s/o math sequence for accelerated student?

[Halcyon]

Can someone give me an idea of their math sequence if their child completed a solid Pre-A course in 5th grade? Thank you.

[lewelma]

Here is the plan, but not set in stone. We are 1 year ahead of you, so not sure if this will help.

 

My ds did AoPS Intro Algebra in 5th and 6th (2+ years to finish because it is much more than algebra 1)

7th AoPS geometry + Number Theory

8th AoPS intermediate Algebra + probability

9th AoPS PreCalc + intermediate probability and number theory (precalc is only a 1/2 year course)

10th AoPS Calc

11th Calculus at university

12th Stats at university

 

He wants to do Calculus in 10th so he can do Calculus based physics in 11th. He plans to major in Math or physics.

 

Ruth in NZ

[wapiti]

Kathy in Richmond's kids' math sequences: http://forums.welltr...t/#entry1559496

 

It's early for us, but dd is in her first of two years of algebra at her middle school, following AoPS Prealgebra in 5th. Plus, she's enjoying math club and she's looking forward to trying out the MathCounts chapter competition in a couple months :)

[8filltheheart]

 

MUS alg/geo

Foersters alg

CD's Houghton Mifflin geo

Foerster's alg 2

AoPS Intro to C&P

AoPS alg 3

AoPS pre-cal

AoPS cal (followed by AP BC exam)

university multivariable cal this semester

next semester--linear alg

next yr--diffEQ, ???, statistics, (he will take diffEQ and whatever next math his advisor recommends at the university and stats wil be either PAH AP or self-studied AP)

 

[regentrude]

DS:

6th AoPS Intro to Algebra ch 1-12

7th AoPS Intro to Algebra ch. 13-15; AoPS Intro to Counting and Probability

8th AoPS Intro to Algebra - finish (1st semester); begin Intro to Geometry in spring

 

DD's sequence (she was pulled out in 6th and could only begin algebra 1 in 7th. She skipped 8th grade):

7th AoPS Intro to Algebra - entire book

8th (which we labeled 9th with the grade skip): AoPS Intro to Geometry, begin Intermetiate Algebra

9th (10th w/ skip): Intermediate Algebra, Precalculus

10th (11th w/skip): AoPS Calculus/ Stewart Calculus

next year we plan multivariable calculus and differential equations

[cassiemc]

So far we've done:

5th grade~TT 7th

6th grade~ Kahn Pre-Algebra, He tested into Algebra 1 after finishing TT 7,but we wanted to make sure he covered what was needed so we did the Kahn lectures thru the summer and then completed TT Algebra 1

7th grade~ He is now working on geometry thru completecurriculum.com

 

He will probably go on to Algebra 2 next year. I am looking at some AOPS courses and then Pre-Calc and Calc

[Halcyon]

Okay so how does this look

5th: Dolciani Pre A

6th: Dolciani Algebra 1

7th: Dolciani Algebra 2

8th: Geometry (not sure which one) or Intro to Number Theory

9th: Pre Calc and Algebra 3

10th: Calc and Statistics

11th: Multivariable Calc

12th: Linear Algebra??

 

I get lost around 10th grade.....

[regentrude]

Okay so how does this look

5th: Dolciani Pre A

6th: Dolciani Algebra 1

7th: Dolciani Algebra 2

8th: Geometry (not sure which one) or Intro to Number Theory

 

 

Geometry is a must. If you don't do it in 8th, you must schedule it for the next year.

 

9th: Pre Calc and Algebra 3

 

 

What is algebra 3??? Precalculus is a one year course that usually contains the contents of college algebra (which some call algebra 3) and trigonometry.

 

10th: Calc and Statistics

 

 

If you do a rigorous calculus course, comparable to AP BC, calc and statistics will be too much.

 

11th: Multivariable Calc

 

Does not need a full year.

 

12th: Linear Algebra??

 

 

Generally, I would not plan this far ahead, but base the upper end of the math sequence on my student's interests.

For example, my DD is interested in physics and wanted to take calc based physics - so she had to have finished calculus before and did not take detours into discrete math. Because differential equations are much more important for physics than linear algebra (the highly specialized topics that actually need linear algebra will not be encountered until grad school), she will take diff eq in senior year.

OTOH, if I had a student interested in computer science, I would most definitely fit in statistics and discrete math, and focus on numerical methods in the calculus course.

A student with an interest in a math major would take number theory and probability.

[kiana]

I would not worry so much about having courses planned for senior high school. There *will* be courses of interest, and the precise courses will depend on your student's intended major. Your student might also need extra time to process, and take more than a year to finish algebra 1. (This is not at all unusual when starting young). That is perfectly okay! Don't try to lock into a progression now.

[8filltheheart]

What is algebra 3???

 

AoPS refers to their Intermediate Alg class as alg 3.

[8filltheheart]

Okay so how does this look

5th: Dolciani Pre A

6th: Dolciani Algebra 1

7th: Dolciani Algebra 2

8th: Geometry (not sure which one) or Intro to Number Theory

9th: Pre Calc and Algebra 3

10th: Calc and Statistics

11th: Multivariable Calc

12th: Linear Algebra??

 

I get lost around 10th grade.....

 

I wouldn't make a plan that far out either. For example, our ds's load w/his other classes was too heavy to manage the pace of AoPS's precal class in a single semester. He spread it over his entire 9th grade yr. He was very glad that he made that decision.

