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Order of Operations By Ashley Bartkowiak Next

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Lesson Overview Objective: to use the order of operations to simplify numerical expressions Why do we need to learn about the order of operations?: Like most things in life, we have a set of rules in mathematics that we must follow so that everyone behaves in the same way. Much like a classroom with no rules, without the order of operations, mathematics problem-solving becomes a bit chaotic; one problem could have three different answers. Thus, the order of operations have been established to keep everyone on the same page when simplifying expressions. NextPrevious

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Order of Operations Homepage Click on the yellow ovals to get to the desired location Previous

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Vocabulary Terms Numerical ExpressionNumerical Expression: a mathematical phrase that includes numbers and operational symbols Example: 3x+2 OperationOperation: a process or action which produces a new value Examples: multiplication, division, subtraction, addition PowerPower: the base and the exponent of an expression Example: 3² (3 is the base and 2 is the exponent) *Click on the vocabulary word to take you back to the page with that term. Home

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Simplifying Expressions To simplify a numerical expression, you replace it with its simplest name.numerical expression » Example 1: The simplest name for the expression 2+3 is 5 (we add 2 and 3 to get a sum of 5). » Example 2: The simplest name for the expression 3² is 9, (3² is the same thing as 3 × 3 and when we multiply 3 and 3 we get a product of 9). HomeNext *Click on the vocabulary term to take you to the definition.

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Lets take a look at the expression 2 + 8 – 6 ÷ 2. John and Sarah both simplified the expression, but they got two different answers. Although their work is different, they are both convinced that they are correct. Johns Work: Sarahs Work: 2 + 8 – 6 ÷ 2 10 – 6 ÷ 2 2 + 8 – 3 4 ÷ 2 10 – 3 2 7 Different Answers HomePreviousNext

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To avoid getting different answers when simplifying the same expression (like with John and Sarah), mathematicians have agreed on an order for doing operations. There are essentially 4 steps that we follow to simplify expressions.operations Order of Operations Step 1: Perform any operation(s) inside grouping symbols (grouping symbols include parentheses ( ), brackets [ ], or absolute value signs||). Note: If there are more than one set of grouping symbols, then start with the innermost set. Step 2: Simplify powers (get rid of the exponents).powers Step 3: Multiply and divide in order from left to right. Step 4: Add and Subtract in order from left to right. PreviousHomeNext

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Helpful Tips A common technique for remembering the order of operations is to simplify each step to one word and use the abbreviation PEMDAS. Parentheses Exponents Multiply Divide Add Subtract You can remember PEMDAS by thinking of the phrase Please Excuse My Dear Aunt Sally. Left to Right PreviousHomeNext

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Order of Operations Going back to our original expression 2 + 8 – 6 ÷ 2, lets use the order of operations to find out who (John or Sarah) found the correct answer.John or Sarah Step 1: There are no parentheses or grouping symbols, so we skip this step and go on to the next. Step 2: There are no exponents, so we skip this step. Step 3: We now multiply/divide from left to right. This means that the first thing we need to do is 6 ÷ 2, which gives us an answer of 3. So, when we re-write the expression, we replace 6 ÷ 2 with 3 to get 2 + 8 – 3. Step 4: We now add/subtract from left to right. So, we are going to start by doing the computation 2 + 8, which gives us 10. If we re-write the expression we now have 10 – 3. When we subtract, we get a final answer of 7. Since we ended up with a final answer of 7, Sarah was correct. Parentheses Exponents Multiply/Divide Add/Subtract PreviousHome

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Order of Operations Example 1: Problem: (17 – 7) ÷ 5 + 1 Solution: 10 ÷ 5 + 1 2 + 1 3 The simplest name for the numerical expression (17 – 7) ÷ 5 + 1 is 3. Parentheses: 17 – 7 = 10 Exponents: No exponents Multiply/Divide (Left to Right) : 10 ÷ 5 = 2 Add/Subtract (Left to Right) : 2 + 1 = 3 HomeNext

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Order of Operations Example 2: Problem: 25 – 8 × 2 + 3² Solution: 25 – 8 × 2 + 9 25 – 16 + 9 9 + 9 18 The simplest name for the numerical expression 25 – 8 × 2 + 3² is 18. Parentheses: No parentheses Exponents: 3² = 3×3 = 9 Multiply/Divide (Left to Right) : 8×2 = 16 Add/Subtract (Left to Right) : First, 25 – 16 = 9 Then, 9 + 9 = 18 PreviousHomeNext

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Order of Operations Example 3: Problem: 3 + |1 – 2| Solution: 3 + 1 4 The simplest name for the numerical expression 3 + |1 – 2| is 4. Parentheses: The absolute value signs are treated as grouping symbols, like parentheses. 1 – 2 = -1 and the absolute value of -1 is 1. Exponents: No exponents Multiply/Divide (Left to Right) : No multiplication or division Add/Subtract (Left to Right) : 3 + 1 = 4 PreviousHomeNext

