Warm Up 12/14/15 Solve the following rebus puzzles

History/Origin of the Order of Operations …In summary, I would say that the rules actually fall into two categories: the natural rules (such as precedence of exponential over multiplicative over additive operations, and the meaning of parentheses), and the artificial rules (left-to-right evaluation, equal precedence for multiplication and division, and so on). The former were present from the beginning of the notation, and probably existed already, though in a somewhat different form, in the geometric and verbal modes of expression that preceded algebraic symbolism. The latter, not having any absolute reason for their acceptance, have had to be gradually agreed upon through usage, and continue to evolve. More at: http://mathforum.org/library/drmath/view/52582.html

The Order of Operations is: Review The Order of Operations is: P.E.M.D.A.S. arenthesis xponents ultiplication ivision ddition ubtraction

Practice 6x4÷2+3 24÷2+3 12+3 Answer: 15

Practice 15÷(6x2-9) 15÷(12-9) 15÷(3) Answer: 5

Practice (32+5)÷7 (9+5)÷7 14÷7 Answer: 2

Practice 7+(6x52+3) 7+(6x25+3) 7+(150+3) 7+(153) Answer: 160

Practice (18+2)÷5 20÷5 (3x6+2)÷5 Answer: 4

Try on your own… 5 + 18 ÷ 3² - 1 3 x ( 12÷ 4) - 5⁰

Two More on Your Own 4² + 48 ÷ (10 – 4) 9 ÷ 3 + 6 x 2³

Tips to Remember: An easy way to remember PEMDAS is: lease xcuse ally ear unt ally

Solve the following Rebus Puzzles Warm Up 12/15/15 Solve the following Rebus Puzzles

Algebraic Expressions

Definitions Variable – A variable is a letter or symbol that represents a number (unknown quantity). 8 + n = 12

Definitions A variable can use any letter of the alphabet. (expect o, and i needs to be lower case) n + 5 x – 7 w - 25

Definitions Algebraic expression – an expression that contains at least one variable, operation, and constant m + 8 r – 3

Definitions A constant is a number that does not change. A coefficient is a number multiplied by a variable. A term is part of the expression separated by addition or subtraction 6x + 5 6 is the coefficient, x is the variable, and 5 is the constant. 6x and 5 are terms.

Definitions Evaluate an algebraic expression – To find the value of an algebraic expression substitute numbers for variables and simplify. m + 8 m = 2 2 + 8 = 10 r – 3 r = 5 5 – 3 = 2

Definitions A term is a part of an expression that is added or subtracted from another term. 5x + 6 – 4x This expression has 3 terms: 5x, 6, and -4x Like terms are terms which have the same variable raised to the same power In the expression above, 5x and -4x are like terms

Definitions Simplify – Combine like terms and complete all operations m + 8 + m 2 m + 8 (2 x 2) + 8 4 + 8 = 12

Words That Lead to Addition Sum More than Increased Plus Altogether

Words That Lead to Subtraction Decreased Less Difference Minus How many more

Write Algebraic Expressions for These Word Phrases Ten more than a number A number decreased by 5 6 less than a number A number increased by 8 The sum of a number & 9 4 more than a number n + 10 w - 5 x - 6 n + 8 n + 9 y + 4

Write Algebraic Expressions for These Word Phrases A number plus 2 A number decreased by 1 31 less than a number A number b increased by 7 The sum of a number & 6 9 more than a number s + 2 k - 1 x - 31 b + 7 n + 6 z + 9

Multiplication Phrases: Product Times Multiply Of Twice or double Triple

Although they are closely related, a Great Dane weighs about 40 times as much as a Chihuahua. An expression for the weight of the Great Dane could be 40c, where c is the weight of the Chihuahua.

Division Phrases: Quotient Divide Divided by Split equally

Write an Algebraic Expression for These Situations Scott’s brother is 2 years younger than Scott The sum of two numbers is 12 The difference between two numbers is 5 s - 2 v + c = 12 m – n = 5

What word becomes shorter when you add two letters to it? Warm Up 12/16/15 What word becomes shorter when you add two letters to it? Short

What word in the English Language is always spelled “incorrectly”?

