OperationsOperationsThe The verbsverbs of mathematics. of mathematics.

SubtractionSubtraction

Same as: Same as: adding a negative numberadding a negative number..

4 - 3 =4 - 3 = 4 + (-3) 4 + (-3)

Convert the following to addition: 5 - 2Convert the following to addition: 5 - 2

MultiplicationMultiplication

Best understood as “Best understood as “repeated additionrepeated addition.” .”

3 x 53 x 5 = 5 + 5 + 5= 5 + 5 + 5

or 3 rows of 5 items.or 3 rows of 5 items.

DivisionDivision Multiplication Multiplication by the inverse or by the inverse or reciprocal of a number.reciprocal of a number.

6126

112

6

1

1

12 26

12

This definition of division is This definition of division is essentialessentialwhen working with fractions!!!when working with fractions!!!

?2

1

6

1

1

2

6

1

3

1

6

2

Convert the following to multiplication:Convert the following to multiplication:2

116

Your turn:Your turn:

1.1. Change this into Change this into addition:addition: 4 – 1 4 – 1

2. Change this into 2. Change this into multiplication:multiplication: 35

3. 3. ?5

7

5

2

PropertiesPropertiesThe The grammargrammar of mathematics. of mathematics.

““I have fun riding my motorcycle.” (English)I have fun riding my motorcycle.” (English)

““To ride a motorcycle fun I have.” (Persian)To ride a motorcycle fun I have.” (Persian)

““Aez savor shodan-e mashin-e xodaem,Aez savor shodan-e mashin-e xodaem, lezaet miboraem.” (Persian)lezaet miboraem.” (Persian)

Order of Operations Order of Operations (PEMDAS)(PEMDAS)““PPlease lease EExcuse xcuse MMy y DDear ear AAunt unt SSally.”ally.”

ParenthesesParenthesesExponentsExponentsMultiplicationMultiplicationDivisionDivisionAdditionAdditionSubtractionSubtraction

1

4

133 2

14

23 2

14

4*3 413

1

4

133 2

14

23 2

1

4

6 2

14

36 1019

Your turn:Your turn:

4. 4.

?43

8523

2

CommutativeCommutative Property of Property of AdditionAddition

2 + 32 + 3 = 3 + 2= 3 + 2

Adding Adding twotwo numbers numbers doesn’t matter doesn’t matter which number comes first.which number comes first.

Commutative Property of Commutative Property of MultiplicationMultiplication

2 x 32 x 3 = 3 x 2= 3 x 2

multiplying multiplying twotwo numbers numbers doesn’t matter doesn’t matter which number comes first.which number comes first.

Associative Property of Associative Property of AdditionAddition2 + 3 + 4

We use PEMDAS (parentheses) to “associate”We use PEMDAS (parentheses) to “associate” the first 2 numbers together.the first 2 numbers together.

(2 + 3) + 4(2 + 3) + 4

= 5 + 4= 5 + 4

= 9= 9

2 + (3 + 4)2 + (3 + 4)

= 2 + 7= 2 + 7

= 9= 9

The property says: when adding 3 or more numbersThe property says: when adding 3 or more numbers together, it doesn’t matter which two of numbers you together, it doesn’t matter which two of numbers you add together first (“associate”), you’ll always get the add together first (“associate”), you’ll always get the same answer. same answer.

Using the Using the commutativecommutative and and associativeassociative properties. properties.

7 + x + 3 + 2x = ?7 + x + 3 + 2x = ?

= 7 + 3 + x + 2x= 7 + 3 + x + 2x Rearrange the order (commutative)Rearrange the order (commutative)

= (7 + 3) + (x + 2x)= (7 + 3) + (x + 2x) Group terms to add togetherGroup terms to add together

= 10+ 3x= 10+ 3x

When doing problems, you don’t need to rewriteWhen doing problems, you don’t need to rewrite the equation to the equation to re-arrangere-arrange the order, or to the order, or to groupgroup terms together, you can do it in your head. terms together, you can do it in your head.

Your turn:Your turn:

5.5. Simplify the following expression using theSimplify the following expression using the commutative (order) and associative (grouping)commutative (order) and associative (grouping) properties. properties.

?353 xx

Associative Property of Associative Property of MultiplicationMultiplication

2 x 3 x 42 x 3 x 4

We use PEMDAS (parentheses) to “associate”We use PEMDAS (parentheses) to “associate” the first 2 numbers together.the first 2 numbers together.

