PreCalc - upping the challenge but not too much

[Matryoshka]

Dd15 has always really liked math, especially the conceptual side (definitely big-picture, not detail-oriented).  She did Singapore through 6A, then we switched to AoPS and she did PreAlg (online) and Intro Algebra (at home). She loved it. This all got a bit interrupted when she went to ps for a bit.  She went for most of 6th, and she managed to continue the AoPS class and start on Intro Alg in spite of that.  In 7th she was back home and finished up most of Intro Alg (except the last few chapters).  I'd heard AoPS Geometry was super-tough, so we switched to Wilson Hill for Geometry, but then 1/2-way through 8th she decided she wanted to try ps again (and go to ps high school).  No Geometry in 8th at ps, she took a placement test for their 'advanced Algebra' and bombed it, so she had to go back and take the 1/2-speed Algebra. :glare:  We'd planned to have her continue with WH online, but it became too much and she dropped it.  The school at least let her take the end-of-year Advanced Algebra (regular speed) test, and this time she aced it, so they let her take Geometry in 9th.  But she ended up hating ps and came back home, and she ended up finishing WH Geometry and to 'get caught up again' also took Derek Owens Algebra 2, which she's just finishing up.

 

So... what to do next.  Derek Owens was very easy for her.  She was already familiar with most of the topics from AoPS.  She has not been challenged, even with doing all the honors problems.  She just finished the last chapter, and mentioned that she finally hit something she didn't immediately understand, and she seemed almost a bit excited about it.  She'd like to take Calculus at the CC probably next fall, but what to do about PreCalc?  She really lights up about cool math concepts like imaginary numbers and logarithms, she says she likes teaching/figuring it out herself (she said she did more looking stuff up online for the DO course than watching his videos...)  I can tell that even though she was happy math was easy this year, she also was a bit bored by it.  She's not into competitive math or working really hard problems for fun, but I do think she'd enjoy stretching more.

 

I thought about returning to AoPS, but where?  It's so off-sequence, I wouldn't know what book to put her in or what sequence to follow to get her ready for Calc at the CC.  I suggested maybe she could stick with DO for PreCalc but we could come up with some challenge problems every week, and she didn't hate that idea, but where from?  AoPS isn't something you can pick and choose from...  I've got Foerster's PreCalc lying around, and Dr. Chung's SAT math book (which is supposed to be full of hard problems, but in relation to what?).  Any other ideas?  I actually have all the materials for Foerster's PreCalc to do at home, but she'd have to keep herself on track and teach herself - even if I felt like re-learning PreCalc, she doesn't want me teaching her... (or grading her tests)... I'm more of a stickler than DO about her showing her work, so she likes him waaay better than me...

[JanetC]

From the recent AOPS discussion... How about an abbreviated AOPS precal based on post #5 here:

 

http://forums.welltrainedmind.com/topic/361425-aops-or-college-algebratrig/

[tarahillmom]

I would go to the AOPS and do the placement tests.  Knowing trigonometry is incredibly important for success in Calculus.  I would also try calling the CC and see what they are using for Calculus texts and if an instructor there has any recommendations for books to best prepare for their classes.   

[regentrude]

I thought about returning to AoPS, but where?  It's so off-sequence, I wouldn't know what book to put her in or what sequence to follow to get her ready for Calc at the CC.  I suggested maybe she could stick with DO for PreCalc but we could come up with some challenge problems every week, and she didn't hate that idea, but where from?  AoPS isn't something you can pick and choose from.

 

 

Actually, the AoPS Precalculus text is pretty much a stand alone. You need basic algebra and geometry, but don't need to have gone through Intermediate Alg to be successful.

The trigonometry section is extensive and starts from scratch. Complex numbers are reviewed; she has seen those in Intro to Algebra.

I'd have her do AoPS precalc.

 

Note that most of this is not really necessary for calculus. In order to be prepared for calc, you need solid algebra skills and some knowledge of trig, but you don't really need most of Intermediate and most or Precalc. 

Sequences and series are important; those have been introduced in Intro to Alg. You can also review with the relevant chapters from Intermediate.

Edited September 26, 2016 by regentrude

[regentrude]

 Knowing trigonometry is incredibly important for success in Calculus. 

 

Some basic rudimentary trig, yes. Knowing since, cosine, tangent - if the course includes trig functiosn (not all calc 1 courses do).

However, for  calc 1, you don't really need fancy trig identities, addition theorems, Moivre formula - you only need a small portion of what is covered in the 8 chapters on trig in AoPS Precalc.

Now, calc 3 and complex analysis, that's when you want to have all the complex representations of trig functions...

 

Far more important is a good understanding of functions, the ability to graph, seeing asymptotes, etc.

But mostly, solid algebra 1 skills and a knowledge of series and sequences so the introduction of the concept of limits makes sense.

[Matryoshka]

From the recent AOPS discussion... How about an abbreviated AOPS precal based on post #5 here:

 

http://forums.welltrainedmind.com/topic/361425-aops-or-college-algebratrig/

 

 

Thank you, I hadn't seen that!  She actually says her preference is to go back to AoPS...

[Matryoshka]

Some basic rudimentary trig, yes. Knowing since, cosine, tangent - if the course includes trig functiosn (not all calc 1 courses do).

