Integers

Integers

What are integers?

• Include all Whole Number and their OPPOSITES

• Integers DO NOT include fractions, or decimals

What are opposites?

The opposite number, is a number that is the same distant from zero on the number line in the opposite direction.

Zero is neither positive or negative and does not have an opposite.

​WHAT IS AN INTEGER?

​An integer is any WHOLE NUMBER (Positive or Negative) and Zero.

Multiple Select

CHOOSE ALL INTEGERS.

• 15

• -2.7

• $$\frac{3}{4}$$  

• -10

Absolute Value

• Is how many spaces a number is from zero.

• It is always POSITIVE

• It is annotated by |#|

Additive Inverses

• Numbers that have the same absolute value but opposite signs.

• eg. -11 and 11 are additive inverses

• The sum of additive inverses is always ZERO

• eg: -11 + 11 = 0

Multiple Select

Pick ALL the correct statements:

• $$\left|5\right|=-5$$  

• Additive Inverse of 7 is -7 $$\left|-10\right|=10$$
  

• Absolute Value of 5 is 5

• Additive Inverse of 3 is 3

• undefined 

Multiple Choice

What is the opposite of -10?

• 0

• 10
  

• $$\frac{1}{10}$$  

• $$\frac{10}{1}$$  

Multiple Choice

What number is the opposite of -7?

• 0

• -7

• 7

• 10

Multiple Choice

• 5 $$>$$ -5

• 5 $$<$$ -5

• 5 $$=$$ -5

• 5 $$\ge$$ -5

Multiple Choice

• -9 $$<$$ -18

• -9 $$>$$ -18

• -9 $$\cong$$ -18

• -9 $$\le$$ -18

Multiple Choice

• -21 $$>$$ -17

• -21 $$\ge$$ -17

• -21 $$\le$$ -17

• -21 $$<$$ -17

Multiple Choice

• -21 $$>$$ -17

• -21 $$\ge$$ -17

• -21 $$\le$$ -17

• -21 $$<$$ -17

Multiple Choice

• -12 $$=$$ 12

• -12 $$\ge$$ 12

• -12 $$<$$ 12

• -12 $$\cong$$ 12

Multiple Choice

What is your basis to get the symbol of the sum?

• Bigger number

• Smaller number

Multiple Choice

How would you represent 500 ft below sea level

• $$\frac{1}{500}$$

• 500

• -500

Multiple Choice

How would you represent the absolute value of -23?

• $$\left|-23\right|=-23$$

• $$\left|-23\right|=0$$

• $$\left|-23\right|=23$$

• $$\left|-23\right|=\left|23\right|$$

Multiple Select

When could you use negative numbers in real life?

• Money

• Distance

• Temperature

• Height

Additive Inverses

• Numbers that have the same absolute value but opposite signs.

• eg. -11 and 11 are additive inverses

• The sum of additive inverses is always ZERO

• eg: -11 + 11 = 0

Multiple Select

Pick ALL the correct statements:

• $$\left|5\right|=-5$$  

• Additive Inverse of 7 is -7 $$\left|-10\right|=10$$
  

• Absolute Value of 5 is 5

• Additive Inverse of 3 is 3

• undefined 

To Add Integers

• Same signs ADD and keep the sign

• Different signs SUBTRACT and take the sign of the bigger number

Multiple Choice

Evaluate the following:    $$-3\ +\ \left(-8\right)$$  

• 5

• -5

• 11

• -11

Multiple Choice

Evaluate the following:    $$-2+5$$  

• 7

• -7

• 3

• -3

5 + 9 = ?

32 + 41 = ?

16 + 21 = ?

​+1

-1

​+1

-1

​+1

​+1

​+1

​+1

​+1

​5

Multiple Choice

• 6 + 2

• (-6) + 2

• 6 + (-2)

• (-6) + (-2)

Multiple Choice

• 6 + 4 = 10

• (-6) + 2 = -4

• 6 + (-4) = -2

• 6 + (-4) = 2

Multiple Choice

Find the sum of 64 + (-38)

• 102

• -102

• 26

• -26

Multiple Choice

Answer the following:

1. (-67) + (-24)=

• -91

• -43

• +91

• +43

Multiple Choice

2. (+45) + (+98) =

• +53

• +143

• -53

• -143

Rule No. 1:

• To add integers having the same signs, add the numbers and copy the common sign to the sum.

