Math 8 Unit 2 Integers NOTES (Blank Copy).notebookmrsjeddore.weebly.com/uploads/.../math8_unit2_notes... · Math 8_Unit 2 Integers_NOTES (Blank Copy).notebook 1 October 16, 2015 Unit

Math 8_Unit 2 Integers_NOTES (Blank Copy).notebook

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October 16, 2015

Unit 2 Integers Review of Grade 7In your text book Yellow is Positive and Red is Negative

In notes, handouts, tests shaded is Positive and non-shaded (white) is Negative

Remember that one Positive and one Negative form a Zero Pair. (+1) + (-1) = 0


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October 16, 2015

Adding Integers with TilesA) (+2) + (+3) =

B) (-3) + (-1) =

C) (+5) + (-4) =


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October 16, 2015

Check. Use tiles to add.D) (+1) + (+3) =

E) (-2) + (-3) =

F) (-4) + (+3) =

G) (+4) + (-2) =

H) (-5) + (+2) =


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October 16, 2015

Adding Integers with Number Lines*Start at zero. Positive integer move right, negative integer move left.I) (+4) + (-5) =

J) (-2) + (+6) =


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Check. Use a number line to add.K) (+7) + (+4) =

L) (+6) + (-4) =


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M) (-11) + (-4) =

N) (+3) + (+5) =


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O) (-8) + (+2) =


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Subtracting Integers with TilesA) (+5) – (+3) =

B) (-4) – (-3) =


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C) (-5) – (-1) =

D) (-2) – (-6) =

We only have 2 negatives, and need to remove 6 negatives. So we ADD 4 zero pairs because adding zero pairs is the same as adding ZERO. Now we can remove 6 negatives, what are we left with?


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E) (-3) - (+1) =

We have 3 negatives, and need to remove 1 positive. So we add 1 zero pair. Now we can remove 1 positive, what are we left with?


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Check. Use tiles to subtract.A) (+2) – (-7) =

B) (-3) – (-4) =

C) (-5) – (-5) =


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I) (+10) – (-4) =

J) (-5) – (+6) =

K) (-3) – (-5) =


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Remember the rule for subtracting integers ?ADD THE OPPOSITE: the first integer stays the same, change the subtract sign to addition sign, then switch the second integer to its opposite.

(-4) – (-3) → (-4) + (+3) =

(-5) – (-1) → (-5) + (+1) =

(-2) – (-6) → (-2) + (+6) =

(-3) – (+1) → (-3) + (-1) =


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October 16, 2015

Check. Use Add the Opposite

L) (+6) – (-5) →

M) (-7) – (+3) →

N) (+9) – (+4) →

O) (-3) – (+5) →


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October 16, 2015

Section 2.1 Using Models to Multiply Integers

Multiplication can be viewed as repeated addition.

From earlier grades:

2 x 3 means 2 sets of 3

3 + 3 = 6

4 x 5 means 4 sets of 5

5 + 5 + 5 + 5 = 20


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Try these:

2 x -3 means ___ sets of ____

4 x - 5 means ___ sets of ____

(+3) x (-5) means ___ sets of ____

(+3) x (-2) means ___ sets of ____


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Write each repeated addition sentence as multiplication.

A) (-6) + (-6) + (-6) + (-6) + (-6)

B) (+4) + (+4) + (+4) +(+4)


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What if the first integer is negative?

In grade 8, we will be taking the idea of repeated addition and combine it with making deposits and withdrawals from a bank as positive and negative.


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Examples

A) (+4) x (+3)

The first tells us toDEPOSIT (put in the bank if positiveWITHDRAWAL (take out of the bank) if negative

The second integer tells us what to "put in" or "take out"

The bank will start as a circle with zero value.


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(+4) x (+3) Solution

Put in4 sets

of +3


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B) (+4) x (-3) Solution

put in of -34 sets


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Try these!

C) (+2) x (+4) Solution

D) (+2) x (-3) Solution


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E) (-4) x (-3)

withdrawal of -3(take out)4 sets

* If the bank starts at zero, how can we take out four sets of -3 ?


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Use zero pairs* make 4 sets of 3 zero pairs

* now take out 4 sets of -3Solution: ____ what is left in the bank

+- +-+- +-+- +-

+- +-+- +-+- +-


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October 16, 2015

F) (-4) x (+3)

take out of +34 sets

Need 4 sets 3 Zero Pairs

Solution:What is left in the bank?