 

Since you don't know what his daily life will look like that many yrs from now, it is impossible to predict what pace he will move through the books.

 

Looking at from this end of the journey, really, there is no need to get this far this fast unless a student does it completely under their own incentive. Ds set his own pace.....I had nothing to do with it.

 

As my kids get older, I like to work w/them in developing a plan that respects their vision--their goals and abilities. What a child is like in 5th grade is nothing like what they are like in high school. They change a lot.

[Beth in SW WA]

Can someone give me an idea of their math sequence if their child completed a solid Pre-A course in 5th grade? Thank you.

 

 

I agree with others not to plan too far ahead. Life happens. This year isn't quite turning out like I originally planned. But I'm rolling with it. Dds thoroughly enjoy our neighborhood ALE where they are now attending on Mon & Wed all day. The math is great review of the basics with a heavy focus on problem solving -- and the girls LOVE it. That gives me 2 full days to work on projects of my own. :)

 

Dd9 didn't do a formal prealg. We used various resources that got the job done.

 

grade 4 --

TT Alg 2

3x/wk algebra tutoring via Cybershala

1x/wk algebra tutoring via Crewton Ramone

 

grade 5 -- finish TT Alg 2, more algebra online classes, local tutors?

 

grade 6 -- public school algebra?

 

grade 7-- public school geometry?

 

 

I have CD & Aops if needed for supplementation. We're in no hurry.

 

HTH! :)

[mumto2]

I agree with what others have said about not planning too far ahead. Your chidren change as they grow and develop. Recently I have had to become more understanding of this.

 

My dd likes to be involved in her own planning at this point. She has a good idea of what she is interested in exploring mathwise next and I plan on letting her be interest led for the next few months. She is finishing LOF calculus in the next week and plans to look at our bookshelf for her next text.

 

Ds needs more direction and is having a harder time because he does not like to write out his math work. It appears to be catching up with him in the jump up to advanced algebra. We are taking a step back starting today to mom led exercises concentrating on writing your work down. Not much forward progress will be made in the immediate future. I am having a hard time with the fact he won't be where dd is in 2 years -- dh pointed out he does not have to be. He does have to be accurate to achieve his goals which means a deeper understanding and not doing it all in his head.

[ccolopy]

My DS is only in 6th, but I can already tell you that the previous posters are right - we've changed our plans several times since DS started algebra 1. We're planning a switch to AoPS, but if that doesn't work out, we'll be back to the drawing board. Right now our sequence/plan is:

 

4th: finished pre-algebra, started Algebra 1 (Foerster)

5th: finished Algebra 1, started Algebra 2 (Foerster) and Geometry (Jacobs)

6th: finish Algebra 2 and Geometry, AoPS Introduction to Counting and Probability

7th: AoPS Geometry, AoPS Introduction to Number Theory

8th: AoPS Intermediate Algebra

9th: AoPS Precalculus

10th: AP Calculus

11th & 12th: CTY or university math courses

[KAR120C]

I aimed for a "middle ground" between too much planning and too little (lol.... I like knowing where we're headed! but I promise I don't stick to plans that aren't working...), so my intention when DS was reaching Algebra very young was to alternate years - one year of "progress" (standard curricular stuff) followed by one year of "tangents" (Counting & Probability, Number Theory, Stats, etc.)

 

It still didn't quite work out that way.... We did algebra, it went well, and then we switched to statistics... which requires a couple topics from algebra 2... no problem... but also we had a great co-op group doing Zome Geometry, so that year ended up being Statistics AND Algebra 2 AND Geometry. I had another year of geometry for him (there's plenty of geometry to go around... lol), but really between what of algebra 2 we picked up for stats and what he picked up to use elsewhere (science, math competitions, etc.) there wasn't much left of Algebra 2 by then. After Geometry we did AoPS Counting & Probability and Number Theory, and then a year that I meant to call Algebra 2 but ended up being more Precalculus.... and at that point, he wanted calc-based physics, so we jumped right into BC Calc and to heck with the plan... lol...

 

Like Regentrude's DD, my DS is interested in the physics applications more than the pure math... so after multivariable I expect he'll do DiffEQ. And after that? no idea. He may fall in love with computer science finally (I keep expecting him to...) and go back for more discrete math... Or he might need something specific for a particular science use (that's usually what drives our direction around here...) and whatever that is... I guess we'll find out when it shows up! If he's not too particular, I'll encourage him to do another year of discrete math and a calc-based stats class... and linear algebra somewhere in there.

 

BTW... my general approach would be to put statistics before Precalc. AP stats doesn't require anything more than Algebra 2, and I don't like to interrupt the Precalc-calc1-2-3 sequence if I can help it. There may be more that should be in that uninterrupted run, but that's as far as I got myself... so that's as far as I can advise.

[boscopup]

We'll be doing prealgebra in 4th and likely algebra in 5th, so I have thought about this, not wanting to get to calculus super early unless something comes up that makes that a good choice.

