Rigorous Precalculus

[Julie of KY]

What's available for a rigorous Precalc course besides AoPS?

 

I've used AoPS Int. Algebra and Precalc for my oldest and I love AoPS. However, I don't think my second son would do well with those books. They'd just frustrate him too much, though he's quite capable of hard math.He will finish the AoPS Intro books and I will fill in a few more algebra 2 topics with somehting (any suggestions?) and then I'm looking at what to do for precalc.

 

I've looked some at Foerster's. One thing I notice is that a lot of problems are dependent on a graphing calculator. Is this a good thing to be teaching? or should I focus on something more like AoPS that doesn't require a calculator to solve problems. What are other rigorous choices?

[TheApprentice]

I have heard that Derek Owens's Pre-Calculus course is pretty good. I have no experience with that particular course. Two of my 3 sons have used some of his other courses though, and I have been very pleased.

[forty-two]

From the New Math of the 60s, Dolciani's "Modern Introductory Analysis" and Allendoerfer and Oakley's "Principles of Mathematics" were both rec'd on the board in past years. (I have them both on my shelf, but haven't used them; my grandpa actually taught from the Dolciani book (I have his teacher's edition) - turns out he was instrumental in getting his school to switch to New Math.)

[GoodGrief]

My daughter used Derek Owens precalculus, followed by his Calc AB.  She has gone on to get high "A"s in Calc 2 and Calc 3 at the local university.  I think it was good prep.

[8filltheheart]

My dd hasn't found DO's to be rigorous.  Adequate, yes.  But it does not compare in difficulty to the level of math she has done in the past.  Still, it has been a decent choice for her since math is her get it done subject.

[Julie of KY]

What about textbooks I can teach myself - I can teach any of the math, just haven't looked at textbooks.

[Mike in SA]

We use Swokowski, fwiw.  It's fairly classic in presentation and coverage, and well written.  We will enrich with AoPS or Gelfand as needed.

[Brad S]

I'm also looking for a rigorous precalculus course, although perhaps not AoPS.  In case you haven't seen it, on a recent thread, there's a brief summary of math books for various levels, http://forums.welltrainedmind.com/topic/542418-homeschool-high-school-math/ , roughly in sequence; if you scroll down a bit, you'll come to precalculus.  While it's a very helpful start, it doesn't really answer the question.

 

Rigorous can mean different things to different people.  I'm not looking for a text heavy on theorem and proof, although some would be nice.  I'm primarily interested in books that will have a lot of realistic problems across the natural sciences, social sciences, and other areas.  My DS likes a variety of real-world problems, and I like ones for him that are not just repetition and require a little thought. 

 

[kiana]

Rigorous can mean different things to different people.  I'm not looking for a text heavy on theorem and proof, although some would be nice.  I'm primarily interested in books that will have a lot of realistic problems across the natural sciences, social sciences, and other areas.  My DS likes a variety of real-world problems, and I like ones for him that are not just repetition and require a little thought. 

 

I think Foerster sounds like a pretty good fit for that.

[Brad S]

Thanks! Foerster seems like one good option.  Do you know how that would compare with Advanced Math: Precalculus with Discrete Mathematics and Data Analysis, 2003, by Richard Brown (ISBN: 0618250379)?  Would that also be reasonable?

[kiana]

Not a clue, sorry. Haven't seen the Brown book although I've seen other places use it.

[Brad S]

Thanks.  In response to some of Julie of KY's earlier questions, I think some use of a graphing calculator is a good thing; there's a lot you can learn playing around with those things.  In term's of self learning, I've heard good things about Lial's precalculus book, but I was under the impression that it wasn't as challenging as Foerster or Brown, let alone AoPS -- I may have understood incorrectly, but I wanted to try to answer Julie's question (and confirm the answer myself).

 

Back to Foerster, that was the best book for calculus I could find for DS (not AoPS, but still reasonably rigorous and with a variety of real-world applications), so I already bought Foerster's Calculus book.

