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You can ABSOLUTELY start algebra without a firm handle (if any) on decimals and fractions.

 

You should be pushing forward and spiraling back at all times anyway.

 

(My son started algebra at 5....and finished at 6.5.)

 

Interested to know which algebra book u used that not require knowing fraction and dicemal left and right, inside out. Most book I have u won't pass chapter 4 without understanding fraction.

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Interested to know which algebra book u used that not require knowing fraction and dicemal left and right, inside out. Most book I have u won't pass chapter 4 without understanding fraction.

By start algebra I simply mean introduce them to the concept of solving an equation, e.g. "3X + 6 = 21....what is the *mystery number* X that makes this true..."

 

Obviously you can't go too far/deep in algebra without fractions and negative numbers, but still it's a sound strategy to introduce them to the concepts as soon as possible.

 

One thing I would strongly advise is to skip the word problems altogether at first. I have my kids/students only do say 4th grade level word problems once they are doing 6th grade level computation. There's almost no better vehicle out there in conventional math curricula for ruining kids for math than *word problems*! It's like riding a bike it's hard at first is because *pedaling* and *balancing* are enemies.

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One thing I would strongly advise is to skip the word problems altogether at first. I have my kids/students only do say 4th grade level word problems once they are doing 6th grade level computation. There's almost no better vehicle out there in conventional math curricula for ruining kids for math than *word problems*! It's like riding a bike it's hard at first is because *pedaling* and *balancing* are enemies.

 

Maybe this depends on the kid, since my dc prefer word problems to straight computations. When my older ds was younger, he would make up story problems for the computation problems in the Singapore Primary Math books.

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By start algebra I simply mean introduce them to the concept of solving an equation, e.g. "3X + 6 = 21....what is the *mystery number* X that makes this true..."

 

Obviously you can't go too far/deep in algebra without fractions and negative numbers, but still it's a sound strategy to introduce them to the concepts as soon as possible.

.

 

I don't think that is what most of us are discussing when referring to alg. I consider that example basic 3rd grade math, not alg. At least from my perspective, when I am referring to alg, I am discussing full-blown high school equivalent courses.

 

But, I am also coming from the perspective that even having taken alg, 1, 2, and 3 that I would not state that a child had finished studying alg.

Edited by 8FillTheHeart
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One thing I would strongly advise is to skip the word problems altogether at first. I have my kids/students only do say 4th grade level word problems once they are doing 6th grade level computation. There's almost no better vehicle out there in conventional math curricula for ruining kids for math than *word problems*! It's like riding a bike it's hard at first is because *pedaling* and *balancing* are enemies.

We had the opposite experience. Well-written word problems were a good introduction to algebraic thinking and problem solving long before I introduced the idea of a variable.

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Maybe this depends on the kid, since my dc prefer word problems to straight computations. When my older ds was younger, he would make up story problems for the computation problems in the Singapore Primary Math books.

 

My son jumps for joy when I bring out word problems :D

 

I don't think that is what most of us are discussing when referring to alg. I consider that example basic 3rd grade math, not alg. At least from my perspective, when I am referring to alg, I am discussing full-blown high school equivalent courses.

 

Agreed, the discussion has been about young children taking a high school level algebra 1 course, not finding x in basic 3rd grade math. My K'er did missing addend problems last year at age 4. While school standards call that â€algebraic thinkingâ€, he wasn't doing algebra. He was doing K level math.

 

An algebra course will require knowledge if decimals and fractions and how to manipulate them. If your â€algebra course†doesn't, it's probably still elementary math, not a high school algebra course. :tongue_smilie:

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One thing I would strongly advise is to skip the word problems altogether at first. I have my kids/students only do say 4th grade level word problems once they are doing 6th grade level computation. There's almost no better vehicle out there in conventional math curricula for ruining kids for math than *word problems*! It's like riding a bike it's hard at first is because *pedaling* and *balancing* are enemies.

 

My kids don't enjoy word problems generally, but I've found that word problems are where they demonstrate their understanding of concepts. If a child can do a three step word problem using several math concepts I know that he understands the concepts rather than just plugging in the numbers into a familiar algorithm.

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By start algebra I simply mean introduce them to the concept of solving an equation, e.g. "3X + 6 = 21....what is the *mystery number* X that makes this true..."

 

Obviously you can't go too far/deep in algebra without fractions and negative numbers, but still it's a sound strategy to introduce them to the concepts as soon as possible.

 

One thing I would strongly advise is to skip the word problems altogether at first. I have my kids/students only do say 4th grade level word problems once they are doing 6th grade level computation. There's almost no better vehicle out there in conventional math curricula for ruining kids for math than *word problems*! It's like riding a bike it's hard at first is because *pedaling* and *balancing* are enemies.

 

 

I think the difference might be in the audience. When I taught PS, I'd have agreed with you, because a lot of kids were tripped up by the language and ended up missing the math, due to being weak readers and to the fact that many math problems were written on a higher reading level than the student's grade level based on vocabulary alone. As a math tutor, in a state where the state test was mostly word problems, the single thing I could do that would help improve test scores was to teach vocabulary specific to the math test.

 

However, on this board, that's usually not an issue. My DD was reading adult level science textbooks before she would have typically entered K. Word problems weren't an issue for her. They were a blessing because they included enough extra information and detail to make math challenging and give her reason to practice. And I agree totally with several other posters who said that word problems were our bridge into algebra as well-it was in SM 3 that I realized just how algebraic my DD's thinking was and we started transitioning to using algebra to set up problems because it simply worked well for her.

 

Last Spring, my DD took Painless Algebra with her on vacation, and her 4th grade cousin commented "Oh, we do algebra at school!". DD looked at her, and asked, "Have you done the quadratic formula yet? I'm having trouble with the graphing part". Her cousin looked at her blankly. Apparently, "algebra" was things like 3x=15, x=5. Which yes, is algebra at a 4th grade level, but wasn't exactly what DD was trying to teach herself.

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My kids don't enjoy word problems generally, but I've found that word problems are where they demonstrate their understanding of concepts. If a child can do a three step word problem using several math concepts I know that he understands the concepts rather than just plugging in the numbers into a familiar algorithm.

