Math Magazine. One of projects this school year 2010-2011. this is arranged as back to back, so if you are going to print this. it would be easy. Good Luck :">

2. 5.
statement that two
algebraic expressions
are equal inequality
 equation
 variable
 polynomial
6.
equation whose terms
contain squares of the
variable and no higher
powers quartic
 quadratic
 cubic
 linear

3. MAERK CHECK (/) FOR THE CORRECT ANSWER
1.
number that multiplies a variable or
variables monomial
 - binomial
 - coefficient  Why is Algebra so Important?
 - trinomial It was not very long ago when good
2. computation skill in arithmetic was the key
a number, a variable or product of numbers and to getting a job, a diploma, or managing a
variables monomial business. Even as late as the 1970's we
expected that most students who finished
 - binomial
high school would be able to calculate
 - coefficient quickly and accurately. We taught algebra
 - trinomial and geometry to our college bound, and
3. just a very few of the math geeks took
letter that stands for an unknown number constant calculus.
 - coefficient
Today the world has radically changed.
 - binomial Algebra has moved down to the 7th and
 - variable 8th grades for many of our students and it
4. is a requirement for high school
a monomial or a sum of two or more graduation, for college graduation, and for
monomials unlike terms most any job in America today. But is this
 - like terms right? Should we be demanding algebra as
a rite of passage?
 - polynomial
 - equation The answer is a resounding yes!


4.  Algebra is not only the essential language
of mathematics it is about two ideas that
are critical to 21st century jobs and
citizenship. The first idea is variable.
Variables are symbols that can represent
not only a number but a quantity that is
changing. A variable can represent a ball
being thrown or an automobile being
driven or the Dow Jones Average. Algebra
lets us manipulate variables like arithmetic
lets us manipulate numbers.
The second fundamental idea, one that is
often called the most important idea in
mathematics, is the concept of function. A
function is a well defined relationship
between two variables so that as the value
of one changes so does the value of the
other. With variables and functions we can
command spreadsheets, we can define the
physical laws that govern our
universe, and we can build patterns that
enable us to understand how our world
works.

5.  This is the solution to algebra number problem 13
as asked by an anonymous user: "A maths test  So algebra is more than a set of rules and
contains 10 questions. Ten points are given for procedures for solving canned problems. It
each correct answer and three points deducted for is a way of thinking. We built Enablemath
an incorrect answer. If Ralph scored 61, how many to be
did he get correct?"
algebraic. Even in the simplest
assignments we have students think
 Ok on this problem you can come up with a
algebraically. Students can not only step
solution faster by just quick trial an error. You
through an example, they can change the
know that if it only counts 3 points off for each
values, treating numbers as variables.
wrong answer and they got a 61 then they had to
They can see how these values are related
have at least a 70 before the points were taken
to each other in the dynamic visualizations
off. If they got a 70 that means they missed 3
that make our presentation of concepts
problems (3 * 3 = 9) 70 - 9 = 61 . So the answer
understandable. The What if Wheel gives
is they got 7 problems right and 3 wrong.
every student the power over
Now to set this up using algebra...
variables, and the screens are populated
10x - 3(10-x) = 61 You are subtracting the wrong
with objects that have functions tied to
answers worth 3 points a piece from the right
these variables.
ones worth 10 a piece. You don't know how many
Our students not only learn the algebra
of each so you are saying there are x amount of
that they need for school and for passing
10 valued answers and (10 - x) number of 3
exams, they learn and intuitively
valued wrong answers.
understand algebra and use the
10x - (30 - 3x) = 61
understanding to apply it to any problem
10x -30 + 3x = 61
that they may find. We believe that no one
13x = 91 --> x = 91/13 = 7 --> so there are 7
else has done this in as rich and
right 10 valued problems and (10-7) or 3 wrong 3
comprehensive a fashion. Yes, algebra is
point negative value ones.....
fundamental, but it also has to be learned
in a new way. We have created
You could have also set it up like this 10(10 - x) -
Enablemath with that in mind. We believe

6.  Elementary algebra
 Elementary algebra is the most basic form
of algebra. It is taught to students who are
presumed to have no knowledge
of mathematicsbeyond the basic principles
of arithmetic. In
arithmetic, only numbers and their
arithmetical operations (such as +, −, ×, ÷)
occur. In algebra, numbers are often
denoted by symbols (such as a, x, or y).
This is useful because:
 It allows the general formulation of
arithmetical laws (such as a + b = b + a for
all a and b), and thus is the first step to a
systematic exploration of the properties of
the real number system.
 It allows the reference to "unknown" Across Down
numbers, the formulation of equations and 1. 10 + 5 2. 79 -28
the study of how to solve these. (For 3. 52 + 32 4. 30 + 15
instance, "Find a number x such that 3x + 1
6. 18 - 7 5. 54 - 10
= 10" or going a bit further "Find a
8. 97 - 40 7. 21 - 9
number x such that ax+b=c". This step
leads to the conclusion that it is not the 10. 51 - 9 9. 98 - 28
nature of the specific numbers that allows 12. 47 - 26 11. 16 + 8
us to solve it, but that of the operations 14. 32 + 14 13. 23 - 10
involved.) 16. 78 - 47 15. 18 +43
19. 9 + 7 17. 26 -12
 It allows the formulation
21. 50 - 9 18. 48 + 4
of functional relationships. (For instance, "If
23. 70 - 42 20. 78 - 15
you sell x tickets, then your profit will be
3x − 10 dollars, or f(x) = 3x − 10, where f is 25. 79 - 47 22. 13 - 3
the function, and x is the number to which 27. 4 + 21 24. 78 + 4
the function is applied.") 28. 52 - 29 26. 50 - 28

7. Example #1 Example  Polynomials
+ #2 -  A polynomial is an expression that is
=? constructed from one or
more variables and constants, using
only the operations of
=? addition, subtraction, and
multiplication (where repeated
multiplication of the same variable is
standardly denoted as exponentiation
with a constant nonnegative integer
exponent). For example, x2 + 2x − 3
is a polynomial in the single
variable x.
 An important class of problems in
algebra is factorization of
polynomials, that is, expressing a
given polynomial as a product of
other polynomials. The example
polynomial above can be factored as
(x − 1)(x + 3). A related class of
problems is finding algebraic
expressions for the roots of a
polynomial in a single variable.

