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Our Lesson:

Review of Integers

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oppositesopposites

Integers are all of the positive whole numbers, their opposites and zero

Example …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

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We use VARIABLES when we don't know the number.

For example 2 yards = x feet

We can also write VARIABLE EXPRESSIONS such as

s + 9

8 n

16 x h

VARIABLES AND ABSOLUTE VALUES

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The VARIABLES in an expression can be replaced with any number.

3 + x

If I substitute a 5 for the x ......

I have 3 + 5 or 8

This is how we Evaluate (or find the value of) the Expression

Therefore we Evaluate Expressions by finding the value of the Expression when we replace the Variable with a Number.

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The ABSOLUTE VALUE of a number isits distance from zero on the number line.

The Absolute Value of 4 is written as follows:

4

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Distance is Positive.

Therefore, since Absolute Value is the distance from zero, Absolute Value is always Positive.

9 = 9

-11 = 11

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1 = 1

-47 = 47

11 = 11-5 = 5

Let’s Find

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We will explore 3 different METHODS

METHOD 1 - ZERO PAIRS

++

The red chips are NEGATIVE.

The grey chips are POSITIVE

-

-

-

-

+

+

ADDING POSITIVE AND NEGATIVE INTEGERS

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- +

1 red chip 1 grey chip+ = 0

We call 1 red chip and 1 grey chip a

ZERO PAIR

Zero Pair

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METHOD 2 - Number Line

We can use a number line to find the sum of 2 integers.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

WE ALWAYS START AT ZERO!

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Now, let's add (-2) + (-8)

1. Start at zero and move to the left (in a negative direction) 2 spaces.

2. Next, move further to the left 8 spaces.

3. Your answer is -10. (-2) + (-8) = -10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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Method 3 - Rules for adding integersAdding the The sum of 2 positive numbers is POSITIVE.

Same sign The sum of 2 negative numbers is NEGATIVE.

Example: 4 + 7 = 11

(-3) + (-6) = (-9)

11 + 8 =

(-9) + (-1) =

1919

-10-10

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Adding Different Signs -

1. Find the absolute value of the 2 numbers.2. Subtract the smaller number from the larger number.3. Take the sign of the larger number for your answer.

Example: (-2) + 9 =

1.) 2 9 Absolute Values

2.) 9 - 2

Step

Step Subtract

Step 3.) +7 Sign of the larger number

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Let us try a few problems Remember your Addition Rules!

1.) (-7) + (-11) = (-18)

2.) 22 + 7 = 29

3.) 3 + (-10) = (-7)

4.) (-8) + 20 = 12

5.) (-15) + (-5) = (-20)

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Two methods for subtracting integers.

METHOD 1 - Zero Pairs

The red chips are NEGATIVE. The grey chips are POSITIVE.

+

+

+-

--

--

Subtracting Positive and Negative Integers

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For example, let's subtract 5 - (-3) =

We begin with 5 POSITIVE (grey) chips.

+ + + + +

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5 - (-3) =

We begin with 5 POSITIVE (grey) chips.

++

+ +

Now, we need to take away (or Subtract) 3 NEGATIVE (red) chips but we don't have any!!

So we must ADD 3 Zero Pairs...............

- - -

+++

+

+

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++++ +

- - -

+++

Now, we can take away (or Subtract) the 3 NEGATIVE (red) chips.

And we are left with 8 POSITIVE (grey) chips.

Therefore, 5 - (-3) = 8

5 - (-3) =

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METHOD 2 - Rule for Subtracting Integers

When we subtract an integer, we add its opposite.

For example, (-10) - 7 =

(-10) + (-7) = (-17)

we add its opposite

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Of course, if we are going to ADD its opposite, we must remember the rules for ADDING INTEGERS.

Adding Different Signs -

1. Find the absolute value of the 2 numbers.2. Subtract the smaller number from the larger number.3. Take the sign of the larger number for your answer.

Adding the Same Sign -

The sum of 2 positive numbers is POSITIVE.The sum of 2 negative numbers is NEGATIVE.

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You try a few.

(-4) - (-4) = (-2) - 4 = 12 - (-8) =

(-4) + 4 = 0 (-2) + (-4) =(-6) 12 + 8 = 20

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Game time

Click here to play a game

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We are very familiar with Addition and it is easy for us to add4 + 4 + 4 = 12

We can also think of this as three groups of 4 OR

3 x 4 = 12

Multiplying and Dividing IntegersMultiplying and Dividing Integers

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So, let's look at

(-4) + (-4) + (-4) = -12

We know from our Addition Rules that when we add NEGATIVE numbers together, our sum is NEGATIVE.

We can also look at this as three groups of (-4) OR

3 x (-4) = -12

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On a number line, this might look like......

3 x (-4) = (-12)

40-4-8-12

123

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Rules for Multiplying Integers

The product of two integers with the SAME sign is always POSITIVE.

The product of two integers with DIFFERENT signs is always NEGATIVE.

6 x 7 = +42

(-5) x (-4) = +20

8 x (-4) = -32

(-9) x 3 = -27

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Let's try a few!