Illustrative Examples 

 1. (+13)   +  (+34)  = 

 Solution:          

 +1 3             

  + +3 4         

 +4 7 The sum is +47 or 47 since the common sign is positive (+). 

Examples

 2. (-411)  +  (-234)  =

  Solution:  

             1     

    -4 1 1  

         + -2 3 9     

    -6 5 0   The sum is -650 since the common sign is negative (-).

 

Multiple Choice

Answer the following:

1. (-67) + (-24)=

• -91

• -43

• +91

• +43

Multiple Choice

2. (+45) + (+98) =

• +53

• +143

• -53

• -143

Rule No. 2:

• To add integers having different signs, disregard first the signs and subtract the numbers. Then, copy the sign of the greater number to the final answer. (Note: Disregard also the sign in identifying the greater number.)  

Illustrative Example 

 

 

 1. (-56)   +  (+24)  =  

Solution: 

 

             (-5 6)

    + (+2 4)         

       -3 2    The final answer is -32. The sign is negative (-) since  the sign of the greater number which is 56 is negative.  

Multiple Choice

Answer the following:

1. (-89) + (+24)=

• -65

• -78

• +65

• +78

Multiple Choice

2. (+75) + (-13) =

• - 28

• -62

• +65

• +62

Multiple Choice

Answer the following:

1. (-89) + (+24)=

• -65

• -78

• +65

• +78

SUBTRACTION OF INTEGERS

Mastery of addition of intergers is a pre-requisite on this lesson.

SUBTRACTION OF INTEGERS

To subtract integers, change the sign of the subtrahend and proceed to addition process.

SUBTRACTION OF INTEGERS

Example 1: Solve : 18 - ( - 21 )

Multiple Choice

Find the difference :  $$45\ -\ \left(-93\right)$$  

• $$-138$$  

• $$-48$$  

• $$48$$  

• $$138$$  

SUBTRACTION OF INTEGERS

Example 2: Solve : - 54 - ( - 33 )

Multiple Choice

$$-78\ -\ \left(-54\right)$$  

• $$-132$$  

• $$-24$$  

• $$24$$  

• 132

SUBTRACTION OF INTEGERS

Example 3

SUBTRACTION OF INTEGERS

Example 3

Multiple Choice

$$-71\ -\ 82$$  

• $$-9$$  

• $$-\ 153$$  

• $$153$$  

• $$9$$  

THANK YOU!!!

Image on intro slide: www.pexels.com, creator: Black Ice

THANK YOU!!!

Image on intro slide: www.pexels.com, creator: Black Ice

​- 6 - (-4) = -2 0r -6 - (-2) = -4​

​-2 + -4 = -6

Subtraction is an Opposite of addition

Subtraction of Integers

​4 + 6 = 10

​10 - 6 = 4 or 10 - 4 = 6

Subtraction is an Opposite of addition

Subtraction of Integers

​10 minuend

- 6 subtrahend

​ 4 difference

​

Change the operation to addition and then change the sign of the subtrahend.

​ 10

+ -6

​ 4

​

​ or 10 + -6 =​ 4

​

Subtraction of Integers: keep, change, change

Subtraction of Integers

​ 1. Keep the first number ( minuend).

2. ​Change the minus sign to plus sign.

3. Change the last number ( subtrahend) to its opposite​

​4. Add the integers

​-10 - 4 =

​- 10 + (-4) = - 14

​You may use the number line to solve/ to find the answer. Check my screen and observe.

Multiple Choice

Negative number + Negative number = 

• postive number
• negative number
• zero

Multiple Choice

-9 + (-5)

• -14
• 14
• -4

Multiple Choice

-14 + (-3)

• -17
• 11
• -11
• 17

Multiple Choice

Is the answer to -23 + 34 is .....

