Adding Rules:

1.

Adding Rules:

Positive + Positive = Positive: 5 + 4 = 9 Negative + Negative = Negative: (- 7) + (- 2) = - 9 Sum of a negative and a positive number: Use the sign of the larger number and subtract (- 7) + 4 = -3 6 + (-9) = - 3 (- 3) + 7 = 4 5 + ( -3) = 2

2.

Subtracting Rules:

Negative - Positive = Negative: (- 5) - 3 = -5 + (-3) = -8 Positive - Negative = Positive + Positive = Positive: 5 - (-3) = 5 + 3 = 8 Negative - Negative = Negative + Positive = Use the sign of the larger number and subtract (Change double negatives to a positive) (-5) - (-3) = ( -5) + 3 = -2 (-3) - ( -5) = (-3) + 5 = 2

3.

Multiplying Rules:

Positive x Positive = Positive: 3 x 2 = 6 Negative x Negative = Positive: (-2) x (-8) = 16 Negative x Positive = Negative: (-3) x 4 = -12 Positive x Negative = Negative: 3 x (-4) = -12

4.

Dividing Rules:

Positive ÷ Positive = Positive: 12 ÷ 3 = 4 Negative ÷ Negative = Positive: (-12) ÷ (-3) = 4 Negative ÷ Positive = Negative: (-12) ÷ 3 = -4 Positive ÷ Negative = Negative: 12 ÷ (-3) = -4

Tips: 1. When working with rules for positive and negative numbers, try and think of weight loss or poker games to help solidify 'what this works'. 2. Using a number line showing both sides of 0 is very helpful to help develop the understanding of working with positive and negative numbers/integers.

This tutorial is designed to help you solve problems correctly by using the 'Order of Operations'. When there is more than one operation involved in a mathematical problem, it must be solved by using the correct order of operations. A number of teachers use acronyms with their students to help them to retain the order. Remember, calculators will perform operations in the order which you enter them, therefore, you will need to enter the operations in the correct order for the calculator to give you the right answer. * In Mathematics, the order in which mathematical problems are solved is extremely important.

Rules 1. Calculations must be done from left to right. 2. Calculations in brackets (parenthesis) are done first. When you have more than one set of brackets, do the inner brackets first. 3. Exponents (or radicals) must be done next. 4. Multiply and divide in the order the operations occur. 5. Add and subtract in the order the operations occur.

Remember to: • Simplify inside groupings of parentheses, brackets and braces first. Work with the innermost pair, moving outward. • Simplify the exponents. • Do the multiplication and division in order from left to right. • Do the addition and subtraction in order from left to right.

Acronyms to Help you Remember How will you remember this order? Try the following Acronyms: Please Excuse My Dear Aunt Sally (Parenthesis, Exponents, Multiply, Divide, Add, Subtract) BEDMAS (Brackets, Exponents, Divide, Multiply, Add, Subtract) Big Elephants Destroy Mice And Snails (Brackets, Exponents, Divide, Multiply, Add,

Subtract) Pink Elephants Destroy Mice And Snails (Parenthesis, Exponents, Divide, Multiply,Add, Subtract) Examples 12 ÷ 4 + 32 12 ÷ 4 + 9 3+9 12

(42 + 5) - 3 21 - 3 18 20 ÷ (12 - 2) X 32 - 2 20 ÷ 10 X 32 2 20 ÷ 10 X 9 - 2 18 - 2 16

Rule Rule they Rule they

3: Exponent first 4: Multiply or Divide as appear 5: Add or Subtract as appear

Rule 2: Everything in the brackets first Rule 5: Add or Subtract as they appear Rule 2: Everything in the brackets first Rule 3: Exponents Rule 4: Multiply and Divide as they appear Rule 5: Add or Subtract as they appear

Does It Make a Difference? What If I Don't Use the Order of Operations? Mathematicians were very careful when they developed the order of operations. Without the correct order, watch what happens: 15 + 5 X 10 -- Without following the correct order, I know that 15+5=20 multiplied by 10 gives me the answer of 200. 15 + 5 X 10 -- Following the order of operations, I know that 5X10 = 50 plus 15 = 65. This is the correct answer, the above is not! You can see that it is absolutely critical to follow the order of operations. Some of the most frequent errors students make occur when they do not follow the order of operations when solving mathematical problems. Students can often be fluent in computational work yet do not follow procedures. Use the handy acronyms to ensure that you never make this mistake again.

