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K5 & R3 September 01, 2014
Properties of Real NumbersK5 R3 &
K5 & R3 September 01, 2014
WARM-UP Properties of Real NumbersK5 R3 &
A. List two real numbers less than 1,000.
Real numbers include:RECALL
whole numbers
positives & negatives
fractions ("pretty" & "ugly")
decimals ("pretty" & "ugly")
square roots ("pretty" & "ugly")
K5 & R3 September 01, 2014
WARM-UP Properties of Real NumbersK5 R3 &
Possible operations:RECALL
Addition (+)
Subtraction (-)
Multiplication (x)
Division (/)
Exponents (x2)
Grouping Symbols () [] {}
B. List five mathematical operations.
K5 & R3 September 01, 2014
DEFINITIONS Properties of Real NumbersK5 R3 &
a relationship that is always TRUE
Example: Distributive Property
a(b+c) = ac + ac
Property
K5 & R3 September 01, 2014
DEFINITIONS Properties of Real NumbersK5 R3 &
have the SAME value for all values of the variable(s)
Example: x + y + z = z + y + x
Equivalent Expressions
K5 & R3 September 01, 2014
DEFINITIONS Properties of Real NumbersK5 R3 &
a process of reasoning logically from given facts to
a conclusion
Example:
FACT - Gravity makes things fall.CONCLUSION - The apple hit my head because of gravity.
Deductive Reasoning
K5 & R3 September 01, 2014
DEFINITIONS Properties of Real NumbersK5 R3 &
an example showing that a statement is FALSE
Example: x + 1 = x
False. CE - x=22 + 1 = 2
Counterexample (CE)
K5 & R3 September 01, 2014
WARM-UP Properties of Real NumbersK5 R3 &
A. For all real numbers a and b,
ab = b + a
1 2 = 2 + 1CE - a = 1, b=2
2 = 3FALSE
K5 & R3 September 01, 2014
WARM-UP Properties of Real NumbersK5 R3 &
B. For all real numbers a, b and c,
(ab)c = b(ac)
See LEVEL 2 Example 2TRUE
K5 & R3 September 01, 2014
Distributive Property
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Distributive Property
Words: Algebra:
Numbers: Example:
a(b+c) = ab + acTo multiply a sum or difference by a number, multiply EACH number inside the group by the number OUTSIDE the group
2(4 + 3) = 2 4 + 2 32(4 - 3) = 2 4 - 2 3
a(b-c) = ab - ac
K5 & R3 September 01, 2014
Distributive Property
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:How many months are in 24 years?
12 24 = 12(20+4) = 12(20) + 12(4) = 240+ 48
= 288 months
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Distributive Property
Words: Algebra:
Numbers: Example:
a(b+c) = ac + acTo multiply a sum or difference by a number, multiply EACH number inside the group by the number OUTSIDE the group
2(4 + 3) = 2 4 + 2 32(4 - 3) = 2 4 - 2 3
How many months are in 24 years? 12 24 = 12(20+4)
= 12(20) + 12(4) = 240 + 48= 288 months
a(b-c) = ac - ac
K5 & R3 September 01, 2014
Commutative Properties
K5 & R3 September 01, 2014
Commutative Property of Addition
Property Act-Outs Properties of Real NumbersK5 R3 &
Performed by: Fischer, Jared & Jason Words: Algebra:
Numbers: Example:
When two real numbers are ADDED, you may change the
ORDER of the numbers
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:13 + 29 + 17
Commutative Property of Addition
13 + 29 + 17 = 13 + 17 + 29 = 30 + 29 = 59
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Commutative Property
Words: Algebra:
Numbers: Example:of Multiplication
When two real numbers are MULTIPLIED, you may change
the ORDER of the numbers
K5 & R3 September 01, 2014
4 13 25 = 4 25 13
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:4 13 25
= 100 13 = 1,300
Commutative Property of Multiplication
K5 & R3 September 01, 2014
Associative Properties
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Performed by: Ainsley, Casey, David, Faye, Hunter, Margaux & Noah
Associative Property
Words: Algebra:
Numbers: Example:of Addition
When a number is ADDED to the SUM of two numbers, you may change the
GROUPING of the numbers
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:(29 + 13) + 17