Remember:SAME SIGNS PositiveDIFFERENT SIGNS Negative

1.) -6 x 44 = (-264)

2.) 100 x -5 = (-500)

3.) (-8) x (-9) = 72

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Let's look at that relationship with POSITIVE and NEGATIVE numbers!

DIVISIONMULTIPLICATION

(-3) X 5 = -15

Remember that a Negativetimes a Positive number equalsa Negative number.

(-15) ÷

5 = (-3)

And, in division, a Negativedivided by a Positive number ALSO equals a Negative number.

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RULES FOR DIVIDING INTEGERS

The quotient of two integers with the same sign is always POSITIVE.

The quotient of two integers with different signs is always NEGATIVE.

24 4 = 6

(-12) (-3) = 4

32 (-8) = -4

(-27) 3 = -9

Remember that theQUOTIENT is theanswer to a divisionproblem

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This is very similar to our MULTIPLICATION rule.

Signs are the same result is POSITIVE

Signs are different result is NEGATIVE

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1.) 27 (-3) = -9

2.) 24 6 = 4

3.) -35 7 = -5

Let's try a few!

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Order of Operations

It tells us which operation to perform first so that everyone gets the same final answer!

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Order of Operations

1. Parentheses

2. Exponents

3. Multiplication Division

4. Addition Subtraction

Simplifying the expressionsinside grouping symbolsexamples: (3+5) or (4*6)Find the value of all powersexamples: 23 or 42

Perform multiplication or division in the order in which it occurs when reading the expression from left to right.

Perform addition or subtraction in theorder in which it occurs when reading the expression from left to right.

P

E

MDA

S

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We can remember the Order of Operations as

PPEEMMDDAASSP E M D A Sarentheses

xponents

ultiplicationivision

dditionubtraction

"Please Excuse My Dear Aunt Sally"

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P E M D A Sarentheses

xponents

ultiplicationivisionddition

ubtraction

Or

"Purple Eggplants Make Delicious Afternoon Snacks!"

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P E M D A Sarenthese

s

xponents

ultiplicati

on

ivision

dditio

n

ubtraction

whichever comes first whichever comes first

"Purple Eggplants Make Delicious Afternoon Snacks!"

10 - 2 + 8 =

Read the expressionfrom left to right.

8 + 8 =

We perform Addition and Subtraction in the order in which they occur.So do the Subtraction first!

Then do the Addition.16

Numerical Expression

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Sometimes we have operations "nested" within the parentheses which must be completed first.

Example:

60 ÷ (12 + 23) x 9 =60 ÷ (12 + 8) x 9 =

60 ÷ 20 x 9 =

3 x 9 =

27

We want to do the Parentheses but first we must take care of the Exponents WITHIN the Parentheses.

Then we can solve the Parentheses.

Division comes before Multiplication when reading the expression from left to right.

Finally we can perform the Multiplication.

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Practice Problems

(26 + 5) x 2 - 15 = 47

52 + 18 ÷ 2 = 34

5 + (2 - 3) = 4

28 - (22 + 14) + 6 = 16

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Algebraic Expressions consist of Variables, Numbers, and Operations.

Example: 4x + 8

Variables are symbols (usually letters) that represent Numbers.

Example: x or y or a or b etc.

Evaluating Expressions

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Algebraic Expressions consist of Numbers, Operations, and Variables.

3 + n is an "ALGEBRAIC EXPRESSION"

Numbers

Operations

Variables

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b 10

x less than 3 3 - x

Word phrases can be translated into VARIABLE EXPRESSIONS.

Word Phrase Variable Expression

2 more than n 2 + n

k times 8 k x 8

a number b divided by 10

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The Variables in an Expression can be replaced with Numbers. Example: In the Expression 4b - 2 , we can replace b with a Number.

We can Evaluate Expressions by finding the value of the Expression when we replace the Variable with a Number.

We can write multiplication as 3 x a OR 3a OR 3 a.

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Let's Evaluate the Algebraic Expression 16 + b if b = 25

We replace b with the number 25

16 + b = 16 + 5

= 21

The Value of the Algebraic Expression when b = 25 is

21

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Let's Evaluate the Algebraic Expression x - y if x = 22 and y = 7

In other words, in the Expression x - y , let's replace x with the number 64 and replace y with the number 27.

x - y = 22 -7

= 15

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Your Turn!

Evaluate each expression if a = -5 and b =3

1.) a + 4 = -1

2.) a - b = -8

3.) a x b = -15

4.) 4a – 3b = -29

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Complete the Table

Algebraic Expressions

Name the Variables

Name the Numbers

Name the Operations

a and b x, y,z

12, 5 2, 12, -5

+, x , ^ x, - and +

12a + 5b2

2x – 12y + (-5z)

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1) A hot iron piece was at 800 degrees. It was left for cooling every two minutes the change of temperature was -20 degrees. What will be the temperature of this iron piece after 15 minutes

650 degrees

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Ben and Frank had a cycle race. The race was conducted in six sections. In the first section Frank gained 10 seconds. After that he gained 20 seconds then lost one minute, gained 15 seconds lost 27 seconds and finally gained 41 seconds over his friend Ben. Who lost the race?

Frank Lost the race by one second

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The sum of two numbers is 1 and their product is -30. What are the two numbers?

-5 and 6

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You had a great lesson today!

Be sure to practice what you learned!