Integers – The Positives and Negatives

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### Integers – The Positives and Negatives

5 =

C

-6 =

D

-2 =

B

9 =

E

2 =

F

-8 =

A

• When adding integers, there is few simple rules to follow.

If the signs are the same

SAME SIGNS – add them and keep the sign

19 + 3 = 22

-19 + - 3 = -22

Opposite Signs
• If the signs are opposite :
• OPPOSITE SIGNS - find the difference and take the sign of the larger number

-15 + 3 = -12

15 + -3 = 12

Subtracting Integers
• We don’t subtract Integers
• We change the sign to positive and change the sign of the number behind the sign

+5 – 6 =

+5 + - 6 =

+5 + - 6 = -1

WHITE BOARDS

-10 + (-4) =

9 - 10 =

-5 - (-2) =

-7 - (-7) =

-10 - 10 =

76 - (-3) =

60 + (-10) =

-18 - (-7) =

-6 + (-10) =

4 - (-4) =  -7 - 4 =

22 + (-5) =

16 - (-8) =

76 + (-6) =

-12 - (-8) =

WHITE BOARDS

-10 + (-4) =  -14

9 - 10 =  -1

-5 - (-2) =  -3

-7 - (-7) =  0

-10 - 10 =  -20

76 - (-3) =  79

60 + (-10) =  50

-18 - (-7) =  -11

-6 + (-10) =  -16

4 - (-4) =  8 -7 - 4 =  -11

22 + (-5) =  17

16 - (-8) =  24

76 + (-6) =  70

-12 - (-8) =  -4

Activity

Directions: In a group of two complete this worksheet.

Directions: Below is listed your starting balance at your bank as well as a series of withdrawals and deposits. Complete the table below by adding or subtracting the given amount and see how much money you have at the end. Starting balance (how much money you have at first) = \$100

• Transaction Current Amount
• You deposit \$10 \$100+10 = \$110
• You write a \$20 check for food _\$110-20 = \$90
• Deposit \$30 _____________
• Write a \$40 check for new shirts _____________
• Write a \$220 check for two pairs of new shoes _____________
• Deposit \$300 (payday at work!) _____________
• Write a \$400 check for this month’s rent _____________
• Write a \$50 check for groceries _____________
• Deposit \$150 (you won a raffle) _____________
• Deposit \$200 (A birthday present) _____________
• What is the current amount in your checking account?______________
• What would your account balance be if your identity was stolen and a \$400 check was written (by the identity thief)?______________
• Afterwards, you were able to convince your bank that you weren’t responsible for writing the \$400 check and the bank therefore deposited \$400 back into your account. What would your balance be now?____________

Activity

Directions: In a group of two complete this worksheet.

Directions: Below is listed your starting balance at your bank as well as a series of withdrawals and deposits. Complete the table below by adding or subtracting the given amount and see how much money you have at the end. Starting balance (how much money you have at first) = \$100

• Transaction Current Amount
• You deposit \$10 \$100+10 = \$110
• You write a \$20 check for food _\$110-20 = \$90
• Deposit \$30 __90 + 30 = 120____
• Write a \$40 check for new shirts _120-40=80_______
• Write a \$220 check for two pairs of new shoes __80-220 = -140___________
• Deposit \$300 (payday at work!) __-140 + 300 = 160___
• Write a \$400 check for this month’s rent _160 – 400 = -240____________
• Write a \$50 check for groceries _-240 - 50 = -290____________
• Deposit \$150 (you won a raffle) __-290 + 150= -140___________
• Deposit \$200 (A birthday present) __-140 + 200 = 60___________
• What is the current amount in your checking account?_____60_________
• What would your account balance be if your identity was stolen and a \$400 check was written (by the identity thief)?___-340___________
• Afterwards, you were able to convince your bank that you weren’t responsible for writing the \$400 check and the bank therefore deposited \$400 back into your account. What would your balance be now?_______60_____

PRACTICE

2.1 Models to Multiply Integers
• Recall that 6 + 6 + 6 = 3 × 6Instead of adding 6 three times, you can multiply 3 by 6 and get 18, the same answer.Similarly,-6 + -6 + -6 + -6 + -6 + -6 + -6 = 7 × -6 = -42
Practice
• Complete pg 29 of workbook
Number Line
• 2 + 2 + 2 + 2 = 4 × 2In algebra, 4 × 2 can be written as (4)(2)You can think of this as 4 groups of 2 or 4 jumps of 2
• The first number tells you how many jumps and the second number tells you how big each jump should beThis situation is shown in the number line below.You basically start at 0 and count by 2\'s until you have put four 2\'s on the number line. You end up at 8 and 8 is positive.