[Julie of KY]

Brad,

Thanks for the reminder that there is a sticky of Homeschool math options at the top of the forum. I'm a math person and I've already taught AoPS Precalc to my oldest so I'm not intimidated to teach from anything else, but I'm looking for what is a good fit for my next son.

 

[Brad S]

Julie: My comments about the graphing calculator ("you can learn" could have been changed to "one can learn") were meant to apply to a kid's learning.  My DS has played around with his (TI-84 Silver) for years and learned some things through it and seems to have figured it on his own, except to write programs, which he needed to be shown.

 

For us, the precalculus text will probably come down to Foerster or Brown, unless a recent interest change in math leads us back to AoPS (their Intro to Algebra book was not enjoyed by DS at 10 y.o., so we changed texts).

[Julie of KY]

I agree you can learn a lot by playing with a graphing calculator.

 

I need to look more carefully at the Brown text. Also, I own the Larson text so I'll take a look at that more closely.

[Brad S]

My understanding is that Larson is less challenging than Foerster or Brown, but I may be wrong.  That doesn't mean it's worse. I also have Larson Precalculus Functions and Graphs (4th ed.).  To me, it doesn't look all that rigorous.  I didn't see anything on limits either.  There are about eight pages on math induction, which is nice to have (I haven't read it though).  I'd like to cover some data analyxis and discrete math during precalc, and it's light there, although I could supplement. I don't remember seeing much theory or proof in Larson, which is harder to supplement and still have things match up well.  To me, it seems like an ok text, but middle of the road rather than rigorous.  Sullivan seems a little more advanced than the version of Larson I have, and I was thinking that Foerster and Brown would be a step up from Sullivan.  Others may have other comments or corrections.

[kiana]

Larson and Sullivan are pretty darn close imo, and Lial is on the same level, as is Bittinger. All of them are very computational. From the edition I saw I'd consider Blitzer even more computational and I didn't care for it.

[EndOfOrdinary]

As someone who taught math at the highschool level, I would say two things about a graphing calculator:

 

1) you need to know how to not use one. Understanding and doing the physical graphing and solving is invaluable. Much like knowing multiplication tables in elementary versus using a calculator, graphing is equivalent. The magic box is only as powerful as the one pushing the buttons. If you do not completely understand what you are doing, the box is useless.

 

2) it is really, stinkin' nice to know how to use one sometimes. Certain tests are timed with the idea kids are using calculators. Certain college professors assign the amount of homework based on the idea of using a graphing calculator. Also, certain tasks are just plain annoying. Again, likening it to multiplication, I let my son do multi-digit decimal multiplication and division with a calculator. Yes, he knows how to do it long hand, but it is tedious. By using the calculator for his chemistry, he understands more of the chemistry and the problems do not take two sheets of paper. as an adult, I rarely write out longer operations; however, I also do quick ones in my head easily without using a calculator. That is the balance you are shooting for.

 

So by no means is it necessary and by no means should it be necessary. It is however one of those skills that can certainly take the pressure off later. If you can teach it, try to. If you can't, do not sweat it. Definitely do not let it become a crutch.

[Mike in SA]

So by no means is it necessary and by no means should it be necessary. It is however one of those skills that can certainly take the pressure off later. If you can teach it, try to. If you can't, do not sweat it. Definitely do not let it become a crutch.

 

Totally agree, especially with the last sentence.  You MUST be able to work without one, then let the tool serve its intended purpose, to free you from mundane tasks.

 

[kiana]

Agree with EoO. I would never do something like 123456789/987 without a calculator although I'm perfectly capable of doing so if it were necessary. But I see my students pick up a calculator to do things like 2 divided by 2. Seriously?!

[Julie of KY]

Totally agree about graphing calculators. My oldest has one and can use it well, but does all his textbook math without a calculator.

 

I'll take a look at the other texts. Thanks for all the suggestions.

