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1 Lesson 2.1.4 Order of Operations

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2 Lesson 2.1.4 Order of Operations California Standards: Algebra and Functions 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process. Algebra and Functions 1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator. What it means for you: Youll see that it matters in which order you evaluate an expression, and youll learn the correct order. Key Words: parentheses exponent multiply and divide add and subtract expression evaluate

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3 Sometimes youll meet expressions containing different combinations of operations (such as ×, ÷, +, and –). Lesson 2.1.4 Order of Operations When you do, you need to be sure to do all of them in the right order. 2 × 4 – 64 – 10 + 7 2 – x ÷ 3 a × 4 + b × 3

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4 The Order In Which You Evaluate Makes a Difference The expression 2 + 3 × 7 looks like it could give two different answers, depending on how you work it out. Lesson 2.1.4 Order of Operations To avoid this situation, this rule is used for all math expressions: Working out 2 + 3 first gives: 2 + 3 × 7 = 5 × 7 = 35 Working out 3 × 7 first gives: 2 + 3 × 7 = 2 + 21 = 23 Always do multiplication before addition, unless parentheses tell you to do otherwise.

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5 Example 1 Evaluate 2 + 3 × 7, and (2 + 3) × 7. Solution follows… Lesson 2.1.4 Order of Operations Solution 2 + 3 × 7 (2 + 3) × 7 = 23 = 35 Work out the multiplication first = 2 + 21 The parentheses tell you to do the addition first = 5 × 7 You do all multiplications and divisions before additions and subtractions (unless parentheses tell you otherwise).

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6 Example 2 Evaluate: (i) 32 – 3 × 8 (ii) 56 – 8 ÷ 2 Solution follows… Lesson 2.1.4 Order of Operations Solution (i)32 – 3 × 8 (ii)56 – 8 ÷ 2 = 8 = 52 Work out the multiplication first = 32 – 24 Work out the division first = 56 – 4

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7 You do multiplications and divisions working from left to right. Lesson 2.1.4 Order of Operations You also do additions and subtractions from left to right after youve finished all the multiplications and divisions. For example, 3 × 7 – 2 + 9 ÷ 3 = (3 × 7) – 2 + (9 ÷ 3) = (21) – 2 + (3) = 19 + 3 = 22

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8 = 47 – 6 Then + and –, from left to right 45 + 34 ÷ 17 – 2 × 3 Example 3 Evaluate: 45 + 34 ÷ 17 – 2 × 3. Solution follows… Lesson 2.1.4 Order of Operations Solution = 45 + 2 – 6 = 45 + 2 – 2 × 3 × and ÷ first, from left to right = 41

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9 Guided Practice Solution follows… Lesson 2.1.4 Order of Operations Evaluate the expressions shown in Exercises 1–4. 1. 4 + 2 × 5 2. 3 + 7 × 4 3. 12 ÷ 2 + 4 4. 8 × 7 + 1 4 + 2 × 5 = 4 + 10 = 14 3 + 7 × 4 = 3 + 28 = 31 12 ÷ 2 + 4 = 6 + 4 = 10 8 × 7 + 1 = 56 + 1 = 57

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10 Guided Practice Solution follows… Lesson 2.1.4 Order of Operations Evaluate the expressions shown in Exercises 5–8. 5. 11 – 1 × 4 6. 88 + 20 ÷ 2 7. (88 – 3) ÷ 5 8. 8 + 9 ÷ 3 × 5 – 3 11 – 1 × 4 = 11 – 4 = 7 88 + 20 ÷ 2 = 88 + 10 = 98 (88 – 3) ÷ 5 = (85) ÷ 5 = 17 8 + 9 ÷ 3 × 5 – 3 = 8 + 3 × 5 – 3 = 8 + 15 – 3 = 23 – 3 = 20

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11 Remember... multiplications and divisions before additions and subtractions unless parentheses tell you otherwise. Lesson 2.1.4 Order of Operations Order of Operations: ( ) [ ] 1. Evaluate anything inside parentheses (or brackets) × ÷ 3. Multiply and divide from left to right y 2 2. Evaluate exponents + – 4. Add and subtract from left to right More generally, the order you should perform operations is as follows:

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12 An easier way to remember the order of operations is to remember the word PEMDAS. Lesson 2.1.4 Order of Operations ( ) [ ] 1. Parentheses × ÷ 3. Multiplication / Division y 2 2. Exponents + – 4. Addition / Subtraction This word stands for:

