Integers Intro

-$1.24-3.4

Integers• Integers are whole numbers

that describe opposite ideas in mathematics.

• Integers can either be negative(-), positive(+) or zero.

• The integer zero is neutral. It is neither positive nor negative, but is an integer.

• Integers can be represented on a number line, which can help us understand the valve of the integer.

Positive Integers

• Are to the right of zero • Are valued greater than

zero.• Express ideas of up, a gain

or a profit.• The sign for a positive

integer is (+), however the sign is not always needed.

• Meaning +3 is the same value as 3.

Negative Integers

• Are to the left of zero • Are valued less than

zero.• Express ideas of

down or a lose.• The sign for a

negative integer is (-). This sign is always needed.

Negative integers are valued less than zero, and are always to the left

of zero.

Zero is neither positive or negative

Positive integers are valued more than

zero, and are always to the right of zero.

- 1- 1

- 4- 4

+ 3+ 3

- 3- 3

+ 2 + 2

Representing Integers

• - 4 using 6 counters

• + 2 using 6 counters

• 0 using 6 counters

• - 3 using 6 counters

The “net worth” of opposite integers is

Opposite Integers

• Opposite integers always have a “net worth” of 0. This is called the ZERO PRINCIPAL.

• Opposite integer have the same “absolute value”, meaning the distance from the points on a number line to zero is the same.

• This can be referred to as the integers magnitude.

Movement on a Number Line

Magnitude and Direction• Every integer represents a

magnitude and a direction.• The integer +3 describes a

movement of 3 units in a positive direction.(right)

• The sign (+) tells you the direction.

• The number (3) indicates how far to move or the MAGNIUDE( a movement of 3 units)

Direction

Magnitude

Which integer has a higher

value?

-4 or -8

Comparing Integers

Use your number line to help you compare each set of number.

(i.e. for the numbers 3 ,and - 2 …. 3 > -2 -2 < 3)

a) - 6, 7 b) 12, 3 c)- 5,- 8 d) 11, - 15

e) - 7, - 4 f) - 3, - 7 g) 7, - 8 h) - 13, -14

Putting Things Together

• What is the greatest valued negative integer?

(4,-2)

(-1,-3)

(-2,1)

(4,5)(4,5)

(-8,+3)(-8,+3)

(-5.-1)(-5.-1)

(-6,3)(-6,3)

(0,-7)(0,-7)

Comparing Integers

Use your number line to help you compare each set of numbers. Copy the question and write two sentences for each pair of numbers.

(i.e. for the numbers 3 ,and - 2 …. 3 > -2 -2 < 3)

a) - 6, 7 b) 12, 3 c)- 5,- 8 d) 11, - 15

e) - 7, - 4 f) - 3, - 7 g) 7, - 8 h) - 13, -14

i) 8, 7 j) - 8, - 7 k) 5, -1 l) 0, -2

m) 0, 3 n) - 5, 0 o) – 14, -10 p) - 9, 0

q) -7, -6 r) -1, 0 s) 4, -4 t) 0, -15

Comparing Integers Again

• For each of the previous questions (a) to (t), write a new mathematical sentence showing how much bigger or smaller the first number is than the second.

• (i.e. 3, - 2 ….. 3 is 5 more than –2)

Comparing Integers

• -5 ___ -8

• 0 ___ -3

• 3 ___ +2

Quadrant l

(4,-5)(4,-5)

(-8,+3)(-8,+3)

(-5,-1)(-5,-1)

Outcomes• A12 represent integers (including zero)

concretely, pictorially, and symbolically, using a variety of models

• B11 add and subtract integers concretely, pictorially, and symbolically to solve problem

• B14 solve and pose problems which utilize addition of integers

• B2 use mental math strategies for calculations involving integers

Lab Performance Evaluation• A – Student is performing beyond

expected level.• B – Student is performing at upper

range of expected level. • C – Student is performing at expected

grade level• D – Student is performing at lower

range of expected level. • E – Student is performing below

expected level.

