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Basics We will meet Monday – Friday from 1 – 4pm There is no class Tue., June 16 Last class is Thur., July 2 If the door is locked, you can call 515-341-3763 Bring paper, pen/pencil, workbook, & calculator

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You will not receive official credit for the class, but … You must master the material and pass the tests/quizzes to continue into Geometry in the fall.

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Notes and assignments will be archived at http://bggoldenbears.org/summeralgebra/ You can e-mail me at burrowd@bishopgarrigan.org If you are gone, you are responsible for what you miss.

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Algebra is one of the oldest branches of mathematics First developed about 4,000 years ago by the ancient Babylonians

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The Hindus in India organized algebra into a formal system.

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During the Middle Ages, the Arabs spread algebra across north Africa and into Spain, which is where we got it from.

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The word algebra literally means “simplifying”. originally an Arabic medical term

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The main thing we do in algebra is to take complicated expressions and simplify them. 3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) x = -26

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Algebra is the language of science. We apply ideas from algebra in almost every field.

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Variable

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Variable A symbol (usually a letter) that can stand for any number

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A variable is different from a constant, which is a symbol that stands for some specific number.

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Numerical Expression Just have numbers 2 + 2 3 5 – 6 3 Algebraic Expression Include variables 3x + 2y n 2 + 7n – 1

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We often use algebraic expressions to change words into symbols, which allows us to solve problems.

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What words mean … + – X =

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sum total increased by more than greater longer farther older

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difference decreased by less shorter closer younger

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product twice half of by

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quotient shared split per

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most verbs

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You can use variables to identify patterns

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1 4 2 7 3 10 4 13 n ?

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1 4 2 7 3 10 4 13 n 3n + 1

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Evaluating Expressions The most common tool for evaluating expressions is the order of operations.

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Order of Operations

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Parentheses AND other grouping symbol

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Parentheses AND other grouping symbols brackets

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Parentheses AND other grouping symbols brackets fraction bar

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Parentheses AND other grouping symbols brackets fraction bar root symbols

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Exponents AND roots

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Multiplication and division (same time – left to right)

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Addition and subtraction (same time – left to right)

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Evaluate 3x + y 2 if x = -2 and y = -3

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Evaluate 3x + y 2 if x = -2 and y = -3 3(-2) + (-3) 2 -6 + 9 = 3

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SETS OF NUMBERS Natural Numbers 1, 2, 3, 4, 5, 6 … numbers you count with positive (not zero) whole numbers

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Whole numbers 0, 1, 2, 3, 4, 5, … the natural numbers, and also zero. No negatives; no fractions

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Integers … -3,-2,-1, 0, 1, 2, 3, … Whole numbers and their opposites

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Each of these sets includes the ones from before Every natural number is also a whole number and an integer.

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Rational numbers“ratio” means fractionRational numbers include anything that can be written as a fraction of integers. _¾, -½, 2¼, -.5,.4, 7.3Integers like 6, -3, and 0 are also rational numbers.

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Rational numbers can always be expressed as a decimal which either terminates (ends) or repeats.

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Irrational NumbersNOT rationalNumbers that CAN’T be written as a fraction of integers“Weird” numbersNon-terminating, non- repeating decimals

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Examples of irrational numbers:Special numbers like Roots that are not whole numbers likeDecimals that don’t repeat the exact same thing like.34334433344433334444…

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Real NumbersALL numbers you know so farBoth rational and irrational numbers together

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Tell which numbers in this set are …NaturalWholeIntegersRational numbersIrrational numbersReal numbers

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Place, or = between each pair of numbers. 2 / 3.666… 3.14

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Place, or = between each pair of numbers. 2 / 3 =.666… >3.14

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If a number is less, it is to the left on a number line or to the bottom on a thermometer. -3 < 2 -20 < -10

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If a number is greater, it is to the right on the number line or to the top on a thermometer. 10 > -40 -5 > -7

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Place, or = 2___ -5 -7___ -3 -5___ 0

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Place, or = 2___ -52 > -5 -7___ -3-7 < -3 -5___ 0-5 < 0

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Place, or = 0___4 7___5 -2___3 |-4|___ | 4 |

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Place, or = 0___40 5 -2___3-2 < -3 |-4|___ |4||-4| = |4|

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Using calculators with negative numbers On graphing calculators, the key marked (-) means “negative”. Press this BEFORE negative numbers.

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On older cheap calculators, the key marked +/- means “negative”. Press this AFTER negative numbers.

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Adding and subtracting signed numbers: Pos – Neg = Pos Neg – Pos = Neg Pos + Pos = Pos Neg + Neg = Neg Other combinations depend on which number is larger.

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Things it’s useful to know about negative numbers: Neg X Neg = Pos Neg Neg = Pos Pos X Neg = Neg Pos Neg = Neg Neg X Pos = Neg Neg Pos = Neg Pos X Pos = Pos Pos Pos = Pos

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… So if you multiply or divide numbers with the same sign, the answer is positive. …If you multiply or divide numbers with opposite signs, the answer is negative.

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Special Cases 0 X anything = 0 0 anything (besides 0) = 0 anything 0 is undefined (can’t do it)

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Properties of real numbersThings that will always be true for all real numbers.

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Commutative Property5 + 4 = 4 + 57 x 3 = 3 x 7You can multiply or add in any order, and it doesn’t change the answer.

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Associative Property(3 + 4) + 1 = 3 + (4 + 1)4(6 x 3) = (4 x 6) x 3You can group together what you want when you add or multiply.

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You can use the associative property to easily do problems like 137 + 45 + 155

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You can use the associative property to easily do problems like 137 + 45 + 155 This is 137 + 200 or 337.

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Distributive Property3(2x + 7) = 6x + 215(3x – 2) = 15x – 10-4(2x – 1) = -8x + 4If you take a number times something in parentheses, multiply what’s in front times each thing in ( ), one at a time.

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We use the distributive property to get rid of parentheses and also to combine like terms. 3x + 5 + 7x – 2 = 10x + 3

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Identity Properties 1x = x x + 0 = x

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Other properties … 0x = 0 -1x = -x

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SUMMARY OF CHAPTER 1 History of algebra Variables & constants Numerical & algebraic expressions Order of operations Evaluating Expressions Sets of Numbers Properties of real numbers

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