However, for  calc 1, you don't really need fancy trig identities, addition theorems, Moivre formula - you only need a small portion of what is covered in the 8 chapters on trig in AoPS Precalc.

Now, calc 3 and complex analysis, that's when you want to have all the complex representations of trig functions...

 

Far more important is a good understanding of functions, the ability to graph, seeing asymptotes, etc.

But mostly, solid algebra 1 skills and a knowledge of series and sequences so the introduction of the concept of limits makes sense.

 

Hmmm... so do you think she could actually skip the AoPS intermediate Algebra altogether?  She actually hasn't had Sequences & Series yet, it was one of the chapters we skipped back in Intro Algebra, but we could go back and do it now.  I'm wondering if that might even be a good way to get her to see if she really likes AoPS as much as she remembers (it's been 2 full years since she's used it) without spending a lot of $$ on new books.

 

Wouldn't she need to have Conic Functions in PreCalc, though?  That appears to be only in the Intermediate Algebra book?  The Sequences and Series in Int Algebra also seems to go further - and there's also Piecewise Defined Functions that's only in that book (I remember older dd doing this in PreCalc)?  Anything else in the Int. Algebra that wouldn't have been touched on yet in a more standard Algebra II course?  I know she's done Polynomial Division and things like Irrational Roots already...

 

ETA: And where is the concept of Limits introduced in AoPS?  I've got to think it's somewhere in there, but Limits aren't mentioned by name in any of the TOC's of Intro or Int. Alg, or PreCalc - where's it hiding?...

Edited September 26, 2016 by Matryoshka

[regentrude]

Hmmm... so do you think she could actually skip the AoPS intermediate Algebra altogether?  She actually hasn't had Sequences & Series yet, it was one of the chapters we skipped back in Intro Algebra, but we could go back and do it now.  I'm wondering if that might even be a good way to get her to see if she really likes AoPS as much as she remembers (it's been 2 full years since she's used it) without spending a lot of $$ on new books.

 

Wouldn't she need to have Conic Functions in PreCalc, though?  That appears to be only in the Intermediate Algebra book?  The Sequences and Series in Int Algebra also seems to go further - and there's also Piecewise Defined Functions that's only in that book (I remember older dd doing this in PreCalc)?  Anything else in the Int. Algebra that wouldn't have been touched on yet in a more standard Algebra II course?  I know she's done Polynomial Division and things like Irrational Roots already...

 

ETA: And where is the concept of Limits introduced in AoPS?  I've got to think it's somewhere in there, but Limits aren't mentioned by name in any of the TOC's of Intro or Int. Alg, or PreCalc - where's it hiding?...

 

I think doing sequences and series will be helpful, because that is how most calculus courses start, since this is a logical area to introduce the concept of limits. Limits are not introduced prior to calculus.

 

You can do precalc without conic functions. I would recommend covering conic functions if she has not done so in her geometry course, since this is very useful. Piecewise defined functions should be covered before calculus because she will encounter such functions which are great examples for functions that may approach different values if you approach the same spot from different sides, a concept that is discussed with limits early in calc. There is nothing TO piecewise functions; she should just have seen some.

 

She won't need all the polynomial stuff in order to do calc 1; a little bit is useful in calc 2 when it comes to the partial fractions method; it is useful to see how to factor polynomials. But most of the myriad of theorems on polynomials she'll never need for anything.

[katilac]

Have her check out the Just in Time Algebra and Trigonometry for Calculus book: 

 

https://www.amazon.com/Just---Time-Algebra-Trigonometry-Calculus/dp/032167104X/ref=sr_1_1?s=books&ie=UTF8&qid=1475031258&sr=1-1&keywords=just+in+time+algebra+and+trigonometry+for+calculus

 

It's meant to teach and refresh algebra & trig topics as they come up in calculus. I would imagine it would be helpful for pre-calc as well. My dd has the transcendental version, so maybe look at both. I have no idea of what the difference is, I just know that the calculus dd is taking has the word "transcendental" in it. 

 

 

[MarkT]

Have her check out the Just in Time Algebra and Trigonometry for Calculus book: 

 

https://www.amazon.com/Just---Time-Algebra-Trigonometry-Calculus/dp/032167104X/ref=sr_1_1?s=books&ie=UTF8&qid=1475031258&sr=1-1&keywords=just+in+time+algebra+and+trigonometry+for+calculus

 

It's meant to teach and refresh algebra & trig topics as they come up in calculus. I would imagine it would be helpful for pre-calc as well. My dd has the transcendental version, so maybe look at both. I have no idea of what the difference is, I just know that the calculus dd is taking has the word "transcendental" in it. 

I borrowed "Just in Time Algebra and Trigonometry for Calculus" from our library.  It looks good overall. It maps the Calculus topic to the corresponding needed Algebra and Trig. The examples are reasonable but some are just Algebra 1 (if you need Algebra 1 help maybe you shouldn't be taking Calculus).  It has a lot of white space for 216 pages and is over-priced at $32 new because it was priced for college bookstores by Pearson.  Buy used or borrow.

 

This type of booklet would better if the authors just direct published via Lulu or similar and priced it around $15 (it's value).

Edited October 5, 2016 by MarkT