 

Very tentative, not at all set in stone, just brainstorming type hypothetical schedule that assumes each course takes one year, though I may do algebra over 2 years:

 

4th - Prealgebra

5th - Algebra

6th - Counting&Probability/Number Theory

7th - Algebra II

8th - Geometry

9th - Algebra III (AoPS definition)

10th - Precalculus

11th - AP Calculus BC

12th - Calc3/Diff Eq. (at university)

 

Again, this is laid out just as a "we could do this..." type deal, not at all what we plan to do. ;) I'm only currently planning for prealgebra starting around April. From there, we'll go whereever I feel like we need to go, based on how he's doing with whatever we're using. I just needed to lay out a possible plan at one point, so that I knew if I was going to be stuck doing calculus in 9th grade or if we could push it out to 11th or 12th. I don't foresee THIS child needing calculus in 9th. We'll see what his career plans end up being (he's wanted to be a "storm chaser" for a long, long time, and I can totally see him being a meteorologist that does research, so he'll need math, but not calc in 9th if he continues on that path).

[jennynd]

DS did

SM 5b 6 a/b with CWP and IP and key to algebra

Now

( current) AOPS algebra ch 1-13 with NEM 1 as review

 

Next

- counting/probability/ part of geometry with NEM 2 as review

- AOPS algebra ch 14-end NEM 3

--- finish NEM 4 and AOPS geometry

 

then....supposed pre cal? not so sure from this point on... I will like to concentrate on algebra based science from this point on.

[Crimson Wife]

Haven't totally decided on where I'm going after this year because while I like Singapore Discovering Math, I'm not sure how long I'm going to be able to teach it.

 

5th: Singapore DM 1 & Horizons Pre-A

6th: Finish DM 1 & Horizons Pre-A, start DM 2 & Horizons Alg. 1

7th: Finish DM 2 & Horizons Alg. 1, ??? start DM 3 or maybe a quick run-through of a geometry text to hit the topics not covered in DM 1 & 2.

 

Beyond that I expect to have to outsource. Cybershala uses NEM so that's an option for continuing with Singapore. However, I'm not sure how favorably that would be viewed on a college application compared to a more well-known program like EPGY or CTY. I do want her to get through the equivalent of the post-AP math course at my alma mater so that she will be a competitive applicant.

 

AOPS wouldn't be a good "fit" because of the discovery approach.

[mathwonk]

Okay so how does this look

5th: Dolciani Pre A

6th: Dolciani Algebra 1

7th: Dolciani Algebra 2

8th: Geometry (not sure which one) or Intro to Number Theory

9th: Pre Calc and Algebra 3

10th: Calc and Statistics

11th: Multivariable Calc

12th: Linear Algebra??

 

I get lost around 10th grade.....

 

 

conceptually, linear algebra belongs before multivariable calculus. i.e. one should learn linear functions of several variables before non linear ones, not just because one should walk before one runs, but because the subject of differential calculus means using linear functions to approximate non linear ones. so it is impossible to understand or to treat properly, multivariable calculus without linear algebra.

 

i am fully aware that is often done, but many wrongheaded things are often done, and this one is an anachronism we have spent decades trying to stamp out in math departments at university.

 

just a remark inspired by your question mark.

[boscopup]

 

 

conceptually, linear algebra belongs before multivariable calculus. i.e. one should learn linear functions of several variables before non linear ones, not just because one should walk before one runs, but because the subject of differential calculus means using linear functions to approximate non linear ones. so it is impossible to understand or to treat properly, multivariable calculus without linear algebra.

 

i am fully aware that is often done, but many wrongheaded things are often done, and this one is an anachronism we have spent decades trying to stamp out in math departments at university.

 

just a remark inspired by your question mark.

 

 

I only vaguely remember linear algebra (or multivariable calculus, for that matter), but I remember thinking, "This is really easy. Why is it after calc3 and diff. eq.?" We did learn parts of it in earlier courses - probably the parts necessary for multivariable calculus. It wasn't all new to me. But I just thought the whole course was ridiculously simple and seemed out of place when I took it.

[jennynd]

 

I only vaguely remember linear algebra (or multivariable calculus, for that matter), but I remember thinking, "This is really easy. Why is it after calc3 and diff. eq.?" We did learn parts of it in earlier courses - probably the parts necessary for multivariable calculus. It wasn't all new to me. But I just thought the whole course was ridiculously simple and seemed out of place when I took it.

 

 

I am really at loss here. When I was in college for mechanical engineering, after calculus, we had "engineering math I and II", which puts everything together. In graduate school in US, we had Math 1 and math II and then numerical mathod. I never had math got seperated like this. what each topic covers??

[boscopup]

 

I am really at loss here. When I was in college for mechanical engineering, after calculus, we had "engineering math I and II", which puts everything together. In graduate school in US, we had Math 1 and math II and then numerical mathod. I never had math got seperated like this. what each topic covers??

 

 

Strange! I went to schools in the US and Canada (I took linear algebra in Canada). I was a EE major. They separated everything out. Engineering majors had to take certain courses, but they were still labeled the type of math it was. So my freshman year I did calc3 and differential equations in the US. I can't remember if I took linear algebra my softmore or junior year. I think those are the only math courses I had to take, actually. It's been a while though, so my memory is quickly fading! :D I know I didn't have to take any math courses my senior year at a school in the US. I did, however, have to take EE100 that year, because "No other school has this course." :rolleyes: That course was otherwise known as "homework period" for me, since I'd taken 400 and 500 level courses in ALL of the topics that were intro'd. Easy 100. :p

[Halcyon]

My husband said math was not separated out for him when he went to school for engineering. That was in Germany though.

