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Slide 2 Order of Operations By Ashley Bartkowiak Next Slide 3 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 Slide 4 Order of Operations Homepage Click on the yellow ovals to get to the desired location Previous Slide 5 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: 3 (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 Slide 6 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 3 is 9, (3 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. Slide 7 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 Slide 8 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 Slide 9 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 Slide 10 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 Slide 11 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 Slide 12 Order of Operations Example 2: Problem: 25 8 2 + 3 Solution: 25 8 2 + 9 25 16 + 9 9 + 9 18 The simplest name for the numerical expression 25 8 2 + 3 is 18. Parentheses: No parentheses Exponents: 3 = 33 = 9 Multiply/Divide (Left to Right) : 82 = 16 Add/Subtract (Left to Right) : First, 25 16 = 9 Then, 9 + 9 = 18 PreviousHomeNext Slide 13 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 Slide 14 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 Slide 15 Order of Operations Example 5: Problem: 9 + [4 (10 9)] Solution: 9 + (4 1) 9 + (4 1) 9 + 3 9 + 27 36 The simplest name for the numerical expression 9 + [4 (10 9)] 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 - 1) like a new expression. To simplify, follow the remaining steps (EMDAS). Exponents: 1 = 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 + 3. We can now move on to the step below. Exponents: 3 = 3 3 3 = 9 3 = 27 Multiply/Divide (Left to Right) : No multiplication or division Add/Subtract (Left to Right) : 9 + 27 = 36 PreviousHome Slide 16 Question 1 Given 3 6 4 2, what is the first step to simplifying the expression? a. b. c. 3 6 4 4 2 Home Next Question Slide 17 Try Again! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home Slide 18 Think Hard! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home Slide 19 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 4. Back to Question Next Question Home Slide 20 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 Slide 21 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 Slide 22 Try Again! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home Slide 23 Sorry! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home Slide 24 Question 3 Given |7 16| 8, what will the next step look like when simplifying the expression? a. b. c. |7| 2 49 16 8 |49 16| 8 Home Next Question Previous Question Slide 25 Think Again! Remember, absolute value signs are treated as grouping symbols. Back to Question Home Slide 26 Take Another Look! Did you do all of the operations inside of the grouping symbols before getting rid of them? Back to Question Home Slide 27 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 (7 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 7. Back to Question Next Question Home Slide 28 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 Slide 29 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 Slide 30 Not Quite! When you have more than one set of grouping symbols what do you do first? Back to Question Home Slide 31 That is Incorrect! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home Slide 32 Question 5 What is the simplest form of the expression 40 2 3? a. b. c. 22 342 28 Home Previous Question Slide 33 Keep Trying! Remember, 3 is the same as 3 3 Back to Question Home Slide 34 Great Job! Order of Operations Parentheses: No parentheses Exponents: 3 = 3 3 = 9 Multiply/Divide (Left to Right): 2 9 = 18 Add/Subtract (Left to Right): 40 18 = 22 Problem: 40 2 3 Solution: 40 2 9 40 18 22 Back to Question Home Slide 35 Dont Give Up! Think of PEMDAS! Parentheses Exponents Multiply/Divide Add/Subtract Back to Question Home Slide 36 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