algebra help

[hollyh]

Hi.  I have a friend using Teaching Textbooks for Algebra.  However, the charter school she uses says that it does not cover all of the necessary Algebra content.  She needs to fill in some gaps and be able to prove through worksheets or what not that she did.  Any ideas on Algebra worksheets/lessons she could print out for specific topics?  

 

Additionally, what are your fave Algebra curriculums - in case she decides to try something new.

[Space station]

The charter school should tell her what THEY think she should use to fill in the gaps! Gee, what are they there for anyway if they can't do that!

 

ALEKS math online would fill in gaps.

[hollyh]

Well the charter school would prefer her to use a different curriculum altogether. If she wants to use this one they have told her that she can if she fills in the gaps.

[Michelle_NC]

DD is using Saxon, 3rd edition and it seems to be going well.

[mathmarm]

1) Get a copy of the Table of Contents of the charters "preferred Algebra 1" textbook*

2) Get a copy of the charters Algebra 1 Syllabus/pacing guide (the doc that says X section on this day)*

3) Compare their syllabus/pacing guide to the TOC of Teaching Textbooks to see what isn't covered.*

4) Find worksheets on those topics via Google.

 

* It would be nice if the stupid charter school would take the initiative to do this themselves. Jeez! How lazy can they be? Its their job to remotely guide education and they won't even look at the darn 'road map'? Sheesh!

[mathwonk]

Thumbnail sketch of elementary algebra:

 

In algebra we try to practice rules for addition, multiplication, subtraction and division that will hold true for all numbers usually used.  E.g. not only is (2)(3) = (3)(2), but also (17)(12) = (12)(17), and (103)(76) = (76)(103),...and so on.  To state this once for all, we say that XY = YX, for all familiar numbers X and Y.  A typical fact true for all familiar numbers X is the equation X^3-1 = (X-1)(X^2+X+1).  Thus (2^3-1) = (2-1)(2^2+2+1), and (5^3-1) = (5-1)(5^2+5+1),  (19^3-1) = (19-1)(19^2+19 +1),..., etc.  To get a better feel for this, work out all these examples, including the one with X’s to see if they are really true.  To appreciate the value of algebra, note how much simpler it is to check the version with X’s compared to working out say (273^3 - 1)(273^2 + 273 + 1), in numerical terms.

 

This begs us to make more clear what are the “familiar†numbers, so we do so as follows:  there are the integers, or whole numbers, positive and negative and zero: ....,-4,-3,-2,-1,0,1,2,3,.......

 

Then there are the rational numbers, of form n/m where n and m are integers and m ≠ 0, and then the “real†numbers, which fill in all the holes on the number line.  Real numbers can be expressed as infinite decimals  like  328.12149836551100911......, where it is possible for all decimal entries to be zero after some point.

 

The rational numbers are a subset of the real numbers and consist of those numbers whose decimal expansion eventually repeats, possibly with all zeroes, such as 1.234565656....., or 317.998200000......

 

Some examples of real numbers that can be shown not to be rational are sqrt(2), sqrt(3), sqrt(3), cubert(5), ....,Ï€,......

 

Since non rational real numbers are so hard to write down, impossible really, since it takes infinitely many decimals, we try to do problems that involve mostly rational numbers, or else irrational numbers that have simple names like sqrt(3).

 

Algebra is used to find answers to certain questions whose answer is known to be a number, but we don’t know just which number it is.  Since we don’t know the answer in advance, we are required to reason on the unknown number using properties that are true of all numbers.

 

E.g. suppose we want to find a number X such that  X^2 - 5X + 6 = 0.  We may reason as follows:  for all numbers X it is true that  X^2 - 5X + 6 = (X-2)(X-3),  so we are trying to find a number X such that (X-2)(X-3) = 0.  But the product of any two numbers is zero only if at least one of those factors is zero, so we are seeking a number X such that either X-2 = 0 or X-3 = 0.  Since adding the same number to two equal numbers again gives equal numbers, we seek X such that either X = X-2+2 = 0+2 = 2, or X-3+3 = 0+3 = 3, i.e.our answer could be either X = 2 or X=3.