+- +-+- +-+- +-

+- +-+- +-+- +-


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October 16, 2015

Try these!

G) (-2) x (-5)

We need ___ sets of ____ zero pairs

Solution: ______What is left in the bank?


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October 16, 2015

H) (-2) x (+4)

We need ___ sets of ____ zero pairs

Solution: ______What is left in the bank?


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Whenever the first integer is negative, we need to "take out", so zero pairs need to be added to the bank first.

Complete Page 68 #'s 5, 6, 9, 10


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Multiplying Integers using Number Lines

* Start at zero* The first integer indicates which direction to face and how many steps to take

Positive = Face Right Negative = Face Left

* The second integer indicates which direction to move and the size of the steps

Positive = Move forward Negative = Move Backward

* Follow these steps to land on the answer


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#1: The temperature fell 3oC each hour for 6 hours. Find the total drop in temperature.

(+6) x (-3)

Face Right Move Backward6 steps of size 3

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0


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# 2: Kyle spends $2 a day for recess at school, 5 days of the week. Use a number line to represent how much Kyle spends a week?

(+5) x (-2)


15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0


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#3: (+3) x (+2)

Face Right Move Forward3 steps of size 2

10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10


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#4: (+3) x (-2)


10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10


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#5: (-3) x (-2)

Face Left Move Backward 3 steps of size 2

10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10


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#6: (-3) x (+2)

Face Left Move Forward3 steps of size 2

10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10


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Try these....

#7: (-4) x (+2)

Face _____ Move __________ steps of size ___

10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10


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#8: (-4) x (-2)


10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10


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#9: (+4) x (-2)


10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10


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IntegerMultiplication

Number Lineon a


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Multiplying Integers

10 2 3 4 5 6 7 8 9 1012345678910

( ) ( )

Tells us which direction the man is facing.

Tells us the number of steps to

take.

Tells us to step

backwards or forwards.

Tells us the size of the steps.

+ 3 x + 2


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10 2 3 4 5 6 7 8 9 1012345678910

(+ 3) x (+ 2)

+6Erase for

answer


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10 2 3 4 5 6 7 8 9 1012345678910

(+ 4) x (+ 2)

Erase for answer +8


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10 2 3 4 5 6 7 8 9 1012345678910

(+ 2) x ( 4)

Erase for answer 8


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10 2 3 4 5 6 7 8 9 1012345678910

( 4) x (+ 2)

Erase for answer 8


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10 2 3 4 5 6 7 8 9 1012345678910

( 5) x ( 2)

Erase for answer +10


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10 2 3 4 5 6 7 8 9 1012345678910

( 3) x (+ 3)

Erase for answer 9


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10 2 3 4 5 6 7 8 9 1012345678910

(+ 3) x ( 2)

Erase for answer 6


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10 2 3 4 5 6 7 8 9 1012345678910

( 3) x ( 1)

Erase for answer +3


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Extra PracticeMultiply the following using number lines.

1. (+2) x (+5)

2. (+3) x (1)

3. (4) x (+2)

4. (2) x (5)

5. (+3) x (3)

6. (4) x (1)

7. (+5) x (2)

8. (3) x (+1)

9. (2) x (2)

10. (1) x (+8)


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Summary

For Multiplying two integers

2 positives =(+2) x (+3) =

2 negatives = (-2) x (-3) =

negative and positive = (-2) x (+3) =

positive and negative =(+2) x (-3) =


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Therefore, If the signs are the SAME, the answer is POSITIVE

If the signs are DIFFERENT, the answer is NEGATIVE

Same Signs (++) (--)PositiveOrOpposite Signs (+-)(-+)Negative


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Some Properties for Multiplying:

Multiplying by Zero

0 x (+3) = 0 x (-3) =

Multiplying an integer by zero results in a product of ________.


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Multiplying by One (Multiplicative Identity)

(-3) x (+1) = (+3) x (+1) =

Since multiplying by 1 does not change the number, we call 1 the multiplicative identity.


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Communative Property

(+3) x (+4) = (+4) x (+3) =

Did our answer change because we changed the order of the integers?


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NO

multiplication is commutative because we can change the order of multiplying the integers but it didn't change the answer.