:iagree:

Until a student can solve word problems, he has not actually mastered algebra, but is performing algorithms without a deep understanding of what it is he is doing.

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By start algebra I simply mean introduce them to the concept of solving an equation, e.g. "3X + 6 = 21....what is the *mystery number* X that makes this true..."

 

FWIW, there are elementary programs that do this, though at lower levels they may label the variable in different ways (e.g., as a blank or as a shape).

 

One thing I would strongly advise is to skip the word problems altogether at first. I have my kids/students only do say 4th grade level word problems once they are doing 6th grade level computation. There's almost no better vehicle out there in conventional math curricula for ruining kids for math than *word problems*! It's like riding a bike it's hard at first is because *pedaling* and *balancing* are enemies.

 

I don't believe this assessment applies across the board to most kids. I agree that the level of word problems a student is capable of may depend as much on their language skills as their math skills. For kids with disparate strengths and weaknesses (such as my kids), as well as young gifted kids whose math strengths happen to outpace their language skills, the teacher may need to make some adjustments, possibly tweaking the presentation of the problems. However, IMO word problems are a skill area to be developed rather than avoided.

 

I strongly disagree that word problems ruin kids for math, though perhaps the key there is your reference to conventional math curricula, as many widespread PS elementary math curricula are not well designed, to be polite. Moreover, in contrast, there are great word problems (e.g., math contest problems) that may bring an entirely new perspective and hint at the fact that arithmetic is barely the tip of the iceberg of real math.

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My kids don't enjoy word problems generally, but I've found that word problems are where they demonstrate their understanding of concepts. If a child can do a three step word problem using several math concepts I know that he understands the concepts rather than just plugging in the numbers into a familiar algorithm.

:iagree:

 

If they can't do word problem, it usually imply the kid just memorize the process rather than actually understand what they are doing

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Thanks for all the warm welcomes!

 

I do want to clarify two things.

 

When I said:

 

One thing I would strongly advise is to skip the word problems altogether at first. I have my kids/students only do say 4th grade level word problems once they are doing 6th grade level computation.

 

I was only talking about a slight time lag because the way I teach math (and they way it SHOULD be taught)....it only takes about 3-4 months to go from 4th grade to 6th grade level computation. And in no way, shape, or form did I imply that one could *finish algebra* without doing and comprehending algebraic word problems.

 

Also, I should say that I personally know more than a little bit about accelerated math.

 

I captained my high school math team to the New England Championship. I have a couple of high-scoring medals from ARML. Etc.

 

And by no means was I born brilliant. I systematically MADE myself into a math genius; and at every level I blew past more and more *gifted* naturals.

 

My son is lazy as sin, and we haven't done much math at all in the past 2 years....yet as a math student he's still 7-8 years ahead of his father because of the foundation I gave him.

 

So I certainly know what I'm talking about when it comes to math....and I'm always willing and eager to share my thoughts, methods, and resources (even if a few people are going to heap abuse on me, misinterpret my comments, etc.).

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So I certainly know what I'm talking about when it comes to math....and I'm always willing and eager to share my thoughts, methods, and resources (even if a few people are going to heap abuse on me, misinterpret my comments, etc.).

 

:lol: well then!

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...the way I teach math(and they way it SHOULD be taught)....it only takes about 3-4 months to go from 4th grade to 6th grade level computation..

 

 

Ah well, this is a core difference of philosophies that will not be resolved at all I should think -- certainly not on a thread. I have noticed that parents here have a variety of math goals, a variety of children, &c., and myself firmly believe that there is more than one right way to teach math even within a given family. Thank you for being so clear about your logical givens! it makes conversation clearer and more productive.

 

My son is lazy as sin, and we haven't done much math at all in the past 2 years....yet as a math student he's still 7-8 years ahead of his father because of the foundation I gave him.

 

 

:) This is another difference of philosophy. I am uncomfortable when I read that, because I value children being respected and the gentle, firm guidance of their characters over their accomplishments.

 

Some common ground is that I value my family and my children's education, too, and am willing to work very hard to achieve our goals, as you clearly are.

 

Blessings to you and yours.

 

ETA: CyberScholar, would you like to start a thread to discuss your methods?

Edited by serendipitous journey
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One thing I would strongly advise is to skip the word problems altogether at first. I have my kids/students only do say 4th grade level word problems once they are doing 6th grade level computation.

 

I was only talking about a slight time lag because the way I teach math (and they way it SHOULD be taught)....it only takes about 3-4 months to go from 4th grade to 6th grade level computation. And in no way, shape, or form did I imply that one could *finish algebra* without doing and comprehending algebraic word problems.

 

Your goals are completely the opposite of mine. My son is set to be a mathematician in life, and my goal is to make sure that math is always HARD for him. I don't want to give him word problems 2 years behind his computational abilities, I want to give him word problems that take him 2 hours to figure out. I want him to have an idea, try it, realize it won't work, think it out over the day, and try something new the next day. Scientists and Mathematicians do not encounter easy problems, they must struggle, and I plan for my son to struggle all the way through. Challenging math makes him persistent and humble and a true problem solver.

 

So I certainly know what I'm talking about when it comes to math....and I'm always willing and eager to share my thoughts, methods, and resources (even if a few people are going to heap abuse on me, misinterpret my comments, etc.).
Just be aware that there are a lot of us that do know what we are talking about, but are just a bit more humble about it.

 

Ruth in NZ

Edited by lewelma
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My oldest child started at the beginning of Alpha at the start of first grade, and raced through MUS. He had finished a third of Alg I by the end of 3rd grade. I was glad that I went along and did not skip. MUS is set up that you should not skip Pre-Algebra. (Though many public school curriculums are set up skipping Pre-Algebra. The content is not the exact same). However, I post because MUS was not a good fit for us anymore. I found that having a gifted child was fixed by moving faster in MUS for the most part. Until Algebra I. He just started 4th grade and we switched to Art of Problem Solving Intro to Algebra. It is working amazingly well. It's harder for my child - but now neither of us are bored! MUS did help prepare ODS for AoPS, but it no longer met his needs for Algebra I.

