Evaluating Expressions Using the Order of Operations

Transcription

1 Section 6. PRE-ACTIVITY PREPARATION Evaluating Expressions Using the Order of Operations Sales of admission tickets to the family concert were as follows: 50 adult tickets sold for $5 each, 00 youth tickets sold for $ each, and 50 senior citizen tickets sold for $4 each. What was the total of ticket sales in dollars? To answer the question, you would simply calculate the revenue from each type of ticket sold (by multiplying) and then add the three results: 50 $5 or $50 from the sale of adult tickets, 00 $ or $900 from youth tickets, and 50 $4 or $00 from senior citizen sales add up to $150. In other words, you would follow a sequence of operational steps to arrive at your answer. As you progress in your knowledge of mathematics, you will find that the processes you encounter most frequently will be the basic operations of addition, subtraction, multiplication, division, and the evaluation of exponential expressions. Your success in mathematics builds on your mastery of these basic skills as well as your ability to apply them in the correct order when several appear in the same mathematical expression. LEARNING OBJECTIVE Evaluate expressions using the correct order of operations. TERMINOLOGY PREVIOUSLY USED evaluate exponent expression operation simplify term NEW TERMS TO LEARN order of operations BUILDING MATHEMATICAL LANGUAGE In mathematical language, you can answer the question in the introduction about ticket sales by setting up and evaluating an expression: 50 $ $ + 50 $4 = $50 + $900 + $00 = $150 58

2 584 Chapter 6 Signed Numbers, Exponents, and Order of Operations That is, you follow a sequence of operational steps, multiplying before adding the three components of ticket sales. In fact, there is a logical and universally agreed upon mathematical sequence, an order of operations, used to evaluate any mathematical expression requiring a series of steps. Using this order of operations ensures the same answer for everyone who evaluates a given expression. It is as follows: Order of Operations To evaluate an expression, the simplification process must follow the Order of Operations: First, simplify the operations within Parentheses. Then, simplify all numbers with Exponents. Then, compute Multiplication and Division, left to right as they occur in the expression. Finally, compute Addition and Subtraction, left to right as they occur in the expression. METHODOLOGY Applying the Order of Operations to Simplify an Expression 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. [75 5 ]+[ ] [4(11 8) ] ( 7 + ) [( 7 + ) ] + [ 5 9] [ 4 ] + [ 1 4] Special Case: The expression is written as a fraction with an expression in either or both the numerator and denominator (see page 588, Model 4)

3 Section 6. Evaluating Expressions Using the Order of Operations 585 Steps in the Methodology Example 1 Example Step Simplify operations in parentheses. Simplify the operation(s) within Parentheses, if there are any, for each term. To assure that you are doing the steps in the correct order of operations, it may be helpful to label each step as you compute it. As each term is simplified to one number, you may drop the brackets surrounding it. [75 5 ]+[ ] [4(11 8) ] =[75 5 ]+[ ] [4() ] P P [( 7 + ) ] + [ 5 9] [ 4 ] + [ 1 4] [( 9) ] + [ 5 9] [ 4 ] + [ 1 4] E [ 81] + [ 5 9] [ 16] + [ 1 4] M&D [ 81] + [ 45] [ 16] + [ ] Step Simplify numbers with exponents. Simplify the numbers with Exponents, if there are any, in each term. =[75 5 ]+[ ] [4() ] =[75 5 ]+ 8 [4 9] E Step 4 Mulitply 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] =[0]+ 8 [6] M & D A&S 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. = = 8 6 = A & S Step 6 Present your final answer. Present the answer. Note: If the answer is in fraction form, reduce it. 11

4 586 Chapter 6 Signed Numbers, Exponents, and Order of Operations Suggested Validation when Applying 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. Keeping track of your steps by labeling each one as you go is an effective way to assure that the order is correct. Validating the accuracy of each computation as you work through the problem can further assure the accuracy of your final answer. The following is a sample validation for Example 1 of the Methodology (11 8) Validation Step 1 [75 5 ] + [ ] [4(11 8) ] of Step Step = [75 5 ] + [ ] [4() ] P + 8 = 11 Step = [75 5 ] + 8 [4(9)] E 15 of Step 4 0 = 15 Step 4 = M & D 15 5 = = 4 of Step 5 Steps 5 & 6 = Answer A & S = 8 8 = 0 Order used: P, E, M & D, A & S

