Introduction to Positive and Negative Numbers

INTERESTING INTERESTING INTEGERS!INTEGERS!

5

-7 -3

-8

4

What You Will Learn:What You Will Learn:

Vocabulary related to integers Rules for adding and subtracting

integers A method for proving that a rule is

true

Are you ready??Are you ready??

Part I: Introduction to Part I: Introduction to IntegersIntegers•VocabularyVocabulary• positive numberpositive number• negative numbernegative number

•Horizontal & vertical number linesHorizontal & vertical number lines•Comparing IntegersComparing Integers•Ordering IntegersOrdering Integers•Vocabulary - continuedVocabulary - continued• opposite numberopposite number• integerinteger

•Real World Applications & ExamplesReal World Applications & Examples• temperaturetemperature• sea levelsea level• money money

Positive number – a number greater than (>) zero

0 1 2 3 4 5 6

Vocabulary:

Hint:Hint:

If you don’t see a negative or positive sign in front of a number, the number is positive.

9 is the same as +9

Negative number – a number less than (<) zero

0 1 2 3 4 5 6-1-1-2-2-3-3-4-4-5-5-6-6

Vocabulary:

Integer Number LineInteger Number Line

Horizontal

Numbers above or right of 0are positive

Numbers below or left of 0

are negative ZER

O

Integer Number LineInteger Number LineV

ertica

lNumbers above

0 are positive

ZERO

Numbers below 0

are negative

Use the number line to compare the following integers with >, <, or =.

-4 -2 1 -3 -5 0

Hint: On a number line, the number to the left is always less than the number to the right.

Comparing IntegersComparing Integers

<

< >

Use the number line to compare the following integers with >, <, or =.

Comparing IntegersComparing Integers

Hint: On a number line, the number on the top is always greater than the

number on the bottom.

-3 -5 -5 0 0 -1>

>

>

Ordering IntegersOrdering Integers

Use the number line to put the following integers in order from least to greatest.-4, 3, 0, and -5 -5, -4, 0, 3

Opposite Numbers – numbers that are the same distance from zero in the opposite direction

0 1 2 3 4 5 6-1-2-3-4-5-6

Vocabulary:

What is the opposite of each integer?

+7 -7

+5

-1

+8

+1

5 -8

Vocabulary:Integers – all the whole numbers and all of their opposites on the number line including zero

0 1 2 3 4 5 6-1-2-3-4-5-6

integers

Now, you’re probably saying, “That’s interesting and everything, BUT where are negative numbers in the real world?? ??

Negative Numbers Are Used to Measure Temperature

Negative Numbers Are Used to Measure Under Sea Level

0102030

-10-20-30-40-50

Positive and negative numbers are used when keeping track of money.

+ Positive +$$ you earn

- Negative -$$ you spend

Positive Numbers are Used to Show Earnings or Assets

When you get paid (or win the lottery), you add that $$ to your account.

Negative Numbers are Used to Show What You Owe or

DebtIf your mom loaned you $10 for pizza, Mom,

I. O. U.$10

The $10 you owe her is described by the integer -10.

Write an integer to describe the real world situation:

Gain 3 pounds:

Withdraw $15:

5 feet below sea level:

Move ahead 4 spaces:

3 or +3

-15

-5

4 or +4

End - Part I: Introduction to End - Part I: Introduction to IntegersIntegers•VocabularyVocabulary• positive numberpositive number• negative numbernegative number

•Horizontal & vertical number linesHorizontal & vertical number lines•Comparing IntegersComparing Integers•Ordering IntegersOrdering Integers•Vocabulary - continuedVocabulary - continued• opposite numberopposite number• integerinteger

•Real World Applications & ExamplesReal World Applications & Examples• temperaturetemperature• sea levelsea level• money money

Part II: Adding IntegersPart II: Adding Integers

Key ConceptsKey ConceptsInteger Addition RulesInteger Addition RulesUsing Number LinesUsing Number Lines

** Key Concepts **** Key Concepts **

The sum of two positive numbers is always positive (+) + (+) = (+)

ex. 5 + 1 = 6

The sum of two negative numbers is always negative (-) + (-) = (-)

ex. -5 + -1 = -6

** Key Concepts **

(+) + (+) = (+) (-) + (-) = (-)