 

This is what I find confusing though. If one calls something Algebra 1, one would expect that to mean something. That there are certain topics always covered in Algebra 1. Apparently that isn't true though. It depends on the book, school, etc.

 

I think I understand why Halycon wants a big picture. She doesn't want to miss anything! At least that is how I feel. Ideally I'd find a series I liked and we'd be able to stick with it through high school. I don't feel comfortable picking a different book from a different series every single year.

 

 

YOu have hit the nail on the head, Wendy. I just want a big picture of what, in general, I will need to cover with him over the years. Seriously, I followed such a different course myself in high school that I have no idea what is standard these days. And yes, I would love to not jump around from series to series if I caan help it, and if we find something we like, such as Dolciani.

[go_go_gadget]

 

I only vaguely remember linear algebra (or multivariable calculus, for that matter), but I remember thinking, "This is really easy. Why is it after calc3 and diff. eq.?" We did learn parts of it in earlier courses - probably the parts necessary for multivariable calculus. It wasn't all new to me. But I just thought the whole course was ridiculously simple and seemed out of place when I took it.

 

The elementary linear algebra course at my university requires calc II (mostly integration, and comes before multivariable), and I also thought it could easily be done without calculus (differential or integral).

[KAR120C]

 

I only vaguely remember linear algebra (or multivariable calculus, for that matter), but I remember thinking, "This is really easy. Why is it after calc3 and diff. eq.?" We did learn parts of it in earlier courses - probably the parts necessary for multivariable calculus. It wasn't all new to me. But I just thought the whole course was ridiculously simple and seemed out of place when I took it.

I never took a whole Linear Algebra course myself, so I don't know what I don't know (lol) but we had what seemed like quite a bit mixed into the calculus sequence as needed...

 

I don't know if it made a difference or not, but my calc sequence was at an engineering school, and maybe even though they called it calculus 1, 2, and 3, it might have been more like the "engineering math" sequence. There were things in the course that weren't in the text, and supplements they added in... But I have no basis for comparison, really.

[kiwik]

 

 

I am really at loss here. When I was in college for mechanical engineering, after calculus, we

had "engineering math I and II", which puts everything together. In graduate school in US, we had Math 1 and math II and then numerical mathod. I never had math got seperated like this. what each topic covers??

 

 

Depends where you go to school. In the US they seem to split it in a mastery system. In New Zealand we have a spiral progression until the last year of highs hook when it is split into calculus and statistics. This year was never required for university but most people do it nowadays. I skipped the last year of highs hook msths and the last two of science and still did fairly well but I had to work harder.

[Crimson Wife]

I agree that there doesn't seem to be a consistent S&S in terms of what gets covered. My h.s. Algebra 2 class included some easy linear algebra and trig was lumped into pre-calc. My husband's h.s. had trig lumped in with Algebra 2 (using Dolciani) and then students went directly into AP Calculus AB. He didn't see any of those linear algebra topics until college.

[NittanyJen]

Here is DS11's current plan, but it is subject to change-- when they are ahead so young, a "slow down" can happen at any point, without having to cause any stress. Sometimes he gets stuck for a day or two, and I have to remind him to not beat himself up. I tell him, "You're 11. In PS you'd be doing 6th grade arithmetic right now, probably practicing multiplication and learning fractions again. Instead you are nearly done with algebra I. YOU ARE FINE! You are doing really well, and it's okay to take your time and learn this carefully right now." We are in no rush.

 

 

~5th-6th grade- LOF Algebra, self-exploring trig on Khan Academy, start algebra II

~6th-7th grade finish LOF Algebra II, (whatever he wants on Khan Academy), probably finish most of geometry

~8th finish LOF geometry, AoPS Number theory, LOF Trig, AoPS Probability

~9th LoF Calc (at least start it-- that one is a tome)

10th University Calc I & II possibly III

11th Will see from there-- we can do Calc III, LOF or University Stats, Linear Algebra, or a world of other math

12th see 11th. We'll know more where his interests lie by the time he gets this far. Math is a really big field :). My husband is a professor of mathematics, and there are branches at his university that he has never even studied.

[LMD]

I'm also lost with all the different 'maths', at my highschool it was just called maths, until 11th & 12th when we were given the additional choices of 'maths methods' and 'specialist maths'

 

I have no idea what I studied! : lol: I'm pretty sure it may have been pre-calc, maths was not my strength!

[Reya]

Mine did:

K-2nd: Singapore 1-6

2nd-3rd: Harold Jacobs' Elementary Algebra

3rd-4th: Harold Jacobs' Geometry

will probably do:

5th: Foerster's Alg II + Trig

5th-6th: Foerster's Precalc + Trig

6th-7th: Foerster's Calc

 

For an older child, I'd go with AoPS. My son really likes the reading level on the Harold Jacobs and Foerster books. He can concentrate on math during math and on reading comprehension during reading. It's rough to have a book written at the 11th grade level while he's trying to learn new concepts.

[boscopup]

Mine did:

K-2nd: Singapore 1-6

2nd-3rd: Harold Jacobs' Elementary Algebra

3rd-4th: Harold Jacobs' Geometry

will probably do:

5th: Foerster's Alg II + Trig

5th-6th: Foerster's Precalc + Trig

6th-7th: Foerster's Calc

 

Is he planning to graduate early? Or do you have something planned for grades 8-12?