Example 1:

Evaluate each expression using the rules for order of operations.

Solution:

Order of Operations Expression Evaluation

Operation

6+7x8

=6+7x8

Multiplication

= 6 + 56

Addition

= 62 16 ÷ 8 - 2

= 16 ÷ 8 - 2

Division

=2-2

Subtraction

=0 (25 - 11) x 3 = (25 - 11) x 3 Parentheses = 14 x 3

Multiplication

= 42 In Example 1, each problem involved only 2 operations. Let's look at some examples that involve more than two operations. Example 2:

Evaluate 3 + 6 x (5 + 4) ÷ 3 - 7 using the order of operations.

Solution:

Step 1: Step 2: Step 3: Step 4: Step 5:

Example 3:

Evaluate 9 - 5 ÷ (8 - 3) x 2 + 6 using the order of operations.

Solution:

Step 1: Step 2: Step 3: Step 4: Step 5:

3 + 6 x (5 + 4) ÷ 3 - 7 3+6x9÷3-7 3 + 54 ÷ 3 - 7 3 + 18 - 7 21 - 7

9 - 5 ÷ (8 - 3) x 2 + 6 9-5÷5x2+6 9-1x2+6 9-2+6 7+6

= = = = =

= = = = =

3+6x9÷3-7 3 + 54 ÷ 3 - 7 3 + 18 - 7 21 - 7 14

9-5÷5x2+6 9-1x2+6 9-2+6 7+6 13

Parentheses Multiplication Division Addition Subtraction

Parentheses Division Multiplication Subtraction Addition

In Examples 2 and 3, you will notice that multiplication and division were evaluated from left to right according to Rule 2. Similarly, addition and subtraction were evaluated from left to right, according to Rule 3. When two or more operations occur inside a set of parentheses, these operations should be evaluated according to Rules 2 and 3. This is done in Example 4 below. Example 4:

Evaluate 150 ÷ (6 + 3 x 8) - 5 using the order of operations.

Solution:

Step 1: Step 2:

150 ÷ (6 + 3 x 8) - 5 = 150 ÷ (6 + 24) - 5 Multiplication inside Parentheses 150 ÷ (6 + 24) - 5 = 150 ÷ 30 - 5 Addition inside Parentheses

Step 3: Step 4:

150 ÷ 30 - 5 5-5

= 5-5 = 0

Division Subtraction

Example 5:

Evaluate the arithmetic expression below:

Solution:

This problem includes a fraction bar (also called a vinculum), which means we must divide the numerator by the denominator. However, we must first perform all calculations above and below the fraction bar BEFORE dividing. Thus Evaluating this expression, we get:

Example 6:

Write an arithmetic expression for this problem. Then evaluate the expression using the order of operations. Mr. Smith charged Jill $32 for parts and $15 per hour for labor to repair her bicycle. If he spent 3 hours repairing her bike, how much does Jill owe him?

Solution:

32 + 3 x 15 = 32 + 3 x 15

= 32 + 45 = 77

Jill owes Mr. Smith $77. Summary:

When evaluating arithmetic expressions, the order of operations is: • •

Simplify all operations inside parentheses. Perform all multiplications and divisions, working from left to right.

•

Perform all additions and subtractions, working from left to right.

If a problem includes a fraction bar, perform all calculations above and below the fraction bar before dividing the numerator by the denominator.

Exponent Problem:

Evaluate this arithmetic expression: 18 + 36 ÷ 32

In the last lesson, we learned how to evaluate an arithmetic expression with more than one operation according to the following rules: Rule 1: Rule 2: Rule 3:

Simplify all operations inside parentheses. Perform all multiplications and divisions, working from left to right. Perform all additions and subtractions, working from left to right.