(-3) x (+4) = (+4) x (-3) =


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Complete Page 73#3, 4, 5, 8 a -e, 15, 17, 18


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Area ModelTwo Integers can also be multiplied using an

area model in combination with distributive property.

Distributive PropertyA property multiplication stating that a product

can be written as a sum of 2 products

For example: 3 x (4 +5)= ( 3 x 4) + (3 x 5) = 12 + 15= 27


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Example A 8 x 26 Same as 8 x (20 + 6)

= (8 x 20) + (8 x 6)= (160) + (48)= 208

Area Model20 6

8 8 x 20 8 x 6 160 48

Answer: 208


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Example B -5 x 36

Area Model Distributive Property**Ignore the signfor area model to start -5 x (30 + 6)

= (-5 x 30) + (-5 x 6)30 6 = -150 + -30

5 5 x 30 5 x 6 = -180 150 30

150 + 30 180

GO BACK TO SIGN5 x -36 = -180


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Example C -25 x -48

Ignore sign for area model...we know are answer is _____

40 820 20 x 40 20 x 8

_____ _____

5 5 x 40 5 x 8_____ _____

25 x 48 = _____ + _____ + _____ + ______

= _____ + _____

= _____

SIGN CHECK (-25) x (-48) = _______


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October 16, 2015

Example D (-72) x (+15)

Ignore sign for area model...we know are answer is _____

10 5

70 70 x 10 70 x 5_____ _____

2 2 x 10 2 x 5_____ _____

72 x 15 = _____ + _____ + _____ + ______

= _____ + _____

= _____

SIGN CHECK (-72) x (+15) = _______


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October 16, 2015

Complete Page 73# 6 a - d# 7 a - d


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Dividing Integersusing

Counters


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October 16, 2015

(+6) ÷ (+2)

+6+2 ?will makeHow many groups of

Dividing Integers


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October 16, 2015

Dividing IntegersThe Answer? ÷

?will makeHow many groups of

(+8) (+4)+2

(+8)


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October 16, 2015

Dividing IntegersThe Answer?

?will makeHow many groups of??


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October 16, 2015



(9) (3)+3


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(6) (3)+2


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(10) (+5)-2


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(6) (+3)-2


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(8) (+2)-4


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(+12) (3)-4


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(+9) (3)-3


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(+8) (2)-5


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Page 81 # 8


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Dividing Integersusing

The Number Line


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Directions:Grab the first number in the division and place it in the first blank in the "Ask yourself..." question. Touch just below the second integer in the question. Look at the sign of the second integer. Grab and move "forward" (for +) or "backward" (for -) and place it in the second blank in the question. Then grab the second number in the division and place it at the end of the question.

Use the correct man to do answer the question on the number line. Then fill in the first blank in the "Answer..." with the number of steps and the second blank with the direction the man is facing (+ or -). This give you the quotient!


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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

(10) (+2)

Ask yourself...

How do I get to stepping of size ?

forwards

backwards

(10) 2

Answer...

To get there I took steps and I am now facing .+

-

= 5Touch me when done!


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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

( ) ( )

Ask yourself...


forwards

backwards

Answer...


-

+8 +4+8 4 = +2Touch me when done!


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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

( ) ( )

Ask yourself...


forwards

backwards

Answer...


-

+6 22+6 = 3Touch me when done!


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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

( ) ( )

Ask yourself...


forwards

backwards

Answer...


-

9 +39 3 = 3Touch me when done!


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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

( ) ( )

Ask yourself...


forwards

backwards

Answer...


-

1212 44 = +3Touch me when done!


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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

( ) ( )

Ask yourself...


forwards

backwards

Answer...


-

+15 3+15 3 = 5Touch me when done!


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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

( ) ( )

Ask yourself...


forwards

backwards

Answer...


-

+10 +5+10 5 = +2Touch me when done!


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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

( ) ( )

Ask yourself...


forwards

backwards

Answer...


-



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October 16, 2015

0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415

( ) ( )

Ask yourself...


forwards

backwards

Answer...


-



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Extra Practice(use numberlines)

1. (+12) (-4)

2. (-8) (-2)

3. (+6) (-3)

4. (-15) (+5)

5. (-15) (-5)

..........

6. (+8) (-2)

7. (+9) (-3)

8. (+10) (+2)

9. (-12) (-3)

10. (-15) (+3)

..........


