Objective - To simplify expressions using commutative and associative properties.

Commutative - Order doesn’t matter! You can flip-flop numbers around an operation.

Commutative Property of Additiona + b = b + a5 + 7 = 7 + 5

10 + 3 = 3 + 10Commutative Property of Multiplication

a b b a• = •8 4 4 8• = •

3 12 12 3• = •

Write an equivalent expression.

2) 3 8•

1) (6 + 4) = (4 + 6)

8 3= •

3)

4) 9 + 7

5) 3 (4 8)+ +

5 12• 12 5= •

= 7 + 9

3 (8 4)= + + or (4 8) 3+ +

Addition and multiplication are commutative.

Is subtraction commutative?

Difficult toprove true!Easy toprove false!

Counterexample - An example that proves aCounterexample An example that proves astatement false.

Counterexample: 8 5 5 8− ≠ −3 3≠ −

Therefore, subtraction is not commutative.

Give a counterexample that shows division isnot commutative.

Counterexample: 4 2 2 4÷ ≠ ÷12 2≠

Therefore, division is not commutative.

State whether each situation below is commutativeor not commutative.

1) Waking up in the morning and going to school.

2) Brushing your teeth and combing your hair.not commutative

commutative

3) Putting on your socks and putting on yourshoes.

4) Eating cereal and drinking orange juice.

not commutative

commutative

commutative

Identities

Identity Property of Additionx + 0 = x

Zero is sometimes called the Additive Identity.

Identity Property of Multiplicationx x• =1

One is sometimes called the Multiplicative Identity.

Lesson 1-6

Algebra Slide Show: Teaching Made Easy As Pi, by James Wenk © 2010

Identify the property shown below.

3) (2 + 10) + 3 = (10 + 2) + 3

4)

1) 6 + 8 = 8 + 6

2)

Comm. Prop. of Add.

Comm. Prop. of Add.

Comm Prop of Mult5 7 4 7 4 5• • • •( ) ( )

(10 4) (4 10)• = • Comm. Prop. of Mult.

4)

5)

6) 7 + 0 = 7

7) 7 1 7• =

Mult. Prop. of Zero

Comm. Prop. of Mult.

Identity Prop. of Add.

Identity Prop. of Mult.

5 7 4 7 4 5• • = • •( ) ( )

7 0 0• =

Associative - Re-grouping is ok! You can re-group numbers together.

Associative Property of Addition(a + b) + c = a + (b + c)

Associative Property

(4 + 2) + 9 = 4 + (2 + 9)

Associative Property of Multiplication(a b) c a (b c)• • = • •

(3 5) 7 3 (5 7)• • = • •

Write an equivalent expression.

2) 3 (4 8)• •

1) (1 + 5) + 9 = 1 + (5 + 9)

(3 4) 8= • •

3)

4) (6 + 2) + 3

5) 3 (4 8)• •

1 (6 7)• • (1 6) 7= • •

= 6 + (2 + 3)

(3 4) 8= • •

Addition and multiplication are associative. Give counterexamples to prove that subtraction and division are not associative.

SubtractionCounterexample: (10 3) 2 10 (3 2)− − ≠ − −

7 2 10 1− ≠ −5 9≠

Therefore, subtraction is not associative.5 9≠

DivisionCounterexample: (16 8) 2 16 (8 2)÷ ÷ ≠ ÷ ÷

2 2 16 4÷ ≠ ÷

Therefore, division is not associative.1 4≠

Commutative vs. AssociativeIdentify each property shown below.

1) 5 + 1 = 1 + 5

2) 3 (4 8) (3 4) 8• • = • •

Comm. Prop. Of Add.

Assoc. Prop. Of Mult.

3)

4) (6 + 2) + 3 = (2 + 6) + 3

7 2 2 7• = • Comm. Prop. Of Mult.

Comm. Prop. Of Add.

Commutative vs. AssociativeCommutative Associative

(2 + 7) + 8 = (7 + 2) + 8 (2 + 7) + 8 = 2 + (7 + 8)

Flip-flop Re-group

(2 + 7) + 8 = 8 + (2 + 7)

Flip-flop

( ) does not imply Associative

Lesson 1-6 (cont.)

Algebra Slide Show: Teaching Made Easy As Pi, by James Wenk © 2010

Commutative vs. AssociativeIdentify each property shown below.

1) (9 + 3) + 1 = (3 + 9) + 1

2) 10 (5 3) (10 5) 3• • = • •

Comm. Prop. Of Add.

Assoc. Prop. Of Mult.

3)

4)

4 8 8 4+ = + Comm. Prop. Of Add.

Comm. Prop. Of Mult.(3 2) 4 4 (3 2)• • = • •

Give the property that justifies each step.

Statement Reasons16 + (27 + 84) Given

16 + (84 + 27) Comm. Prop. of Add.

(16 + 84) + 27

(100) + 27 Simplify

127 Simplify

Assoc. Prop. of Add.

Use the commutative and associative propertiesto simplify each expression.1) 2) 12 + (29 + 8) 25 (37 4)• •

12 + (8 + 29)

(12 + 8) + 29

25 (4 37)• •

(25 4) 37• •

(20) + 2949

(100) 37•

3700

Distributive Propertya(b c) a b a c+ = • + •

a(b c) a b a c− = • − •or

3(8 2)− 3(6)= 18=Order of Operations( ) ( )

DistributiveProperty

3(8) 3(2)−24 6−

18It works!

Why use the distributive property?