 

I would not skip word problems they help to show if the child understands the logical part of math, which is very important.

 

Also, keep in mind that there's a math maturity so you may come to a roadblock from time to time with a young gifted student. (Actually, I saw this teaching kids of ALL abilities when I was a public high school math teacher). When that happens, then it's time to take a break from what you're doing and go off on another tangent.

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hoping its not too late to jump in with another version of the original question.

 

My youngest will be 9 in september. He really hates curriculum, so we havent done much. He was in public school in Kindergarten and started asking about negative numbers, so he had those down cold by the end of 1st. We have read some murderous maths, done Primary Challenge, done some random worksheets, finished LOF fractions and started LOF decimals

 

He has been somewhat behind in all verbal skills - spoke and read late, and isnt really writing yet. We have done much of our math out loud. For the summer he is doing some worksheets, but he is slow and wont do very many problems at a time. Right now we are doing Kaleidoscope math, which he fairly willingly does 2 problems a day. Because of the way we ran through topics, I felt that he needed practice just doing rows of problems with a pencil, so thats what we are doing.

 

i have been really wondering what to do next. Obviously we will finish LOF decimals, but then what? Challenge math? i dont own it but it looks kinda advanced. LOF pre-algebra? I already have LOF preA/Bio which i'd bought for his older brother. AoPS?

 

I am not so sure my little one is ready for AoPS. We did the sample chapter of BA and he got frustrated. He is not used to doing hard problems. He did start to really internalize the fact that doing the easy problems first made the hard problems easier - oh, he also has some behavior/defiance issues, and a lot of perfectionism and anxiety. So its a balancing act.

 

Part of me thinks LOF preA and THEN AoPS prealgebra . .. i've never seen an AoPS book in person, so I was thinking - do the fun LOF (mostly out loud) and then try AoPS next . . . maybe when he's 10, so he's hopefully mature enough to handle something different.

 

can anyone give me feedback on the idea of using AoPS as a second pass through PreA? With a young kid? Ok, i guess 10 isnt even that young for preA . . . but we'd be hitting LOF PreA at 9, which i think is early?

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can anyone give me feedback on the idea of using AoPS as a second pass through PreA? With a young kid? Ok, i guess 10 isnt even that young for preA . . . but we'd be hitting LOF PreA at 9, which i think is early?

 

Go to the AoPS website and sign him up for Alcumus. It's a free online site and asks them math questions. He can have it focus on Pre-Algebra questions. That will give you some idea if he's ready. What I saw was that my oldest would be in tears if he couldn't get all of MUS correct. He would get frustrated if he didn't understand it immediately. But he's been a LOT more willing to struggle to work through with AoPS. There's a greater sense of satisfaction now when he gets a problem correct, and that helps him work harder when it's not easy. ODS did MUS Pre-Algebra, then MUS Algebra for a third of the book - but I would have been find doing MUS Pre-Algebra and then AoPS Algebra. I just hadn't realized yet that MUS was going to stop working well for us. In hindsight, I wish I had switched to AoPS for Pre-Algebra so he would have been adequately challenged. But taking two weeks to just do Alcumus before going into AoPS did help him adjust to type of question better.

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Did you finish all basic math first (i.e. addition, subtraction, division, multiplication, fractions, decimals) or did you just address any gaps as they came up?

 

Yes. Dd9 finished basic math before starting alg. She is rusty on percents and forgets lengthy long div problems once in a while. We review as needed.

 

 

Also, keep in mind that there's a math maturity so you may come to a roadblock from time to time with a young gifted student. (Actually, I saw this teaching kids of ALL abilities when I was a public high school math teacher). When that happens, then it's time to take a break from what you're doing and go off on another tangent.

 

:iagree:

Coming at a topic with different resources has helped here. I also took the advice from parents here to re-do alg 1 with a 'harder/deeper' program the second time around. Dd9 used TT Alg 1 and a few chapters at the end of MUS Alg 1 (using the blocks) along with other resources last year. I consider that her gentle intro to algebra. I didn't have an agenda and I knew she would have gaps on some concepts because of her age. Thankfully she had a blast (most days :)) and is eager to learn more. She would still prefer to read mythology than do math. She's not one to sit and wrestle with problems. She likes to be taught which is why aops doesn't suit her at this point.

 

Now she's going back to the beginning of algebra 1 starting with graphing and working through quadratics via Kinetic Books. KB will be a challenge for her and we may skip some of the problems/games if she's not ready. A year of maturity will open doors and insight that she didn't experience the first time through her 'light' intro to alg.

 

It's a blessing to have so many fabulous options.

 

Her online tutor, Rachna (Cybershala), is working with her on topics that I didn't hit until mid-high school. She is now working with Ben (Crewton Ramone) online weekly for a completely different take on algebra. Very non-traditional and radically different than Rachna. Abi loves it all. She is doing 'completing the square' problems with Ben in a hands-on conceptual way. KB lessons on completing the square will be great for reinforcement a practice. It's a win-win.

 

Following KB 1 (in one year) she will do TT Alg 2 and begin TT Geometry. That could change and dd may end up attending our local STEM school. For now, she's enjoying what she's learning in a home setting.

 

HTH!! I never imagined when we began homeschooling a couple years ago that our math lessons would look quite like this. I'm taking it month by month, year by year and following dd's interest and abilities. So thankful for the advice and wisdom here. :)

 

ETA: After I wrote the above, I noticed Ben put a picture of Abi on his FB page today.

Edited by Beth in SW WA
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We did TT5 for my 10yrs old last year and they were bored to tears. I think Math U SEE would bore them to tears as well.

 

We also did Key To Fractions 1-4 and also Key to Decimals 1-2 too. Some Key to Percents. I will have to do more white board stuff with them but one is more ahead than the other.

They don't like the Key to series much. Some is hard and some is just plain boring they say.

 

They played Rightstart math games with younger sister who was doing Rightstart B. Also we play Mastermind.

 

We did some Hands On Equation-computation was easy, but the word problems killed them.

