Properties of Real Numbers - My Math...

K5 & R3 September 01, 2014

Properties of Real NumbersK5 R3 &

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")

Possible operations:RECALL

Addition (+)

Subtraction (-)

Multiplication (x)

Division (/)

Exponents (x2)

Grouping Symbols () [] {}

B. List five mathematical operations.

DEFINITIONS Properties of Real NumbersK5 R3 &

a relationship that is always TRUE

Example: Distributive Property

a(b+c) = ac + ac

Property

have the SAME value for all values of the variable(s)

Example: x + y + z = z + y + x

Equivalent Expressions

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

an example showing that a statement is FALSE

Example: x + 1 = x

False. CE - x=22 + 1 = 2

Counterexample (CE)

A. For all real numbers a and b,

ab = b + a

1 2 = 2 + 1CE - a = 1, b=2

2 = 3FALSE

B. For all real numbers a, b and c,

(ab)c = b(ac)

See LEVEL 2 Example 2TRUE

Distributive Property

Property Act-Outs Properties of Real NumbersK5 R3 &

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

Example:How many months are in 24 years?

12 24 = 12(20+4) = 12(20) + 12(4) = 240+ 48

= 288 months

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

Commutative Properties

Commutative Property of Addition

Performed by: Fischer, Jared & Jason Words: Algebra:

Numbers: Example:

When two real numbers are ADDED, you may change the

ORDER of the numbers

Example:13 + 29 + 17

Commutative Property of Addition

13 + 29 + 17 = 13 + 17 + 29 = 30 + 29 = 59

Commutative Property

Words: Algebra:

Numbers: Example:of Multiplication

When two real numbers are MULTIPLIED, you may change

the ORDER of the numbers

4 13 25 = 4 25 13

Example:4 13 25

= 100 13 = 1,300

Commutative Property of Multiplication

Associative Properties

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

Example:(29 + 13) + 17

Associative Property of Addition

(29 + 13) + 17 = 29 + (13 + 17)= 29 + 30= 59

Associative Property

Words: Algebra:

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

(13 4) 25 = 13 (4 25)

Example:(13 4) 25

= 13 100 = 1,300

Associative Property of Multiplication

IdentityProperties

Performed by: Cameron, Cindy & Eitan

Identity Property

Words: Algebra:

Numbers: Example:of Addition

Any number ADDED to ZERO equals ITSELF

Example:SIMPLIFYING

(See LEVEL 2 Example 2)

Identity Property of Addition

Identity Property

Words: Algebra:

Any number MULTIPLIED by ONE equals ITSELF

Example:1

Identity Property of Multiplication

3+ = 1x

+ = x 22

1 x23x2

Other ImportantProperties...

Zero Property

Words: Algebra:

Any number MULTIPLIED by ZERO equals ZERO

Example:SIMPLIFYING

Zero Property of Multiplication

Multiplication Property

Words: Algebra:

Numbers: Example:of -1

Any number MULTIPLIED by NEGATIVE ONE equals the OPPOSITE of that number

Example:SIMPLIFYING

of -1 Multiplication Property

Level Check

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

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

LEVEL 2: justifyExample 1:

Simplify. Justify each step.

5(-1x)

5(-1x)(5 -1)x-5x

Associative Prop. of Multiplication

Multiplication Prop. of -1

0a + b + 0

0a + b + 0 b + 0

bZero Prop. of Multiplication

Identity Prop. of Addition

LEVEL 3: true or false

1 + x + y = x + y + 1

Example 1:

1 + x + y = x + y + 1

Example 1:

True!1 + x + y x + 1 + y x + y + 1

Commutative Prop. of Addition

a(b-c) = (ab) - c

Example 2:

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

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

Properties of Real Numbers - My Math...

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