[Reya]

 

Is he planning to graduate early? Or do you have something planned for grades 8-12?

 

 

Yes and yes! :-)

 

Intellectually, he will be at an average freshman level by 12 or 13. He'll probably graduate at 15 or 16, though, but with 2 years of college credits under his belt.

 

What I'm not saying is that Jacobs went from being quite a challenge--not the MATH part, but the reading and writing--to one of his easiest subjects, so my 5th/6th/7th may end up getting faster by a good deal. If it doesn't:

 

8th, fall: Calc III

8th, spring: Linear Alg.

9th, fall: Dif Eq

9th, spring: college-level programming course

 

Keep in mind that he wants to be a computer scientist.

 

For science, we have the following:

 

4th: Bio I, Bio II, Honors Bio I, which sounds weird but really isn't--it's a reading level issue

5th: Chem I, Physics I, bit o' programming

6th: AP Chem, bit o' programming, maybe AP Physics C (half)

7th: AP Physics C (rest of course), maybe first college-level programming course

8th: college-level programming

9th: college-level programming

 

For history, we're all messed up:

 

4th: college-level history course on China, by request, pror to trip to China

5th: jumping back to middle-grades/high school survey of early modern world and modern world, plus a one-area college-level focus in preparation for family trip to that country

6th+: a series of 200/300-level college-level courses covering the entirety of written history of the world, with AP requirements for courses that he needs credit for worked in there somewhere so he can take the tests

 

In literature, the "great literature" is going to start either in 5th or 6th grade. Depends on his reading level and speed at the end of this year, really.

 

Meanwhile, he's doing a grade level text in composition! HA!

[boscopup]

Yes and yes! :-)

 

Intellectually, he will be at an average freshman level by 12 or 13. He'll probably graduate at 15 or 16, though, but with 2 years of college credits under his belt.

 

Gotcha! :D

 

Meanwhile, he's doing a grade level text in composition! HA!

 

I know how that is... :lol:

[kiana]

If he continues to want to be a computer scientist, I'd look into a discrete math/programming math course rather than (or in addition to) diffeq. This course is a rude awakening to some CS majors because it's so very different than what they've studied before, and the material really isn't included pre-college in most sequences.

[onaclairadeluna]

My son is doing more of a discrete math sequence. We definitely have not planned this out but I can try to recreate what he has done.

3rd grade AOPS Probability and Counting (there was no pre algebra and he was not yet ready for Alg) He took the class online, didn't do challenge problems. Loved it. After this he only used the books.

4th grade Elements of Mathematics Book 0 chapters 1-7 I think. Um...Maybe some of the P and C challenge problems. I think he started AOPS Intro to Algebra this year.

5th AOPS Algebra + Elements of Mathematics Book 1

6th AOPS Geometry + AOPS Number Theory + Elements of mathematics Book 2

7th AOPS Geometry + Elements of Mathematics Book 3

8th Problems from AOPS Book 1 and 2 (the compilation books...he skipped around) Elements of Mathematics Book 4 (At this point the lack of an answer key became too challenging) Also MIT Mathematics for Computer Science (about the first half of this course..this is a really good course! It was recommended to me by Kathy in Richmond)

9th grade AOPS Intermediate Algebra EPGY Number Theory.

 

As you can see he has accelerated way more radically in the discrete math field than in the traditional math sequence. This is a rough sketch.He absolutely loves discrete proofy type math.

[FairProspects]

I didn't realize several of you had radically accelerated so much. Very intriguing.

[quark]

Experience and son's math tutor have taught me not to plan ahead too much. But I like to plan anyway. :laugh:

This is what we did for math (if memory serves me right).

 

PreK-2nd: misc. workbooks from Amazon, MEP (jumped around a lot from Years 1-4, 5-6 just a little), EPGY, SM (jumped around a lot) and living math plus a lot of our own made-up math problems on the whiteboard up to very basic trig level

 

3rd-4th: LOF fractions, LOF d&p, some EPGY, MM fractions-decimals-percents units, lots of self-exploration and our made-up whiteboard problems but this time integrating some basic physics calculations and more real-life concepts. He only learned to read the clock accurately at this age! 3 months after he turned 8 we started with Dolciani's Algebra 1 because we just couldn't put it off any longer. He was obviously too cranky with the boring, easy math. He finshed the course in 9 months. Half-way through Algebra 1, started and finished AOPS Counting and Probability. Started AOPS Number Theory but haven't finished it. Did a Coursera cryptography course (half the course) in 4th and learned a little about math in computer security/ hacking. It was a very good intro to working on one subject dedicatedly for approx. 3 hours a day. He developed a lot of stamina and note-taking skills.

 

4th - 5th: AOPS Intro to Algebra on a part-discovery, part-review sort of track. Jurgensen Geometry with his tutor. This is where I have had to stop planning because his tutor is taking all sorts of bunny trails with geometry and kiddo adores it. They have started on trig, analytic geometry and a few other things that I did not learn in high school. I was going to try MUS Precalc but dropped it in favor of geometry bunny trails and self-exploration. Lots of practice in proof writing.

 

Summer between 4th and 5th: A research-based online math summer camp for middle and high school students (Camp Euclid) where he learned to work on unsolved problems and "write math" by maintaining a math wiki. Also took an 8-week online modular arithmetic/cryptography camp.