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13 Example 4 Evaluate: 3 2 × 5 – (20 – 2) ÷ 3 2. Solution follows… Lesson 2.1.4 Order of Operations Solution 3 2 × 5 – (20 – 2) ÷ 3 2 = 3 2 × 5 – 18 ÷ 3 2 = 9 × 5 – 18 ÷ 9 = 43 Then exponents Parentheses first = 45 – 18 ÷ 9 × and ÷, from left to right = 45 – 2 + and –, from left to right

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14 Example 5 Alan and Jessica are each trying to calculate the expression 24 ÷ 3 – 22. Their work is shown below. Who has the correct answer? Solution follows… Lesson 2.1.4 Order of Operations Alan: 24 ÷ 3 – 2 2 = 8 – 2 2 = 6 2 = 36 Jessica: 24 ÷ 3 – 2 2 = 24 ÷ 3 – 4 = 8 – 4 = 4

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15 Example 5 Lesson 2.1.4 Order of Operations Jessica evaluated the exponent first, then performed the division, then the subtraction. Solution Alan performed the division, then the subtraction, then evaluated the exponent. Alan: 24 ÷ 3 – 2 2 = 8 – 2 2 = 6 2 = 36 Jessica: 24 ÷ 3 – 2 2 = 24 ÷ 3 – 4 = 8 – 4 = 4 That is incorrect he should have found the exponent first. This is the right order, so Jessica has the right answer.

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16 Guided Practice Solution follows… Lesson 2.1.4 Order of Operations Evaluate the expressions shown in Exercises 9–18. 9. 8 + (10 – 2) ÷ 4 10. 4 – 2 × 16 11. 5 × 7 + 7 × 5 12. 7 × 3 – 8 ÷ 2 + 6 13. 3 2 + 4 × 9 8 + (10 – 2) ÷ 4 = 8 + 8 ÷ 4 = 8 + 2 = 10 4 – 2 × 16 = 4 – 32 = –28 5 × 7 + 7 × 5 = 35 + 7 × 5 = 35 + 35 = 70 7 × 3 – 8 ÷ 2 + 6 = 21 – 8 ÷ 2 + 6 = 21 – 4 + 6 = 17 + 6 = 23 3 2 + 4 × 9 = 9 + 4 × 9 = 9 + 36 = 45

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17 Guided Practice Solution follows… Lesson 2.1.4 Order of Operations Evaluate the expressions shown in Exercises 9–18. 14. 2 3 – 9 ÷ 3 15. (4 + 1) 2 – 12 ÷ 4 – 1 16. 3 × (4 2 – 5) + 11 17. (6 2 – 3 2 ) ÷ (6 – 3) 2 18. (15 ÷ 3 – 2) × (3 3 – 5 × 4) 2 3 – 9 ÷ 3 = 8 – 9 ÷ 3 = 8 – 3 = 5 (4 + 1) 2 – 12 ÷ 4 – 1 = 5 2 – 12 ÷ 4 – 1 = 25 – 12 ÷ 4 – 1 = 25 – 3 – 1 = 21 3 × (4 2 – 5) + 11 = 3 × (16 – 5) + 11 = 3 × 11 + 11 = 33 + 11 = 44 (6 2 – 3 2 ) ÷ (6 – 3) 2 = (36 – 9) ÷ (6 – 3) 2 =27 ÷ 3 2 = 27 ÷ 9 = 3 (15 ÷ 3 – 2) × (3 3 – 5 × 4) = (5 – 2) × (27 – 20) = 3 × 7 = 21

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18 Order Rules Also Apply to Expressions with Variables The order rules apply to all types of expressions, including those with variables. Lesson 2.1.4 Order of Operations

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19 Example 6 Evaluate x + yz when x = –3, y = 9, and z = 10. Solution follows… Lesson 2.1.4 Order of Operations Solution Substitute in your values for x, y, and z There are no parentheses or exponents Carry out the multiplication Carry out the addition, giving the answer –3 + 9 × 10 –3 + 90 87

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20 Example 7 Evaluate 19 – f 2 × ( w + q ) when f = 3, w = 5, and q = 2. Solution follows… Lesson 2.1.4 Order of Operations Solution Substitute in your values for f, w, and q Evaluate the parentheses Evaluate the exponents Carry out the multiplication Carry out the subtraction, giving the answer 19 – 3 2 × (5 + 2) 19 – 3 2 × 7 19 – 9 × 7 19 – 63 –44