 

 

So a basic algebra skill to learn next is how we knew that X^2 - 5X + 6 = (X-2)(X-3).  This is called “factoringâ€, and is worth some practice. A few basic examples are worth learning by heart:

 

(X-A)(X- B ) = X^2 - (A+B)X + AB,  

X^2 - A^2 = (X-A)(X- B ), 

X^3-A^3 = (X-A)(X^2+AX+A^2), 

X^4-A^4 = (X-A)(X^3 + X^2A + XA^2 + A^3),

.....

(X^3+A^3) = (X+A)(X^2-AX+A^2),

X^5 + A^5 = (X+A)(X^4 + X^3A + X^2A^2 + XA^3 + A^4),

....

Actually this is more than I knew when I won the 1959 Tenn. mid-state algebra competition.

 

BASIC CONCEPTS and RESULTS:

 

Dividing polynomials (expressions in X).

 

Degree of a polynomial.

 

Root/factor theorem:  If a polynomial f(X) = 0 when X = A, then f(X) can be factored as

f(X) = (X-A)g(X), where g(X) is a polynomial of degree one less than f(X), (and vice versa).

 

solving equations by the quadratic formula;

 

graphing linear and quadratic equations;

 

solving linear equations involving one, two, and three unknowns.

 

Exponential notation:  i.e.  X^3.X^5 = X^8, and in general X^n.X^m = X^(n+m).

 

More advanced:  rational root theorem:  If f(X) is a polynomial with integer coefficients, and lead term X^n (i.e. with coefficient 1), then the only possible rational roots of the equation f(X) = 0, are integers, and indeed are integer factors of the constant term.

 

Consequence:  Since no integer factor of -2 is a root (solution) of X^2 -2 = 0, there is no rational root, so sqrt(2) is not a rational number.

 

 

If your child knows all this, then he/she surpasses my average entering college calculus student.

[kiana]

1) Get a copy of the Table of Contents of the charters "preferred Algebra 1" textbook*

2) Get a copy of the charters Algebra 1 Syllabus/pacing guide (the doc that says X section on this day)*

3) Compare their syllabus/pacing guide to the TOC of Teaching Textbooks to see what isn't covered.*

4) Find worksheets on those topics via Google.

 

* It would be nice if the stupid charter school would take the initiative to do this themselves. Jeez! How lazy can they be? Its their job to remotely guide education and they won't even look at the darn 'road map'? Sheesh!

 

I would agree with this.

[maize]

 

Thumbnail sketch of elementary algebra:

 

In algebra we try to practice rules for addition, multiplication, subtraction and division that will hold true for all numbers usually used. E.g. not only is (2)(3) = (3)(2), but also (17)(12) = (12)(17), and (103)(76) = (76)(103),...and so on. To state this once for all, we say that XY = YX, for all familiar numbers X and Y. A typical fact true for all familiar numbers X is the equation X^3-1 = (X-1)(X^2+X+1). Thus (2^3-1) = (2-1)(2^2+2+1), and (5^3-1) = (5-1)(5^2+5+1), (19^3-1) = (19-1)(19^2+19 +1),..., etc. To get a better feel for this, work out all these examples, including the one with X’s to see if they are really true. To appreciate the value of algebra, note how much simpler it is to check the version with X’s compared to working out say (273^3 - 1)(273^2 + 273 + 1), in numerical terms.

 

This begs us to make more clear what are the “familiar†numbers, so we do so as follows: there are the integers, or whole numbers, positive and negative and zero: ....,-4,-3,-2,-1,0,1,2,3,.......

 

Then there are the rational numbers, of form n/m where n and m are integers and m ≠ 0, and then the “real†numbers, which fill in all the holes on the number line. Real numbers can be expressed as infinite decimals like 328.12149836551100911......, where it is possible for all decimal entries to be zero after some point.

