6.3 SIMPLIFYING EXPRESSIONS USING THE ORDER OF OPERATIONS

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1 6.3 SIMPLIFYING EXPRESSIONS USING THE ORDER OF OPERATIONS Brian has decided to open a savings account, where he will earn 4% interest, compounded semi-annually. He will deposit $500 to open the account. He plans to leave the account alone for 1 year, making no additional deposits and no withdrawals. At the end of that year, he will withdraw only the interest he has earned. That amount can be determined by evaluating the following expression: $500 ( ) $500 Complete the withdrawal slip at right, filling in the amount Brian will withdraw from his account in one year. 0.0 Assess your readiness to complete this activity. Rate how well you understand: Not ready Almost ready Bring it on! how to apply the Order of Operations to simplify an expression with multiple operations the purpose for an agreed upon Order of Operations validation techniques for Order of Operations Simplifying expressions accuracy documentation of steps 65

2 Chapter 6 Signed Numbers, Exponents, and Order of Operations Order of Operations To simplify an expression, the process must follow the Order of Operations: First, simplify all operations within Parentheses. Then, simplify all factors and terms with Exponents. Then, compute Multiplication and Division, left to right as they occur in each term. Finally, compute Addition and Subtraction, left to right as they occur in the expression. While Example 1 is worked out, step by step, you are welcome to complete Example as a running problem. Space has been left for you to do precisely that. Example 1: Simplify: (11 8) Example : Simplify: (7 + ) Try It! Steps in the Methodology Example 1 Example Step 1 Identify the terms. Use brackets, [ ] or { }, to identify the terms of the expression. Recall that addition and subtraction signs separate the terms of an expression. [( 7 + ) ] + [ 5 9] [75 5 ]+[ 3 ] [4(11 8) ] [ 4 ] + [ 1 4] Special Case: The Expression is Written as a Fraction with an Expression in Either or Both the Numerator and Denominator (see Model 4) Step Simplify operations in parentheses. Simplify the operation(s) within Parentheses, if there are any, for each term. [75 5 ]+[ 3 ] [4(11 8) ] =[75 5 ]+[ 3 ] [4(3) ] P P [( 9) ] + [ 5 9] [ 4 ] + [ 1 4] As each term is simplified to one number, you may drop the brackets surrounding it. To ensure that you are doing the steps in the correct order of operations, it may be helpful to label each step as you compute it. 66

3 Activity 6.3 Simplifying Expressions Using the Order of Operations Steps in the Methodology Example 1 Example Step 3 Simplify numbers with exponents. Simplify the numbers with Exponents, if there are any, in each term. =[75 5 ]+[ 3 ] [4(3) ] 3 3 =[75 5 ]+ 8 [4 9] E E [ 81] + [ 5 9] [ 16] + [ 1 4] Step 4 Multiply and Divide left to right. Compute Multiplication and Division, left to right as they are situated in each term. =[75 5 ]+ 8 [4 9] =[15 ]+ 8 [4 9] =[30]+ 8 [36] M & D M&D [ 81] + [ 45] [ 16] + [ 3] Step 5 Add and Subtract left to right. Compute Addition and Subtraction of the simplified terms, left to right as they are situated in the expression. = = = A & S A&S Step 6 Present the answer. Present your final answer. Note: If the answer is in fraction form, reduce it. 113 Model 1 Simplify 4.6 (0.5) + (8 1.5) Validation: Step 1: Identify the terms: [4.6 ] [(0.5) ] + [ (8 1.5)] Step : P = [4.6 ] [(0.5) ] [ (6.5)] of Step = 8 Step 3: E = [4.6 ] [0.5] + [ (6.5)] Step 4: M & D = of Step 4.3 = = Step 5: A & S left to right as they occur = = of Step = =.30 Order: P, E, M&D, A&S Step 6: Answer:

4 Chapter 6 Signed Numbers, Exponents, and Order of Operations Model Simplify 5 ( 4) (6 1) 3 7 ( ) Validation: Step 1: Identify the terms: [5( 4)] [(6 1) 3 ] [7 ( )] Step : P = [5( 4)] [(5) 3 ] [7 ( )] of Step = 6 Step 3: E = [5( 4)] 15 [7 ( )] Step 4: M = 0 15 ( 14) of Step 4 0 ( 4) = ( ) = +7 Step 5: S change all subtraction to addition = 0 + ( 15) + (+14) = (+14) = 131 of Step ( 14) + 15 = = 0 Order: P, E, M, S Step 6: Answer: 131 Model 3 Simplify Validation: Step 1: Step : skip this step no operations inside Parentheses 68 Step 3: E = Step 4: M = of Step = = = = 5 6 Step 5: A & S, left to right as they occur = = 1 = Step 6: Answer: = = = + = = of Step = = = = 1 4 Order: E, M, A & S

