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Precalculus Scope and Sequence


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Ds is working through Larson's Precalculus with Limits this year. We're using the generic Chalkdust materials. He has done really well with the material (homework and tests in the high 90%s) and is likely headed into a math intensive major.  (We're aren't in a check the box with higher level math and move on situation.) He spent some intensive time prepping for the ACT this past month  (which I don't regret) but we're now extending his school year into summer or we're cutting material. I'm ok either way.

My question is this: Is Analytical Geometry typically covered in precalculus? The remaining chapters in the book are: Topics in analytical geometry, Analytical geometry in three dimensions and Limits and an introduction to calculus.  He's already covered functions and trigonometry and we're heading into sequences, series, and probability. Is there an easy way to cut material? Is that a good idea? When I look at how others have handled precalculus topics, it seems like there's a roughly 50/50 split on whether they continue on to analytical geometry. 

He's chosen to use Larson's Calculus of a Single Variable next year so I'm leaning towards making him complete the entire book including the analytical geometry chapters this year. Ideas? There is NO one in my local homeschooling group I can talk about this with, so thank you for your patience if there is an obvious answer. 

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So I've just gone and checked the TOC. (went with this one -- https://www.sjaweb.org/ourpages/auto/2014/5/27/63078308/Precalculus with Limits _ 2E 2010 ISBN 9781439049099 _ Ron Larson_ Robert P_ Hostetler.pdf -- in case your numbering is different). 

First of all, he won't be crippled in calc if you just stop where you are. Other people may vary, but I'm going to give my opinion just as someone who sees what my calculus students struggle with. 

I assume (based on what you say about him) that he is solidly competent in basic algebra skills such as negative/fractional exponents, factoring, rational expressions/complex fractions, and graphical representation of functions. If he weren't, my advice would be more to review those. 

I've listed each chapter individually below, but I've summarized all the sections here. The ones I'd really hate to miss are 9.5 - 9.7, 10.1 - 10.4, 10.6, 11.1. The last two, just so that there is some basic exposure to coordinate systems beyond the standard cartesian plane. When I say "skim", I mean read the section, work through the examples and try to understand the big picture, and do at least the beginning computational problems. 

  • Chapter 9: I'd  look at arithmetic + geometric sequences and series -- they'll be re-done in calc 2 as part of a much broader overview but calc 2 students find sequences + series horribly difficult in general. Some base knowledge is a good idea, especially becoming familiar with the sigma notation for series. Very few precalc students are going to get anything out of the mathematical induction section -- I'd look through it (it's an early exposure to proof), but not worry about doing a lot of the homework. You can return to this one later if there's time. Binomial theorem I *would* do, and counting + probability -- it's not so necessary in calculus, but imho every functional adult should have some basic understanding of counting and probability. The binomial theorem is actually a huge help if you need to expand something like (x + y)^4. I also think I'd skim the "proofs in mathematics" at the end of the chapter.
  • Chapter 10: For analytic geometry, I would definitely cover lines, parabolas, ellipses (I assume circles fit in there as a special case?), and hyperbolas. Lack of understanding of these really hurts a lot of calculus students. Parametric equations and polar coordinates are also something that I'd like to include although the calculus textbook should also teach it. I'd at least skim the graphs of polar equations. I wouldn't worry about the rotation of conics and the polar equations of conics -- again, you can return to these. I don't even know if I'd skim them. We don't do a lot with them in calculus class, other than 90 degree rotations and polar equations of circles, and those usually aren't terribly difficult for people who understand the basic levels. 
  • Chapter 11: It's good to have some exposure to 3d geometry but it is customarily taught from scratch in calc 3 and/or linear (whichever comes first at the school) because many high schools don't teach it or don't teach it well. I would make sure to do the first section just so he has an idea that the cartesian coordinate system goes beyond two dimensions, and after that I would skim.
  • Chapter 12: The limits + intro to calculus will be completely retaught in calculus. This is the last chapter in the book. I think at least skimming it would be a good idea just to ease the transition to calculus, but if by then you are just ready to be Done it won't hurt him to stop. It'll just make the beginning of calculus a little bit easier if he's seen it once before. 

HTH. 

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