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Order of Operations with Exponentsand Integers

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Number Sense and Numeration and Algebra

Order of Operations with Exponents and Integers

Listen & LearnPRESENTED BY MARSHALLMathematics, Grade 8


Introduction

• Welcome to today’s topic

• Parts of Listen & Learn

Presentation, Q&A

• Housekeeping

Your questions

Satisfaction meter

Downloading slides


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What you will learn

After viewing this presentation, youwill be able to

• apply the Order of Operationsrules (BEDMAS) to correctlysimplify an expression thatcontains exponents andintegers


Agenda

• The importance of Order of Operations

• Review of concepts

• Order of Operations

Evaluating expressions with exponents

Evaluating expressions with integers

Evaluating expressions with exponentsand integers

• Recap

• Putting it all together


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Importance

BEDMAS is used in all mathcalculations involving morethan one operation

Figure 1

Mathematicians agreed on a specificway to simplify expressions so thateveryone gets the same answer.Math problems with 2 or moredifferent operations must be solvedwith BEDMAS. The Order ofOperations with BEDMAS rules havemany real-life applications such as:

• building a fence, a house or almost any other object

• a manager ordering food for a restaurant based on the number of visitors expected for the day

• balancing your bank account


Importance

Real-life Application

John works for a company that pays himonce every 2 weeks. His paychequealways equals $1003. Within that 2-weektime period, John has many expenses tocover.


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Importance

Expenses over 2 weeks

Clothing $234Gas $56 for each fill (he fills up 3 times)

Groceries $170Savings 10% of paycheque put into savings account

Rent $400 (for 2 weeks)Cellphone $35 (for 2 weeks)

How much does John have left over?


Importance

Real-life Application

Developing the equation:

$1,003 John’s income

– $234 Subtract clothing expenses

– ($56 x 3) Subtract gas (must be multiplied by 3)

($56 x 3 = $168)– $170 Subtract groceries

– ($1003 x 0.10) Subtract how much goes into savings

($1003 x 0.10 = $103)– $400 Subtract rent

– $35 Subtract cellphone bill


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Importance

What this equation looks like:$1003 - $234 – ($56 x 3) – $170 – ($1003 x .10) – $400 – $35

BEDMAS: $1003 – $234 – ($168) – $170 – ($103) – $400 – $35BEDMAS: No exponentsBEDMAS: No further division or multiplicationBEDMAS: $1003 – $234 – $168 – $170 – $103 – $400 – $35

BEDMAS: – $107

John does not have enough money to coverhis expenses.


Agenda







• Recap



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Review of ConceptsBEDMAS MnemonicB

E

D

M

A

S

Brackets

Exponents

Division

Multiplication

Addition

Subtraction


Review of ConceptsRules according to BEDMASB

E

D

M

A

S

Do all operations inside brackets first.

Evaluate all exponents.

Divide and

multiply, in order from left to right.

Add and

Subtract, in order from left to right.


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Review of ConceptsWorking with Exponents

62Let’s review how to solve an exponent.Example:


Review of ConceptsExponents

The parts of a power are thebase and the exponent.

base

exponent

62


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Simplify the power:

62

The base is multiplied by ITSELF the number oftimes noted in the exponent.

Therefore to simplify this power you need to multiply6 x 6

NOT 6 x 2

Therefore 62 = 36

Review of ConceptsSolving the Exponent


Review of ConceptsSolving IntegersWhen simplifying expressions youmay encounter integers within theexpression.

You are encouraged to have acalculator with a +/– button on itto assist you with integers ifnecessary.


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Review of ConceptsIntegersPositive Integers are whole numbers suchas +17, +2, +1. They are numbers thatare greater than 0.

Negative Integers are whole numberssuch as -5, –28, –1. They are numbersthat are less than 0.

Any positive integer is greater than anynegative integer. For example, +20 isgreater than –700.


Review of ConceptsAdding IntegersTo add a positive integer, move right on anumber line. (–3) + (+6)Start at –3 on a number line and move 6 units to the right.

To add a negative integer, move left onthe number line. (–1) + (–2)Start at –1 on the number line and move 2 units to the left.


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Review of ConceptsSubtracting IntegersThere is a trick for subtracting integers.

Think about subtracting one wholenumber from another (such as 3 from 9).Instead of just subtracting the numbersfrom each other, ask yourself “What can Iadd to get from 3 to 9?” The answer is 6.Therefore 9 – 3 = 6.

To subtract an integer from another (suchas (–3) – (+9)), ask “What can I add to+9 to get to –3?”


Review of ConceptsSubtracting IntegersSubtracting an integer is the sameas adding the opposite integer.(–6) – (+7) = –13

Or

(–6) + (–7) = –13


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Review of ConceptsMultiplying IntegersPositive x Positive = Positive

(+3) x (+3) = +9

Negative x Negative = Positive

(–4) x (–4) = +16

Positive x Negative = Negative

(+5) x (–3) = –15

Negative x Positive = Negative

(–6) x (+3) = –18


Review of ConceptsDividing IntegersPositive / Positive = Positive

(+15) / (+3) = +5

Negative / Negative = Positive

(–16) / (–4) = +4

Positive / Negative = Negative

(+15) / (–3) = –5

Negative / Positive = Negative

(–20) / (+5) = –4


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Review of ConceptsThe Understood OperationIn mathematics, we sometimes usebrackets instead of a multiplicationsign.45 + 21 (36) – 17

Multiplication is the understood operation here.


Agenda







• Recap



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Evaluating Expressions withExponentsLet’s first look at an expressionthat contains an exponent.

