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[quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=pettifogger][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous]Yes, you should complement. Have your child enroll in RSM or AoPS either concurrently or shortly before. That's what we did. School instruction is insufficient in multiple ways. First, students aren't doing any problem solving in school (all they do is textbook worksheets and SOL prep); second, they don't do any mathematical writing in school; third, the school curriculum is abridged for an advanced student [b](e.g., no complex numbers in Algebra 1, no linear programming[/b]). Fourth, school math is much less fun.

In short, if your child is gifted and interested in math then you cannot rely on the school curriculum.[/quote]

Is this a joke?  Why would there be complex numbers in Algebra I?  You sound nutty.[/quote]
When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's [url=https://s3.amazonaws.com/aops-cdn.artofproblemsolving.com/products/intro-algebra/toc.pdf]textbook does so in Chapter 12[/url]. [/quote]

That's not The Correct Way handed down from God.
Chapter 10 is factoring, which has the same problem (some problems don't have solutions) that the quadratic formula has (chapter 13), until i is introduced.

i is very much a teaser in AoPS Algebra 1. Algebra 2 reintroduces i and covers it in a lot more detail. [/quote]
I know :-)

Still, one of the last writing problems in Algebra 1 was factoring z^4+1=0 using elementary means. Which I thought was really cool.
[/quote]

do you mean z^4 - 1 = 0?  That can be factored using alg1 and introduces imaginary numbers.  z^4 + 1 is much harder to factor.[/quote]

Nope, I meant z^4+1=0. (Now that I think about it, it was actually z^4 + 4 = 0, which gives nice integer solutions.) Don't use the polar form, though. Set z=a+bi and see where this gets you.[/quote]
Setting z = a+bi will work, but it's a quite lengthy to work through the algebra (expanding by binomial theorem, forming two separate equations and solving each of the cases). Much simpler is to rewrite the expression as the square of a binomial by completing the square, i.e:

z^4 + 4 = (z^2 + 2)^2 - 4z^2

Now apply difference of squares to turn the above into a product of two quadratics. The four roots are easily extracted with the quadratic formula, two for each of the quadratics. (Bonus for students who have studied polar/exponential form of complex numbers: Plot the roots of z^4 + 4 in the complex plane, and also plot the roots of z^4 - 4... what interesting observation can you make? Can you explain how they are related to each other and exactly why this is the case?)[/quote]

That's un-completing a square while completing another square.

Much simpler is to use the already complete square, and go directly to difference of squares, analogy to the more elementary z^2-4=0:

z^4 + 4
 = z^4 - -4
 = (z^2)^2 - (2i)^2
 = (z^2 + 2i)(z^2 - 2i)

z^2 + ±2i = 0
(a + bi)^2 = 0 + ±2i
a^2 - b^2 + 2abi = 0 + ±2i

real: a^2 = b^2, so a = ±b
imaginary: ab=±1
z = ±1 + ±1i

All 4 roots solved simultaneously and symmetrically,
no guessing a square to complete, 
no trinomial quadratics to solve, 
no fractions or division or subtraction, 
no integers besides {0,1,2} after immediately removing the 4s from the problem.

[/quote]
Yes, that's how my 5th grader did it.  Also remember that the problem was posed after introducing complex numbers but before exhaustive treatment of the quadratics. I love how AoPS includes so many deep problems in their curriculum.

By contrast, and this is true, when they took Algebra 1 in 6th, their Algebra teacher refused to include [b]any kind of derivation[/b] of the quadratic formula because it's not part of Virginia's SOL requirements (and thus optional in their mind).  Going back to the topic of this thread, this is why we cannot do without supplementation.[/quote]

Bruh, my DS derived the quadratic formula in 7th grade.  This is a teacher issue, not a public school issue.  No, kids do not need supplementation.  We can do without it. [/quote]

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