 

We are now going through Singapore Math 3a/3b and Process Skills 3 when it is ordered and Challenging Word Problems 3. Most of the computational stuff is review but these kids are new to mental math and bar method so that is why I backed it up to 3a/3b. (They were in PS previously)

 

One child is doing Life of Fred Fractions and the other is doing Beast Academy 3a.

 

MEP 3 here and there.

 

I am going through Key To Algebra slowly on the white board before they attempt to even start doing the workbooks on their own.

 

Plus we have Dreambox going on here as well as Education Unboxed and Crewton Ramone with blocks and c-rods. They are completely the square, factoring polynomials--it is just for fun right now.

 

I just don;t know what else to do.....how to blend it better....

 

I do want to do the Righstart Geometry program. That looks great.

 

I do have PreAlg-AOPs sitting here but not ready to really open it yet. Oh and Kinetic books look great too. Oh and that Charlotte Mason own your petstore, books store, sports store looks great too.

 

But does any of this sound ridiculously redundant? Can someone help me focus?

Edited by happycc
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Just for fun, as a tool for comparing books on algebra, I have written a short 4 page book on algebra, which I will post here in 4 segments. (Just what you always wanted hey??!) In a nutshell, what I look for in a calculus student is whether they know the "factor theorem" from algebra or not. Many do not.

 

 

Basic “high school” Algebra:

I. In mathematical parlance there is no such thing as “pre-algebra”. I.e. we don’t use this word, so I don’t know what this topic consists of. In traditional elementary mathematics there is arithmetic, geometry, algebra, and calculus. That’s it. So presumably pre - algebra is a combination of the abstract parts of arithmetic with the easier parts of algebra.

 

So what is the content of elementary arithmetic and algebra? It is first the study of elementary types of numbers, primarily integers and rational numbers (fractions of integers), and operations on those numbers, +, -, *, /. The operation * (multiplication) leads to the operation of exponentiation, a^b, at least for a >0.

 

Then we study the properties of those operations, such as commutativity and distributivity, and the exponent laws, which hold no matter what numbers they are applied to. This may be the beginnings of algebra, since even to state these properties we muse use letters to represent arbitrary numbers. I.e. we say things like this: ”for all integers n and m, we always have n+m = m+n”; and “for all integers n,m,p, we have n(m+p) = nm + np”. Or “for all positive rational numbers a, and all rational numbers b and c, we have a^(b+c) = a^b *a^c”. Hence this wonderful operation “changes addition into multiplication”.

 

Finally we try to solve equations, in which we seek numbers that have certain required properties when our operations are applied to them. E.g. we could seek numbers x and y such that x+y = 10, and xy = -24, or equivalently we could seek all numbers x such that x^2 -10x -24 = 0.

 

To gain skill in answering such questions we practice manipulating expressions containing letters which stand for numbers, i.e. which have the same properties as numbers when our operations are applied to them. We call such expressions polynomials. More precisely a “monomial in X” is a “power of X”, or something like X^n, where n is a positive integer. Then a polynomial is a sum of monomials, each multiplied by a number. The multipliers are called coefficients.

 

E.g. 6X^3 – 7X + 14, is a polynomial in X with integer coefficients, while the polynomial X^3 – (7/6)X + 7/3 in X has rational coefficients. The highest exponent of X which occurs is called the “degree” (in X) of the polynomial, so these examples both have degree 3. A polynomial of degree 1 is called linear, and one of degree 2 is called quadratic, while one of degree 3 is called cubic, (quartic, quintic, etc…)

 

A polynomial is called “monic” if the coefficient of the highest power of X occurring (the “leading” coefficient) is 1. The second cubic polynomial above is monic but not the first. Since the second polynomial was obtained by dividing the first one by 6, we have made it monic at the cost of introducing fractions as coefficients. In every problem with polynomials, one should always say whether a polynomial with rational coefficients is allowable as a solution, or whether one wants a solution with integer coefficients.

Edited by mathwonk
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II. A basic rule of arithmetic is the division law for positive integers, which says if d and n are positive integers, then there are unique non negative integers q,r such that n = qd + r, and 0 ≤ r < d. q is called the “quotient” of n by d, and r is the remainder.

 

E.g. if n = 75 and d= 12, then q = 6 and r = 3, i.e. 75 = 6*(12) + 3.

If n = 12 and d = 75, then q = 0 and r = 12, i.e. 12 = 0*(75) + 12.

 

If r = 0 we say “d divides n” (evenly), or d is a divisor, or factor, of n.

Thus you can divide n by d so that the remainder r is either zero or at least smaller than d.

 

It is often of interest to write an integer as a product of positive integer factors as small as possible, but greater than 1. E.g. 243 = 3*3*3*3*3 = 3^5;

2310 = 2*3*5*7*(11).

 

An integer n ≥ 2 which cannot be factored as a product of smaller factors greater than 1 is called “prime” or “irreducible”.

E.g. 2,3,5,7,11,13,17,19,23,29,31, are all prime, as is 197 and also 65,537.

 

Euclid proved in his book the Elements, that there are infinitely prime integers. In particular Euclid contains much more than just elementary plane geometry.

 

Factoring large integers into prime factors can be quite challenging. E.g. can you factor the integer 1,507,351 into primes? (Hint: its factors occur among the examples above.)

 

The next important concept about integers is the greatest common divisor (gcd) of two integers, and the crucial fact that the gcd of two integers can always be written as the sum of (positive or negative) integer multiples of those integers. In geometric terms, given two measuring sticks, the largest third measuring stick that will measure them both evenly, equals the smallest length that can be measured using the two original sticks together repeatedly. (In Euclid, the Greek word for “divides” is “measures”.) Not all arithmetic courses treat this fundamental topic.

 

This can be illustrated by fun word puzzles such as trying to figure out how to measure one quart of water using two different size buckets, holding say 18 quarts and 7 quarts. This is possible because the gcd of 18 and 7 is 1. But using buckets holding 18 and 33 quarts one can not measure less than 3 quarts.

 

The basic result linking gcd’s and division is that for integers n,m,k, if the gcd(n,m) = 1, and if n divides mk, then n divides k.