Associative Property of Addition
(29 + 13) + 17 = 29 + (13 + 17)= 29 + 30= 59
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Associative Property
Words: Algebra:
Numbers: Example:of Multiplication
When a number is MULTIPLIED to the PRODUCT of two numbers, you may
change the GROUPING of the numbers a(bc) = (ab)c
3 (5 2) = (3 5 ) 2
K5 & R3 September 01, 2014
(13 4) 25 = 13 (4 25)
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:(13 4) 25
= 13 100 = 1,300
Associative Property of Multiplication
K5 & R3 September 01, 2014
IdentityProperties
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Performed by: Cameron, Cindy & Eitan
Identity Property
Words: Algebra:
Numbers: Example:of Addition
Any number ADDED to ZERO equals ITSELF
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:SIMPLIFYING
(See LEVEL 2 Example 2)
Identity Property of Addition
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Identity Property
Words: Algebra:
Numbers: Example:of Multiplication
Any number MULTIPLIED by ONE equals ITSELF
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:1
6
Identity Property of Multiplication
13+
16
13+ 1
61
3+ = 1x
16
13
+ = x 22
16
1 x23x2
+ =
16
26
+ =
36
1 2
= =
K5 & R3 September 01, 2014
Other ImportantProperties...
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Zero Property
Words: Algebra:
Numbers: Example:of Multiplication
Any number MULTIPLIED by ZERO equals ZERO
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:SIMPLIFYING
(See LEVEL 2 Example 2)
Zero Property of Multiplication
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Multiplication Property
Words: Algebra:
Numbers: Example:of -1
Any number MULTIPLIED by NEGATIVE ONE equals the OPPOSITE of that number
K5 & R3 September 01, 2014
Property Act-Outs Properties of Real NumbersK5 R3 &
Example:SIMPLIFYING
(See LEVEL 2 Example 1)
of -1 Multiplication Property
K5 & R3 September 01, 2014
Level Check
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 1: identify properties
1.25 + (y + .28) = (1.25 + y) + .28Example 1:
Example 2:37 x 0 x 18 = 0 x 37 x 18
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 1: identify properties
1.25 + (y + .28) = (1.25 + y) + .28Example 1:
Example 2:37 x 0 x 18 = 0 x 37 x 18
Associative Property of Addition
Commutative Property of Multiplication
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 2: justifyExample 1:
Simplify. Justify each step.
5(-1x)
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 2: justifyExample 1:
Simplify. Justify each step.
5(-1x)(5 -1)x-5x
Associative Prop. of Multiplication
Multiplication Prop. of -1
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 2: justifyExample 2:
Simplify. Justify each step.
0a + b + 0
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 2: justifyExample 2:
Simplify. Justify each step.
0a + b + 0 b + 0
bZero Prop. of Multiplication
Identity Prop. of Addition
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 3: true or false
1 + x + y = x + y + 1
Example 1:
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 3: true or false
1 + x + y = x + y + 1
Example 1:
True!1 + x + y x + 1 + y x + y + 1
Commutative Prop. of Addition
Commutative Prop. of Addition
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 3: true or false
a(b-c) = (ab) - c
Example 2:
K5 & R3 September 01, 2014
Level Check Properties of Real NumbersK5 R3 &
LEVEL 3: true or false
a(b-c) = (ab) - c
Example 2:
False!
4(3-2) = (4 3) - 24 1 = 12 - 24 = 10
C.E. - a=4, b=3, c=2
K5 & R3 September 01, 2014
Quiz Return
K5 & R3 September 01, 2014
Quiz Return
Things to Remember...
Fill out form to RETAKE before FRIDAY!
12 = 1 x 1 = 1
Groups First! (3 + 2)2 = 32+22
(3 + 2)2 = 52 = 25
"Show" does not mean give one example; means to show for ALL cases using variables