546 #8, 9, 11, 13, p.549 #5, 6, 7, p.554 #12, 13, 18, 20. A calculator will be useful.
11/16: Quiz tomorrow! Open everything. Good practice problems are p.554 #14, 16, 17, 19 (these are the special triangles described in my handout and the hardcover), p.580 #54-56, 2 (you can keep the answer exact, not rounding required), p. 582 #4 (don't worry about Pythagorean triples), 7-10 (if it's a right triangle, say so. Don't worry about acute/obtuse), 13, p. 585 #13
11/17: Work on the angle trisection proof that I handed out earlier in the week. Bring the handout and your work to class tomorrow.
11/18: Try to prove the other case (the top one) on the angle trisection handout. I'll collect your work on Monday. Also, try to find the flaw in the supposed proof that all triangles are equilateral. Remember that it has something to do with the diagram. It may be a good idea to draw a more accurate one (Geogebra is perfect for this, or good old straightedge and compass manual work will do) and question EVERY single conclusion in the proof- one of them is clearly wrong!
11/28: Short quiz tomorrow, similar to the one from last week (Pythagorean Theorem, special triangles). We'll pick up with Book II on Wednesday.
11/30: Finish the "FOIL" problems that I wrote on the board. Tomorrow we start Book III, so bring your copy of Euclid.
12/1: Try to follow the method demonstrated in class to complete the square for the expressions i) x^2+4x+4, ii) x^2+8x-5, and iii) x^2+2x+4.
Remember that x^2 is read "x squared". Tomorrow, for those of you with the Dover books, bring the second volume of Elements.
12/2: Complete the square for i) x^2+10x-3, ii) x^2-4x-9. Challenge problem: represent competing the square for 2x^2+5x+8 geometrically (also do it algebraically).
12/5: Quiz tomorrow on distributing ("FOILing") and completing the square.
12/6: Bring your copy of volume 2 of Elements to class every day, or the Green Lion edition if you have that. Tonight, read through Euclid's proof of Proposition III.3
12/8: Read through the rest of the proof of proposition III.7. We started it in class and we left off with the hint that proposition I.24 will help. FOR THE FINAL, the best way to study is to look through all your previous quizzes. It will consist of one partial proof, similar to the previous quizzes. You will be able to use your notes and Elements, but you should know all the major theorems we've covered (congruence theorems, side/angle relationships, parallel line theorems, Pythagorean Theorem, etc.). My Book One notes will help here. The other part will be more computational, including completing the square and distributing problems.
12/10: Good problems to look over in the hardcover: p.114 #27, 28, p.146-147 #11-26, p.154 #20-27, p.161 #15-24, p.538-539 #7-30 (don't worry about Pythagorean Triples- we didn't mention them), p.546 #8-13, p.554 #9-20. Also make sure you know how to complete the square.