(6 + 10.8) + 10.4 x 102

Recall: BEDMAS

Exponent


Evaluating Expressions withExponents(6 + 10.8) + 10.4 x 102

Simplify: BEDMAS – Brackets= 16.8 + 10.4 x 102


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Simplify: BEDMAS – Exponents= 16.8 + 10.4 x 102

= 16.8 + 10.4 x 100





= 16.8 + 10.4 x 100

Simplify: BEDMAS – Division & Multiplication= 16.8 + 10.4 x 100= 16.8 + 1040


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= 16.8 + 10.4 x 100

Simplify: BEDMAS – Division & Multiplication= 16.8 + 10.4 x 100= 16.8 + 1040

Simplify: BEDMAS – Addition & Subtraction= 16.8 + 1040= 1056.8



= 16.8 + 10.4 x 102

= 16.8 + 10.4 x 100= 16.8 + 1040= 1056.8


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Evaluating Expressions withIntegersLet’s look at an expression that contains aninteger.

200 + 6 [40/(–4)]Since we use curved brackets for integers, forexample (–4), we will use square brackets togroup terms together.

To keep our expressions “clean” we do not addthe (+) positive sign in front of positivenumbers. We write them as whole numbers.

Integer


Evaluating Expressions withIntegers200 + 6[40/(–4)]

Simplify: BEDMAS – Brackets

= 200 + 6[40/(–4)]

= 200 + 6(–10)

Note: The BEDMAS bracket is signified by squarebrackets. The curved bracket used to indicate anegative integer does not have an operation toperform inside the bracket. Integer operation issolved just like any question involving an integer.


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Evaluating Expressions withIntegers200 + 6 [40 / (–4)]Simplify: BEDMAS – Brackets (with integer)= 200 + 6 [40 / (–4)]= 200 + 6 (–10)

Simplify: BEDMAS – ExponentsThere are no exponents to solve.

Simplify: BEDMAS – Division & Multiplication= 200 + 6 (–10)= 200 + (–60)

Simplify: BEDMAS – Addition & Subtraction= 200 + (–60)= 140

Integer multiplication

Integer division

Integer addition


Evaluating Expressions withExponents and IntegersEvaluate an expression that containsexponents and integers.

(4) x (–10) – 10

(–5)2


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Evaluating Expressions withExponents and IntegersEvaluate an expression thatcontains integers and exponents.

(4) x (–10) – 10

(–5)2

This expression is the same as

[(4) x (–10) – 10] ÷ (–5)2

So you would work on the expression inthe brackets first (numerator, thendenominator).


Evaluating Expressions withExponents and IntegersEvaluate:

(4) x (–10) – 10

(–5)2

Work through the numerator (the top sectionof the fraction) first!

(4) x (–10) – 10

= (–40) – 10

= –50


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Evaluating Expressions withExponents and IntegersEvaluate:

–50

(–5)2

Now that we have solved the numerator,let’s solve the denominator.

(–5)2

Remember this is (–5) x (–5)= 25


Evaluating Expressions withExponents and IntegersOur expression simplified:

–50

25

The last part is finally completing thedivision:

–50/25

= –2


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Evaluating Expressions withExponents and Integers

Write an expression for the followingstatements and then evaluate eachexpression:

a) Multiply 5 and 2, then subtract 32

b) Multiply the sum of –8 and –16 by –2.


Evaluating Expressions withExponents and Integersa) Multiply 5 and 2, then subtract 32


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Evaluating Expressions withExponents and Integersb) Multiply the sum of –8 and –16

by –2


Agenda







• Recap



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Recap

When there is no operation symbolbetween numbers and one numberis in brackets, it is understood thatyou perform:

a) addition

b) division

c) multiplication

d) subtraction


Recap

When there is no operation symbolbetween numbers and one numberis in brackets, it is understood thatyou perform:

a) addition

b) division

c) multiplication

d) subtraction


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Recap

Multiplication is always used whenthere is no symbol between thenumbers in an expression and onenumber is in brackets.


Recap

(3 x 2)2 – 10 + 9

–7

a) 5

b) –5

c) 35

d) –245


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Recap

(3 x 2)2 – 10 + 9

–7

a) 5

b) –5

c) 35

d) –245


Recap

(3 x 2)2 – 10 + 9 –7

= (6)2 – 10 + 9 –7

= 36 – 10 + 9 –7

= 26 + 9 –7

= 35 –7

= –5 (Solution)

Numerator first BEDMAS(Bracket)

BEDMAS (Exponent)

BEDMAS (Nomultiplication/divisionin numerator)

BEDMAS(Addition/Subtraction)

BEDMAS(Addition/Subtraction)

Denominator – dividenumerator by denominator


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Agenda







• Recap



Let’s Put It All Together!

(5 – 3)2 x (–4) – 10 + 18 / 6 = ?

a) –23

b) 0

c) 9

d) –5


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(5 – 3)2 x (–4) – 10 + 18 / 6 = ?

a) –23

b) 0

c) 9

d) –5



(5 – 3)2 x (–4) – 10 + 18 / 6

= (2)2 x (–4) – 10 + 18 / 6

= 4 x (–4) – 10 + 18 / 6

= (–16) – 10 + 3

= (–26) + 3

= –23

BEDMAS

BEDMAS

BEDMAS

BEDMAS


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Resources

OERB: Ontario Educational Resource Bankhttp://resources.elearningontario.ca• Order of Operations 1: Grade 6, 7,8 Math Resource Bank ID: ELO1020530

• Exponents Grade 8 Math Resource Bank ID: ELO1033350

• Integers…Positive or Negative? Grade 8 Tutorial Resource Bank ID: ELO1033460

Math Goodieshttp://www.mathgoodies.com/lessons/vol7/order_operations.html(Note: this website uses PEDMAS, the US version of BEDMAS. “P”stands for “parentheses,” which means the same as brackets.)

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