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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.

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Developmental Math II

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  1. Developmental Math II Al Grocia

  2. Chapter 1 Module 0 – Numeric and variable expressions

  3. Numeric expressions

  4. numbers • What is a number? • What does a number “do”? • What can we “do” with numbers? • How many numbers are there here? 5 – 12

  5. Building numbers – Operators Geometric view Algebraic view: abstract Algebra defines 2 operators Add – combine terms Multiply – combine factors with 9 properties Identity Inverse Commutative Associative Distributive • Adding -Line segments • Multiplying – rectangles Algebra meets geometry Number lines and algebra tiles

  6. Examples: 4 + 2 : 8 – 3 : 5(6) : 15÷3 geometric algebraic Number lines/tiles

  7. Examples:7 – 12 : -3 + -8 : (-3)(2) : (-6)÷(-2) geometric algebraic • ????? Number lines/tiles

  8. Examples geometric algebraic Number lines/tiles

  9. 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”

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. Building numbers with 2 or more operators • Order of operations (pemdas - gema) • Multiplication first – combine factors - product • Addition last – combine terms – sum • Ex:

  16. 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|

  17. Examples

  18. Fractions • Geometric view:

  19. Fractions • Algebraic view:

  20. 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

  21. Variable expressions

  22. 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

  23. Evaluating variables

  24. Chapter 2 Module 1 – linear equations, inequalities, functions

  25. Single variable Variable equations

  26. Single variable Inequalities and number line graphs

  27. Two or more variables Variable equations

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