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Classroom Tools Series: Mathematics

Interactive Algebra Tutorials

Mathematician Stanley Gudder once said, “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” Indeed, mathematics can only be as complicated as a student sees it to be. But with an open mind and constant practice in solving equations and problems, students can better appreciate the wonders and applications of mathematics.

Here are some algebra lessons and exercises that teachers and students can use in math classes!

Algebraic Terms and Expressions

Term

Definition

Constant

A symbol whose domain contains ONLY ONE element
Example: In 4x2 + x + 3, 3 is the constant

Variable

A symbol used to represent any element of a given domain
Examples: x or a or y or b or z or c….

Coefficient

Any factor of a product is said to be the coefficient of the other factors
Examples:
In 5x, 5 is the coefficient of x
In 5xy, 5x is the coefficient of y

Algebraic
Term

A term of a polynomial is a constant or a constant multiplied by nonnegative-integer powers of variables
Examples:
3 -  a constant
3x – a constant multiplied by an integer
3x2 – a constant multiplied by nonnegative-integer powers of variables

Like terms

Terms that differ only in their constant coefficients
Examples: 3x2 and 7x2 OR 15y4 and 6y4

Algebraic
Expression

Combination of a constants and variables, involving operations (MDAS, powers, roots, etc.)
Examples:
4x2y5
4x4y5 + 3x – 6
√x4 + 2y5(6x)

Polynomial

An algebraic expression involving only nonnegative-integer powers of one or more variables
Examples:
4x2y5
4x4y5 + 3x – 6y + z
√x4 + 2y5(6x) – 8x

Monomial

A polynomial with only ONE term
Example: 4x2y5

Binomial

A polynomial with TWO terms
Example: 4x4y5 + 3x OR 3x – 6

Trinomial

A polynomial with THREE terms
Example: 4x4y5 + 3x – 6

Degree
of polynomial

The degree of the term with the highest exponent value
Examples:
In a2 + b3 + c4, degree = 4
In x + 3, degree = 1

Try these links below for more notes and exercises on algebraic terms and expressions:

Online Notes on Polynomials – with definitions, examples, and solutions
Writing Algebraic Expressions – with examples and quiz on how to write and translate expressions

Rational Expressions, Special Products, and Factoring Polynomials

Operation

Formula

Product of two binomials

(x+a)(x+b) = x2 + (a+b)x + ab
(ax+by)(cx+dy) = acx2 + (ad+bc)xy + by2

Square of a sum

(x+y)2 = x2 + 2xy + y2

Square of a difference

(x-y)2 = x2 - 2xy + y2

Product of a sum and
difference

(x+y)(x-y) = x2 – y2

Cube of a sum

(x+y)3 = x3 + 3x2y + 3xy2 + y3

Cube of a difference

(x-y)3 = x3 - 3x2y + 3xy2 - y3

Removing a Common
Monomial Factor

ax + ay + az = a(x + y + z)

Difference of Two Squares

x2 – y2 = (x+y)(x-y)

Factoring Trinomials

x2 + (a+b)x + ab = (x + a)(x + b)

Perfect-Square Trinomial

x2 + 2xy + y2 = (x + y)2

Factoring Cubes

x3 + y3 = (x+y)(x2 – xy + y2)
x3 - y3 = (x-y)(x2 + xy + y2)

Try these links below for more notes and exercises on Rational Expressions, Special Products, and Factoring Polynomials:

Integer Exponents – with simplified formulas on rules of exponents, and practice problems
Rules of Exponents – with lessons, examples, and quiz
Operation with Rational Expressions - with easy-to-read notes and practice problems
Rational Expressions - with definitions, examples, and solutions
Factoring Special Products – with easy-to-read notes and practice problems
Factoring Polynomials - with definitions, examples, and solutions
FOIL Method – animated illustration of the FOIL method
Complex Numbers – with comprehensive notes and practice problems

Equations and Inequalities

Linear equations can be applied in solving word problems like investment problems, work problems, dimension problems, age problems, and uniform-motion problems. Here’s a simple step-by-step guide on how to solve story or word problems:

  1. Read and understand the problem.
  2. Re-read the problem. Take note of key algebraic sentences (e.g. the sum of, twice a number, less than a number, etc.)
  3. Determine the known and unknown quantities.
  4. Use a variable to represent the known and unknown quantities.
  5. Determine the relation of the known and unknown quantities with each other. (If applicable, sketch a figure to illustrate the situation, as in dimension or distance problems)
  6. Construct an equation for the problem. Make use of known applicable formulas for the problem.
  7. Solve the equation.
  8. Write the final answer with its correct unit of measurement.
  9. Check the final answer.

For more lessons and exercises on solving equations and inequalities, check out the links below:

Linear Equations (Slope of a Line, System of Linear Equations, Applications of Equations)
Solving Equations and Inequalities
Translating Word Problems into Equations
Interactive Quizzes
Linear Inequalities

For more online resources, visit the sites listed below. Have fun with Math!

Internet4Classrooms (Secondary Math Internet Resources)
Math Nerds
Ask Dr. Math
For teachers: Algebra Lesson Plans from Math Forum
Homeschool Math (Online Math resources, quizzes, lesson plans, tutorials, software)

Sources:

Leithold, L. “College Algebra”. California: Addison-Wesley Publishing Co., Inc., 1991. pp. 26-127
“Paul’s Online Math Notes”. Lamar University Math Tutorial. Retrieved on October 2, 2007 from http://tutorial.math.lamar.edu/
“Topics in Pre-Algebra”. Mrs. Glosser’s Math Goodies. Retrieved on October 2, 2007 from http://www.mathgoodies.com/lessons/toc_vol7.html
“Interactive Math Lessons”. Math Warehouse. Retrieved on October 2, 2007 from http://www.mathwarehouse.com/