 

The rational numbers are a subset of the real numbers and consist of those numbers whose decimal expansion eventually repeats, possibly with all zeroes, such as 1.234565656....., or 317.998200000......

 

Some examples of real numbers that can be shown not to be rational are sqrt(2), sqrt(3), sqrt(3), cubert(5), ....,Ï€,......

 

Since non rational real numbers are so hard to write down, impossible really, since it takes infinitely many decimals, we try to do problems that involve mostly rational numbers, or else irrational numbers that have simple names like sqrt(3).

 

Algebra is used to find answers to certain questions whose answer is known to be a number, but we don’t know just which number it is. Since we don’t know the answer in advance, we are required to reason on the unknown number using properties that are true of all numbers.

 

E.g. suppose we want to find a number X such that X^2 - 5X + 6 = 0. We may reason as follows: for all numbers X it is true that X^2 - 5X + 6 = (X-2)(X-3), so we are trying to find a number X such that (X-2)(X-3) = 0. But the product of any two numbers is zero only if at least one of those factors is zero, so we are seeking a number X such that either X-2 = 0 or X-3 = 0. Since adding the same number to two equal numbers again gives equal numbers, we seek X such that either X = X-2+2 = 0+2 = 2, or X-3+3 = 0+3 = 3, i.e.our answer could be either X = 2 or X=3.

 

 

So a basic algebra skill to learn next is how we knew that X^2 - 5X + 6 = (X-2)(X-3). This is called “factoringâ€, and is worth some practice. A few basic examples are worth learning by heart:

 

(X-A)(X- B ) = X^2 - (A+B)X + AB,

X^2 - A^2 = (X-A)(X- B ),

X^3-A^3 = (X-A)(X^2+AX+A^2),

X^4-A^4 = (X-A)(X^3 + X^2A + XA^2 + A^3),

.....

(X^3+A^3) = (X+A)(X^2-AX+A^2),

X^5 + A^5 = (X+A)(X^4 + X^3A + X^2A^2 + XA^3 + A^4),

....

Actually this is more than I knew when I won the 1959 Tenn. mid-state algebra competition.

 

BASIC CONCEPTS and RESULTS:

 

Dividing polynomials (expressions in X).

 

Degree of a polynomial.

 

Root/factor theorem: If a polynomial f(X) = 0 when X = A, then f(X) can be factored as

f(X) = (X-A)g(X), where g(X) is a polynomial of degree one less than f(X), (and vice versa).

 

solving equations by the quadratic formula;

 

graphing linear and quadratic equations;

 

solving linear equations involving one, two, and three unknowns.

 

Exponential notation: i.e. X^3.X^5 = X^8, and in general X^n.X^m = X^(n+m).

 

More advanced: rational root theorem: If f(X) is a polynomial with integer coefficients, and lead term X^n (i.e. with coefficient 1), then the only possible rational roots of the equation f(X) = 0, are integers, and indeed are integer factors of the constant term.

 

Consequence: Since no integer factor of -2 is a root (solution) of X^2 -2 = 0, there is no rational root, so sqrt(2) is not a rational number.

 

 

If your child knows all this, then he/she surpasses my average entering college calculus student.

Thank you for the nutshell summary!

[OneStepAtATime]

1) Get a copy of the Table of Contents of the charters "preferred Algebra 1" textbook*

2) Get a copy of the charters Algebra 1 Syllabus/pacing guide (the doc that says X section on this day)*

3) Compare their syllabus/pacing guide to the TOC of Teaching Textbooks to see what isn't covered.*

4) Find worksheets on those topics via Google.

 

* It would be nice if the stupid charter school would take the initiative to do this themselves. Jeez! How lazy can they be? Its their job to remotely guide education and they won't even look at the darn 'road map'? Sheesh!