Areas of Evaluation• Organization into activity• Following directions• Presenting work neatly• Completion of work• Representing Integer sentences in

words• Your ability to discover and

represent Integer Rules• Making use of the Integer mat• Working quietly and cooperative

Net Result

Positive 9

Net Result

Positive 9

(+5) + (+4) = +9Or

(+4) + (+5) = +9

(+5) + (+4) = +9Or

(+4) + (+5) = +9

Finding The Sum of Positive Integers

• When finding the sum of positive integers you add the magnitudes and keep the positive sign.

Net Result

Negative 10

Net Result

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

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

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

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

Finding The Sum of Negative Integers

• When finding the sum of negative integers you add the magnitudes and keep the negative sign.

Net Result

Positive 2

Net Result

Positive 2

(+7) + (-5) = +2Or

(-5) + (+7) = +2

(+7) + (-5) = +2Or

(-5) + (+7) = +2

Finding The Sum of a Positive and a Negative

Integer

• When finding the sum of a positive and a negative integer you subtract the magnitudes and keep the sign of the integer with the largest magnitude.

Net Result

(+5) + (-5) = 0

Or (-5) + (+5) =

(+5) + (-5) = 0

Or (-5) + (+5) =

Integer Recap

• Positive symbol means

• Negative symbol means

You Haveor

You’ve Earned

You Haveor

You’ve Earned

You OweYou Owe

• (+3) + (-7)

• (-5) + (-2)

• (-3) + (-6) + (+4)

• (+3) + (-2) + (+2)

• (+50) + (-100)

• (-25) + (+10)

• -60 + -20

• -20 + 15

• 30 + -5

Rules For Adding IntegersPositive Integers

To add two positive integers you add the magnitude and keep the positive sign.

Negative IntegersTo add two negative integers you add the magnitude and keep the negative sign.

A Negative and a Positive IntegerTo add a positive and a negative integer you subtract the magnitudes and keep the sign of the integer with the largest magnitude.

(+5) – (+3) =

(+5) – (+3) = +2

(-6) – (-2) =(-6) – (-2) = (-6) – (-2) = -4

(-6) – (-2) = -4

(+3) – (+5) =

(+3) – (+5) = -2

(-2) – (-6) =(-2) – (-6) = (-2) – (-6) = +4

(-2) – (-6) = +4

(+3) – (-2) =

(+3) – (-2) = +5

(+1) – (+4) =

(+1) – (+4) = -3

(-5) – (+3) =

(-5) – (+3) = -8

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

(-2) – (-5) = +3

Try These• (-8) – (-3) = • (+4) – (-5) = • (-4) – (-5) =• (+1) – (-6) =• (-5) – (+6) =• (-2) – (-3) =• (-20) – (-10) =• (+30) – (-3) =• (-20) – (-30) =

Try These• (-3) – (-2) = • (+6) – (-2) = • (-1) – (-4) =• (+3) – (-2) =• (-5) – (+2) =• (-2) – (-4) =• (-30) – (-20) =• (+50) – (-10) =• (-20) – (-30) =

Try These

1. (-5) + (+2) = 2. (+6) + (-2) = 3. (-2) – (-6) = 4. (+7) + (-2) = 5. (-5) + (+2) = 6. (+8) + (-4) = 7. (-3) – (+6) = 8. (+50) – (-10) = 9. (-20) + (-30) =

Try These

1. (-5) + (+2) = -32. (+6) + (-2) = +43. (-2) – (-6) = +4 4. (+7) + (-2) = +55. (-5) + (+2) = -36. (+8) + (-4) = +47. (-3) – (+6) = -98. (+50) – (-10) = +609. (-20) + (-30) = -50

Site: www.aplusmath.com

• Go to Flashcards• Go to Non-Java Flashcards• Go to Adding, Subtracting, Multiplying and

Dividing With Negative Numbers• Click on Multiplying (One by One) Use the

site to help you complete the chart• Then, Go To Division (One by One)

(+2) x (+4) =

(+2) x (+4) = +8

This means you have two sets of four positive tiles or you have

earned two groups of four dollars.