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III. Next we try to imitate these basic operations and rules also for polynomials. The first rule of polynomial algebra is the division algorithm:

 

Given two polynomials with integer coefficients f(X), g(X), with f monic, there exist unique polynomials q(X),r(X), also with integer coefficients, such that g(X) = q(X)*f (X)+ r(X), and either r(X) = 0 or degree(r(X)) < degree(f(X)).

 

Thus you can divide any g by a monic f so that the remainder r is either zero or at least has lower degree than f. (If rational coefficients are acceptable for q and r, then f need not be monic, just f≠0 is enough.)

 

E.g. if we divide g = 14X^4 – X^3 + X^2 -100 by f = X^3 + 6X + 1, I got q = 14X – 1, and r = -83X^2 -8X -99.

I.e. I hope 14X^4 –X^3 +X^2 -100 = (14X-1)*(X^3+6X+1) +(-83X^2 -8X -99).

 

If r = 0, we say f is a factor of g, or f divides g (evenly). Again we are interested in factoring polynomials into factors whose degrees are as small as possible. A polynomial with rational coefficients that cannot be factored into a product of two polynomials of positive degree with rational coefficients, is called irreducible over the rationals.

 

(Factoring with integer polynomials is more complicated since we might be able to factor out an integer factor. Thus 2X-2 is factorable over the integers as 2(X-1) but irreducible over the rationals.)

 

E.g. since the degrees of the factors add up to the degree of the product, a polynomial of degree one must be irreducible . Thus the most extreme way to factor a polynomial would be as a product of linear factors. However that is often not possible as many rational polynomials, e.g. X^2 – 2, do not have positive lower degree rational linear factors.

 

The divison algorithm leads immediately to a basic tool for finding linear factors of polynomials, the “factor theorem” or “root/factor theorem”, the first basic result in elementary algebra.

 

Theorem: If f(X) is a polynomial over the rationals, then the monic linear polynomial (X-c) is a factor of f (over the rationals) if and only if f© = 0, i.e. if and only if c is a “root” of f.

 

Proof: If f(X) = (X-c)*q(X), then setting X=r on both sides gives f© = (c-c)*q© = 0. Conversely, since by dividing we get f(X) = (X-c)*q(X)+r, where either r = 0 or degree® < degree (X-c) = 1, then r is a constant. Hence f© = (c-c)*q©+r = r. So the value f© equals the remainder after dividing f(X) by (X-c). Hence f©= 0 if and only if X-c divides f(X) evenly. QED.

 

This result is used all the time in elementary calculus. Any course that does not contain this result is not an adequate elementary algebra course. However I have regularly had calculus classes whose members seemed mostly innocent of this result.

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IV. There is one other basic result that goes with this one, the ”rational root theorem”. I.e. since the result above shows how to go from rational roots to rational factors, it is useful to have a tool for locating rational roots.

 

Theorem: If f(X) is a monic integer polynomial, all rational roots are integers, and the only possible such integer roots must be among the integer factors of the constant term.

 

E.g. the only possible rational roots of X^3 –X - 6, are the integer factors of 6, namely ±1, ±2, ±3, ±6. In fact X=2 works, so (X-2) is a factor.

 

If the integer polynomial is not monic, then there can be rational roots obtained by dividing some factor of the constant term by some factor of the leading coefficient.

E.g. the only possible rational roots of 9X^3 – 7X + 2, are fractions of form p/q where p divides 2 and q divides 9. I think X = 1/3 works for instance, hence (X-1/3) is a factor, or equivalently (3X-1).

 

The proof of this theorem is more advanced since it uses the properties of greatest common divisors of integers, a topic that may not have been taught in the arithmetic course.

 

This result is very strong; it implies for example that there is no rational number r with r^2 = 2. I.e. the only possible rational solutions of X^2 – 2 = 0, by this theorem are integer factors of 2, namely ±1, ±2, but these do not work.

 

Thus if a book includes this theorem but then claims without proof that sqrt(2) is irrational, implying it is hard, then something is odd.

 

The last topic in elementary algebra would be learning to complete the square and solve, hence factor, all quadratic equations, at least if we allow numbers formed from square roots of rational combinations of the coefficients.

 

The fundamental fact about quadratic equations is this, given a quadratic equation X^2 – bX + c, the sum of the roots is b and their product is c.

 

Thus if p,q are the roots then b = p+q and c = pq. To solve it note that (p-q)^2 = (p+q)^2 - 4pq = b^2 – 4c. Hence (p-q) = sqrt(b^2-4c),

 

so 2p = (p+q)+(p-q) = b + sqrt(b^2-4c), and 2q = (p+q) – (p-q) = b – sqrt(b^2-4c).

 

E.g. the roots p,q of X^2 – X - 1 = 0 are (1/2)(1±sqrt(5)).

 

If we introduce the new number i = sqrt(-1) we can express solutions of all quadratic equations using it if necessary, i.e. if b^2-4c < 0. (In fact then all polynomial equations have roots that theoretically can be expressed using i in combination with real numbers, but these solutions cannot usually be expressed as formulas involving just simple roots of the coefficients.)

 

End of basic algebra. If it does not look like much, I assure you I would have been delighted if my calculus classes knew this much.

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Thank you! To me that list is mostly computational arithmetic (sections 2,3,4,5,7,8), plus graphing lines (section 6), and apparently memorizing and plugging into a few geometry formulas (section 9), unless 9.5 ("find" areas circumferences, etc of circles), really means understand the formulas. This is a compendium of the basic facts we were taught from 3rd grade through 8th grade in the 1950's, with the fun stuff possibly omitted, like the sum of a geometric series, unless that is in section 5.2, for changing infinite repeating decimals to fractions.

 

Then there are a few manipulative exercises involving adding and multiplying polynomials (section 10). The gcd is apparently not even mentioned, and the lcm is mentioned in a manipulative way, presumably to use when adding fractions, (which is actually unnecessary for that task).

 

There is no mention of the most powerful tool for finding gcd's and lcm's by repeated alternate division, namely the Euclidean algorithm (contained also in his famous Elements), so I don't how (or if) the student is supposed to be able for example to "find" the gcd and lcm of say 234,567 and 291,403. No, after re - reading it, I'm pretty sure he/she is not.