(+) + (-) = sometimes (+) = sometimes (-) = sometimes 0

AND

Integer Addition Integer Addition RulesRules Rule #1 – If the signs are the same,

add the numbers and then put the sign of the addends in front of your answer.

b) -9 + -5 = -14

a) -9 + -5 =

SolveSolve the Problems the Problems

-3 + -5 = 4 + 6 = +3 + (+4) = -6 + -7 = 5 + 9 = -9 + -9 =

-8

-1814-137

10

Rule #2 – If the signs of the addends are DIFFERENT, start at the location of the first integer on the number line and: a) move RIGHT to add a positive integer

Integer Addition RulesInteger Addition Rules

-5 + 3 =-2

1 2 3

ex. (-6) + 5 = -1Start here at -6

0 1 2 3 4 5 6-1-2-3-4-5-6

then count forward or right 5 spaces+

Adding Integers Using a Number LineAdding Integers Using a Number Line* adding a * adding a positive integer *integer *

Solve the ProblemsSolve the Problems

• 8 + 6 =

• (-9) + 5 =

• (–11) + 11 =

• (–8) + 16 =

14

0

8

-4

Rule #2 – If the signs of the addends are DIFFERENT, start at the location of the first integer on the number line and: b) move LEFT to add a negative integer

Integer Addition RulesInteger Addition Rules

4 + -3 =1

123

0 1 2 3 4 5 6-1-2-3-4-5-6

-

ex. +3 + (-5) = -2Start here at +3

Then count back or left 5 spaces

Adding Integers Using a Number Adding Integers Using a Number LineLine

* adding a * adding a negative integer *integer *

Solve the ProblemsSolve the Problems

• 2 + (-12) =

• –8 + (-5) =

• 14 + (-7) =

• 15 + (-15) =

-10

7

-13

0

Part III

Part III: Subtracting IntegersPart III: Subtracting Integers

** Key Concept **** Key Concept **

To subtract an integer, add its opposite

ex. 5 – 2 = 5 + (-2) = 3

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EP

CH

AN

GE

CH

AN

GE

ex. -1 – (-2) is the same as -1 + (+2) and -1 + 2 = 1

Subtracting a negative number is the same as adding a positive. Change the signs and add.

Integer Subtraction Rule

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EP

CH

AN

GE

CH

AN

GE

-3 – 4 is the same as-3 + (-4) and -3 + (-4) = -7

More Examples

2 – (-7) is the same as2 + (+7) and 2 + 7 = 9

KEEP the sign of the 1st integer the sameCHANGE the operation ( + to – or – to +)CHANGE the sign of the 2nd integer

More Examples

12 – (-8) is the same as 12 + (+8) and 12 + 8 = 20

-3 – (-11) is the same as-3 + (+11) and -3 + 11 = 8

KEEP the sign of the 1st integer the sameCHANGE the operation ( + to – or – to +)CHANGE the sign of the 2nd integer

Problems to Solve8 – (-12) is the same as 8 + (+12)and 8 + 12= 20

22 – (-30) is the same as22 + (+30)and 22 + 30= 52

KEEP the sign of the 1st integer the sameCHANGE the operation ( + to – or – to +)CHANGE the sign of the 2nd integer

Problems to Solve

-17– (-3) is the same as -17 + (+3)and -17 + 3= -14

-8 – 3 is the same as-8 + (-3)and -8 + -3 = -11

KEEP the sign of the 1st integer the sameCHANGE the operation ( + to – or – to +)CHANGE the sign of the 2nd integer

Part IV

How do we know that “Subtracting a negative number is the same as adding a positive” is true?We can use the same method we use to check our answers when we do regular subtraction.

When you subtract a – b it equals c a – b = c ex. 5 – 2 = 3

To check if your answer is correct, add b and c a = b + c ex. 5 = 2 + 3

If a – b = c, and….

2 – (-5) is the same as

2 + (+5), which equals 7,

Then let’s check with the negative numbers to see if it’s true…

Here are some examples:

a – b = c a = b + c9 – 5 = 4 9 = 5 + 4

a – b = c a = b + c20 – 3 = 17 20 = 3 + 17

If the method for checking subtraction works, it shouldalso work for subtracting negative numbers.

a – b = c a = b + c2 – (-5) = 7 2 = -5 + 7

It works!

a – b = c a = b + c-11 – (-3) = -8 -11 = -3 + -8

YES!

Aren’t integersinteresting?