# 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.

## Presentation on theme: "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."— Presentation transcript:

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

 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.

 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.

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

 The Hindus in India organized algebra into a formal system.

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

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

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

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

Variable

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

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

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

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

What words mean … + – X  =

sum total increased by more than greater longer farther older

difference decreased by less shorter closer younger

product twice half of by

quotient shared split per

most verbs

You can use variables to identify patterns

1  4 2  7 3  10 4  13 n  ?

1  4 2  7 3  10 4  13 n  3n + 1

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

Order of Operations

Parentheses AND other grouping symbol

Parentheses AND other grouping symbols  brackets

Parentheses AND other grouping symbols  brackets  fraction bar

Parentheses AND other grouping symbols  brackets  fraction bar  root symbols

Exponents AND roots

Multiplication and division (same time – left to right)

Addition and subtraction (same time – left to right)

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

… 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.

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

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

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

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.

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

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

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.

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

Identity Properties 1x = x x + 0 = x

Other properties … 0x = 0 -1x = -x

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

Download ppt "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."

Similar presentations