 

possibly also in 5th: Advanced geometry/ precalc (Coxeter's) and more number theory. He is getting excellent intros to higher math (some game theory topics, number theory topics and a mix of several other topics) through our math circle. We might enrol in eimacs (he qualifed for both the math and CS higher level courses) and/or move on to AoPS Algebra 2 section of Intro to Algebra. AOPS problem solving volumes will figure in there somewhere too.

 

Summer between 5th and 6th: eimacs course, improving problem-solving speed (he did well enough in AMC8 but is not satisfied because he was still stymied by the time requirement)

 

6th: possibly AOPS Intermediate Algebra and some more Precalc if he needs it. eimacs course as available. He might increase practice in researching and writing math articles (wants to be a math major so this will be a useful skill to learn).

 

6th-7th: possibly calculus and the stuff that comes after calculus. eimacs course as available. Math research and writing.

 

8th: I have no idea at all!

 

His science sequence:

 

PreK-4th: Lots and lots of living science books and experiments from the Happy Scientist. Lots of astronomy, kitchen science and a fixation on epidemiology and chemistry topics. Teaching Company physics courses. He learned a LOT of scientific theory at this stage.

 

4th-5th: Honors Physics with Derek Owens (taking it slow and loving it very much). High school level biology readings and Thinkwell biology course (he does this for fun). Periodic high school level chemistry experiments using The Home Scientist's all lab no lecture book as well as other kits but without lab reports just yet. He will most likely be preparing for the Physics SAT subject test in 2013.

 

6th: Either Bio or Chemistry. Haven't decided. Might do AP level for Bio.

 

7th: Either Bio or Chemistry. (If he does Bio in 6th, he'll take Chem in 7th). Possibly also, a geology and astronomy course if interested.

 

8th: AP Physics (that's the plan anyway)

 

A lot of things happen that I may or may not count as school unless we really need the documentation: e.g. electronics projects with dad, building and taking apart things, robotics.

[Reya]

If he continues to want to be a computer scientist, I'd look into a discrete math/programming math course rather than (or in addition to) diffeq. This course is a rude awakening to some CS majors because it's so very different than what they've studied before, and the material really isn't included pre-college in most sequences.

 

 

My husband is a computer scientist, and I was taking the sequence from what I could remember from the university he'll be attending. I honestly didn't go and look it up again--I've got a firm view of the next ear and a loosey-goosey one longer out! (There is really only one option of university because I am NOT letting this kid live on campus until age 18. :) ) After some introductory stuff, he'll be doing programming courses toward his major, too.

[Reya]

I didn't realize several of you had radically accelerated so much. Very intriguing.

 

 

I don't accelerate HIM, he accelerates himself. :-) I just make sure his work matches his level as well as it can. Honestly, he's working quite a bit below his intellectual level, but he has other skills that he need to work on--self-discipline, handwriting, attention span. And I'm ok with him now not doing his "best" in one area to let others develop. With more hand-holding, he could have been done with geometry a year ago.

 

I used to be more uptight, worrying about how everything will work out, worrying about whether I'm doing him a disservice by either not giving him enough or unknowingly "pushing" him. Part of it was the fact that he's 2E, and part of it is that I was new to all this.

 

Now, I'm so laid back, I'm pretty much horizontal.

[Dmmetler]

Has anyone ever come up with a post-Algebra sequence that doesn't go through the typical high school stuff first? Right now, DD is saying that she wants to go back to PS for high school, and if she does, she'll have to have at least two high school math classes she hasn't yet taken before she's eligible for Dual Enrollment, and will have to be able to pass the exit exams for whatever high school classes we're listing on her transcript before high school so she won't have to start with Algebra I. Because of this, DH (who's our resident mathematician-I'm certified as a math teacher, but he went a lot farther in math than I did) really wants to take the next few years and focus on other areas of math, then go back in Middle school and do the first several years of the traditional US sequence, sitting for the end of course exams locally so she gets high school credit. He likes the Elements of Mathematics sequence as laid out on the website, so we're going to try the intro course after Christmas, but I'm also looking for other ideas. We've taken a scattershot approach to math overall for years, where we've had a couple of main resources and a bunch of additional ones, so the idea of putting too much faith in a web-based class that isn't even fully published yet is downright terrifying (especially since some of the areas of math he wants DD to cover are ones I haven't actually taken a class in myself).

[regentrude]

Has anyone ever come up with a post-Algebra sequence that doesn't go through the typical high school stuff first?

 

 

AoPS has some excellent books about topics not typically included in the math sequence. Intro to Counting and Probability and Intro to Number theory can each be completed in a semester, but you can also easily stretch each topic over a year by including other resources. Both books require algebra 1, but no other prerequisites. (I do not know about Intermediate C&P)

[wapiti]

dmmetler, how about the Aops books for Counting and Probabilty and Number Theory?

 

I am hoping that my ds9 will be interested in at least one of those, though I think he will need at least some algebra first. He will finish Aops prealgebra by the end of this year, and will probably attend a private middle school starting in 6th that will allow independent study for math once he is done with alg 1. Ds keeps trying to skip ahead in Alcumus to the counting topics under prealgebra, so I'm hopeful that he will at least look at the C&P book.

 

eta, I am slow on my phone. I agree with regentrude

[kiana]

Elements of Math would be great. So would AOPS discrete math books. AOPS Problem solving books in general. You might also find Mathematics: a Human Endeavor interesting (easy to pick and choose topics of interest). I will say, though, that she will probably find any high-school level classes very trivial if she's worked through these harder courses.

