Developmental Math II

Al Grocia

Chapter 1

Module 0 – Numeric and variable expressions

NUMERIC EXPRESSIONS

numbers

• What is a number?

• What does a number “do”?

• What can we “do” with numbers?

• How many numbers are there here? 5 – 12

Building numbers – Operators

Geometric view• Adding -Line segments • Multiplying – rectangles

Algebraic view: abstract• Algebra defines 2 operators Add – combine terms Multiply – combine factors• with 9 properties Identity InverseCommutative Associative Distributive

Algebra meets geometry

Number lines and algebra tiles

Examples: 4 + 2 : 8 – 3 : 5(6) : 15÷3

geometric algebraic

Number lines/tiles

Examples:7 – 12 : -3 + -8 : (-3)(2) : (-6)÷(-2)

geometric• ?????

algebraic

Number lines/tiles

Examples

geometric algebraic

Number lines/tiles

Basic operations • Adding properties: order does not matter but signs do 0 is called the additive identity and does not change the value of a number when added to it opposites are numbers with the same units but different signs and opposites add to zero – these numbers are sometimes called additive inverses (discuss opposites vs subtraction vs negatives)• Multiplication properties Always use / for division sign and write division with fraction notation so that order doesn’t matter but numerator vs denominator does 1 is called the multiplicative identity and does not change the value of a number when multiplied by it Reciprocals are numbers with the numerator and denominator switched (flipped) and these numbers multiply to one – these numbers are sometimes called multiplicative inverses • Math “bee”

Exponent notation

• Actually not an operator in algebra – exponent notation is a functional notation for repeated multiplication-

• Exponent – a physically small number written to the right and raised higher than its base – the exponent counts the factors of a multiplication problem (or follows the ^ sign)

• Base – the repeated factor • Power – the answer to the problem

Prime factorization

• It is sometimes useful to view a number “unsimplified”

• You can break a number into terms or factors• For now we will explore factoring

vocabulary

• Integral factor – an integer (not a fraction/decimal) which is a factor of the given number (ie: divides exactly into it)

• Prime – a number that has exactly 2 integral factors

• Composite – a number that has more than 2 integral factors

• NOTE: 0 and 1 are neither prime nor composite • Note: 2 is the only even prime number

factoring

• Writing a number as the product of 2 or more other numbers is called factoring

ex: 56 = 8(7) 56 = (-2)(-28) 56 = (.5)(112)• Writing a number as the product of prime

numbers is called prime factoring 56 = 2(2)(2)(7) = 23 • 7

Tools for finding prime factoring• Hints: Be familiar with prime numbers : {2, 3, 5, 7, 11, 13…} Know some division tricks : Even numbers- divisible by 2 Ends in 0 or 5 – divisible by 5 Digits add up to 3 or 9 – divisible by 3 • Factor tree

• Factor tower

Building numbers with 2 or more operators

• Order of operations (pemdas - gema)• Multiplication first – combine factors - product

• Addition last – combine terms – sum • Ex:

Grouping symbols• Parenthesis, brackets, braces, radical signs, fraction bar, absolute value

bars…• Brackets and braces were used to make grouping in long nested problems

easier to track. They are seldom used any more therefore parenthesis must be “tracked”

ex: 4 – ( 9 – (8 + 3(-7) – 12)) + 15• Fraction bar is a division sign that groups the numerator and denominator ex: 6 + 21/3 • Absolute value bars: a grouping symbol that is also a function – | #### | - it

means “distance” and therefore turns the number inside to a positive value. ex: |2 – 8| |8 – 2|

Examples

Fractions

• Geometric view:

Fractions

• Algebraic view:

Fraction - summary• Multiplying and dividing fractions: reciprocals cancel, order doesn’t matter,

denominators do NOT need to match showing work: 1. factor numerator and denominator 2. rearrange order to show factors of 1 3. remove factors of 1• Adding and subtracting fractions: the addition must be grouped by using

COMMON DENOMINATORS and changing the division expression showing work: 1. show inserted factor that make the denominators the same 2. show regrouping 3. show added value 4. simplify if necessary – show factor of one that is removed

VARIABLE EXPRESSIONS

Variable – a number that changes value

• Variables have several uses 1. a blank to fill in with a given number 2. a way to maneuver an unknown number 3. a way to explore how one number affects other number

• a variable is used whenever the value of a number is unknown, unimportant, or unnecessary to answering a question

Evaluating variables

Chapter 2

Module 1 – linear equations, inequalities, functions

VARIABLE EQUATIONSSingle variable

INEQUALITIES AND NUMBER LINE GRAPHS

Single variable

VARIABLE EQUATIONSTwo or more variables