 

In 3.1, is a student expected to know the basic rules for "casting out" 3's and 9's to test divisibility by those numbers? Of course everyone presumably learns to test divisibility by 2 and 5, but what about the easy rule for divisibility by 11? Is a student expected to know quickly that a number like 739 is not divisible by 7? Is it explained also that in factoring say 389, one need only test primes less than 20?

 

It is not clear to me to what depth these topics are expected to be known for "mastery". So I myself would find it a little hard to make sure my child met these criteria fully. But there seems a lack of mention of any proofs, or explanations of "why". In sections 6.6 and 6.7 it seems odd to omit solving two simultaneous linear equations in two unknowns. Exponents in general seem to receive short shrift. It is not clear to me whether the meaning of a basic concept like area is discussed. I.e. what does it mean to say the "area" of a circle of radius r is πr^2?

 

In college we prefer that a student knows what area is, rather than memorizing a formula supposedly giving it. I suspect it is because some students do not even know that volume refers to the number of little cubes that fit inside something, that some have trouble remembering whether the volume of a sphere is a multiple of r^3 (r "cubed") or of r^2 (r "squared").

 

And if area is the number of little squares that fit inside something, then what does it mean to say the area of sphere is 4πr^2, when no squares at all fit onto a spherical surface? Or even that the area of a circle is πr^2, when in fact you cannot fill up a circle with squares. In fact what does it mean to take "π" squares, since π is not even a rational number? Do students think π is 22/7? I.e. do they realize that rather than circumference being defined in terms of π and r, that it is actually the other way around, and π is defined by circumference versus radius? And what does the "length" of circle mean anyway? You can't measure it with a straight measuring stick, i.e. how do you lay off copies of the radius along the circumference?.....????? Are these questions discussed?

 

I.e. in teaching for "mastery", are the reasons behind the formulas discussed, and the meanings of the concepts? In section 5.2 do students learn to convert infinite decimals like 3.141414.... into fractions?

 

Based on recommendations, I just bought my granddaughter a copy of AoPS pre algebra but have not seen it myself. I wonder, does it cover roughly the things on this list? Does it address some of the concerns I have noted?

 

Thanks for all your help!

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Pre-Algebra mastery checklist http://cty.jhu.edu/sebin/a/y/prealg.pdf

 

Dd9 isn't proficient with everything on that list. I doubt my older dc were before they started algebra years ago. This week dd9 is multiplying and simplifying radical expressions with polynomials -- yet....she still needs work on percents, metric conversions and exponents. I'm not worried about gaps since she is young. Layering different topics and skills seems to work for a lot of students who have an interest to start algebra early.

 

I think it is Sharon in Austin who uses aops algebra and elementary math with her 'middle' girl. I thought that was a fabulous idea when I read that here about a year ago.

 

To the OP, I am not an expert like mathwonk and others here. Our family enjoys math so it seemed natural to progress despite the gaps. My dds think 'math is fun' which is my ultimate goal. Every day is slightly different and we love it this way. Time will tell if this grand experiment worked. :)

 

They came to me with these notes last week.

post-530-13535087205543_thumb.jpg

post-530-13535087205543_thumb.jpg

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For the curious, here is an example of changing standards in algebra over 250 years. In 1765, the great Euler wrote the following algebra book reportedly for his butler, who knew no mathematics.

 

http://archive.org/details/elementsalgebra00lagrgoog

 

"'[A] work which, though purely elementary, displays the mathematical genius of its author, and is still considered one of the best works of its class. Euler was one of the very few great mathematicians who did not deem it beneath the dignity of genius to give some attention to the recasting of elementary processes and the perfecting of elementary textbooks, and it is not improbable that modern mathematics is as greatly indebted to him for his work along this line as for his original creative work' (B. F. Finkel, 'Leonhard Euler', American Mathematical Monthly, Vol. 4, Dec. 1897, pp. 297-302. "

 

It begins in part 1 with whole numbers, after a short explanation of what mathematics is! Within less than 50 pages he has introduced complex ("imaginary") numbers. By part 4, chapter IX, page 244, he discusses quadratic equations, and less than 20 pages later, chapter XII, page 262, he gives Cardano's method for solving cubic equations, a topic probably not taught in any high school in the US, I would guess.

 

Although I have "taught" this topic to graduate students, I never really understood it clearly and simply until I read Euler's version. After reading it here I taught it last summer to 10 year olds. Shortly later Euler also discusses solving quartic equations, but I haven't read that yet. And this is about halfway through the book! For a young future mathematician, I would be curious, and optimistic, about the impact of this level of algebra treatment. (And it's free!)

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Here is an introduction to solving cubics for those who understand solving quadratics: (I wrote this as an online introduction for the more advanced epsilon campers last summer, to have something to do before camp.)

 

Cubic formula

Suppose we have x = a+b. Then cube both sides, and we get x^3 = (a+b)^3 = a^3 + b^3 + 3ab^2 + 3a^2b = a^3 + b^3 + 3ab(a+b) = a^3 + b^3 + 3abx.

Thus x= a+b is always a solution of the special cubic equation

X^3 = 3abx + (a^3+b^3).

 

In fact, this equation isn’t so special.

I.e. if we are given an equation x^3 = fx + g, we can try to solve it by finding a,b, so that f = 3ab, and g = a^3 +b^3. I.e. then x = a+b will be a solution. We claim such a and b can always be found.

 

E.g. given x^3 = 3x + 2. We can look for a,b so that 3ab = 3, and a^3 + b^3 = 2.

In this case, a=b=1 works. Thus x = 1+1 = 2 solves the equation, i.e. 2^3 = 3(2) + 2.

 

Another easy one is x^ 3 = 9x + 28. We want a,b so that 3ab = 9, and a^3 + b^3 = 28.

But a=1, b=3, works. Thus x = 1+3 = 4 solves the equation.

 

Try x^3 = 12x + 16. If you don’t guess it, notice that we want 3ab = 12, so we want ab = 4. And we want a^3 + b^3 = 16. Now this implies that a^3b^3 = 4^3 = 64. So if we try first to find a^3 and b^3, we know their sum is 16 and their product is 64.