 

Other ideas: I notice she's 8. When I was that age, I *loved* math for smarty pants and the I hate mathematics book. Awesome introductions to a lot of stuff.

[Crimson Wife]

Topology would be an interesting "off-the-beaten-path" course. I'm not sure what the pre-reqs would be beyond basic H.S. geometry as it's not a course I ever took. Isn't there a math professor on this board? That would be the person to ask.

[regentrude]

Topology would be an interesting "off-the-beaten-path" course. I'm not sure what the pre-reqs would be beyond basic H.S. geometry as it's not a course I ever took. Isn't there a math professor on this board? That would be the person to ask.

 

I am not a math professor - but since topology is about spatial relationships, I would assume that a knowledge of geometry should be an essential prerequisite.

[Dmmetler]

I have a friend who is a topologist, who I can ask for recommendations beyond some of the basic things (which letters are topologically equivalent type stuff). Because DH did his grad work in math, we know a lot of working mathematicians, even though DH used his 2nd BS and diverged into industry.

 

 

[kiana]

Topology would be an interesting "off-the-beaten-path" course. I'm not sure what the pre-reqs would be beyond basic H.S. geometry as it's not a course I ever took. Isn't there a math professor on this board? That would be the person to ask.

 

A solid course in proof-based geometry would be good. Some exposure to set theory would be a good idea as well. Here's a few books with low prerequisites, written for high school students. (disclaimer: I have not read them, but none are unduly expensive and they have been recommended by people whose opinion I respect.)

 

http://www.amazon.com/Shape-Space-Chapman-Applied-Mathematics/dp/0824707095/ref=sr_1_1?ie=UTF8&qid=1355543513&sr=8-1&keywords=weeks+shape+of+space

 

http://www.amazon.com/Intuitive-Topology-Mathematical-World-Vol/dp/0821803565/ref=sr_1_1?ie=UTF8&qid=1355543404&sr=8-1&keywords=prasolov+intuitive+topology

 

http://www.amazon.com/First-Concepts-Topology-Mathematical-Library/dp/0883856182/ref=sr_1_2?s=books&ie=UTF8&qid=1355543984&sr=1-2&keywords=steenrod+topology

 

Topology, imo, would be an excellent elective for a mathematically mature high school student who was aimed at a career in STEM and potentially mathematics. It would also really set them up for success in real analysis.

[mathwonk]

topology is a huge subject, that can be studied at many different levels, some requiring calculus of several variable, and others not requiring anything at all. I t is the study of "connectivity" in geometry, independent of size and specific shape. The joke is that a topologist is someone who cannot tell the difference between a doughnut and a cup of coffee, because both have one hole. At that simple level, it can be taught to a very young child. The origin is sometimes said to be Euler's analysis of the problem of the 7 bridges of Konigsberg. This concerned an array of bridges spanning some pieces of land and maybe one or two islands, and the question was something like whether a person walking could traverse every bridge exactly once and return to his original place.

 

The reason this is a problem in topology is that the solution does not depend on the length or size of the bridges, merely on the way they are interconnected. I'll look for a link to that problem.

here is a basic one:

http://mathforum.org...s/bridges1.html

 

I know one excellent book on this subject for high school students, called maybe first concepts in topology, by chinn and steenrod, but it is a little formalistic, suited mostly for math geeks.

here is a link (also linked in the previous post):

http://www.abebooks.... steenrod&sts=t

 

 

there should be some good ones out there for more naive students. come to think of it, the fantastic book, geometry and the imagination, by hilbert and cohn-vossen, contains a lot about topology, with "no prerequisites at all ".

 

http://www.abebooks....hn-vossen&sts=t

 

 

heres an elementary topology book i just ran across on amazon. it might suit a student who has had a high school calculus course:

 

http://www.amazon.co...entary topology

 

 

one of my professors said that topology is the geometry of continuity. the first theorem in topology that students usually encounter is the "intermediate value theorem" in calculus. basically it says that if a continuous curve in the x,y plane, contains a point below the x axis and a point above the x axis, then it must contain some point on the x axis.

 

the power of the result is that it has so few assumptions, but the weakness of the result, which is typical of topology, is the fact that it gives no help at all in actually finding the specific point on the x axis that it claims to exist. i.e. since finding actual solutions to many problems is so difficult, topology focuses on easier problems that assert only the existence of a solution with trying to actually find one.

[wapiti]

As usual, Amazon thanks you all. I just ordered Rubber Bands, Baseballs and Doughnuts, a Book about Topology (to add to the collection of Young Math books I started recently thanks to Quark).

[mathwonk]

the sequence of math courses is somewhat flexible, with some exceptions.

 

first one should learn arithmetic of integers, whole numbers and their addition, subtraction, multiplication and division. for gifted children one could include the concepts of prime numbers, unique prime factorization, and greatest common divisors, but these usually come later.

 

next one should ideally learn euclidean geometry, the interplay of lines and circles, including the concepts of angles between lines, and area of plane figures, as well as similarity of plane figures.

