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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
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
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.
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
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
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
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
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 = ?