The reasoning is the same; Instead of adding -3 two times, you can just multiply -3 by 2.To model this on the number line, just start at 0 and put 2 groups of -3 (2 x -3) of the number line or make 2 backwards jumps. You end up at -6.

The first number tells you what direction to look and the second number tells you to walk forwards or backwards

• (4) x (-2) = face the positive direction and make 4 jumps of 2 backwards
• (-4) x (2) = face the negative direction and make 4 jumps of 2 forwards
• (4) x (2) = face the positive direction and make 4 jumps of 2 forwards
Practice
• Complete workbook pg 30
Tiles can be used to model as well
• The first number is how many groups and the second number is the quantity.
• If the first number is positive, you will be ‘PUTTING IN’ the tiles
• If the first number is negative, you will be ‘TAKING OUT’ the tiles
• In order to take out negative tiles, you need to have enough zero pairs to balance the question.
Tiles
• To model the multiplication of an integer by a positive integer, you can insert integer chips of the appropriate colour. (Black is Positive, Red is Negative – Hence Black Friday, or in the Red)
• (+2)(-3) = - 6 (put in 2 groups of -3)
Tiles
• To model the multiplication of an integer by a negative integer, you can remove integer chips of the appropriate colour from zero pairs.
• (-2)(-3) = 6 (Remove 2 groups of -3 – ensure to have zero pairs to remove negatives)
Another Example
• (-2)(5)
• Take 2 groups of +5 out
Practice
• Complete pg 31 in the booklet
2.2Rules to Multiply Integers
• Did you notice any patterns from yesterdays homework?
• Complete the multiplication chart and number 1 in workbook pg 32
• What is the sign of the product when you multiply 2 integers?
• If they are both positive
• If one integer is positive and the other integer is negative
• If both integers are negative
Sign Rule
• The product of two integers with the same sign is positive
• The product of two integers with different signs is negative
Practice
• Complete workbook pg32 and pg 33
2.3 Dividing Integers with Number Lines
• Remember that division is the inverse of multiplication
• 10 ÷ 2 = ? Is the same as __ x 2 = 10 (you are looking for how many jumps it takes)
Division with Number Lines
• Positive ÷ Positive
• (8) ÷ (2) We need to find how many jumps of 2 make +8. The jump size is +2, is positive, so we walk forward. Start at 0 and take jumps forward until you end at +8.
• We took 4 jumps. We are facing the positive end of the line so (8) ÷ (2) = +4
Division with Number Lines
• Negative ÷ Negative
• (-8) ÷ (-2) We need to find out how many jumps of 2 make -8. The jump size, -2, is negative, so we jump backward. Start at 0. Take jumps backward to end at -8.
• We took 4 jumps. We are facing the positive end of the number line so (-8) ÷ (-2) = +4
Division with Number Lines
• Negative ÷ Positive
• (-8) ÷ (2) We need to find out how many jumps of 2 make -8. The jump size, 2, is positive, so we jump forward. Start at 0. Take jumps forward to end at -8.
• We took 4 jumps. We are facing the negative end of the number line so (-8) ÷ (2) = -4
Dividing with Number Lines
• Positive ÷ Negative
• (8) ÷ (-2) We need to find out how many jumps of 2 make 8. The jump size, -2, is negative, so we jump backward. Start at 0. Take jumps backward to end at 8.
• We took 4 jumps. We are facing the negative end of the number line so (8) ÷ (-2) = -4
Practice
• Try workbook pg 35 and 36
2.4 Rules to Divide Integers
• Is there a pattern?
• Complete pg 37 - 38 of workbook #1 – 2
• Yes – The Sign Rule Applies to Division
• The product of two integers with the same sign is positive
• The product of two integers with different signs is negative
• Finish workbook