 

an alternate concept that could be learned next, and which is useful to know before reading chapter II of euclid’s elements (of geometry), is elementary algebra, which is the

science of arithmetic operations on unknown quantities, which elucidates the fundamental properties of arithmetic that apply to all calculations regardless of the value

of the quantities in them, such as a(b+c) = ab +ac, no matter what numerical values are assigned to a,b,c. this is sometimes called algebra I. It usually includes the study of linear equations, i.e. solution of equations such as 3x+2 = 7, or even pairs of simultaneous such equations in 2 variables, such as 2x+y = 3 and x-4y = 6. One should learn to add, subtract, multiply and divide polynomials f(x) in one variable. This could be considered the beginnings of linear algebra.

 

next one could advance to three dimensional geometry, the study of pyramids, cubes, cones, and platonic solids like tetrahedra, dodecahedra, and icosahedra, all treated later in Euclid, and more deeply in Archimedes.

 

or one could advance to algebra involving quadratic equations, such as x^2 – 2x + 1 = 0. Euclid's Elements explains how to solve quadratic equations entirely geometrically, but it could help in understanding Euclid to also know the algebraic version. Indeed Euler’s elements of algebra even explains the solution of cubic equations, such as

x^3 = 9x+28, but typical algebra books never touch this topic.

 

[for those of you following along and doing the exercises, let me sketch how simple euler makes solving cubic equations seem. if you set x = p+q and expand x^3 = (p+q)^3 = p^3 + q^3 + 3pq(p+q) = 3pqx + (p^3 + q^3), you may conclude that if x = p+q then x solves the cubic equation, x^3 = 3pqx + (p^3+q^3). Hence to solve x^3 = 9x + 28, it suffices to solve for p and q, such that p^3 + q^3 = 28 and 3pq = 9. In general this is done by solving the quadratic equation t^2 -28t +3 = 0, for t = p^3, q^3, and taking cube roots, but here we can solve by inspection for p = 1, q = 3, so x = 4 is at least one solution. the moral is that if one can solve quadratics, one can also solve cubics. (all cubics can be put in the special form x^3 = ax + b.)]

 

A general algebraic result that is universally useful is the factor theorem, a polynomial f(x) has x=a as a root, i.e. f(a) = 0, if and only if (x-a) divides f(x) evenly.

 

another related topic is trigonometry, which relates circles, angles, and algebra, but someone who has studied euclid will have already encountered the fundamental law of cosines in geometric form in book II of the elements. It is useful to know the concept of the basic trig functions, cos, sin, tan and the simplest addition formulas involving them. e.g. sin(a+ B) = sin(a)cos( B) + sin(b)cos(a), cos(a+ B) = cos(a)cos( B)-sin(a)sin( B). if I recall them correctly. I never learned the addition law for tangent. [i don't know why my b's are coming out as smiles.]

 

as taught in trigonometry courses, the law of cosines is just some complicated unmotivated formula to memorize, but in Euclid it answers the very natural question of how the pythagorean theorem changes when the angle in the triangle is no longer a right angle. I.e. when that angle gets larger, the "hypotenuse" gets longer and therefore so does the square of which it is a side, but the other two sides do not get longer so their squares do not. thus the sum of the squares on the two legs of a triangle no longer add up to the square on the third side when that third side is opposite an angle greater than 90degrees. Euclid explains exactly how much larger that square is, which is equivalent to the formula we know as the law of cosines. It gets larger by exactly twice the area of a certain rectangle, which he proves. Neither I nor other college professors I knew were aware that all this is in Euclid, actually quite near the beginning.

 

Not teaching from Euclid has occasioned an enormous loss in mathematical preparation for children trying to learn basic and higher mathematics over the past 100 years since it was banished from the US curriculum. I would actually link that change roughly to the decline of mathematical instruction in the US.

 

Elementary topology, e.g. graph theory (an elaboration of euler’s 7 bridges problem), and/or elementary number theory (like why 5 is a sum of 2 squares but 7 is not), can be studied at almost any time after basic arithmetic and geometry, and are fun and stimulating. One should also learn about complex (“imaginaryâ€) numbers as early as possible after elementary algebra.

 

After learning algebra, geometry, and perhaps (but not necessarily) trigonometry, one may profitably study calculus of one variable. There are two basic concepts in elementary differential calculus, continuity and differentiability. Preliminary to both is the concept of a function. precalculus is the name usually given to the course in which one learns what a function is. roughly it is a correspondence between numbers, such that to every input there corresponds a specific output. on should also learn the concept of the graph of a function.

 

after understanding the idea of a function of one variable, a continuous function is one whose output changes only a little when its input changes only a little. a differentiable function is one that can be approximated near every fixed input by a linear function, i.e. one whose graph is a line. the slope of the line that approximates a function f near an input a, is called the derivative of f at a. Every differentiable function is also continuous but a continuous function need not be differentiable. Strictly speaking differentiable functions are the topic of study in calculus, and continuous functions belong to topology, but it is usual to study both in a calculus course.

 

The first theorem in differential “calculus†(actually topology) is the intermediate value theorem, that continuous function on an interval [a,b] which is negative at a and positive at b must equal zero somewhere between a and b.

 

The second theorem is the Rolle theorem, that a differentiable function that takes the same values at a and at b, must have derivative zero somewhere in between.

 

The third theorem is the mean value theorem, which implies that a differentiable functino with derivative zero everywhere is constant.

 

The other main type of calculus is integral calculus, which can be studied before or after differentiable calculus. I will continue this survey of the sequence of math courses later.