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Algebraic Symmetries
Radical Expressions and Equation
The Exponential Function
Math 1010-3 Exam #3 Review Guide
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Are You Ready for Math 65?
Solving Simultaneous Equations Using the TI-89
Number Theory: Fermat's Last Theorem
Course Syllabus for Intermediate Algebra
Solving Inequalities with Logarithms and Exponents
Introduction to Algebra Concepts and Skills
Other Miscellaneous Problems
Syllabus for Calculus
Elementary Linear Algebra
Adding and Subtracting Fractions without a Common Denominator
Pre-Algebra and Algebra Instruction and Assessments
Mathstar Research Lesson Plan
Least Common Multiple
Division of Polynomials
Counting Factors,Greatest Common Factor,and Least Common Multiple
Real Numbers, Exponents and Radicals
Math 115 Final Exam Review
Root Finding and Nonlinear Sets of Equations
Math 201-1 Final Review Sheet
Powers of Ten and Calculations
Solving Radical Equations
Factoring Polynomials
Section 8
Declining Price, Profits and Graphing
Arithmetic and Algebraic Structures
Locally Adjusted Robust Regression
Topics in Mathematics
Syllabus for Mathematics
The Quest To Learn The Universal Arithmetic
Solving Linear Equations in One Variable
Examples of direct proof and disproof
Algebra I
Quadratic Functions and Concavity
More on Equivalence Relations
Solve Quadratic Equations by the Quadratic Formula
Solving Equations and Inequaliti
MATH 120 Exam 3 Information
Rational Number Ideas and Symbols
Math Review Sheet for Exam 3
Linear Algebra Notes
Factoring Trinomials
Math 097 Test 2
Intermediate Algebra Syllabus
How to Graphically Interpret the Complex Roots of a Quadratic Equation
The General, Linear Equation
Written Dialog for Problem Solving
Radian,Arc Length,and Area of a Sector
Internet Intermediate Algebra
End Behavior for linear and Quadratic Functions
Division of Mathematics
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General linear equations
Algebraic Symmetries
Math 20A Final Review Outline
Description of Mathematics
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Basic Operations with Decimals: Division
Mathematics Content Expectations
Academic Systems Algebra Scope and Sequence
Syllabus for Introduction to Algebra
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Environmental Algebra
More Math Practice Problems
Intermediate Algebra
Syllabus for Linear Algebra and Differential Equations
Intermediate Algebra
Rational Expressions and Their Simplification
Course Syllabus for Intermediate Algebra
GRE Review - Algebra
Foundations of Analysis
Finding Real Zeros of Polynomial Functions
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Real Numbers
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Final Review Solutions
Exponential and Logarithmic Functions

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Pre-Algebra and Algebra Instruction and Assessments

Connecting Arithmetic to Algebra

Here is how you read algebraic expressions

sum n + 3 “n plus 3” difference n – 3 “n less 3”
product 3 x n or 3n “3 times n” quotient n÷3 “n divided by 3”

In the Dog House

Moving from Expressions to

• Problem: The left pan of a set of scales
contains 5 identical boxes of noodles,
and the right pan contains 3 identical
boxes and two 2-kg weights. The scales
are balanced. How much does each box

Equation: An equality with a variable.
Root: Each value of the variable that makes the statement a
true equality.

The scale each part shows the total weight of the shapes on that scale ,The same shapes
have the same weight in each of the pictures, Find the weight of each shape ,
HINT: You may begin with any one of the four pictures that will help you start.

Give your own values to the figure in A, B, and C. The same figures in A, B,
and C will have the same values ,Different figure will stand for different
values ,Remember to keep the scale in A ,B ,and C balanced!

17.Expressions with a variable
problem.5 truckloads of sand were
x truckloads were delivered. Each truck held 3
tons of sand . How many tons of sand were
delivered to the playlot?
5 + t truckloads
Each truckload contained 3 tons of sand
The total amount of sand delivered is 3 X (5 + t)
3 (5 + t) is an expression with a variable
If t is 2, than 3 X (5 + 2) = 21
If t is 5, than 3 X (5 + 5) = 30

Variables and Properties

• A variable represents a number–even
though its value may not be given.

• Expressions with variables satisfy all the
properties of the number system, such
as the commutative, associative, and
distributive properties.

The Real Numbers

2 = 2 + 5

Commutative Property of
5 + t = t + 5

Commutative Property of
3 X 7 = 7 X 3

Commutative Property of
3 X t = t X 3

Commutative Property of
3 X 7 X 2 = 7 X 2 X 3

Associative Property of
5 X 3 X t = t X 3 X 5

Associative Property of
3 + 5 + 2 = 5 + 2 + 3

Associative Property of
3 + 5 + t = 5 + t + 3

Associative Property of
3 X (5 + 2) = 3 X 5 + 3 X 2

Distributive Property of
3 X (5 + t) = 3 X 5 + 3 X t

Distributive Property of

Key Definitions: Expressions
with More than One Variable

1. Solution of an equation: A value (or an ordered pair of values)
that satisfies the equation
2. Equivalent equations: Equations that have the same solution
3. Linear equation: An equation equivalent to one of the form ax
+ by = c with a2 + b2 not equal to 0
4. Function: A rule connecting two sets that assigns to each
element of one set (or input) one and only one element of the
second set (or output)
5. Graph of an equation in two variables: Points in the plane
whose coordinates satisfy the equation
6. Sequence: A function from the positive integers to the real

Expressions involving Two or
More Variables…

• Adhere to the same commutative,
associative, and distributive properties:

x(4x3 + 2y3) = x . 4 . x3 + x . 2 . y3
= 4 . x . x3 + 2 . x . y3
= 4x4 + 2xy3

A Historical Statement on
Translating Word Problems

• In solving a word problem by setting up
equations, the student translates a real
situation into mathematical terms: he has an
opportunity to experience that mathematical
concepts may be related to realities, but such
relations must be carefully worked out. Here is
the first opportunity afforded by the curriculum
for this basic experience.

Process for Translating Word
Problems to Algebra Problems

1. Students verbally read and explain what an
expression/equation means,
2. Students formulate a verbal instruction as an
algebraic expression,
3. Students translate components of word problems
into mathematical expressions,
4. Students construct word problems associated with a
particular algebraic expression
5. Students define variables, and
6. Students explicitly solve problems

Introduction to Functions

Functions Can Be Introduced
Early without Formal Definitions


A rule between two
sets such that there
is a unique element
in the second set
assigned to each in
the first set

rule of correspondence



f(x) = x + 4
{(1, 2), (3, 2), (5, 1)}


  3y + 5x
{(1, 2), (1, 3), (4, 0)}

Counter Examples