Pre-Calculus (Advanced Math) (Saxon)

This is David A. Wheeler's site for the Sovereign Grace Church (SGC) co-op Pre-Calculus class ("Advanced Math"), which uses the textbook Saxon "Advanced Math" (2nd edition, copyright 1996). This site has the latest versions of all class materials, including my detailed notes and examples for every lesson. This site is not officially sponsored by SGC or the co-op, but is provided in the hope that it will help the students in my class. Questions from my students are welcomed and encouraged!

I offer these notes to anyone who is using this textbook, in the hope that they will be helpful. I am delighted that others are finding these notes helpful - so if you find them helpful, let me know! Here's what one happy reader said: "I have been using your notes (for circular permutations [lesson 55] they were a big help). We live in a remote area of Florida and I don't know any others in this area that are working on Advanced Math." [MFurey]. One of the main reasons I teach the class is because I believe it is important for people to learn the basics of math... I cannot teach everyone, but this is a small way I can help.

Sadly, if you're not in my class (or at least SGC), please don't send me math questions - I'd like to help, but I simply can't answer math questions sent from everyone in the world (sorry!). The co-op is a service provided by SGC to church members; Sovereign Grace Church (SGC) Home Ed Blog (HEA) and HEA Blog co-op documents have the latest info.

A word about data formats: I normally edit my class notes in OpenDocument format (ODF), which is the international open standard for office documents (ISO/IEC 26300:2006). Files ending in ".odt" are OpenDocument text (e.g., word processing format). There are many programs that can read and write this format; two good free ones are LibreOffice and, both of which are available for MS Windows, Apple Macintosh, Linuxes, *BSDs, and Unix. (I'm told that Microsoft Office can also read them, but I have not checked.) If you just want to read the notes, you can read the PDF version (PDF is also internationally standardized); there are lots of PDF readers, many free. As a general rule I strongly encourage the use of open data standards (formats that aren't controlled by any one vendor), such as ODF, PDF, and HTML. I also use the international standard format for dates (ISO 8601), YYYY-MM-DD, which helps to avoid confusion in international communication - and it sorts well too.

Syllabus [PDF] | Syllabus [ODF] (It hasn't changed)

Calendar 2013-2014 [PDF] | Calendar 2013-2014 [ODT]

Suggested Schedule [PDF]

Mathematics: What & Why [PDF] | Mathematics: What & Why [ODF] (What Math? is a nice additional essay.)

Show your work (William Mulholland): [PDF] | [ODF]

Notes on Lessons (latest versions):

  1. Lessons 9-12 (plus summaries of 1-8) [PDF] [ODF]
  2. Lessons 13-16 [PDF] [ODF]
  3. Lessons 17-20 [PDF] [ODF]
  4. Lessons 21-24 [PDF] [ODF]
  5. Lessons 25-28 [PDF] [ODF]
  6. Lessons 29-32 [PDF] [ODF]
  7. Lessons 33-36 [PDF] [ODF]
  8. Lessons 37-40 [PDF] [ODF]
  9. Lessons 41-44 [PDF] [ODF]
  10. Lessons 45-49 [PDF] [ODF] (5 not 4 lessons)
  11. Lessons 50-53 [PDF] [ODF] (extra lesson from next week; here we transition to a format where I always present material before you encounter it in the text.)
  12. Lessons 54-57 [PDF] [ODF] (see also lesson 53 in previous handout)
  13. Lessons 58-61 [PDF] [ODF]
  14. Lessons 62-65 [PDF] [ODF]
  15. Lessons 66-69 [PDF] [ODF]
  16. Lessons 70-73 [PDF] [ODF]
  17. Lessons 74-77 [PDF] [ODF]
  18. Lessons 78-81 [PDF] [ODF]
  19. Lessons 82-85 [PDF] [ODF]
  20. Lessons 86-89 [PDF] [ODF]
  21. Lessons 90-93 [PDF] [ODF]
  22. Lessons 94-97 [PDF] [ODF]
  23. Lessons 98-101 [PDF] [ODF]
  24. Lessons 102-105 [PDF] [ODF]
  25. Lessons 106-109 [PDF] [ODF]
  26. Lessons 110-113 [PDF] [ODF]
  27. Lessons 114-117 [PDF] [ODF]
  28. Lessons 118-121 [PDF] [ODF]
  29. Lessons 122-125 [PDF] [ODF]
  30. Intro to Calculus [PDF] [ODF]
  31. Calculus Examples [PDF] [ODF]

I typically have notes on the lessons ready the night before, and make minor revisions within the day after class (based on comments during class).

