Algebra I, geometry, algebra 2

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

Complex numbers have never been part of Algebra I. If you want to pay money to have someone teach your DC ahead, fine, but it's not "abridged" or "appropriate".

You have to realize the division between algebra 1 and 2 is somewhat arbitrary, although it’s true that national curriculums like common core omit complex numbers in elementary algebra.

On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems.

Most good students can handle complex numbers when doing quadratic equations and I thinks it’s even recommended to do so, since it’s a more complete treatment of the topic.

Also, some people choose enrichment to go deeper, broader and different then the worksheets kids do at school.

College algebra has a prerequisites of a high school diploma or GED, and/or remedial Algebra class. A high school diploma has a prequisite of....Algebra 1.

GED includes solving quadratics but not imaginary/complex numbers/roots.

We’re not talking about students struggling to pass GED, you might want to check the name of the forum, it’s “advanced academic programs”. There’s no credit for college algebra, usually it’s taken as remedial class to catch up class prerequisites. It’s the same material as high school math, but it’s better structured and presented more consistently.

You don’t need to follow any particular book for your child supplementation, you don’t even have to supplement at all, totally your choice. But don’t throw a fit if people give their own motivations for doing so.

[Anonymous]

Anonymous wrote:no prep ever if your kid struggles drop down a course

enough with the math arms race. It's pathetic

Why do you even care what other peoples kids are doing for their math class? It’s pathetic.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

Complex numbers have never been part of Algebra I. If you want to pay money to have someone teach your DC ahead, fine, but it's not "abridged" or "appropriate".

You have to realize the division between algebra 1 and 2 is somewhat arbitrary, although it’s true that national curriculums like common core omit complex numbers in elementary algebra.

On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems.

Most good students can handle complex numbers when doing quadratic equations and I thinks it’s even recommended to do so, since it’s a more complete treatment of the topic.

Also, some people choose enrichment to go deeper, broader and different then the worksheets kids do at school.

College algebra has a prerequisites of a high school diploma or GED, and/or remedial Algebra class. A high school diploma has a prequisite of....Algebra 1.

GED includes solving quadratics but not imaginary/complex numbers/roots.

We’re not talking about students struggling to pass GED, you might want to check the name of the forum, it’s “advanced academic programs”. There’s no credit for college algebra, usually it’s taken as remedial class to catch up class prerequisites. It’s the same material as high school math, but it’s better structured and presented more consistently.

You don’t need to follow any particular book for your child supplementation, you don’t even have to supplement at all, totally your choice. But don’t throw a fit if people give their own motivations for doing so.

You're tying yourself in knots now.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

Complex numbers have never been part of Algebra I. If you want to pay money to have someone teach your DC ahead, fine, but it's not "abridged" or "appropriate".

On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems.

Can you name one such book?

Stewart, College Algebra
Openstax, College Algebra

All college algebra books, I’m not aware of a single one that doesn’t use complex numbers in the treatment of quadratics.

The division between algebra, geometry and pre calculus is also quite arbitrary, with many areas of overlap. Of course arithmetic, equations and systems belong in algebra, shapes, angles, lines belong to geometry, and vectors and matrices traditionally taught in precalculus. But functions are thought in both algebra and precalculus, analytical geometry are taught in algebra, geometry and precalculus, etc. trigonometry is another one that can be taught alone, or almost any high school math class.

I found it somewhat amusing when posters decree that complex number are taught in Algebra 2. What exactly is Algebra 2? Then why are they taught again in precalculus. There’s an apt analogy that math is a spiral, it’s up to the individual how fast one goes around and the breath and depth of the material studied.

Those are college algebra books, not elementary algebra books. Those are two different math classes.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

Complex numbers have never been part of Algebra I. If you want to pay money to have someone teach your DC ahead, fine, but it's not "abridged" or "appropriate".

On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems.

Can you name one such book?

Stewart, College Algebra
Openstax, College Algebra

All college algebra books, I’m not aware of a single one that doesn’t use complex numbers in the treatment of quadratics.

The division between algebra, geometry and pre calculus is also quite arbitrary, with many areas of overlap. Of course arithmetic, equations and systems belong in algebra, shapes, angles, lines belong to geometry, and vectors and matrices traditionally taught in precalculus. But functions are thought in both algebra and precalculus, analytical geometry are taught in algebra, geometry and precalculus, etc. trigonometry is another one that can be taught alone, or almost any high school math class.

I found it somewhat amusing when posters decree that complex number are taught in Algebra 2. What exactly is Algebra 2? Then why are they taught again in precalculus. There’s an apt analogy that math is a spiral, it’s up to the individual how fast one goes around and the breath and depth of the material studied.

Those are college algebra books, not elementary algebra books. Those are two different math classes.

It’s the same content, my kids school is using Stewart as textbook for Algebra 1 and 2, although they skip a lot of sections.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

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.

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.

You mean -1?

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

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.

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.

You mean -1?

Objection: Asked and answered.

[Anonymous]

To answer the OP, we do this every summer, using Khan Academy. It keeps the kid from forgetting the entirety of everything they learned the previous year, and keeps them moving forward. By the time school rolls around, they don’t remember everything, but since it’s the second time they’ve seeing it, has the advantage of triggering the memory instead of learning for the first time. Since they’re doing this at home, and it is self paced, we don’t get through everything, just enough that the school year is less stressful.

It came in really handy this year, because the math teacher was terrible and had a really confusing way of introducing material, but she could fall back on Sal Khan instead. She only realized this when they covered something in class she had never seen before, and ended up using Khan for actually learning the content.

[Anonymous]

pettifogger wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

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.