 

 

 

Thumbnail sketch of elementary algebra:

 

In algebra we try to practice rules for addition, multiplication, subtraction and division that will hold true for all numbers usually used.  E.g. not only is (2)(3) = (3)(2), but also (17)(12) = (12)(17), and (103)(76) = (76)(103),...and so on.  To state this once for all, we say that XY = YX, for all familiar numbers X and Y.  A typical fact true for all familiar numbers X is the equation X^3-1 = (X-1)(X^2+X+1).  Thus (2^3-1) = (2-1)(2^2+2+1), and (5^3-1) = (5-1)(5^2+5+1),  (19^3-1) = (19-1)(19^2+19 +1),..., etc.  To get a better feel for this, work out all these examples, including the one with X’s to see if they are really true.  To appreciate the value of algebra, note how much simpler it is to check the version with X’s compared to working out say (273^3 - 1)(273^2 + 273 + 1), in numerical terms.

 

This begs us to make more clear what are the “familiar†numbers, so we do so as follows:  there are the integers, or whole numbers, positive and negative and zero: ....,-4,-3,-2,-1,0,1,2,3,.......

 

Then there are the rational numbers, of form n/m where n and m are integers and m ≠ 0, and then the “real†numbers, which fill in all the holes on the number line.  Real numbers can be expressed as infinite decimals  like  328.12149836551100911......, where it is possible for all decimal entries to be zero after some point.

 

The rational numbers are a subset of the real numbers and consist of those numbers whose decimal expansion eventually repeats, possibly with all zeroes, such as 1.234565656....., or 317.998200000......

 

Some examples of real numbers that can be shown not to be rational are sqrt(2), sqrt(3), sqrt(3), cubert(5), ....,Ï€,......

 

Since non rational real numbers are so hard to write down, impossible really, since it takes infinitely many decimals, we try to do problems that involve mostly rational numbers, or else irrational numbers that have simple names like sqrt(3).

 

Algebra is used to find answers to certain questions whose answer is known to be a number, but we don’t know just which number it is.  Since we don’t know the answer in advance, we are required to reason on the unknown number using properties that are true of all numbers.

 

E.g. suppose we want to find a number X such that  X^2 - 5X + 6 = 0.  We may reason as follows:  for all numbers X it is true that  X^2 - 5X + 6 = (X-2)(X-3),  so we are trying to find a number X such that (X-2)(X-3) = 0.  But the product of any two numbers is zero only if at least one of those factors is zero, so we are seeking a number X such that either X-2 = 0 or X-3 = 0.  Since adding the same number to two equal numbers again gives equal numbers, we seek X such that either X = X-2+2 = 0+2 = 2, or X-3+3 = 0+3 = 3, i.e.our answer could be either X = 2 or X=3.

 

 

So a basic algebra skill to learn next is how we knew that X^2 - 5X + 6 = (X-2)(X-3).  This is called “factoringâ€, and is worth some practice. A few basic examples are worth learning by heart:

 

(X-A)(X- B ) = X^2 - (A+B)X + AB,  

X^2 - A^2 = (X-A)(X- B ), 

X^3-A^3 = (X-A)(X^2+AX+A^2), 

X^4-A^4 = (X-A)(X^3 + X^2A + XA^2 + A^3),

.....

(X^3+A^3) = (X+A)(X^2-AX+A^2),

X^5 + A^5 = (X+A)(X^4 + X^3A + X^2A^2 + XA^3 + A^4),

....

Actually this is more than I knew when I won the 1959 Tenn. mid-state algebra competition.

 

BASIC CONCEPTS and RESULTS:

 

Dividing polynomials (expressions in X).

 

Degree of a polynomial.

 

Root/factor theorem:  If a polynomial f(X) = 0 when X = A, then f(X) can be factored as

f(X) = (X-A)g(X), where g(X) is a polynomial of degree one less than f(X), (and vice versa).

 

solving equations by the quadratic formula;

 

graphing linear and quadratic equations;

 

solving linear equations involving one, two, and three unknowns.

 

Exponential notation:  i.e.  X^3.X^5 = X^8, and in general X^n.X^m = X^(n+m).