[Brad S]

Foerster Precalculus: I've looked over Foerster's Precalculus (2^nd ed.), and I really like the presentation and coverage of topics (thanks, kiana!) and bought the text. I'm not sure yet if it will work as a text to be read from start to finish by DS, but it looks promising. It's not rigorous in the sense of being the perfect text for someone who's sure to be a college math major, but that doesn't match DS, and I agree with others above who think it's a step up in rigor from Larson, Sullivan, and many other precalculus books. It'd be “rigorous enough†for DS in my book, esp. with careful coverage of the two appendices (field axioms and math induction).

 

Another text, FYI: FYI, I did come across another precalculus book which seems to be at approximately a similar level and with similar coverage as Foerster, and which is used in the local schools as a supplement to lectures (Prentice Hall's Advanced Mathematics: A Precalculus Approach, 1993, by Ryan, Doubet, Fabricant, and Rockhill), but I won't use that one.

 

A wrinkle in our plans: To complicate matters, however, when I sat down with DS to discuss the math plan and to double check if he wanted to use AoPS, DS said he'd rather use AoPS. So, possibly back to the drawing board. Except when DS was 10, and he used AoPS Intro to Algebra for only chapter 1 before we learned he didn't have the maturity to struggle with the AoPS problems, we haven't used AoPS, so there would definitely be a transition. Using the AoPS placement tests, DS would be “ready†for AoPS Precalculus, but Intermediate Algebra would not be “only a review.†My inclination is to use AoPS Intermediate Algebra, and then AoPS Precalculus, but that stretches out the sequence for someone who's not likely to be a college math major. I was planning on having the AP Calculus BC test taken in 10^th grade; it'd be worth delaying for a year for AoPS, but I'd really like for DS to have the AP Calculus BC test taken by 11^th grade to include in a college application. I'm not sure how long it takes to cover these books, but I'm going on the assumption that (1) Intermediate Algebra is about a year, (2) a combined Intro to Counting and Probability and Intro to Number Theory would take about a year, and (3) AoPS Precalculus would be about a year. If so, we're probably left with a “pick two†at this point: (1) and (3) or (2) and (3). Or some other plan?

[Arcadia]

Using the AoPS placement tests, DS would be “ready†for AoPS Precalculus, but Intermediate Algebra would not be “only a review.†My inclination is to use AoPS Intermediate Algebra, and then AoPS Precalculus, but that stretches out the sequence for someone who's not likely to be a college math major. ... ... I'm not sure how long it takes to cover these books, but I'm going on the assumption that (1) Intermediate Algebra is about a year, (2) a combined Intro to Counting and Probability and Intro to Number Theory would take about a year, and (3) AoPS Precalculus would be about a year. If so, we're probably left with a “pick two†at this point: (1) and (3) or (2) and (3). Or some other plan?

(2) can be done over the summer or you can let your son use Alcumus for those. Intermediate C&P took longer for my older. So it is possible to do all of the above in two years. We did math everyday but took a week off here and there for vacations.

 

For placement test, I use Alcumus for that instead. it is more comprehensive then the placement tests. My boys use my account originally but now have their own.

 

AoPS calculus is a thin book in contrast to the other books in the series. It is about half as thick. My kids cover data analysis with statistics books. I actually like the Dover books for discrete math topics (ETA: graph theory, Boolean algebra) but it is fun enrichment for us so we aren't following any syllabus.

 

As for graphing calculator, my kid wants one to play at programming it than for math. He use Desmos to graph after he is done with the graphing problems for the fun of it.

 

What I do like about AoPS is that my boys can go on autopilot for math while I am down with seasonal allergies. Sorry if I don't sound coherent. My nose is running faster than my typing.

[raptor_dad]

From the New Math of the 60s, Dolciani's "Modern Introductory Analysis" and Allendoerfer and Oakley's "Principles of Mathematics" were both rec'd on the board in past years. (I have them both on my shelf, but haven't used them; my grandpa actually taught from the Dolciani book (I have his teacher's edition) - turns out he was instrumental in getting his school to switch to New Math.)

 

If you are looking at these books, you should also consider Serge Lang's  "Basic Mathematics"... A pre-calc text written by a Bourbaki member... hard to be more new math than that.