3(x 2)− 3(x) 3(2)= − 3x 6= −

Use the distributive property to simplify.1) 3(x + 7)

2) 2(a + 4)

3) 7(8 + m)

6) x(a + m)

7) 4(3 + r)

8) 2(x + 8)

3x

2a

ax + mx

12 + 4r

+ 21

+ 83) 7(8 + m)

4) 3(4 + a)

5) (3 + k)5

8) 2(x + 8)

9) 7(2m + 3y + 4)

10) (6 + 2y + a)3

56

12

15

2x + 16

14m + 21y + 28

18 + 6y + 3a

+ 7m

+ 3a

+ 5k

Opposite of a Sum or Difference

-(x + y) = -x +

-1(x + y) = -1x + -1y

-yOpposite of a sum

-(x - y) = -x +

-1(x - y) = -1x + 1y

yOpposite of a difference

Simplify the following.1) (x 5)− +

2) (y 3)− −

3) (m y 2)− + −

4) (a 2b 7)− − +x−

y−

m y 2− − +

a 2b 7− + −

5−

3+

Lesson 1-6 (cont.)

Algebra Slide Show: Teaching Made Easy As Pi, by James Wenk © 2010

Subtracting a Quantity

1) -(x + 6)

2) -(2x - 8)

6) -(3a + 1)

7) -(-3x + 2y - 7)-x - 6

-2x + 16

-3a - 1

+3x - 2y + 73) 10 - (4m + 3)

4) 2(x - 5) - (x - 3)

8) -12 - (3y - 8)

9) 4(3k - 5) - (2k + 9)

10 - 4m - 3 - 4m + 7

2x - 10 - x + 3 x - 7

-12 - 3y + 8 - 3y - 4

12k - 20 - 2k - 9 10k - 29

Use the distributive property to help simplify thefollowing without a calculator.1) 2)5(9.96)

5(10) 5(0.04)−

5(10 0.04)−

50 0 20

7(8.2)

7(8) 7(0.2)+

7(8 0.2)+

56 1 4+50 0.20−

49.8056 1.4+

57.4

Use the distributive property to help simplify thefollowing without a calculator.3) 4)8($11.30)

8($11) 8($0.30)+

8($11 $0.30)+

$88 $2 40+

7 5.95×

7(6) 7(0.05)−

7(6 0.05)−

42 0 35$88 $2.40+

$90.4042 0.35−

41.65

Closure PropertyA set of numbers is said to be ‘closed’ if the

numbers produced under a given operation are also elements of the set.

Tell whether the whole numbers are closed under the given operation If not give a counterexamplethe given operation. If not, give a counterexample.1) Addition

2) Subtraction

3) Multiplication

4) Division

Closed

Not Closed 5 - 7 = -2

Closed

Not Closed 2 8 0.25÷ =

A set of numbers is said to be ‘closed’ if the numbers produced under a given operation are

also elements of the set.

Tell whether the integers are closed under the given operation If not give a counterexample

Closure Property

1) Addition

2) Subtraction

3) Multiplication

4) Division

Closed

Closed

Closed

Not Closed

given operation. If not, give a counterexample.

2 8 0.25÷ =

Field Properties (Axioms) Used in ProofsThe Closure Properties

If a and b are rational, then a + b is rational.

The Commutative Propertiesa + b = b + a

The Associative Properties

If a and b are rational, then a b is rational.

, a b = b a

The Identity Properties(a + b) + c = a + (b + c) , (a b) c = a (b c)

a + 0 = a , a 1 = aThe Inverse Properties

(- ) 0+ =a a 1, • 1 (where a 0)= ≠a aThe Distributive Property( )+ = +a b c ab ac

Lesson 1-6 (cont.)

Algebra Slide Show: Teaching Made Easy As Pi, by James Wenk © 2010

Additional Properties (Axioms) Used in ProofsAddition Property of Equality

If a = b, then a + c = b + c.

Subtraction Property of EqualityIf a = b, then a - c = b - c.

Multiplication Property of Equality

Subtraction Property of Equality

If a = b, then a c = b c.

If a = b, then a c = b c.÷ ÷

Other Properties

Reflexive Propertya = a

Symmetric PropertyIf a = b, then b = a.If a b, then b a.

Transitive PropertyIf a = b and b = c, then a = c.

Example of Direct Proof (Deductive)Prove: If a = b, then -a = -b.

Statement Reasona = b Given

a + (-b) = b + (-b) Addition Property of Equalitya + (-b) = 0 Inverse Property( ) p y

(-a) + [a + (-b)] = 0 + (-a) Addition Property of Equality[(-a) + a] + (-b) = 0 + (-a) Associative Prop. of Addition

0 + (-b) = 0 + (-a) Inverse Property-b = -a Identity Property of Addition-a = -b Symmetric Property

Provide a reason that justifies each step.

Statement ReasonGivenDistributive Property

Commutative Prop. of Add.

3x 5(2 x)+ −3x 10 5x+ + −3x 5x 10+ − + p

Distributive Property

Simplify

3x 5x 10+ +(3 5)x 10+ − +

2x 10− +

Provide a reason that justifies each step.

Statement ReasonGivenDistributive Property

Addition Property of Equality

5(2x 3) 35− =10x 15 35− =

10x 15 15 35 15− + = + p y q y

Inverse Property

Multip. Property of Equality

Inverse Property

10x 15 15 35 15+ +10x 50=

1 110x 5010 10• = •

x 5=

Provide a reason that justifies each step.

Statement ReasonGiven

Distributive Property

Associative Property of Add.

2a 3(a 4) 13+ − =2a (3a 12) 13+ − =(2a 3a) 12 13+ − =

Distributive Property

Addition Property of Equality

Inverse Property

(2 3)a 12 13+ − =

5a 12 12 13 12− + = +5a 25=

1 15a 255 5• = •a 5=

Multip. Property of Equality

Inverse Property

Lesson 1-6 (cont.)

Algebra Slide Show: Teaching Made Easy As Pi, by James Wenk © 2010