What fast food restaurant do pirates like to eat at? ARR-by’s

Algebra 3-1 Writing Equations When writing equations, use variables to represent the unspecified numbers or measures referred to in the sentence or problem. Then write the verbal expressions as algebraic expressions. Writing equations Harbour

Expressions for equals sign Algebra 3-1 Writing Equations Writing Equations Some verbal expressions that suggest the equals sign are listed below: is equals is equal to is the same as is as much as is identical to Expressions for equals sign Harbour

When a number is multiplied by four and the result is decreased by six, the final result is 10. 4 n - 6 = 10

Three less than the number x is 12. - 3 = 12 Remember, you have to be very careful when you see the words ‘less than’. x – 3 = 12

b divided by three is equal to six less than c. Translate this sentence into an equation. A number b divided by three is equal to six less than c. b divided by three is equal to six less than c. Answer: The equation is . Example 1-1a

15 z 6 y 2 11 Translate this sentence into an equation. Fifteen more than z times six is y times two minus eleven. Fifteen more than z times six is y times two minus eleven. 15 z 6 y 2 11 Answer: The equation is . Example 1-1b

Translate each sentence into an equation. a. A number c multiplied by six is equal to two more than d. Answer: The equation is . b. Three less than a number a divided by four is seven more than 3 times b. Answer: The equation is . Example 1-1c

Warm Up 12/17/15 http://www.wimp.com/mathapp/

Solving 1+2 Step Equations

Solving 1 Step Equations When you do something to one side of an equation,

Solving 1 Step Equations you have to do exactly the same thing to the other side.

Solving 1 Step Equations To solve an equation means to find every number that makes the equation true.

Solving 1 Step Equations In the equation, x is the number you add to 7 to get 15.

Solving 1 Step Equations What is the number? 8. So we would write the solution x = 8

Solving 1 Step Equations

Try the following on your paper. 10 = X + 3 5 + X = 9 6 = 3 + X

Answers!!! X = 7 X = 4 X = 3

Try the following on your paper. X - 7 = -1 X - 4 = 66 8 = X - 2 -8 = X - 2

Answers!!! X = 6 X = 70 X = 10 X = -6

Multiplication of Equations 3X = -9 3X/3 = -9/3 X = -3 -5X = -40 -5X/-5 = -40/-5 X = 8 When solving multiplication equations, you divide both sides by the number attached to the variable. Be sure to use the same sign.

Try these on your paper. 4X = -16 -1X = 9 6 = 5X 12X = 3

Answers!!! X = -4 X = -9 X = 6/5 X = 1/4

Division of Equations 1/2X = 4 (2/1)1/2X = 4(2) X = 8 -3/4X = 6 Multiply both sides by the reciprocal. You must keep the sign of the fraction with the variable.

Try these on your paper. X/3 = 9 -X/4 = 7 1/3X = -1 4 = X/4

Answers!!! X = 27 X = -28 X = -3 X = 16

Solving 2 Step Equations In this equation, the solution is not so obvious.

Solving 2 Step Equations Many times we need a way to solve equations. Especially if there is more than one step to solve.

Solving 2 Step Equations 1) Undo the addition or subtraction operation first, by using inverse operations. 2) Undo the multiplication or division operation second, by using inverse operations.

Solving 2 Step Equations REMEMBER

Solving 2 Step Equations To solve any equation, you want to get the variable all by itself. You get the variable alone by using inverse operations. Inverse Operations are like Opposites.

I am going to show you 2 ways to solve this equation. Solving 2 Step Equations I am going to show you 2 ways to solve this equation. 3x + 4 = 16 Hard way Easy way Undo Multiplication or Division 1st Undo Addition or Subtraction 1st

Solving 2 Step Equations Hard way Easy way Undo Multiplication or Division 1st Undo Addition or Subtraction 1st 3x + 4 = 16 3x + 4 = 16 3 3 -4 -4 3x 4 16 + = 3x = 12 3 3 3 3 3 4 16 x + = 3 3 x = 4 4 4 - - 3 3 12 x = 3 x = 4

Solving 2 Step Equations 1) GOAL: Get the variable on one side of the equation. 2) You always perform the same operation to both sides of an equation. 3) To undo an operation you perform its opposite operation to both sides of the equation. 4) It is easier if you undo addition or subtraction before you undo multiplication or division.