As you do your lesson problems, do one problem and then check the answer (don't do all the problems at once). Otherwise you might practice doing it wrong. Also, if you make a mistake, work hard to figure out exactly why, and then figure out what to change so that you never make that mistake again.

Turn in tests on Sunday in the "HEA Tests" mailbox (lower rightmost box); mailboxes are in the foyer behind the information desk. Parents: Please do a preliminary grading; put a big "check" or "elongated C" (correct) mark by the correct ones, and a big "X" by the incorrect ones. That way, I can concentrate on figuring out partial credit, and you can have immediate information on how well they're doing. Here are some test-taking tips:

  1. Show your work! Show each step to the answer, not just the answer. I can't give partial credit with only the answer. And in the "real world", you'd need to show your work to some sort of peer review, to make sure that the answer is right.
  2. Use smaller steps, and more paper. If you try to do too many steps at once, you're much more likely to make mistakes, and it'll be harder for you to check your work later. It's also harder for me to help you. Paper's cheap, mistakes aren't; feel free to use lots of paper. I typically do only 1-2 problems per page. You don't need to use that much paper, but please don't worry about conserving paper; concentrate on getting the right answer.
  3. Check your work! If you solved an equation, plug it back in and see if you got it right. At least examine each step and make sure each one is correct. If you're worried about time, do all the problems and then go back and check your work... but check your work! This also means you need to circle your answer, so I can find it. This will mean that your work will take more paper, but as noted above, paper's cheap.
  4. Read and then skip problems you're not sure how to do, and come back to them later. That way, you can be thinking about them while you solve problems you do know how to solve. But be careful - don't forget to do them later! I write the problem number at the top of a blank page, and then start a new blank page. Before I turn it in, I count through each question (blank pages are really easy to spot), so that I'm sure I answered them all.
  5. For creating proofs, see my lesson 9 hints on proofs. Write down what you know, try to derive what you can "easily" derive, and then see if you can bridge the gap to what you need to show.

After this class, you may be able to answer some of the challenges on this page.

Related - here's a quick cheatsheet on Negative numbers (PDF) | Negative numbers (ODT). If you need this, you shouldn't be in the Advanced Math class - I created this for a junior high school student.

Algebra II is the leading predictor of college and work success.

Modern mathematics has been able to prove many amazing relationships, but like everyone mathematicians can make mistakes. There are efforts to formalize mathematics so that computers can automatically check math proofs to ensure they are correct. You don't need to know about these in detail, but I think you might find them interesting. One of the more intriguing projects to me is metamath. Metamath starts with the extreme basics: set theory (using the usual Zermelo-Fraenkel definitions), propositional calculus (which lets you prove statements about true/false statements), and predicate calculus (which lets you prove statements about objects). From there, they have formally proven that numbers exist and lots of properties about them. You can learn a lot about math just by looking at their proofs, in part because if there is some part you do not understand, you can click on that part to learn more (and repeat as necessary). For example, here is a proof that the product of two negative numbers is positive. They use common math notation:

Legal stuff: The Saxon book is copyrighted, so I cannot post extensive portions of the book. To avoid running afoul of copyright law, I create my own explanations and examples - which means these notes are my own copyrighted work, and are not a derivative of Saxon's work. Having things explained in a different way, with different examples, is better for teaching anyway (because it gives my students more examples to learn from). I do use the same order of instruction as Saxon's book - I don't see how I could avoid doing so while using Saxon as the textbook! I believe this is de minimus (so small as to not matter), but even if it isn't, this use of Saxon's outline is clearly fair use. After all, it is for the purpose of teaching, it's for nonprofit educational purposes, it's clearly transformative (I create new examples and explain things differently instead of just posting the outline), it emphasizes mathematical facts (which cannot be copyrighted), its content is quite different from Saxon's (it merely follows the same order), and it does not diminish Saxon's potential market (if anything, it increases it). Fair use is a critically important part of U.S. law.

You can also view my home page.