8.  Abstract algebra
 Abstract algebra extends the familiar concepts
found in elementary algebra
and arithmetic of numbers to more general
concepts.
 Adding or Subtracting
 Sets: Rather than just considering the different
types of numbers, abstract algebra deals with the
Rational Expressions
more general concept of sets: a collection of all with Like Denominators
objects (called elements) selected by
property, specific for the set. All collections of the
 To add or subtract rational
familiar types of numbers are sets. Other expressions with like
examples of sets include the set of all two-by- denominators, add or
two matrices, the set of all second-
degree polynomials (ax2 + bx + c), the set of all
subtract their numerators
two dimensional vectors in the plane, and the and write the result over
various finite groups such as the cyclic the denominator.
groups which are the group of
integers modulo n. Set theoryis a branch Then, simplify and factor
of logic and not technically a branch of algebra. the numerator, and write
 Binary operations: The notion of addition (+) is the expression in lowest
abstracted to give a binary operation, ∗ say. The terms. This is similar to
notion of binary operation is meaningless without
the set on which the operation is defined. For two
adding two fractions with
elements a and b in a set S, a ∗ b is another like denominators, as in.
element in the set; this condition is
called closure. Addition (+), subtraction (-
), multiplication (×), and division (÷) can be binary
operations when defined on different sets, as is
addition and multiplication of

9. Rational Expressions
 Identity elements: The numbers zero and
 When we discuss a rational expression in this one are abstracted to give the notion of
chapter, we are referring to an expression whose an identity element for an operation. Zero is
numerator and denominator are (or can be written the identity element for addition and one is
as) polynomials. For example, and are rational the identity element for multiplication. For a
expressions. general binary operator ∗ the identity
element e must satisfy a ∗e = a and e ∗ a = a.
 To write a rational expression in lowest This holds for addition as a + 0 = a and 0
+ a = a and multiplication a × 1 = a and 1
terms, we must first find all common factors × a = a. Not all set and operator combinations
(constants, variables, or polynomials) or have an identity element; for example, the
the numerator and the denominator. Thus, positive natural numbers (1, 2, 3, ...) have no
we must factor the numerator and the identity element for addition.
denominator. Once the numerator and the  Inverse elements: The negative numbers
denominator have been factored, cross out give rise to the concept of inverse elements.
any common factors. For addition, the inverse of a is −a, and for
 Example 1: Write n lowest terms. multiplication the inverse is 1/a. A general
inverse element a−1 must satisfy the property
Factor the numerator: 6x 2 -21x - 12 = 3(2x 2 - 7x - 4) that a ∗ a−1 = e and a−1 ∗ a = e.
= 3(x - 4)(2x + 1) .  Associativity: Addition of integers has a
property called associativity. That is, the
Factor the denominator: 54x 2 +45x + 9 = 9(6x 2 + grouping of the numbers to be added does
5x + 1) = 9(3x + 1)(2x + 1) .
not affect the sum. For example: (2 + 3) + 4 =
Cancel out common factors: = 2 + (3 + 4). In general, this becomes (a ∗ b)
∗ c = a ∗ (b ∗ c). This property is shared by
most binary operations, but not subtraction or
 Example 2: Write in lowest terms.
division or octonion multiplication.
Factor the numerator: x 3 - x = x(x 2 - 1) = x(x + 1)(x - 1) .
 Commutativity: Addition and multiplication of
4 3
Factor the denominator: 6x +2x -8x = 2x 2 2(3x 2 + x - 4) = real numbers are both commutative. That
2x 2(x - 1)(3x + 4) . is, the order of the numbers does not affect
Cancel out common factors: = .
the result. For example: 2+3=3+2. In
general, this becomes a ∗ b = b ∗ a. This
 property does not hold for all binary
operations. For example,matrix
multiplication and quaternion
multiplication are both non-commutative.

10. Groups
 Combining the above concepts gives one of the most important
structures in mathematics: a group. A group is a combination of
a set S and a single binary operation ∗, defined in any way you
choose, but with the following properties:
 An identity element e exists, such that for every
member a of S, e ∗ a and a ∗ e are both identical to a.
 Every element has an inverse: for every member a of S, there
exists a member a−1 such that a ∗ a−1 and a−1 ∗ a are both
identical to the identity element.
 The operation is associative: if a, b and c are members
of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c).
 If a group is also commutative—that is, for any two
members a and b of S, a ∗ b is identical to b ∗ a—then the
group is said to be abelian.
 For example, the set of integers under the operation of addition
is a group. In this group, the identity element is 0 and the
inverse of any element a is its negation, −a. The associativity
requirement is met, because for any integers a, b and c, (a + b)
+ c = a + (b + c)
 The nonzero rational numbers form a group under
multiplication. Here, the identity element is 1, since 1 × a = a ×
1 = a for any rational number a. The inverse of a is
1/a, since a × 1/a = 1.
 The integers under the multiplication operation, however, do not
form a group. This is because, in general, the multiplicative
inverse of an integer is not an integer. For example, 4 is an
integer, but its multiplicative inverse is ¼, which is not an
integer.
 The theory of groups is studied in group theory. A major result
in this theory is the classification of finite simple groups, mostly