Download Developmental Math II

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
Algebra meets geometry
Number lines and algebra tiles
Algebraic view: abstract
• Algebra defines 2 operators
Add – combine terms
Multiply – combine factors
• with 9 properties
Identity
Inverse
Commutative Associative
Distributive
Examples:
4 + 2 : 8 – 3 : 5(6) : 15÷3
geometric
Number lines/tiles
algebraic
Examples:
7 – 12 : -3 + -8 : (-3)(2) : (-6)÷(-2)
geometric
• ?????
Number lines/tiles
algebraic
Examples
geometric
Number lines/tiles
algebraic
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
6+21
ex: 6 + 21/3
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
Single variable
VARIABLE EQUATIONS
Single variable
INEQUALITIES AND NUMBER LINE
GRAPHS
Two or more variables
VARIABLE EQUATIONS