This means you have two sets of four positive tiles or you have

earned two groups of four dollars.

(+2) x (-4) =

(+2) x (-4) = -8

This means you have two sets of four negative tiles or you have two bills

that you owe,each bill is for four dollars.

This means you have two sets of four negative tiles or you have two bills

that you owe,each bill is for four dollars.

(-2) x (-4) =(-2) x (-4) = (-2) x (-4) = +8

(-2) x (-4) = +8

This means you don’t have two sets of four negative tiles or you

don’t owe two bills, each bill is for four dollars.

This means you don’t have two sets of four negative tiles or you

don’t owe two bills, each bill is for four dollars.

(-2) x (+4) =

(-2) x (+4) = -8

This means you don’t have two sets of four positive tiles or you don’t have two groups of four

dollars.

This means you don’t have two sets of four positive tiles or you don’t have two groups of four

dollars.

Try These

• (+3) x (-2) = • (-2) x (-2) =• (+5) x (-2) =• (-3) x (+2) =• (+3) x (+4) =• (+3) x (-2) =

Try These

• (-91) x (-101) =• (+152) x (-21) =• (-19) x (+203) = • (-69) x (-102) =• (-62) x (-11) =• (-128) x (+12) =

Try These

• (-91) x (-101) =• (+152) x (-21) =• (-19) x (+203) = • (-69) x (-102) =• (-62) x (-11) =• (-128) x (+12) =

Multiplying Integers

FACTOR FACTOR PRODUCT

Dividing Integers

DIVIDEND DIVISOR QUOTIENT

Try These

• (-1) x (+1) x (-1) =• (+1) x (+1) x (-1) =• (-1) x (-1) x (+1) =• (-1) x (-1) x (-1) =• (-1) x (-1) x (+1) x (-1) x (+1) =• (-1) x (+1) x (+1) x (-1) x (+1) =

Short Cuts For Multiplying Several Integer Factors

a. (-1) x (+1) x (-1) = +1

b. (+1) x (+1) x (-1) = -1

c. (-1) x (-1) x (+1) = +1

d. (-1) x (-1) x (-1) = -1

If there is an even number of

negative signs, the product is

positive

If there is an odd number of

negative signs, the product is

negative

Short Cuts For Multiplying Several Integer Factors

a. (-1) x (+1) x (-1) x (+1) =

b. (+1) x (+1) x (-1) x(-1) =

c. (-1) x (+1) x (-1) x (-1) x (+1) =

d. (-1) x (-1) x (-1) x (-1) x (+1) x (-1) =

e. (1) x (+1) x (-1) x (-1) x (+1) x (-1) =

f. (-1) x (-1) x (-1) x (-1) x (-1) x (-1) =

g. (-2) x (-3) x (-2) x (+1) =

h. (-1) x (-3) x (-2) x (-2) x (-3) =

Try These

• (-2) x (+2) x (-1)(-3)=• (+1) x (+4) x (-5) =• (-17) x (-2) x (+2) =• (-2) x (-3) x (-6) x 4 =• (-2) x (-3) x (-3)

(+2) x (+4) =

(+2) x (+4) = +2

Positive and Negative Integers

• For each of the following numbers, write down an example of where it could be used and what it means in that situation.

• -3 -100m +15• +3050m -$45.83

Order of Operations With Integers

3 x (–7) + 4 x (-5)

15 + (+5)2 x 2

(-18) -- 32 – 9 x 2

Practice for Problem Solving

• Fiona spends $5 per week on bus fare. How much does she spend in 2 weeks?

• Lucy spends 2 per week on snacks. How much does she spend in 4 weeks?

• Anton earns $8 each week for baby-sitting. How much does he earn in 3 weeks?

Practice for Problem Solving

• Lional pays $3 per day for bus transportation. How much does she pay in a school week?

• Jill has $100 in the bank. She owes 3 of her friends $10 dollars each. What is her net worth?

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