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0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415


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Summary for Dividing Integers

Dividing Two Positive Integers

(+8) ÷ (+2) =(+6) ÷ (+3) =(+12) ÷ (+3) =What do you notice?

Dividing Two Negative Integers

(8) ÷ (2) =(6) ÷ (3) =(10) ÷ (2) =What do you notice?

Dividing a Positive by a Negative

(+8) ÷ (2) =(+6) ÷ (3) =(+12) ÷ (3) =What do you notice?

Dividing a Negative by a Positive

(8) ÷ (+2) =(6) ÷ (+3) =(10) ÷ (+2)=What do you notice?


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The rules for Dividing Integers are the same as Multiplying Integers.

Same Signs (++) (--)PositiveOrOpposite Signs (+-)(-+)Negative


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Terminology(+15) ÷ (+3) = +5

Dividend Divisor QuotientWe can also write a division statement as a fraction:


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October 16, 2015

Practice: DivideA) (-8) ÷ (-4)

B)=

C) (-12) ÷( ____ ) = +2

D)


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October 16, 2015

E) Shannon made withdrawals of $7 from her bank account, for a total withdrawal of $63? Use integers to find how many withdrawals she made.


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F) Chris and his 3 friends together owe $12. They agree to share the debt equally. What is each person‛s share of the debt?


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G) The temperature in Nain is dropping 2oC each hour. How many hours did it take to fall 10oC?


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Complete Pages 87 – 88 #‛s 4, 5, 7a, 9, 10, 11, 12


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Section 2.5: Order of Operations

To ensure everyone gets the same answer for a question like 9 × 6 + 36 ÷ 4 1 we follow the proper order of operations.

B – Brackets FirstE – Exponents (not in grade 8)D – Division M MultiplicationA AdditionS – Subtraction

}}

Whichever comes first in the problem

Whichever comes first in the problem


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Examples A) 9 × 6 + 36 ÷ 4 1


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B) [(+9) – (2)] × (3)


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C) [(6) + (2)] ÷ ( 4) + (5)


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D)


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October 16, 2015

E)


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F) (2 × 105) – (2 +8) ÷ 2


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G)


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October 16, 2015

H) 15 ÷ [ 10 ÷ (2)]


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I) Kathy completed the following calculations on a math test. Is she correct? Explain.


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October 16, 2015

Complete Page 92 #'s 3, 5, 8, 9ad


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Word Problems

Words/Phrases that indicate MULTIPLY Words/Phrases that indicate DIVIDE

Multiply byProduct ofTimesDoubleTripleAs muchByFind the total

Quotient ofDivided byDivided equallyPerHow many (much) eachSplitEqual piecesShared


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1: You have no money, instead you borrow $2 each day for 3 days. What is the total debt by the end of the third day?


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2: Matthew has agreed to donate to a local charity for 2 years. If he gives $25 per month, how much has he donated at the end of the 2 years?


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3: An oil rig is drilling a well at 2m per minute. How deep is the well after the first 8 minutes?


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4: The temperature decreased 16oC over a 4 hour period. Assuming the temperature decreased at a constant rate, how much did the temperature decrease each hour?


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5: The temperature rose 4oC each hour for 5 hours. Use integers to find the total change in temperature.


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6: The water level in a well dropped 5cm each hour. The total drop in the water level was 30cm. Use integers to find out how long it took for the water level to change.


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7: A glacier retreated about 2m per day for 7 days. Use integers to find the total change in the length of the glacier.


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8: Winnie used the money in her savings account to pay back a loan from her Mother. Winnie paid back her Mother in 12 weekly payments. Over the 12 weeks, the balance in Winnie’s savings account decreased by $132. By how much did her balance change each week?


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9: The product of two integers is 20. The sum of the same two integers is 1. What are the two integers?


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October 16, 2015

10: Without evaluating the quotients, which one will have the least value? Explain.(1428) ÷ (+84)(+1428) ÷ (+84)(1428) ÷ (84)

Documents

Math 8 Unit 2 Integers NOTES (Blank Copy).notebookmrsjeddore.weebly.com/uploads/.../math8_unit2_notes... · Math 8_Unit 2 Integers_NOTES (Blank Copy).notebook 1 October 16, 2015 Unit