5 Section 6. Evaluating Expressions Using the Order of Operations 587 MODELS Model 1 Simplify 4.6 (0.5) + (8 1.5) Validation: Step 1: Identify the terms: [4.6 ] [(0.5) ] + [ (8 1.5)] of Step Step : P = [4.6 ] [(0.5) ] + [ (6.5)] = Step : E = [4.6 ] [0.5] + [ (6.5)] of Step 4. = 4.6 Step 4: M & D = = Step 5: A & S, left to right as they occur of Step 5 = = = =.0 Order: P, E, M&D, A&S Step 6: Answer: Model Simplify 5 ( 4) (6 1) 7 ( ) Validation: Step 1: Identify the terms: [5( 4)] [(6 1) ] [7 ( )] of Step Step : P = [5( 4)] [(5) ] [7 ( )] = Step : E = [5( 4)] 15 [7 ( )] of Step 4 0 ( 4) = +5 Step 4: M = 0 15 ( 14) 14 ( ) = +7 Step 5: S, change all subtraction to addition of Step 5 = 0 + ( 15) + (+14) 11 + ( 14) + 15 = (+14) = = 0 = 11 Order: P, E, M, S Step 6: Answer: 11

6 588 Chapter 6 Signed Numbers, Exponents, and Order of Operations Model 1 Simplify: Step 1: Step : skip this step no operations inside Parentheses Step : E Step 4: M Step 5: A & S, left to right as they occur = = 1 Step 6: = = + 4 = = Answer : = = = = + ( ) = = Validation: of Step = 10 = 5 of Step = = = = 1 4 Order: E, M, A & S Model 4 Special Case: The Expression is Written as a Fraction with an Expression in Either or Both the Numerator and Denominator A Simplify: 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: Identify the terms: Step : P Step : no Exponents, skip this step = 0 6 Step 4: M = ( 6) Step 5: S change to addition = Step 6: Reduce: 6 =+ Answer: ( ) = 6 1 Validation: of Step = + = 5 of Step 4 0 ( 5) = +4 6 = of Step = = 1 + = 10 Order: P, M, S

7 Section 6. Evaluating Expressions Using the Order of Operations 5 ()+ + 7 B Simplify: Validation: Step 1: Identify the terms: + + ( ) [ 14] ( 1) 589 Step : skip this step no operations inside Parentheses Step : E Step 4: M Step 5: A & S Step 6: Reduce: = = = 1 = = [ 14] ( 1) = Answer : of Step = 1 ( 7) = + ( 1) = + of Step 5 ( 1) 9 = + (+1) + ( 9) =4 + ( 9) = ( ) = 14 Order: E, M, A & S ADDRESSING COMMON ERRORS Issue Incorrect Process Resolution Correct Process Validation Incorrectly identifying the terms of an expression Simplify: 5 ( + 9) = 5 (6) = 5 (6) = (6) = 18 Insert brackets to separate the terms. The terms of an expression are separated by addition and subtraction signs. Simplify each term separately. Simplify: [5 ] [ ( + 9)] = [5 ] [ (6)] P 5 5 = 5 [ (6)] E = 5 1 M = 1 S 6 9 = 1 6 = 1 +1 = 5 PEMS, correct order Not multiplying and dividing in the correct order when simplifying a term in which they both occur Simplify: ( 4) 6 (4) = 7 4 = When multiplication and division occur within the same term, follow the order of operations for that term, multiplying and dividing left to right. Simplify: ( 4) 6 (4) This is a single term (no addition or subtraction signs). Left to right: = 7 6 (4) ( 4) = 1 6 ( 4) = 7 ( 4) = = 1 (4) = 48

8 590 Chapter 6 Signed Numbers, Exponents, and Order of Operations Issue Incorrect Process Resolution Correct Process Validation Not adding and subtracting from left to right in the final step Evaluate: = = 6 1 = 15 } Once the terms have all been simplified to one number each, addition and subtraction must be done left to right as they occur to comply with the Order of Operations. At this point, unless you convert each subtraction to addition of the opposite, you cannot use the Associative Property for Addition. Evaluate: Left to right: = 6 + ( 1) + 8 = = = 1 + ( 8) + 1 = = 6 That is, for example, (1+8) Making arithmetic and/or sign errors Simplify: ( ) + 6 = (5) + 6 = = 4 Making arithmetic and/ or sign errors in any step will, of course, ultimately result in an incorrect answer. Validate each step as you work through the problem, so as not to carry through an earlier error. Simplify: ( 8 + ) + 6 [ ( 8 + )] + 6 = [ ( 5)] + 6 P = M = 16 A 5 = 5 + ( ) = 8 10 ( 5) = 16 6 = 10 Order: P, M, A PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with order of operations problems the purpose for an agreed upon Order of Operations how to apply the Order of Operations to simplify an expression with multiple operations validation techniques for Order of Operations problems

