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- INTERMEDIATE ALGEBRA
- Course Syllabus for Algebra I
- Mid-Plains Community College
- FRACTION OF A WHOLE NUMBER
- Systems of Linear Equations
- MATH FIELD DAY
- Course Outline for Finite Mathematics
- Calculus
- Algebra Final Examination
- Math 310 Exam #2
- Review of Trigonometric Functions
- Math 118 Practice test
- Precalculus Review
- Section 12
- Literal Equations
- Calculus Term Definitions
- Math 327A Exercise 2
- Public Key Algorithms II
- Maximizing Triangle Area
- Precalculus I Review for Midterm
- REVIEW OF A FIRST COURSE IN LINEAR ALGEBRA
- Math 6310 Homework 5
- Some Proofs of the Existence of Irrational Numbers
- ALGEBRAIC PROPERTIES OF MATRIX OPERATIONS
- Math 142 - Chapter 2 Lecture Notes
- Math 112 syllabus
- Math 371 Problem Set
- Complex Numbers,Complex Functions and Contour Integrals
- APPLICATIONS OF LINEAR EQUATIONS
- Week 4 Math
- Fractions
- Investigating Liner Equations Using Graphing Calculator
- MATH 23 FINAL EXAM REVIEW
- Algebra 1
- PYTHAGOREAN THEOREM AND DISTANCE FORMULA
- Georgia Performance Standards Framework for Mathematics - Grade 6
- Intermediate Algebra
- Introduction to Fractions
- FACTORINGS OF QUADRATIC FUNCTIONS
- Elementary Algebra Syllabus
- Description of Mathematics
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- College Algebra - Applications
- A Tip Sheet on GREATEST COMMON FACTOR
- Syllabus for Elementary Algebra
- College Algebra II and Analytic Geometry
- Functions
- BASIC MATHEMATICS
- Quadratic Equations
- Language Arts, Math, Science, Social Studies, Char
- Fractions and Decimals
- ON SOLUTIONS OF LINEAR EQUATIONS
- Math 35 Practice Final
- Solving Equations
- Introduction to Symbolic Computation
- Course Syllabus for Math 935
- Fractions
- Fabulous Fractions
- Archimedean Property and Distribution of Q in R
- Algebra for Calculus
- Math112 Practice Test #2
- College Algebra and Trigonometry
- ALGEBRA 1A TASKS
- Description of Mathematics
- Simplifying Expressions
- Imaginary and Complex Numbers
- Building and Teaching a Math Enhancement
- Math Problems
- Algebra of Matrices Systems of Linear Equations
- Survey of Algebra
- Approximation of irrational numbers
- More about Quadratic Functions
- Long Division
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- MATH 101 Intermediate Algebra
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- WRITTEN HOMEWORK ASSIGNMENT
- Description of Mathematics
- Rationalize Denominators
- Math Proficiency Placement Exam
- linear Equations
- Description of Mathematics & Statistics
- Systems of Linear Equations
- Algebraic Thinking
- Study Sheets - Decimals
- An Overview of Babylonian Mathematics
- Mathematics 115 - College Algebra
- Complex Numbers,Complex Functions and Contour Integrals
- Growing Circles
- Algebra II Course Curriculum
- The Natural Logarithmic Function: Integration
- Rational Expressions
- QUANTITATIVE METHODS
- Basic Facts about Rational Funct
- Statistics
- MAT 1033 FINAL WORKSHOP REVIEW
- Measurements Significant figures
- Pre-Calculus 1
- Compositions and Inverses of Functions
Algebraic Thinking
Outline
1 What is Algebra?
2 Algebra and Graphs
3 Algebraic Manipulations
4 Conclusion
Examples of Algebra
Many people find the thought of algebra, equations, and variables
intimidating. But it is just generalized arithmetic.
Algebraic Examples
Consider the following common uses of algebra.
•Formulas – (C = πd)
Relation between two or more variables
•Equations – (5x = 30)
Finding an unknown value
•Identities – (sin2x + cos2x = 1)
An expression true for any x
•Property – (a + b = b + a)
Expression of a general rule
•Function – (f (x) = 3x + 1)
An input (independent) and output (dependent) variable
Examples of Using Algebra
Let’s look at a few examples from this list.
Formulas |
The formula
relates temperatures in Farenheit to temperatures in Celsius. Use this formula to convert 20 degrees cesius to farenheit and 41 degrees farenheit to celsius. |
Equations |
Fill in the blank.
|
Properties |
Use symbols to express the fact that every number
has an additive inverse. |
Algebra as a Study of Structure
There are many different ways to view algebra.
Study of Structure | ||||||||
Algebra can be seen as a study of structure. That
is, what is the structure of arithmetic? How does it work? |
||||||||
Example | ||||||||
For all real numbers a, b and c the following
laws hold:
|
Algebra as a Study of Relationships
Another way to view algebra is a method to express relationships.
Relationships Between Quantities |
Algebra can be seen as a study of the
relationship between quantities. The concept of a function is important here as there is usually an “input” quantity and an “output” quantity. |
Example |
A phone card has a connection fee of $0.25 plus a
$0.05/minute charge for the actual time of the call. Describe the price of the call as a function of the number of minutes spent on the call. |
Use the following methods to express the relationship above |
•a table •a graph •a function rule |
Using Graphs to Visualize Algebraic Relationships
As we saw in the previous example, graphs can be an
important
way to visualize a relationship between quantities which can be
expressed algebraically.
Example |
A beautician charges $15 for haircuts. Each week
she has fixed expenses of $150. Express her profit as a function of teh number of haircuts she gives. Use a graph to describe this function. |
Question |
Are the table, graph, and formula we used to
answer the previous question really accurate? In particular, is it possible to give haircuts? How is this problem seen in the graph? |
Interpreting Graphs
Many times a graph is enough to describe a relationship.
Example |
Below are descriptions of three runners in a
race. Match each description to the correct graph and explain your choice.
•Alex started slowly, then ran a bit faster, and then
ran even faster at the end of the race. |
Recalling Algebraic Rules
There are several symbolic rules to working with algebra
which you
have learned in the past. Let’s review some of these rules.
Adding Polynomials |
When adding polynomials group like terms together
and use the distributive property (a(b + c) = ab + ac) to combine like terms. |
Example |
Add the following polynomials.
|
Multiplication and Factoring
Finally, we will briefly review multiplying and factoring.
Example |
Multiply the following polynomials.
|
Example |
Factor the following polynomials completely.
|
Important Concepts
Things to Remember from Section 2.2 |
1 Various uses of algebra 2 Solving equations 3 Representing relationships using fuctions and graphs 4 Adding, multiplying, and factorying polynomials |