 

From our study of quadratic equations, this means that a^3 and b^3 are solutions of the quadratic equation t^2 – 16t + 64 = 0. But t = 8,8 works for this. So we have a^3 = 8, and thus a = 2, and b = 2, work. Thus x = 2+2 = 4 should solve our equation, and it does so, since 4^3 = 64 = 12(4) + 16 = 48+16.

 

In fact this solution procedure always works, i.e. every cubic of form x^3 = fx+g, can always be written as x^3 = 3abx+(a^3+b^3) for some a and b. We find a and b by solving a quadratic equation like the one above.

 

To see it, we just have to show that if we are given f and g, then we can always solve 3ab = f and a^3+b^3 = g, for a and b. Lets start by solving for a^3 and b^3.

 

If f = 3ab, then f/3 = ab, so f^3/27 = a^3b^3, and thus we are given a^3b^3 = f^3/27, and a^3+b^3 = g, and we want to solve for a^3 and b^3. But we know from our study of quadratic equations, that if we are given the sum and the product of two numbers, we can find the numbers!

 

So we can always find a^3 and b^3 so that a^3b^3 = f^3/27, and a^3 + b^3 = g, by solving the quadratic equation t^2 – gt + f^3/27 = 0.

 

E.g. in the case of x^3 = 18x + 35, we have f = 18, g=35, so f/3 = 6, and f^3/27 = 6^3 = 216. So to find a^3 and b^3, we want to solve t^2 – 35t + 216 = 0.

 

Since 216 = 6^3 = 2^3.3^3 = 8.27, we see that this equation factors as (t-8)(t-27) = 0, so a^3 = 8 is one solution, and we can take a = 2, and then b = 3. So x = 2+3 = 5 is a solution. Indeed then 5^3 = 125 = 18(5) + 35.

 

Try this one: x^3 = 24x + 72.

And this one: x^3 = 30x + 133.

And this one: x^3 = 18x + 217.

Edited by mathwonk
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Finally, every cubic can be transformed into one of that form, i.e. without an x^2 term, just as a quadratic can be changed into a perfect square, i.e. one without an x term. The reason is that just as the coefficient u, of x^2 –ux+v, is the sum of the roots of the quadratic, so also the coefficient u of x^3 – ux^2_+vx-w, is the sum of the roots of the cubic.

 

Thus if we subtract u/3 from each root, their new sum will be zero. I.e. let x –u/3 = y, or x = y+u/3, and multiply out x^3 - ux^2 + vx -w = (y+u/3)^3 – u(y+u/3)^2 +v(y+u/3) – w, we should get a cubic in y with no y^2 term. I.e. the y^2 terms come from the first two terms, the first one has y^2 term y^2.u, while the second term has y^2 term –y^2.u. So yep, they cancel.

 

Next we solve our equation of form y^3 – fy –g = 0, or y^3 = fy+g, by the method explained above, and then we take x = y+u/3 to solve the original equation.

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Mathwonk thank you for sharing your math expertise with all of us. I have to admit it has been a long time since I did algebra in school. The pre-algebra mastery list is more of a check list I use as a guide to prepare my dd9 for algebra (in about 2yrs). I am not in any hurry. She is working her way through Singapore, LOF and TOP Math (fun hands on approach to probability, graphing, measurements/supposedly aimed at middle school levels) series. The TOPS math has added a depth my dd9 needed to get at the "why" study/use math? We got to measuring probability of coin tosses (50), using ratios, converting into percents, in one lesson, which was more effective than any boring worksheet on ratios and percents. We have a copy of AoPS Pre-Algebra and I will be using this as my resource. I am not a mathematician and much of the curriculum we have used thus far has not expressed math terminology which is the real switch I feel coming as we dive deeper.

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You are very welcome and I think you quite patient to put up with all my math jargon. Please feel free to post any math language or concept questions that come up here where everyone can share them.

 

Checklists are sometimes meant, at least where I come from, as a political or face saving tool, to claim that a lot is getting covered, although that one looked pretty straight forward, and not overly ambitious. I have seen some here that were ridiculously inclusive and really made it impossible for anyone to prepare from them. The result was the placement and competency tests were restricted to a tiny part of the official syllabus, but hard for the test taker to know that in advance.

 

 

My point is just that there really isn't much substance to elementary algebra. Forgive me for repeating myself:

 

First one just has to realize what the general rules for adding and multiplying are, and pretend that they apply also to letters standing for numbers. The hard part of this in practice is the endless lists of "simplification" exercises with algebraic expressions, but these can be useful and fun, if not overdone. In particular one should learn the 3 basic rules:

 

ab = ba, a(b+c) = ab + ac, and a^(b+c) = a^b*a^c, maybe also (ab)^x = a^x*b^x. Also the easiest way to add x/a + y/b, is not to memorize the rule (xb+ya)/ab, but just to multiply through by (b/b) and (a/a), getting (bx)/ab + (ay)/ab, and then add.

 

E.g. to simplify 1/(1/a + 1/b) just multiply by (ab)/(ab), getting (ab)/(b+a).

 

Secondly, one needs to understand that "solving" polynomial equations for a "root" c, is exactly the same thing as factoring out a factor of form (X-c). This is the all important "factor theorem" that follows immediately from the basic division principle. This must be mastered.

 

After that, there is the rational root theorem, and really that's about it, unless you want to throw in the binomial theorem. Of course setting up word problems is a valuable exercise in reading and thinking.

 

Somehow many of my entering college students managed to take years of "pre algebra", "algebra", and "pre calculus" courses without coming away even with these few skills and facts.

Edited by mathwonk
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You are very welcome and I think you quite patient to put up with all my math jargon. Please feel free to post any math language or concept questions that come up here where everyone can share them.

 

Checklists are sometimes meant, at least where I come from, as a political or face saving tool, to claim that a lot is getting covered, although that one looked pretty straight forward, and not overly ambitious. I have seen some here that were ridiculously inclusive and really made it impossible for anyone to prepare from them. The result was the placement and competency tests were restricted to a tiny part of the official syllabus, but hard for the test taker to know that in advance.