 

I wouldn't recommend any of these for the OP. I would view any of these as being at least comparable to AoPS.

[MarkT]

If you are looking at these books, you should also consider Serge Lang's  "Basic Mathematics"... A pre-calc text written by a Bourbaki member... hard to be more new math than that.

 

I wouldn't recommend any of these for the OP. I would view any of these as being at least comparable to AoPS.

On sale now 

http://www.springer.com/us/book/9780387967875

[kiana]

Foerster Precalculus: I've looked over Foerster's Precalculus (2^nd ed.), and I really like the presentation and coverage of topics (thanks, kiana!) and bought the text. I'm not sure yet if it will work as a text to be read from start to finish by DS, but it looks promising. It's not rigorous in the sense of being the perfect text for someone who's sure to be a college math major, but that doesn't match DS, and I agree with others above who think it's a step up in rigor from Larson, Sullivan, and many other precalculus books. It'd be “rigorous enough†for DS in my book, esp. with careful coverage of the two appendices (field axioms and math induction).

 

Another text, FYI: FYI, I did come across another precalculus book which seems to be at approximately a similar level and with similar coverage as Foerster, and which is used in the local schools as a supplement to lectures (Prentice Hall's Advanced Mathematics: A Precalculus Approach, 1993, by Ryan, Doubet, Fabricant, and Rockhill), but I won't use that one.

 

A wrinkle in our plans: To complicate matters, however, when I sat down with DS to discuss the math plan and to double check if he wanted to use AoPS, DS said he'd rather use AoPS. So, possibly back to the drawing board. Except when DS was 10, and he used AoPS Intro to Algebra for only chapter 1 before we learned he didn't have the maturity to struggle with the AoPS problems, we haven't used AoPS, so there would definitely be a transition. Using the AoPS placement tests, DS would be “ready†for AoPS Precalculus, but Intermediate Algebra would not be “only a review.†My inclination is to use AoPS Intermediate Algebra, and then AoPS Precalculus, but that stretches out the sequence for someone who's not likely to be a college math major. I was planning on having the AP Calculus BC test taken in 10^th grade; it'd be worth delaying for a year for AoPS, but I'd really like for DS to have the AP Calculus BC test taken by 11^th grade to include in a college application. I'm not sure how long it takes to cover these books, but I'm going on the assumption that (1) Intermediate Algebra is about a year, (2) a combined Intro to Counting and Probability and Intro to Number Theory would take about a year, and (3) AoPS Precalculus would be about a year. If so, we're probably left with a “pick two†at this point: (1) and (3) or (2) and (3). Or some other plan?

You don't have to do C+P/NT at all, or you could easily save them for a fun senior year elective while having done the calculus as a junior for college applications.

[HollyDay]

Oldest used Lial's pre-calculus with the cdroms.  She found it to be a good fit. 

[Homeschoolmom3]

I have heard great things about Foerster's talking with math majors...:)

[Mike in SA]

I have heard great things about Foerster's talking with math majors... :)

 

You'll also hear not-so-great things about it from math majors.  ;) 

 

Different strokes for different folks.

[Julie of KY]

I've got Forester's and I like the looks of it, however it annoys me every time I pick it up to try a chapter from it. It seems that for much of the book (or at least what I've looked at) it requires a calculator and the problems either can't be done by hand or are exceedingly annoying to do by hand.

 

I like the look of the Brown book. It may or may not be quite as rigorous, but I can always add to it if I want.

[Kathy in Richmond]

I've got Forester's and I like the looks of it, however it annoys me every time I pick it up to try a chapter from it. It seems that for much of the book (or at least what I've looked at) it requires a calculator and the problems either can't be done by hand or are exceedingly annoying to do by hand.

 

I like the look of the Brown book. It may or may not be quite as rigorous, but I can always add to it if I want.

 

I feel the same as Julie.  A few years ago I taught out of his precalculus text. While there was nothing wrong with it per se (not every kid needs AoPS type math), Foerster drove me insane with his parts to whole approach and the level of calculator usage required (calculate the answer to 5 decimal places...egad).