9 Section 6. ACTIVITY Evaluating Expressions Using the Order of Operations PERFORMANCE CRITERIA Evaluating expressions accuracy documentation of steps CRITICAL THINKING QUESTIONS 1. What is the order of operations to follow when evaluating 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.. How can you identify the terms of an expression? Do all expressions have 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. 591

10 59 Chapter 6 Signed Numbers, Exponents, and Order of Operations 5. In computing a series of addition and subtraction operations, in what order must they be done and why? 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 operations in parentheses are done before exponents and the exponents before multiplication and division? 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. Why do you think multiplication and division are done before addition and subtraction when simplifying an expression? This is universally accepted as the Order of Operations. I think that in an expression, the terms should be simplifi ed before the terms are added or subtracted.

11 Section 6. Evaluating Expressions Using the Order of Operations 59 TIPS FOR SUCCESS Separate terms with brackets and/or highlight the addition and subtraction signs separating the terms. Once identified, terms can be simplified simultaneously (for example, if several terms have an exponent to evaluate). To assure accuracy, bring the entire expression down to the next line with each simplification. Label which step you are doing to document and assure the correct order. (P, E, M or D, A or S) Validate each step for computational accuracy. When you simplify an expression that begins as a fraction of two expressions, if the answer can be converted to an integer, it must be presented as an integer. For example, 4 = if the answer simplifies to a negative number over a negative number, the final answer is positive. For example, =+ or (A negative number divided by a negative number is always positive.) DEMONSTRATE YOUR UNDERSTANDING Simplify each of the following expressions: Expression Validation (optional) 1) 16 4 () ) )

12 594 Chapter 6 Signed Numbers, Exponents, and Order of Operations Expression Validation (optional) 4) ) 6 (9) (1 8) 6) (4 + 6) 5 ( 8) 7) (5. 1.) + (.) 1 8) (7.1) ( ) + (0.) 1.

13 Section 6. Evaluating Expressions Using the Order of Operations 595 Expression Validation (optional) 9) ) ) ) 14 5 ( 1) 10

14 596 Chapter 6 Signed Numbers, Exponents, and Order of Operations TEAM EXERCISES 1. Simplify: Answer: 10. Change the same sequence of numbers and signs by adding one set of parentheses so that the simplified answer to your new expression is (1 18) 5. Devise a real-life problem that would require applying the Order of Operations to solve. answers will vary IDENTIFY AND CORRECT THE ERRORS 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: (5 + 4) Did not follow order of operations P, E, M&D, A&S. Added and before multiplying. The two terms are and (5+4). 5 ( + 4) = [( 5 + 4)] = [()] 9 P = 7 M = + ( 7) A = 5 Answer: 5 Order P, M, A 9 4 = = = +

15 Section 6. Evaluating Expressions Using the Order of Operations 597 Worked Solution What is Wrong Here? Identify the Errors Correct Process Validation (optional) ) Simplify: 4 ( ) ( 4) Work all multiplications OR divisions working from left to right. ) Simplify: (16 9) + 6 Perform addition and subtraction in order from l eft to right. 4) Simplify: Perform order of + 5 ( ) operations PEMDAS. 16 ( 4)+ Exponents should be done first.

16 598 Chapter 6 Signed Numbers, Exponents, and Order of Operations Worked Solution What is Wrong Here? Identify the Errors Correct Process Validation (optional) 5) Simplify: (0.04) + 6. (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. 6) Simplify: There needs to be a single sign in the ( 7) ( 5) 1 final answer. ( )+ 5( 6)

17 Section 6. Evaluating Expressions Using the Order of Operations 599 ADDITIONAL EXERCISES Simplify each of the following expressions: (5 9) 5 ( 6) 7. 4 ( 7) 57 ( ) ( ) (8 10) 4 5. (4 6) + (1 + ) (14 11) (0.) ( ) (.1) ( ) ( 7) ( ) + ( 6) (5 8) ( ) 46