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.

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.

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.

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?)

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.

[Anonymous]

Why are you spamming the thread?

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

Complex numbers have never been part of Algebra I. If you want to pay money to have someone teach your DC ahead, fine, but it's not "abridged" or "appropriate".

On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems.

Can you name one such book?

Stewart, College Algebra
Openstax, College Algebra

All college algebra books, I’m not aware of a single one that doesn’t use complex numbers in the treatment of quadratics.

The division between algebra, geometry and pre calculus is also quite arbitrary, with many areas of overlap. Of course arithmetic, equations and systems belong in algebra, shapes, angles, lines belong to geometry, and vectors and matrices traditionally taught in precalculus. But functions are thought in both algebra and precalculus, analytical geometry are taught in algebra, geometry and precalculus, etc. trigonometry is another one that can be taught alone, or almost any high school math class.

I found it somewhat amusing when posters decree that complex number are taught in Algebra 2. What exactly is Algebra 2? Then why are they taught again in precalculus. There’s an apt analogy that math is a spiral, it’s up to the individual how fast one goes around and the breath and depth of the material studied.

Those are college algebra books, not elementary algebra books. Those are two different math classes.

It’s the same content, my kids school is using Stewart as textbook for Algebra 1 and 2, although they skip a lot of sections.

No it's not.
https://www.knewton.com/wp-content/uploads/2020/04/alta-Elementary-Algebra-v2-TOC.pdf
https://openstax.org/details/books/elementary-algebra-2e

Notice the lack of imaginary numbers.

[Anonymous]

Anonymous wrote:

pettifogger wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

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.

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.

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.

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.

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?)

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.

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 any kind of derivation 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.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

Complex numbers have never been part of Algebra I. If you want to pay money to have someone teach your DC ahead, fine, but it's not "abridged" or "appropriate".

On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems.

Can you name one such book?

Stewart, College Algebra
Openstax, College Algebra

All college algebra books, I’m not aware of a single one that doesn’t use complex numbers in the treatment of quadratics.

The division between algebra, geometry and pre calculus is also quite arbitrary, with many areas of overlap. Of course arithmetic, equations and systems belong in algebra, shapes, angles, lines belong to geometry, and vectors and matrices traditionally taught in precalculus. But functions are thought in both algebra and precalculus, analytical geometry are taught in algebra, geometry and precalculus, etc. trigonometry is another one that can be taught alone, or almost any high school math class.

I found it somewhat amusing when posters decree that complex number are taught in Algebra 2. What exactly is Algebra 2? Then why are they taught again in precalculus. There’s an apt analogy that math is a spiral, it’s up to the individual how fast one goes around and the breath and depth of the material studied.

Those are college algebra books, not elementary algebra books. Those are two different math classes.

It’s the same content, my kids school is using Stewart as textbook for Algebra 1 and 2, although they skip a lot of sections.

No it's not.
https://www.knewton.com/wp-content/uploads/2020/04/alta-Elementary-Algebra-v2-TOC.pdf
https://openstax.org/details/books/elementary-algebra-2e

Notice the lack of imaginary numbers.

It is the same content, some books introduce concepts earlier, some omit them etc. an honors class will go more in depth, introduce more topics etc. as you correctly point some books include complex numbers some don’t. Both Elementary Algebra and College Algebra books from openstax are one semester introductory courses that have some differences.

Essentially the student should choose based on interest and capability, if you can handle go for the more difficult version. If not the lighter version is good too.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

pettifogger wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

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.

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.

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.

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.

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?)

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.

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 any kind of derivation 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.

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.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote: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 (e.g., no complex numbers in Algebra 1, no linear programming). 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.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

Complex numbers have never been part of Algebra I. If you want to pay money to have someone teach your DC ahead, fine, but it's not "abridged" or "appropriate".

On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems.

Can you name one such book?

Stewart, College Algebra
Openstax, College Algebra

All college algebra books, I’m not aware of a single one that doesn’t use complex numbers in the treatment of quadratics.

The division between algebra, geometry and pre calculus is also quite arbitrary, with many areas of overlap. Of course arithmetic, equations and systems belong in algebra, shapes, angles, lines belong to geometry, and vectors and matrices traditionally taught in precalculus. But functions are thought in both algebra and precalculus, analytical geometry are taught in algebra, geometry and precalculus, etc. trigonometry is another one that can be taught alone, or almost any high school math class.

I found it somewhat amusing when posters decree that complex number are taught in Algebra 2. What exactly is Algebra 2? Then why are they taught again in precalculus. There’s an apt analogy that math is a spiral, it’s up to the individual how fast one goes around and the breath and depth of the material studied.

Those are college algebra books, not elementary algebra books. Those are two different math classes.

It’s the same content, my kids school is using Stewart as textbook for Algebra 1 and 2, although they skip a lot of sections.

No it's not.
https://www.knewton.com/wp-content/uploads/2020/04/alta-Elementary-Algebra-v2-TOC.pdf
https://openstax.org/details/books/elementary-algebra-2e

Notice the lack of imaginary numbers.

It is the same content, some books introduce concepts earlier, some omit them etc. an honors class will go more in depth, introduce more topics etc. as you correctly point some books include complex numbers some don’t. Both Elementary Algebra and College Algebra books from openstax are one semester introductory courses that have some differences.

Essentially the student should choose based on interest and capability, if you can handle go for the more difficult version. If not the lighter version is good too.

It's clearly not the same content, as elementary algebra does not include complex numbers even though you (or the person whose point you're arguing) claimed otherwise.
"On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems" -> unproven and now demonstrably false.