Play Integer BINGO

• Fill in your boxes with the integers from -20 to +20. Each integer can only be used once
2.5 Order of Operations
• B rackets
• E xponents
• D ivision
• M ultiplication
• A ddition
• S ubtraction
Which Operation Would You Do First

1. -4 × 32 + 6

2. 3 × (-2)3 ÷ 6

3. (6 + 2) – 15 ÷ 5 × 2

4. 4(13 – 6)

5. 8 – 4(2 + 52) ÷ 12

1. -4 × 32 + 6

2. 3 × (-2)3 ÷ 6

3. (6 + 2) – 15 ÷ 5 × 2

4. 4(13 – 6)

5. 8 – 4(2 + 52) ÷ 12

White Boards

42 ÷ 6 + 5

64 ÷ 4(2 - 6)

4(-12 + 6) ÷ 3

-122 ÷ 4 – 3 × 24

42 ÷ 6 + 5

7 + 5

12

64 ÷ 4(2 - 6)

64 ÷ 4 (-4)

64 ÷ (-16)

-4

4(-12 + 6) ÷ 3

4(-6) ÷ 3

-24 ÷ 3

-8

-122 ÷ 4 – 3 × 24

144 ÷ 4 – 3 × 16

36 – 3 × 16

36 – 48

-12

Try Some More

6 × 8 - (42 + 2) + 72 ÷ 8

62 + 14 ÷ 2 – 8

9 ÷ 3 + 7 × 4 ÷ 2

12 ÷ 6 + 52 × 3

-4(1+ 5)2 ÷ 6 – (42+5)

7(5 + 3) ÷ 4(9 - 2)

6 × 8 - (42 + 2) + 72 ÷ 8

6 × 8 - (16 + 2) + 72 ÷ 8

6 × 8 - (18) + 72 ÷ 8

48 – (18) + 9

30 + 9

39

62 + 14 ÷ 2 – 8

36 + 14 ÷ 2 – 8

36 + 7 – 8

43 – 8

35

9 ÷ 3 + 7 × 4 ÷ 2

3 + 28 ÷ 2

3 + 14

17

12 ÷ 6 + 52 × 3

12 ÷ 6 + 25 × 3

2 + 25 × 3

2 + 75

77

-4(1+ 5)2 ÷ 6 – (42+5)

-4(6)2 ÷ 6 – (42+5)

-4(6)2 ÷ 6 – (47)

-4(36) ÷ 6 – (47)

-4(36) ÷ 6 – (47)

-144 ÷ 6 – (47)

-24 – (47)

-24 + 47

-71

7(5 + 3) ÷ 4(9 - 2)

7(8) ÷ 4(9 - 2)

7(8) ÷ 4(7)

56 ÷ 4(7)

56 ÷ 28

2

Practice
• Workbook pg 39-41
Integer

the numbers …-3, -2, -1, 0, 1, 2, 3 …

1, 2, 3, etc are positive integers

-1, -2, -3, etc are negative integers

0 is neither positive nor negative

Quotient
• A result obtained by dividing one quantity by another.
Zero Pair
• The result of adding any number to it\'s opposite
• ex: -2 + 2 = 0
Commutative Property
• commutatively is the property that changing the order of something does not change the end result
• Examples of Commutative Property
• 2 + 3 = 3 + 2. Whether you add 3 to 2 or you add 2 to 3, you get 5 both ways.
• 4 × 7 = 7 × 4, Whether you multiply 4 by 7 or you multiply 7 by 4, the product is the same, 28.
• Solved Example on Commutative
Zero Property
• The sum of any number and zero is that number (2 + 0 = 2). The product of any number and zero equals zero (3 * 0 = 0).
Order of Operations
• The rules of which calculation comes first in an expressionThey are:Do everything inside parentheses first: ()then do exponents: x2then do multiplication and division from left to rightlastly do the addition and subtraction from left to right
The Skinny of it