OperationsThe verbs of mathematics.

Subtraction

Same as: adding a negative number.

4 - 3 = 4 + (-3)

Multiplication

Best understood as “repeated addition.”

3 x 5 = 5 + 5 + 5

or 3 rows of 5 items.

Division Multiplication by the inverse or reciprocal of a number.

6126

112

6

1

1

12 2

6

12

This definition of division is essentialwhen working with fractions!!!

?2

1

6

1

1

2

6

1

3

1

6

2

Your turn:

1. Change this into addition: 4 – 1

2. Change this into multiplication: 35

3. ?5

7

5

2

PropertiesThe grammar of mathematics.

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

“Motorcycle my riding fun I have.” (Persian)

Order of Operations (PEMDAS)“Please Excuse My Dear Aunt Sally.”

ParenthesesExponentsMultiplicationDivisionAdditionSubtraction

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:

4.

?43

8523

2

Commutative Property of Addition

2 + 3 = 3 + 2

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

Commutative Property of Multiplication

2 x 3 = 3 x 2

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

Associative Property of Addition

2 + 3 + 4Can you add 3 numbers at the same time?

Pick 2 of the 3 numbers, add them together.

Add the 3rd number to the sum of the 1st two.

2 + 3 = 5

5 + 4 = 9

Associative Property of Addition2 + 3 + 4

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

(2 + 3) + 4

= 5 + 4

= 9

2 + (3 + 4)

= 2 + 7

= 9

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

Using the commutative and associative properties.

7 + x + 3 + 2x = ?

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

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

= 10+ 3x

Your turn:

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

?353 xx

Associative Property of Multiplication

2 x 3 x 4

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

(2 x 3) x 4

= 6 x 4

= 24

2 x (3 x 4)

= 2 x 12

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

Your turn:

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

?253 yy

Distributive Property of Addition over Multiplication

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

= 6 + 8= 14

2 ( 7 )14

This property is important when variables are involved.

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

= 2x + 8

Your turn:

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

?)42(5 x

Your turn:

Identify the property that allows the step indicated.

393)45(345 8.

9. 435345

10. )35()4*5()34(5 xx

Equality Properties

Addition Property of Equality

Subtraction Property of Equality

Multiplication Property of Equality

Division Property of Equality

Inverse Property of Addition

23 + x = 0

“What number do you add so the sum equals zero

x = ?

-25 + x = 0 x = ?

We will use this property to solve equations.

Inverse Property of Multiplication

10

110

10

1

1

10

What number do you multiply byso the product is 1 (one)?

10 * x = 1 x = ?

110

10

10 times its “reciprocal” equals 1

10 divided by itself equals 1

We will use this property to solve equations.

Addition Property of Equality

a = b

a + 1 = b + 1Equivalent Equations

Solving an Equation

x – 1 = 5

x = 6

+ 1

Inverse Property of Addition

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

+ 1

x =

Identity Property of Addition

Subtraction Property of Equality

a = b

a - 1 = b - 1Equivalent Equations

x + 1 = 5

x = 4x =- 1

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

Solving an Equation

- 1

Inverse Property of Addition

Identity Property of Addition

Multiplication Property of Equality

a = b

a * 2 = b * 2Equivalent Equations

Solving an Equation

= 5

x = ?

* 2

Inverse Property of Multiplication

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

x = 10

Identity Property of Multiplication

2

x

2

x

Division Property of Equality

a = b

a ÷ 2 = b ÷ 2Equivalent Equations

Solving an Equation

3x = 15

x = 5

Inverse Property of Multiplication

Division Property of Equality: whateverwe divide the leftside of the ‘=‘ sign by, we must divide the right side of the equation..÷ 3

Identity Property of Multiplication

2

x

÷ 3

11. 2 = 3 + x

Your turn:

12. -27 = x - 3

13. 12 = 3x 14. = -27

x

Combinations

512

x

“Un-doing” operations

Use “reverse” PEMDAS.

What do you do 1st:subtraction or multiplication?

- 1

2

x

- 1

4

* 2

x

* 2

= 8

x = ?

15. 12 = 3 + 3x

Your turn:

16. -8 = - 5

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

2x

3

x