5 Activity 6.3 Simplifying Expressions Using the Order of Operations Model 4 Simplify Special Case: The Expression is Written as a Fraction with an Expression in Either or Both the Numerator and Denominator 4( 7) The fraction bar indicates that the numerator and denominator are to be treated as two separate expressions. Simplify each expression separately, following the Order of Operations procedure; then reduce the resulting fraction, paying careful attention to the correct sign of the answer. Step 1: Step : P = Step 3: no Exponents, skip this step 0 6 Step 4: M = 10 1 = Step 5: S = 10 + ( ) + 1 change to addition 6 13 Validation: of Step = 3 + = 5 of Step 4 0 ( 5) = = of Step = = 1 + = 10 Order: P, M, S Step 6: Reduce: 6 13 =+ Answer: Validation and the Order of Operations When simplifying an expression with the Order of Operations, you can be fully confident in your answer only when you apply the correct order as well as do each computation accurately. Models 1 through 4 demonstrate that keeping track of steps by labeling each one as you go is an effective way to ensure that the order is correct. Validating the accuracy of each computation as you work through the problem (as demonstrated in the models) can further ensure the accuracy of your final answer. 69

6 Chapter 6 Signed Numbers, Exponents, and Order of Operations Make Your Own Model Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of. Answers will vary. Problem: 1. What is the order of operations to follow when simplifying an expression? PEMDAS: Perform operations in parenthesis fi rst; then do all exponents; then do all multiplications or divisions from left to right; then do all additions or subtractions, from left to right.. What is the purpose of the Order of Operations? It serves as a standard to evaluate an expression and to simplify arriving at an answer. 70

7 Activity 6.3 Simplifying Expressions Using the Order of Operations 3. How do you identify the terms of an expression? Give an example of an expression with four terms. The terms of an expression are separated by addition (+) and subtraction ( ) signs. Every expression has at least one term. 4. When computing a series of multiplication and division operations within a single term, in what order must they be done? Multiplications and divisions must be done in order from left to right as they come in order. Divisions are done fi rst if they are the fi rst operations beginning at the left of the expression to be evaluated. 5. In the final step of the Order of Operations, why must addition and subtraction be computed from left to right as they occur in the expression? The operation which comes fi rst when working from left to right is what is to be done fi rst. It could be the subtraction that is done fi rst. This is done this way because that is the way it is stated in the Order of Operations. 6. What is a strategy you can use to validate an order of operations problem? It is easier to validate each step as you do it. 7. Why do you think the Exponents must be computed before the Multiplication and Division step is computed? This is universally accepted as the Order of Operations. Probably parentheses and exponents are done fi rst because they are more complicated to work out and more than one process is usually involved. 8. What aspect of the model you created is the most difficult to explain to someone else? Explain why. Answers will vary. 71

8 Chapter 6 Signed Numbers, Exponents, and Order of Operations Simplify each of the following expressions: Expression Validation (optional) 1) () ) ) ) (5. 1.3) + (.) 1 5) (7.1) ( ) + (0.) 1.3 7

9 Activity 6.3 Simplifying Expressions Using the Order of Operations Expression Validation (optional) 6) ) ) ) ( 1) 10 Simplify each of the following expressions: (5 9) 5 (3 6) 7 3. (0.) ( ) (.1) ( ) ( 7)

10 Chapter 6 Signed Numbers, Exponents, and Order of Operations Identify the error(s) in the following worked solutions. If the worked solution is correct, write Correct in the second column. If the worked solution is incorrect, solve the problem correctly in the third column. You can validate your work in the fourth column. Worked Solution What is Wrong Here? Identify the Errors Correct Process Validation (optional) 1) Simplify: 3 (5 + 4) Did not follow order of operations P, E, M&D, A&S. Added and 3 before multiplying. The two terms are and 3(5+4). -3(5+4) =-[3(5+4)] =-[3(9)] P = - 7 M =+(-7) A =-5 Answer: 5 Order P, M, A 9-4=5 7 9 = = + ) Simplify: 4 ( 3) ( 4) Work all multiplications OR divisions working from left to right. 3) Simplify: ( 4)+ Perform order of operations PEMDAS. Exponents should be done first. 74

11 Activity 6.3 Simplifying Expressions Using the Order of Operations Worked Solution What is Wrong Here? Identify the Errors Correct Process Validation (optional) 4) Simplify: (0.04) (15 4.7) The decimal point is placed incorrectly in the product of (.04). Line up the decimal points and trailing zeros when subtracting 4.7 from 15. 5) Simplify: ( 3)+ 5( 6) There needs to be a single sign in the final answer. 75