Case by case.

Maybe instead think about putting them into groups of four and then crossing off one from each group? Maybe that would make more sense to him?

Going off your further descriptions, I think AOPS sounds like a decent fit. AOPS uses intro to alg for their alg 1 and 2 courses and intermediate for their alg 2. So the intro book might be good reviewing/cementing/extension and furthermore mesh with his learning style as a 'I can figure it out myself durnit' type of person. There are discrete math for CS type books but most of them require more algebra than he's had. The AOPS discrete courses would (imo) be excellent for someone who's interested in CS as discrete has more applications in the CS area than calculus. I'd go ahead and buy one for a try-out -- ask him which one he prefers. If it goes well, you can buy the other.

For the Jacobs course, I would call it Math for Liberal Arts as that's what it's commonly used for at universities. Is there anything specific the student is interested in, or are you just gathering possibilities?

I don't think the triangle they are constructing the circles around is the triangle drawn in the pictures. If the vertices refer to the triangle in the pictures, then the diameter of the circle is equal to 14cm instead of the radius. I think the vertices that they are talking about are supposed to be the centers of the circles, and it is a confusing picture. ETA: In the picture, the distance from the vertex of the drawn triangle to the circumcenter of the triangle CANNOT be a radius of the circle in the picture. To solve it (if what I am thinking is correct), let C be the center of the circle (which is ALSO one of the vertices of the original triangle), and D the circumcenter of the triangle. AC is 14 because it is a radius of the circle. CD is 14 by definition, because it is the distance from the vertex of the triangle to the circumcenter of the triangle. Then continue with regentrude's solution.

I really don't think there's anything to worry about a 4th grader not being able to make the conceptual jump to algebra from bar diagrams. Many, many 4th graders are just not quite developmentally ready. If he still can't do it in 7th grade then I would be concerned.

I would go ahead and start in math -- not all of their curricula needs to move in lockstep. I wouldn't do a six week break, though -- I would do one lesson a week through those six weeks.

Besides shelving Singapore, another option might be doing, say, 5-10 problems of mixed review every day. Something like the evan-moor daily math practice?

!

But in my viewpoint, the mathematical thinking and reasoning is still to solve a problem, even if it is not explicitly posed as a mathematics problem.

FWIW, I agree with regentrude.

And what happened to the students who did not want to join the arithmetic class?

I really wouldn't rush math. She's a lot more likely to do well on the exam if she has a solid knowledge of SM 2a/b than if she has a rushed knowledge of 2a/b/3a. Especially for a child who struggled some in level 1, I would not skip. She will probably need the consolidation of the review. What you can do is: 1) work year-round, 2) work on saturdays. This will allow you to accelerate without skipping.

Is your nephew going to take the precalc course at his high school or try to test out? If he's going to try to test out, I would definitely recommend finding out what textbook they use and what chapters they cover and focusing on those.

FWIW, Bill I think that if you'd said something more like "I'd really recommend at least getting the workbook so that you have it available if your child ends up needing it. Some very bright children may do better without it, but I think it's beneficial for most people" it would have been fine. This would clearly have been you expressing your personal opinion, and there's nothing wrong with that. Where things get really contentious is where you seem to feel the inexplicable urge to proclaim "This is my opinion. And btw, anyone who doesn't agree with my opinion is wrong. Don't listen to them." If you worked on shifting your posts to the first format, not only would you find more listeners (because there are things worth listening to in many of your posts, once you get past the tone), but I strongly believe you would find less argumentation as well. FWIW, there were four of us in my family. Only one of us would sit down and do easy problems because they were easy and quick. As a matter of fact, she would have done well in Saxon. The oldest two of us did much, much better with minimal practice problems and then the really hard stuff. As a matter of fact, the school thought I was stupid in math because I'd daydream when the problems were too easy. I'm not sure if SM + IP would have been enough, because what she ended up doing with me was pulling me out of second grade and going directly to a pre-pre-algebra course (using an old edition of this book -- http://www.pearsonhighered.com/educator/product/Fundamental-College-Mathematics/9780321613424.page). The year after that I did pre-algebra, the year after that algebra 1 + 2, the year after that geometry + precalculus. I would rarely, rarely recommend this course to anyone. I have occasionally mentioned it on the rare occasions when someone has a child who seems to be in a similar situation.

Definitely these should be options. I especially think that a lot of 18 year olds would benefit from a few years working while deciding what they want to major in. At the university, I see so many students come in as 'undecided', when they are really there because it is the next step after high school. Quite a few of them fail out and leave with nothing other than $5k of debt and some bad grades on their permanent record (we're low-cost and most students have Pell). For these students, a job would have been a much superior idea. However, parents of academically capable kids should not decide for their kids, especially not while still in middle school, that they're going for a trade and therefore don't need a college-prep education. IOW, doors should be open as long as possible. Please note, I did say academically capable -- I have no issues whatsoever altering the educational trajectory for less capable students.

Math in focus is not Saxon -- the Saxon math your sister remembers predates math in focus by a long while. I guess you're looking at Saxonhomeschool and they do sell both their original program and Math in Focus (MIF is a pretty decent program)

I would definitely skip doing pages of 6-4 if he can do stuff like 65-18.

I would second the recommendation to try ALEKS, if online is desired. However, at his age, something like the Power Basics 'Basic Math' book might also work, as it was designed for students like him.

Singapore's good. If he likes math, I also highly recommend Math for Smarty Pants as a supplement. I love that book.

You could give a grade based on the chapter tests and daily work. If it's at all a recent edition, you can get chapter tests online which are multiple-choice -- for example, here's the 2005 one -- http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-845773-4&chapter=15&headerFile=6&state=

I am confused. Why does the lack of tests mean you can't put geometry on there?

Frankly I don't see it as a good investment for anyone. I think that someone who spends 50,000 on 'college coaching' to make sure they're 'fit' for ivy leagues is quite likely to end up overmatched and railroaded into a career they don't like either.

"Good college" varies. If you're going for Ivy League, well, you really do have to have something outstanding and even then it's a crapshoot. If you're fine with a decent state college, or community college then transfer, it really, really isn't necessary. For some kids it may be counterproductive -- the ones who are best matched at a state college but pushed to 'aim higher' -- they graduate and are just burned out.

You need hold music with a vuvuzela.