 

 

My point is just that there really isn't much substance to elementary algebra. Forgive me for repeating myself:

 

First one just has to realize what the general rules for adding and multiplying are, and pretend that they apply also to letters standing for numbers. The hard part of this in practice is the endless lists of "simplification" exercises with algebraic expressions, but these can be useful and fun, if not overdone. In particular one should learn the 3 basic rules:

 

ab = ba, a(b+c) = ab + ac, and a^(b+c) = a^b*a^c, maybe also (ab)^x = a^x*b^x. Also the easiest way to add x/a + y/b, is not to memorize the rule (xb+ya)/ab, but just to multiply through by (b/b) and (a/a), getting (bx)/ab + (ay)/ab, and then add.

 

E.g. to simplify 1/(1/a + 1/b) just multiply by (ab)/(ab), getting (ab)/(b+a).

 

Secondly, one needs to understand that "solving" polynomial equations for a "root" c, is exactly the same thing as factoring out a factor of form (X-c). This is the all important "factor theorem" that follows immediately from the basic division principle. This must be mastered.

 

After that, there is the rational root theorem, and really that's about it, unless you want to throw in the binomial theorem. Of course setting up word problems is a valuable exercise in reading and thinking.

 

Somehow many of my entering college students managed to take years of "pre algebra", "algebra", and "pre calculus" courses without coming away even with these few skills and facts.

 

I've been reading and rereading this post, hoping to understand 5% of it. :D

 

I need to go study now. :blush:

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Theorem: If f(X) is a polynomial over the rationals, then the monic linear polynomial (X-c) is a factor of f (over the rationals) if and only if f© = 0, i.e. if and only if c is a “root†of f.

 

Proof: If f(X) = (X-c)*q(X), then setting X=r on both sides gives f© = (c-c)*q© = 0. Conversely, since by dividing we get f(X) = (X-c)*q(X)+r, where either r = 0 or degree® < degree (X-c) = 1, then r is a constant. Hence f© = (c-c)*q©+r = r. So the value f© equals the remainder after dividing f(X) by (X-c). Hence f©= 0 if and only if X-c divides f(X) evenly. QED.

 

This result is used all the time in elementary calculus. Any course that does not contain this result is not an adequate elementary algebra course. However I have regularly had calculus classes whose members seemed mostly innocent of this result.

 

But they all know the @#%(%#@ quadratic formula. They have no idea what it means, but they know that it gives 'the answer'. Even if there's no equals sign, if you just plug everything into the quadratic formula, you get 'the answer'!

 

:rant:

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Yes, elementary algebra is relatively simple and my dd9 has already learned many concepts within the list I posted and the examples you have given on the previous post. The area I need to work on is teaching math terminology/language. Math is not difficult for my dd9 but she lacks the "wow-I love math" attitude. She is not motivated to do more math when she does well on a worksheet/test. The only time I have really seen her excited about math is when she has done the TOPS math probability. These hands on math assignments make math come alive for her. In her mind, she is finally "doing math" for it has a purpose, an outcome to measure rather than a completion to a worksheet. So we are taking math conceptual detours and getting to basic elementary algebra in a round about way. I would rather foster a love for math than rush ahead. Thank you mathwonk for being apart of this board, I am sure you will be much utilized by all of us:001_smile:

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But they all know the @#%(%#@ quadratic formula. They have no idea what it means, but they know that it gives 'the answer'. Even if there's no equals sign, if you just plug everything into the quadratic formula, you get 'the answer'!

 

:rant:

 

:lol:

 

I would have been pleased if they knew the quadratic formula.

I had one last year (developmental math...so basically prealgebra) who didn't k ow the standard algorithm for multiplication. She could only do lattice.

I hate the (expletive) everyday math!

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Jewel, I think you are very wise to foster the love of the subject foremost.

 

 

As to the quadratic formula, I explained it above in the way LaGrange explains it in his algebra book. When I saw that (last year( it was the first time I really understand why the formula looks the way it does.

 

 

I.e. we are trying to find two numbers a and b, and the whole problem is to start from two things, their sum a+b and their product ab, and somehow find a and b themselves.

 

Why is this the problem? Well if a and ba re the roots of a quadratic equation, then the equation factors as (X-a)(X-b), and when we multiply this out we see that we get X^2 -(a+b)X + ab.

 

Thus, by thinking backwards, given any quadratic equation X^2 - pX + q, we know going in that p must be the sum of the desired roots, and q is their product.

 

I.e. solving a quadratic equation means factoring it, or writing X^2 - pX + q as (X-a)(X-b), and to do this means finding a and b such that a+b = p and ab = q.

 

 

Ok, so now we have reformulated the problem. Given the sum and product of two numbers, how do we find the two numbers?

 

Well. simple, just find their difference. i.e. if we know a+b and can also find a-b, tha by adding and subtracting we will get 2a and 2b.

 

I.e. (a+b)+(a-b) = 2a, and (a+b)-(a-b) = 2b. then dividing by 2 will give a and b.

 

(continued later after dinner.... can you see what part of the quadratic formula is a-b?)

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As to the quadratic formula, I explained it above in the way LaGrange explains it in his algebra book. When I saw that (last year( it was the first time I really understand why the formula looks the way it does.

 

That's pretty interesting, not a way I would have thought to attack it. I can see since we know ab we're going to get (a-b)^2 by subtracting 4ab from (a+b)^2.

 

It always made sense to me as completing the square with an arbitrary quadratic.

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Well I suspect you're just better than me at understanding completing the square. Everybody has his own "aha!" experience. I was always turned off by this formula because completing the square worked but did not explain why the terms looked the way they did, at least I didn't get it.

 

But this approach just uses the old basic formula (a-b)^2 = (a+b)^2 - 4ab as you say.

 

 

i.e. if we write x for both a and b, then when p = a+b and q = ab, we get (a-b)^2 = p^2 -4q.

 

Thus 2x = (a+b) ± (a-b) = p ± sqrt(p^2 - 4q).

 

 

That speaks to me more simply and more clearly than the bare formula. Of course you are right, the same thing is visible in completing the square, if you think about what the terms mean, but I didn't.

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