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Order of Operations Example 4: Problem: 6[13 – 2 (4 + 1)] Solution: 6(13 – 2 × 5) 6 (13 – 10) 6 × 3 18 The simplest name for the numerical expression 6[13 – 2 (4 + 1)] is 18. Parentheses: There are two sets of grouping symbols, so we start with the innermost 4 + 1 = 5. Then, go to the second set of parentheses. Treat what is inside (13 – 2 × 5) like a new expression. To simplify, follow the remaining steps (EMDAS). Exponents: There are no exponents Multiply/Divide: 2 × 5 = 10. Add/Subtract: 13 – 10 = 3. We have taken care of what is in the parentheses and are left with the remaining expression 6 × 3. We can now move on to the step below. Exponents: No exponents in 6 × 3 Multiply/Divide (Left to Right) : 6 × 3 = 18 Add/Subtract (Left to Right) : No addition or subtraction PreviousHomeNext

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Order of Operations Example 5: Problem: 9 + [4 – (10 – 9)²]³ Solution: 9 + (4 – 1²)³ 9 + (4 – 1)³ 9 + 3³ 9 + 27 36 The simplest name for the numerical expression 9 + [4 – (10 – 9)²]³ is 36. Parentheses: There are two sets of grouping symbols, so we start with the innermost 10 – 9 = 1. Then, go to the second set of parentheses. Treat what is inside (4 - 1²) like a new expression. To simplify, follow the remaining steps (EMDAS). Exponents: 1² = 1 Multiply/Divide: No multiplication or division Add/Subtract: 4 – 1 = 3 We have taken care of what is in the parentheses and are left with the remaining expression 9 + 3³. We can now move on to the step below. Exponents: 3³ = 3 × 3 × 3 = 9 × 3 = 27 Multiply/Divide (Left to Right) : No multiplication or division Add/Subtract (Left to Right) : 9 + 27 = 36 PreviousHome

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Question 1 Given 3 × 6 – 4² ÷ 2, what is the first step to simplifying the expression? a. b. c. 3 × 6 4² 4² ÷ 2 Home Next Question

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Try Again! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home

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Think Hard! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home

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Way to go! Following the order of operations (PEMDAS), parentheses come first. However, there are no parentheses in the given expression, so we move on to exponents. That means that we need to start by simplifying 4². Back to Question Next Question Home

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Question 2 Given 2(5 + 9) – 6, what is the first step to simplifying the expression? a. b. c. 5 + 9 2 × 5 9 – 6 Home Next Question Previous Question

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That is correct! Following the order of operations (PEMDAS), parentheses come first. That means that we need to start by simplifying 5 + 9. Next Question Back to Question Home

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Try Again! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home

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Sorry! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home

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Question 3 Given |7² – 16| ÷ 8, what will the next step look like when simplifying the expression? a. b. c. |7²| – 2 49 – 16 ÷ 8 |49 – 16| ÷ 8 Home Next Question Previous Question

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Think Again! Remember, absolute value signs are treated as grouping symbols. Back to Question Home

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Take Another Look! Did you do all of the operations inside of the grouping symbols before getting rid of them? Back to Question Home

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You Did It! Following the order of operations (PEMDAS), parentheses come first. That means that we need to start by looking at the expression inside of the parentheses (7² – 16). If we think of this as a new expression, we follow the steps following the parentheses (EMDAS). This means that we start by simplifying the power 7². Back to Question Next Question Home

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Question 4 Given [3(7 + 4) – 2]6, what will you need to do first to simplify the expression? a. b. c. Simplify what is in the brackets [ ] Multiply 11 and 3 Simplify what is in the parentheses ( ) Home Next Question Previous Question

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Nice Work! Following the order of operations (PEMDAS), parentheses come first. Since we have two sets of parentheses, we simplify the expression in the innermost set first. That means that we would start with 7 + 4. Back to Question Next Question Home

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Not Quite! When you have more than one set of grouping symbols what do you do first? Back to Question Home

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That is Incorrect! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home

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Question 5 What is the simplest form of the expression 40 – 2 × 3²? a. b. c. 22 342 28 Home Previous Question

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Keep Trying! Remember, 3² is the same as 3 × 3 Back to Question Home

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Great Job! Order of Operations Parentheses: No parentheses Exponents: 3² = 3 × 3 = 9 Multiply/Divide (Left to Right): 2 × 9 = 18 Add/Subtract (Left to Right): 40 – 18 = 22 Problem: 40 – 2 × 3² Solution: 40 – 2 × 9 40 – 18 22 Back to Question Home

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Dont Give Up! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home

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Additional Resources and Tutorials http://www.wisc-online.com – Instruction as well as practice problems are provided for extra help http://www.wisc-online.com http://www.learnalberta.ca – Test your knowledge of the steps that you need to follow to simplify an expression http://www.learnalberta.ca http://www.funbrain.com – For those that are up for a challenge, see if you can arrange these numbers to get the answer shown using the order of operations http://www.funbrain.com http://www.hippocampus.org – Click on Algebra then, Number Operations and follow along with an interactive tutorial of the order of operations http://www.hippocampus.org http://www.mathgoodies.com – Read about and practice using the order of operations to simplify expressions http://www.mathgoodies.com http://www.free-training-tutorial.com – Test your skills with several of these different games http://www.free-training-tutorial.com Home

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