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21 Guided Practice Solution follows… Lesson 2.1.4 Order of Operations Evaluate the expressions shown in Exercises 19–21, given that f = 2, j = 3, and g = 19. 19. g – j × 8 20. ( f + g ) × j + 4 21. ( j 2 – f 2 ) ÷ 5 19 – 3 × 8 = 19 –24 = –5 (2 + 19) × 3 + 4 = 21 × 3 + 4 = 63 + 4 = 67 (3 2 – 2 2 ) ÷ 5 = (9 – 4) ÷ 5 = 5 ÷ 5 = 1

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22 Guided Practice Solution follows… Lesson 2.1.4 Order of Operations Evaluate the expressions shown in Exercises 22–24, given that s = 10, k = 3, and q = 1. 22. s × k + qs + 23 23. ( k 2 – 1) ÷ 4 q 24. q – sk 3 10 × 3 + 1 × 10 + 23 = 10 × 3 + 10 + 23 = 30 + 10 + 23 = 63 (3 2 – 1) ÷ 4 × 1 = (9 – 1) ÷ 4 × 1 = 8 ÷ 4 = 2 1 – 10 × 3 3 = 1 – 10 × 27 = 1 – 270 = –269

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23 Independent Practice Solution follows… Lesson 2.1.4 Evaluate Exercises 1–4, given that w = 2, b = 8, c = –0.5. 1. b × ( w – c ) 2. w × b + w × c 3. (6 – w ) 2 4. ( b + w + 2 × c ) ÷ ( b – 2 × c ) Order of Operations 20 15 16 1

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24 Independent Practice Solution follows… Lesson 2.1.4 Order of Operations Felipe is correct. Felipe: 2 × 18 – 6 = 36 – 6 = 30 Felipe: 2 × 18 – 6 = 36 – 6 = 30 Sylvia:2 × 18 – 6 = 2 × 12 = 24 5. Felipe and Sylvia are trying to evaluate the expression 2 × 18 – 6. Who is correct? What has the other person done wrong? Sylvias mistake was to do subtraction before multiplication.

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25 Independent Practice Solution follows… Lesson 2.1.4 Which of the expressions in Exercises 6–11 are true, and which are false? For those that are false, explain what is wrong. 6. (2 + 3) × 6 = 2 + 3 × 6 7. (7 + 5) – 3 = 7 + 5 – 3 8. (100 ÷ 2) ÷ 25 = 100 ÷ 2 ÷ 25 9. 10 ÷ 2 × 5 = 10 ÷ (2 × 5) 10. 3 + (6 × 9) = 3 + 6 × 9 11. 6 × 5 2 = (6 × 5) 2 Order of Operations False. On left, + is first equals 30. But on right, × is first equals 20. True both sides equal 9. True both sides equal 2. False. On left, ÷ is first equals 25. But on right, × is first equals 1. True both sides equal 57. False. On left, 5 2 is first equals 150. But on right, × is first equals 900.

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26 Independent Practice Solution follows… Lesson 2.1.4 12.Carly wants to exercise for a total of 50 hours this month. She exercised 1.5 hours for each of the first eight days. Then she exercised 2 hours for each of the next four days. Write and evaluate an expression for the remaining number of hours she has to exercise. Order of Operations 50 – (1.5 × 8) – (2 × 4) = 30

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27 Independent Practice Solution follows… Lesson 2.1.4 13.Mr. Chang earns $12 per hour baking bread at the bakery. He worked 7 hours on Thursday and c hours on Friday. Write an expression of the form a × ( b + c ) that can be used to calculate how much Mr. Chang earned on Thursday and Friday. 14.Given that Mr. Chang worked for 8.5 hours on Friday, find the value of the expression you wrote for Exercise 13. Order of Operations 12 × (7 + c ) 12 × (7 + 8.5) = $186

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28 Independent Practice Solution follows… Lesson 2.1.4 Which of the expressions in Exercises 15–19 are true, and which are false? For those that are false, explain what is wrong. 15. h – r + q = h – ( r + q ) 16. a × (5 + g ) = a × 5 + g 17. 16 × f ÷ 3 + 2 = 16 × f ÷ (3 + 2) 18. 5 + j – w = (5 + j ) – w 19. 18 y – 2 j = (18 – 2) × ( y – j ) Order of Operations False. On left, subtraction is first. But on right, + is first. False. On left, + is first. But on right, × is first. False. On left, order is ×, ÷, +. But on right, order is +, ×, ÷. True False completely different equation.

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29 Lesson 2.1.4 Order of Operations Round Up Hopefully you can now see how important the order of operations is when evaluating expressions. Youll need to be able to get the order right from now on so make sure you memorize it.

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