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What is Algebra?

A generalization of arithmetic. Mathematics Dictionary, 4 th Ed. by James/James Therefore, you need to know and understand arithmetic!

What is Algebra? A generalization of arithmetic in which symbols, usually letters of the alphabet, represent numbers or members of a specified set of numbers and are related by operations that that obey specified laws. The American Heritage Dictionary, 2 nd College Edition

What is Algebra? A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set. www.dictionary.com

Terminology & Notation Mathematical Vocabulary Precise Communication Notation Symbols instead of words Semantics & Syntax Multiplication & Division in Algebra? Yes, you are expected to be able to read, understand, and use correct mathematical terminology and notation.

Terminology & Notation Example …

Terminology Variable A symbol used to represent a value. Usually a letter (Latin or Greek) The value is … – unknown – to be determined – assigned different quantities

Terminology Constant A specific value. – Integers – Fractions – Decimals – Special

Terminology Algebraic Expression A legal combination of variables, constants, operators, and grouping symbols. AKA: Expression

Terminology Equivalent Expressions Expressions that are equal for all values of their variables. Simplifying: Changing expressions into simpler equivalent expressions.

Terminology Term A constant, variable, or product of a constant and one or more variables. When grouping symbols are removed, all expressions are a sum of terms. Coefficient – The constant part of a term Linear Term – A constant times a variable. x = 1x-x = -1x

Terminology Factor The constants or variables that are multiplied together in a term. Common Factor – A constant or variable that is a factor of each term of an expression.

Terminology Identities additive identity: 0 multiplicative identity: 1

Terminology Inverses Opposite – The negative of the number. – aka: Additive inverse Reciprocal - One divided by the number. – aka: Multiplicative inverse

Terminology Commutative Order of addition and multiplication may be reversed.

Terminology Associative Sums and products may be grouped in any way. What about subtraction and division?

Terminology Distributive (involves both operations) “3 is distributed over the sum x+5.” What about subtraction and division?

Terminology Equation A mathematical statement that two expressions are equal. Three possibilities … – always true  “identity” – always false  “fallacy” – neither  “conditional”

Terminology Inequality A mathematical statement that one expression is... > < ≥ ≤ … a second expression.

Terminology Solution A value for the variable that makes an equation or inequality a true statement. Solution Set: All such values.

Terminology Absolute Value How far a number is from zero. AKA: Magnitude

Terminology Real Numbers The set of all numbers that correspond to the points on a number line. – Can be thought of as directed distances from zero. 0 1xy

Terminology Integers The set of positive and negative counting numbers and zero.

Terminology Positive Integers The set of counting numbers. AKA: Natural Numbers

Terminology Negative Integers The set of opposites of the positive integers.

Terminology Non-Negative Integers The set of counting numbers and zero. AKA: Whole Numbers

Terminology Rational Numbers Set of numbers that can be written as a quotient of two integers. Decimal forms will either be terminating decimals or repeating decimals.

Terminology Irrational Numbers All Real numbers that are not Rational.

Properties of Zero

Addition Signed Number Arithmetic Think Position on # Line Start @ 1 st # –Negative  to left of 0 –Positive  to right of 0 Add 2 nd # –Negative  move left –Positive  move right Are you moving towards or away from zero? Think Money in Your Pocket Start w/ 1 st # –Negative  in debt (owe) –Positive  cash on hand Add 2 nd # –Negative  spending –Positive  receiving Are you … –increasing your debt? –increasing your savings? –reversing your state? “Rules” on Page 28.

Subtraction Signed Number Arithmetic Think … –Add the opposite! “Rule” on Page 32. SEE THINK

Multiplication Signed Number Arithmetic Multiplying by -1 “Rules” on Page 37. So … –Multiply numbers ignore signs –Even # of negatives Positive result –Odd # of negatives Negative result

Division Signed Number Arithmetic Treat as a Fraction Therefore … reduce! “Rules” on Page 38. Division is … –multiplication by the reciprocal –sign of the result is just like multiplication.

Division Signed Number Arithmetic Warning: Be careful w/ Cancelling “Rules” on Page 38.

“Value of an Expression” The Question… Solution Replace x with its value and do the arithmetic! (be careful when x is negative)

Order of Operations Grouping symbols { }, [ ], ( ), — may be nested Exponents Multiplication & Division left to right Addition & Subtraction left to right

Simplify! Remove Grouping Symbols Distributive & Associative Laws

Simplify! Remove Grouping Symbols Distributive & Associative Laws Group Like Terms Commutative & Associative Laws Combine Like Terms Distributive Law (i.e. add coefficients) Complete All Arithmetic

Translate: Words  Algebra Addition –added to; sum of; plus; more than; increased by Subtraction –subtracted from; difference of; minus; less than; decreased by Multiplication –multiplied by; product of; times; twice; of Division –divided by; quotient of; divided into; ratio of; per Table of Terms w/ Examples on Page 50. … and combinations of these (e.g. twice the sum of …)

Problem Solving: The 4P’s Problem Solving? Analyzing a situation Organizing information Choosing a strategy Implementing the strategy The 4P’s Prepare –Read (not a novel, it’s technical) –What do you know? –What are you trying to find? Plan (i.e. strategy) –Guess & check? –Patterns? –Algebraic? Process (i.e. implementation) Ponder –Did you answer the question? –Does the solution make sense? –Units?