 

More advanced:  rational root theorem:  If f(X) is a polynomial with integer coefficients, and lead term X^n (i.e. with coefficient 1), then the only possible rational roots of the equation f(X) = 0, are integers, and indeed are integer factors of the constant term.

 

Consequence:  Since no integer factor of -2 is a root (solution) of X^2 -2 = 0, there is no rational root, so sqrt(2) is not a rational number.

 

 

If your child knows all this, then he/she surpasses my average entering college calculus student.

 

Both of those posts should help. Hopefully the charter school will step up to the plate and give specifics.  Whether they do or not, I would also suggest your friend look at the Key to Algebra workbooks to use alongside TT.  That might fill in the gaps in a fairly easy to implement way.  

[Lolly]

Hi.  I have a friend using Teaching Textbooks for Algebra.  However, the charter school she uses says that it does not cover all of the necessary Algebra content.  She needs to fill in some gaps and be able to prove through worksheets or what not that she did.  Any ideas on Algebra worksheets/lessons she could print out for specific topics?  

 

Additionally, what are your fave Algebra curriculums - in case she decides to try something new.

 

I would suggest using Aleks. Any topics that the student knows will be immediately tested out of. Topics that have been covered, but not mastered, will be repeated. Non covered topics can be learned and proven. If the student has mastery of the subject, they will be finished with it in a very short time, with proof. She does need to realize that just because they do not show immediate mastery of everything on the first assessment, that it doesn't mean TT did not cover the material. Also, if a basic concept is not passed (we do sometimes make silly errors), everything from that point will show up as work to be done. The assessment does not test more complex material until everything needed to work those problems has been passed.

[justasque]

Hi.  I have a friend using Teaching Textbooks for Algebra.  However, the charter school she uses says that it does not cover all of the necessary Algebra content.  She needs to fill in some gaps and be able to prove through worksheets or what not that she did.  Any ideas on Algebra worksheets/lessons she could print out for specific topics?  

 

Additionally, what are your fave Algebra curriculums - in case she decides to try something new.

 

 

Well the charter school would prefer her to use a different curriculum altogether. If she wants to use this one they have told her that she can if she fills in the gaps.

If the charter school will lend her the curriculum they prefer, then she can use it for any "gaps".  One approach would be to have the student do the end-of-chapter review for each chapter in the book.  This would help to see both the "gaps" in the TT curriculum, and also any gaps in understanding of what the TT has presented.  She could then use the charter's text to teach those (hopefully few) concepts, and use the corresponding tests to assess the learning.

 

My favorite algebra text (McDougal Littell, by Larson et.al.) has a wide selection of worksheets to supplement the text.  If your friend can get hold of these (they come on a CD-ROM that can sometimes be found inexpensively second-hand, or her charter might have access to it or something similar), she will have a wealth of resources from which to choose.  Worksheets include a "reteaching" page, practice pages at three levels, and a variety of other things for each lesson, plus 2-3 quizzes per chapter, plus tests at three levels.  I have often used the highest level test as a review, then used the middle level test for assessment.

[In The Great White North]

 

1) Get a copy of the Table of Contents of the charters "preferred Algebra 1" textbook*

2) Get a copy of the charters Algebra 1 Syllabus/pacing guide (the doc that says X section on this day)*

3) Compare their syllabus/pacing guide to the TOC of Teaching Textbooks to see what isn't covered.*

4) Find worksheets on those topics via Google.

 

* It would be nice if the stupid charter school would take the initiative to do this themselves. Jeez! How lazy can they be? Its their job to remotely guide education and they won't even look at the darn 'road map'? Sheesh!

 

They need to at least tell her what the gaps are that they require her to fill.  Specifically.

[Murrayshire]

Once you find out what those gaps are........ Have you seen Math Mammoth's Algebra worksheets? They are pretty reasonable in price...... They are only for practice though, no teaching involved. I would also suggest finding a used algebra textbook to go along with MM. http://www.mathmammoth.com/worksheets/algebra_1.php