Solving 2 Step Equations 2x + 4 = 10 Get the variable alone. - 4 - 4 Subtract 4 from both sides. 2x = 6 __ __ 2 2 +4 and -4 cancel out Divide both sides by 2 x = 3 2 divided by 2 cancel to 1 2(3) +4 = 10 Show your check. 6 + 4 = 10 10 = 10 Finish your check.

Solving 2 Step Equations 3m - 10 = 17 Get the variable alone. + 10 +10 Add 10 to both sides. 3m = 27 __ __ 3 3 +10 and -10 cancel out Divide both sides by 3 m = 9 3 divided by 3 cancel to 1 3(9) – 10 = 17 Show your check. 27 – 10 = 17 Finish your check. 17 = 17

Solving 2 Step Equations x 3 – 4 = 8 Get the variable alone. +4 +4 Add 4 to both sides. x 3 = 12 + 4 and -4 cancel out. x 3 Multiply both sides by 3. 3 ( ) = 3 12 3 divided by 3 cancel to 1 x = 36 Show your check. 36 3 – 4 = 8 Finish your check. 12 – 4 = 8 8 = 8

Solving 2 Step Equations x 2 + 6 = 14 Get the variable alone. - 6 - 6 Subtract 6 from both sides. x 2 = 8 + 6 and -6 cancel out. x 2 Multiply both sides by 2. 2 ( ) = 2 8 2 divided by 2 cancel to 1 x = 16 16 2 Show your check. + 6 = 14 8 + 6 = 14 Finish your check. 14 = 14

Solving 2 Step Equations What Operation Do You Do? FIRST SECOND 1) 3x + 8 = 24 Divide by 3 Subtract 8 2) 5x - 10 = 19 Divide by 5 Add 10 3) x - 4 = 20 2 Add 4 Multiply by 2 4) - 30 = -5x + 5 Divide by -5 Subtract 5

Solving 2 Step Equations

Solving 2 Step Equations

Solving 2 Step Equations Think about what you need to do to isolate the variable. Get rid of addition or subtraction first. Second get rid of multiplication or division.

Solving 2 Step Equations The End!

Warm Up 12/18/15 Dry Ice Bubble Activity

Domain and Range

Definition: A relation is any set of ordered pairs.

Ordered pairs are used to locate points in a coordinate plane. y-axis (vertical axis) 5 -5 5 x-axis (horizontal axis) -5 origin (0,0)

In an ordered pair, the first number is the x-coordinate In an ordered pair, the first number is the x-coordinate. The second number is the y-coordinate. Graph. (-3, 2) 5 • -5 5 -5

Domain: In a set of ordered pairs, (x, y), the domain is the set of all x-coordinates. Range: In a set of ordered pairs, (x, y), the range is the set of all y-coordinates.

Ex:{(2,3),(-1,0),(2,-5),(0,-3)} Domain: {2,-1,0} Range: {3,0,-5,-3} The set of ordered pairs may be a limited number of points. Given the following set of ordered pairs, find the domain and range. Ex:{(2,3),(-1,0),(2,-5),(0,-3)} If a number occurs more than once, you do not need to list it more than one time. Domain: {2,-1,0} Range: {3,0,-5,-3}

Practice: Find the domain and range of the following sets of ordered pairs. 1. {(3,7),(-3,7),(7,-2),(-8,-5),(0,-1)} Domain:{3,-3,7,-8,0} Range:{7,-2,-5,-1}

x y 5 6 11 8 State the domain and range of the following Relation (represented by a mapping diagram) x y 5 6 8 11 1 2 3 x y 5 6 11 8

State the domain and range of the following relation. x y 4 2 -3 8 6 1 -1 9 5

Defn: A function is a relation where every x value has one and only one value of y assigned to it. State whether or not the following relations could be a function or not. x y 4 2 -3 8 6 1 -1 9 5 x y 1 3 2 5 -4 6 4 x y 2 3 5 7 8 -2 -5 function not a function function

Functions and Equations. State whether or not the following equations are functions or not. x y -3 5 7 -2 -7 4 3 x y 2 4 -2 -4 16 3 9 -3 x y 1 -1 4 2 -2 function function not a function