integers order of operations substitution · integers order of operations substitution math 7...

Integers

Order of Operations

Substitution MATH 7 REVIEW DAY 1

Ex: Consider the addition 3 + (-2)

We can illustrate the addition using solid and hollow dots

3 + -2 = 1

Ex: Now consider the addition -3 + 2

Again illustrate the addition using solid and hollow dots

-3 + 2 = -1

To recap: 3 + 2 = 5 same sign addends

(-3) + (-2)= (-5) = -5

3 + (-2) = 1 different sign addends

-3 + 2 = - 1

Can we describe a general rule for adding integers?

We see two cases: same sign addends

different sign addends

Addition of Integers

When the addends have the same sign:

Add the numbers. Keep the sign.

When the addends have different signs:

Which sign is bigger? Use that sign for the sum.

Subtract the numbers.

Subtraction of Integers

Let a and b be integers.

Then a – b = a + (-b).

Change subtraction to

addition and change the sign

of what follows.

Subtraction of Integers

We can also use the number line and direction arrows to illustrate

subtraction of integers. Let a positive number be represented by a

right-facing arrow and a negative number

be represented by a left-facing arrow.

The operation of subtraction acts to flip the direction of

the number being subtracted’s arrow.

positive

negative

Ex: Model the subtraction 3 – (-2) using the number line to find the

difference.

0

Start at zero and draw the first addend, 3

Positive

From where the first arrow ends, draw the

second addend, - 2 Negative

Where the second arrow ends is the difference

5

Remember, subtraction flips the arrow!

Ex: Model the subtraction -3 – (-2) using the number line to find the

difference.

0

Start at zero and draw the first addend, -3 Negative

From where the first arrow ends, draw the second addend, - 2

Negative

Where the second arrow ends is the difference

-1

Remember, subtraction flips the arrow!

Multiplying Integers:

3 x 2 = 6 same sign factors

-3 x (-2) = 6

-3 x 2 = -6 different sign factors

3 x (-2) = -6

Can we describe a general rule for multiplying integers?

We see two cases: same sign factors positive

different sign factors negative

Dividing Integers:

6/3 = 2 the same sign

-6/(-3) = 2

-6/3 = - 2 different sign factors

6/(-3) = - 2

Can we describe a general rule for dividing integers?

We see two cases: same sign factors positive

different sign factors negative

Order of

Operations

( ) +

X -

43

Please Excuse My Dear

Aunt Sally

This will help

to you to

remember

the order of

operations.

Add +

Subtract -

Multiply x

Divide

Please Excuse My Dear Aunt Sally

P

E

M

D

A

S

Parentheses ( )

Exponents 43

Please Excuse My Dear Aunt Sally

Parentheses ( )

Always do

parentheses

1st.

Please Excuse My Dear Aunt Sally

Exponents 43

Always do

Exponents

2nd.

Multiply x

Divide

Please Excuse My Dear Aunt Sally

Do

multiplication

and division

3rd, from left to

right.

Add +

Subtract -

Please Excuse My Dear Aunt Sally

Do addition

and

subtraction

4th, from left

to right.

PEMDAS

3+23- (9+1)

3+23- 10 3+8-10

11-10

1

PEMDAS

3 (9+1) + 62

3(10)+62 3(10)+36

30+36

66

PEMDAS

4+5 x (6-2)

4+5 x 4

4+20

24

PEMDAS

4+ 10 x 23 -16 4+10 x 8 -16

4+ 80 -16

84-16 68

PEMDAS

21 + 102 10

21+10010 21 + 10

31

PEMDAS

10+72-2 x 5

10+49–2 x 5 10+49- 10

59 - 10

49

PEMDAS

64 (9 x 3-19) 64(27 –19)

64 8

8

Evaluate a Variable Expression – write the expression,

substitute a number for each variable, and simplify

the result.

Value of a Variable – A number that may be

substituted or assigned to a particular variable; such

as n = 3; or x = 5.

Example 1: Evaluate each expression when n = 4

Substitute 4 for n. Simplify

Simplify (means to solve the problem or perform as

many of the indicated operations as possible.)

7

343

nSolution:

3 b. n Substitute 4 for n. Simplify

1

343

nSolution:

3 a. n

Example 2: Evaluate each expression when x = 8

Substitute 8 for x. Simplify

Simplify (means to solve the problem or perform as

many of the indicated operations as possible.)

Solution:

4 b. x

2

484

xSolution:

x5 a.

40

)8(55

x

Note: No operation sign

between a variable and

number– indicates

multiplication problem.

Using parenthesis is the preferred method

to show multiplication. Additional ways to show

multiplication are:

85 ;85 ;85);8)(5(

Substitute 8 for x. Simplify

Recall that division problems are also

fractions – this problem could be

written as:

44

2;

4

8

4

xx

because

x

Example 3: Evaluate each expression when x = 4, y = 6, z = 24.

xy5 a.Substitute 4 for x; 6 for y. simplify

solution

Recall: No

operation sign

between

variable(s) and

a number–

indicates

multiplication

problem.

Xy means 4(6);

5xy means

5(4)(6)

)6()4)(5(5 xy

)6()20(

120yz b.

Solution: 624 yz

4

Recall that:

46

24624

so,

y

zyz

Evaluate each expression when a = 6, b = 12, and c = 3

ac4 1.

ca 2.

cba 3.

ba 4.

cb 5.

bc 6.

A

A

A

A

A

A

Evaluate each expression when a = 6, b = 12, and c = 3

ac4 1.

)3()6)(4(4 ac

Notice that all the numbers and letters

are together and that there are no

operation symbols which indicates

that this is a multiplication problem.

Substitute the value for a = 6 and c = 3

into the problem and multiply

)3()24(

72

multiply

Simplified

Click to return to

“You try it” slide

Click in the

middle of the

window to view

each answer

Evaluate each expression when a = 6, b = 12, and c = 3

ca 2.

36ca

Division Problem

Substitute the value for a = 6 and c = 3

into the problem and divide

2 Simplified

Click to return to

“You try it” slide

Another way to

solve division

problems is to

write them as

fractions and

simplify. 23

6

c

aca

Click in the

middle of the

window to view

each answer

Evaluate each expression when a = 6, b = 12, and c = 3

cba 3.

3126 cba

Addition problem

Substitute the value for a = 6, b=12,

and c = 3 into the problem, then add

318

Simplified 21

Add

Click to return to

“You try it” slide

Click in the

middle of the

window to view

each answer

Evaluate each expression when a = 6, b = 12, and c = 3

ba 4.

)6)(12(ba

multiplication problem

Substitute the value for b=12 and a = 6

into the problem, then multiply

72 Simplified

Click to return to

“You try it” slide

Click in the

middle of the

window to view

each answer

Evaluate each expression when a = 6, b = 12, and c = 3

cb 5.

312cb

Subtraction problem

Substitute the value for b=12 and a = 3

into the problem, then Subtract

9 Simplified

Click to return to

“You try it” slide

Click in the

middle of the

window to view

each answer

Evaluate each expression when a = 6, b = 12, and c = 3

bc 6.

123bc

Division problem

Substitute the value for c=3 and b = 12 into

the problem, then Divide

Note: It is better to rewrite this division

problem as a fraction.

This fraction can now be reduced to its

simplest form.

12

3

Simplified

3

3

12

3

4

1

Divide both

numerator and

denominator by

the GCF = (3) to

reduce this

fraction.

It is OK to have a fraction

as an answer.

Click in the

middle of the

window to view

each answer

Click to return to

“You try it” slide