However, the problem above includes an exponent, so we cannot solve it without revising our rules. Rule 1: Rule 2: Rule 3: Rule 4:

Simplify all operations inside parentheses. Simplify all exponents, working from left to right. Perform all multiplications and divisions, working from left to right. Perform all additions and subtractions, working from left to right.

We can solve the problem above using our revised order of operations. Problem: Solution:

Evaluate this arithmetic expression 18 + 36 ÷ 32 18 + 36 ÷ 32 = 18 + 36 ÷ 9 Simplify all exponents (Rule 2) 18 + 36 ÷ 9

= 18 + 4

Division (Rule 3)

18 + 4

= 22

Addition (Rule 4)

Let's look at some other examples that involve our new rules for order of operations. Example 1: Solution:

Evaluate 52 x 24 52 x 24 25 x 24

= 25 x 24 Simplify all exponents, working from left to right (Rule 2) = 25 x 16

25 x 16 = 400 Example 2: Solution:

Example 3:

Multiplication (Rule 3)

Evaluate 289 - (3 x 5)2 289 - (3 x 5)2 = 289 - 152

Simplify all operations inside parentheses (Rule 1)

289 - 152

= 289 - 225 Simplify all exponents (Rule 2)

289 - 225

= 64

Evaluate 8 + (2 x 5) x 34 ÷ 9

Subtraction (Rule 4)

Solution:

8 + (2 x 5) x 34 ÷ 9 = 8 + 10 x 34 ÷ 9

Simplify all operations inside parentheses (Rule 1)

8 + 10 x 34 ÷ 9

= 8 + 10 x 81 ÷ 9 Simplify all exponents (Rule 2)

8 + 10 x 81 ÷ 9

= 8 + 810 ÷ 9

8 + 810 ÷ 9

= 8 + 90

8 + 90

= 98

Perform all multiplications and divisions, working from left to right (Rule 3) Addition (Rule 4)

Example 4:

An interior decorator charges $15 per square foot to lay a carpet, and an installation fee of $150. If the room is square and each side measures 12 feet, how much will it cost to carpet it?

Solution:

If one side of the square-shaped room is 12 feet, then the area of the room is (12 feet)2. 15 x 122 + 150

Answer:

= 15 x 144 + 150 Simplify all exponents (Rule 2)

15 x 144 + 150 = 2,160 + 150

Multiplication (Rule 3)

2,160 + 150

Addition (Rule 4)

= 2,310

It will cost $2,310 to carpet this room.

Writing Algebraic Equations Problem:

Solution:

Answer:

Jeanne has $17 in her piggy bank. How much money does she need to buy a game that costs $68? Let x represent the amount of money Jeanne needs. Then the following equation can represent this problem: 17 + x = 68 We can subtract 17 from both sides of the equation to find the value of x. 68 - 17 = x x = 51, so Jeanne needs $51 to buy the game.

In the problem above, x is a variable. The symbols 17 + x = 68 form an algebraic equation. Let's look at some examples of writing algebraic equations. Example 1: Write each sentence as an algebraic equation.

Sentence

Algebraic Equation

A number increased by nine is fifteen. y + 9 = 15 Twice a number is eighteen.

2n = 18

Four less than a number is twenty.

x - 4 = 20

A number divided by six is eight. Example 2: Write each sentence as an algebraic equation. Sentence

Algebraic Equation

Twice a number, decreased by twenty-nine, is seven.

2t - 29 = 7

Thirty-two is twice a number increased by eight.

32 = 2a + 8

The quotient of fifty and five more than a number is ten. Twelve is sixteen less than four times a number.

12 = 4x - 16

Example 3: Write each sentence as an algebraic equation. Sentence

Algebraic Equation

Eleni is x years old. In thirteen years she will be twenty-four years old.

x + 13 = 24

Each piece of candy costs 25 cents. The price of h pieces of candy is $2.00.

25h = 200 or .25h = 2.00

Suzanne made a withdrawal of d dollars 350 - d = 280 from her savings account. Her old balance was $350, and her new balance is $280. A large pizza pie with 15 slices is shared among p students so that each student's share is 3 slices. Summary:

An algebraic equation is an equation that includes one or more variables. In this lesson, we learned how to write a sentence as an algebraic equation.