(2 x 3) x 4(2 x 3) x 4

= 6 x 4= 6 x 4

= 24= 24

2 x (3 x 4)2 x (3 x 4)

= 2 x 12= 2 x 12

= 24= 24The property says: when multiplying 3 or more numbersThe property says: when multiplying 3 or more numbers together, it doesn’t matter which two of numbers you together, it doesn’t matter which two of numbers you multiply together first (“associate”), you’ll always get the multiply together first (“associate”), you’ll always get the same answer. same answer.

Your turn:Your turn:

6. Simplify the following expression using the6. Simplify the following expression using the commutative (order) and associative (grouping)commutative (order) and associative (grouping) properties. properties.

?253 yy

Distributive PropertyDistributive Property of of Addition over MultiplicationAddition over Multiplication

2(3 + 4)2(3 + 4) = (2 * 3)= (2 * 3) ++ (2 * 4)(2 * 4)

= 6 + 8= 6 + 8= 14= 14

2 ( 7 )2 ( 7 )

1414This property is important when variables are involved.This property is important when variables are involved.

2(x + 4)2(x + 4) = (2 x)= (2 x) ++ (2 * 4)(2 * 4)

= 2x + 8= 2x + 8

Your turn:Your turn:

7. Simplify the following expression using the7. Simplify the following expression using the distributive property distributive property of “additional over mulitplication”.of “additional over mulitplication”.

?)42(5 x

Your turn:Your turn:

Identify the property that allows the step indicated.Identify the property that allows the step indicated.

3)45(345 8.8.

9.9. 435345

10.10. )3*5()4*5()34(5 xx

Equality PropertiesEquality Properties

Addition Property of EqualityAddition Property of Equality

Subtraction Property of EqualitySubtraction Property of Equality

Multiplication Property of EqualityMultiplication Property of Equality

Division Property of EqualityDivision Property of Equality

Solving an EquationSolving an Equation

x – 1 = 5x – 1 = 5

x = 6x = 6

+ 1

Inverse Property Inverse Property of Additionof Addition

Addition Property Addition Property of Equalityof Equality: whateverwe added to the leftside of the ‘=‘ sign, wemust add to the right side of the equation..

+ 1

x =x =

Identity Property Identity Property of Additionof Addition

x + 1 = 5x + 1 = 5

x = 4x = 4x =x =

- 1- 1

Subtraction Property Subtraction Property of Equalityof Equality: whateverwe subtracted fromthe left side of the ‘=‘ sign, we must subtractfrom the right side of the equation..

Solving an EquationSolving an Equation

- 1- 1

Inverse Property Inverse Property of Additionof Addition

Identity Property Identity Property of Additionof Addition

Solving an EquationSolving an Equation

= 5

* 2

Inverse Property Inverse Property of Multiplicationof Multiplication

Multiplication Property Multiplication Property of Equalityof Equality: whateverwe multiply the leftside of the ‘=‘ sign by, we must multiply the right side of the equation..* 2

x = 10

Identity Property Identity Property of Multiplicationof Multiplication

2

x

x2

1

Solving an EquationSolving an Equation

3x = 15

x = 5

Inverse Property Inverse Property of Multiplicationof Multiplication

Division Property Division Property of Equalityof Equality: whateverwe divide the leftside of the ‘=‘ sign by, we must divide the right side of the equation..

÷ 3

Identity Property Identity Property of Multiplicationof Multiplication

2

x

÷ 3 (Or: mult. by ⅓)(Or: mult. by ⅓) (Or: mult. by ⅓)(Or: mult. by ⅓)

11. 11. 2 = 3 + x

Your turn:Your turn:

12. 12. -27 = x - 3

13. 13. 12 = 3x 14. 14. = -27

x

CombinationsCombinations

512

1x

“Un-doing” operations

Use “reverse” PEMDAS.

What do you do 1st:subtraction or multiplication?

- 1

x2

1

- 1

4

* 2

xx

* 2

= 8= 8 x = ?

Or: get the variable term all by itself.

15. 15. 12 = 3 + 3x

Your turn:Your turn:

16. 16. -8 = - 5

17. 17. 24 - x = 3x 18. 18. - 4 = -85

2x

3

x