• 0

• negative

• positive

Multiple Choice

7 + (-3) = ?

• -4
• 4
• 10
• -10

Multiple Choice

2 + (-8) = ?

• 6
• 10
• -16
• -6

Multiple Choice

-5 + 9 = ?

• -4
• 4
• -14
• 14

Subtracting Integers

• We will rewrite all problems to addition to make it easier!

• REMEMBER: SAME, Change, OPPOSITE

• Keep the first number the same!

• Change the subtraction sign to addition

• Write down the opposite number!

Multiple Choice

Using additive inverse, write this problem as an addition problem.

-22 - (-7)

• -22 + 7

• -22 + (-7)

• 22 + 7

• 22 + (-7)

Multiple Choice

Which question is the SAME as
-4 - (-2)       ?

• -4 + 2
• -4 - 2
• 4 + 2
• 4 - 2

Multiple Choice

How would you rewrite -10 - 5?

• 10 + (-5)
• -10 + 5
• -10 + (-5)
• 10 + 5 

Multiple Choice

$$-7-3$$  

• $$4$$  

• $$-4$$  

• $$-10$$  

• $$10$$  

Multiple Choice

solve the following problem.

(-15) - (-5)

• -10

• -2

• -23

• -7

Multiple Choice

3 -9

• 6

• 12

• -12

• -6

Week 2 Assignments!

• IXL lessons! C.9 and B.6

• Week 2 Video: Watch in class Wednesday/Thursday

• Quiz has been posted. Take on Friday!

Week 2 Assignments!

• IXL lessons! C.9 and B.6

• Week 2 Video: Watch in class Wednesday/Thursday

• Quiz has been posted. Take on Friday!

​Let's Review Adding Integers!

Keep the sign and add their absolute values

​

-5 + -7 = -12

​

5 + 7 = 12​

Same Signs

Multiple Choice

Evaluate -50 + -25

• 75

• -75

• 25

• -25

​Let's Review Adding Integers!

Determine which number has the biggest absolute value. That will be the sign of your final answer. Then, subtract the absolute values of your integers.​

​

12 + -7 = 5

-15 + 10 = -5

-20 + 10 = -10​

Different Signs

Multiple Choice

Evaluate -6 + 8

• -14

• 14

• 2

• -2

Now Let's Learn How to

Subtract Integers!

• Keep the sign of the first number.

• Change the subtraction sign to an addition sign.

• Change the sign of the last number to its opposite.​

• Follow the addition rules.

​

-10 - 5 ----> -10 + -5 = -15​

​

Keep Change Opposite -----> KCO

Subtraction is the Adding the Opposite.

​Let's try these together....

1) 6 - 9

​

2) -9 - 4

​

3) -8 - 10​

​

Keep Change Opposite -----> KCO

Multiple Choice

How do you rewrite 6 - 12 using KCO?

• -6 - 12

• 6 + -12

• 6 - -12

• -6 - -12

Multiple Choice

Now evaluate 6 - 12

• 18

• -18

• 6

• -6

Multiple Choice

Evaluate 5 - 7

• 2

• -2

• 12

• -12

Subtracting a Negative is like Adding a Positive

​If it's getting less cold, it's warming up.

If you are getting less negative, you are getting more positive.​

If you owe less money, you are increasing in wealth.​

​10 - -2​ ----> 10 + + 2 = 12

​

Subtracting a Negative is like Adding a Positive

​Let's try these together...

1) 9 - - 10

​

2) -5 - -4

​

3) -8 - -12​

​

Keep Change Opposite -----> KCO

Multiple Choice

How would you rewrite 9 - -3 using KCO?

• 9 - 3

• -9 - 3

• 9 + + 3

• -9 + 3

You are on a hike 120 feet above sea level. Your camping site is at 10 feet below sea level. How far apart are you from your camping site? Draw a picture/number line to help you think through this situation.

​

120 - -10 = ​

​

​Real World Problem