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**The Order of Operations**

Lesson 1.1.3

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations California Standard: Algebra and Functions 1.2 What it means for you: You’ll learn about the special order to follow when you’re deciding which part of an expression to evaluate first. Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2. Key Words: Parentheses Exponents PEMDAS

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**“add 7 to 3 and multiply the sum by 2.”**

Lesson 1.1.3 The Order of Operations When you have a calculation with more than one operation in it, you need to know what order to do the operations in. E.g. if you evaluate the expression 2 • by doing… 2 • = 13 “multiply 2 by 3 and add 7,” …you’ll get a different answer from someone who does… 2 • = 20 “add 7 to 3 and multiply the sum by 2.” For example, if you evaluate the expression 2 • by doing “multiply 2 by 3 and add 7,” you’ll get a different answer from someone who does “add 7 to 3 and multiply the sum by 2.” So the order you use really matters. So the order you use really matters. There’s a set of rules to follow to make sure that everyone gets the same answer. It’s called the order of operations — and you’ve seen it before in grade 6.

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**Order of operations — the PEMDAS Rule**

Lesson 1.1.3 The Order of Operations The Order of Operations is a Set of Rules An expression can contain lots of operations. When you evaluate it you need a set of rules to tell you what order to deal with the different bits in. Order of operations — the PEMDAS Rule ()[]{} Parentheses First do any operations inside parentheses. x2 y7 Exponents Then evaluate any exponents. × ÷ Multiplication or Division Next follow any multiplication and division instructions from left to right. + – Addition or Subtraction Finally follow any addition and subtraction instructions from left to right.

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations When an expression contains multiplication and division, or addition and subtraction, do first whichever comes first as you read from left to right. 9 ÷ 4 • 3 Divide first, then multiply. 9 • 4 ÷ 3 Multiply first, then divide. 9 + 4 – 3 Add first, then subtract. 9 – 4 + 3 Subtract first, then add. Following these rules means that there’s only one correct answer. Use the rules each time you do a calculation to make sure you get the right answer.

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations You Can Also Use GEMA For the Order of Operations GEMA is another way to remember the order of operations: First evaluate anything grouped by parentheses, fraction bars or brackets ()[]{} Grouping x2 y7 Exponents Then evaluate any exponents. × ÷ Multiplication or Division Next follow any multiplication and division instructions from left to right. Addition or Subtraction Finally follow any addition and subtraction instructions from left to right. + – You can use either PEDMAS or GEMA — whichever one you feel happier with.

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Example 1 What is 8 ÷ 4 • 4 + 3? Solution 8 ÷ 4 • 4 + 3 There are no parentheses or exponents = 2 • 4 + 3 Do the division first You do the division first as it comes before the multiplication, reading from left to right. = 8 + 3 Then the multiplication = 11 Finally do the addition to get the answer Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Guided Practice Evaluate the expressions in Exercises 1–6. 1. 3 – – 1 – 2 + 7 5. 40 – 10 ÷ 5 • 6 2. 6 ÷ 2 + 1 • 10 ÷ 10 6 4 12 52 28 6 Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Always Deal with Parentheses First When a calculation contains parentheses, you should deal with any operations inside them first. 2(4 • 52) + 1 2(4 • 52) + 1 You still need to follow the order of operations when you’re dealing with the parts inside the parentheses. 2(4 • 25) + 1 2(100) + 1

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Example 2 What is 10 ÷ 2 • (10 + 2)? Solution The order of operations says that you should deal with the operations in the parentheses first — that’s the P in PEMDAS. 10 ÷ 2 • (10 + 2) First write out the expression = 10 ÷ 2 • 12 Do the addition in parentheses You do the division first here because it comes first reading from left to right. = 5 • 12 Then do the division = 60 Finally do the multiplication Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Guided Practice Evaluate the expressions in Exercises 7–14. 7. 10 – (4 + 3) 9. 10 ÷ (7 – 5) • (2 + 4) – 3 13. 6 • (8 ÷ 4) + 11 8. (18 ÷ 3) + (2 + 3 • 4) – (4 + 2 – 3) 12. (5 – 7) • (55 ÷ 11) • (16 ÷ 2) 3 20 5 38 57 –10 23 48 Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations PEMDAS Applies to Algebra Problems Too The order of operations still applies when you have calculations in algebra that contain a mixture of numbers and variables. Do the addition in parentheses, then the multiplication 3 • (2 + 4) Do the addition in parentheses, then the multiplication a • (2 + 4) Do the addition in parentheses, then the multiplication 3 • (b + 4)

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Example 3 Simplify the calculation k • (5 + 4) + 16 as far as possible. Solution k • (5 + 4) + 16 First write out the expression = k • Do the addition in parentheses = 9k + 16 Then the multiplication Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Guided Practice Simplify the expressions in Exercises 15–20 as far as possible. • x 17. 3 • (y – 2) (4 • 2) • t a • 4 – 1 ÷ (3 + 2) – r 20. p + 5 • (–2 + m) 5 + 7x 4a + 1 3y – 6 2 – r 8t + 20 p – m Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Independent Practice 1. Alice and Emilio are evaluating the expression • 4. Their work is shown below Explain who has the right answer. Alice 5 + 6 • 4 = 11 • 4 = 44 Emilio 5 + 6 • 4 = = 29 Emilio has the right answer because he has used the correct order of operations: he has done the multiplication before the addition. Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Independent Practice The local muffler replacement shop charges $75 for parts and $25 per hour for labor. 2. Write an expression with parentheses to describe the cost, in dollars, of a replacement if the job takes 4 hours. 3. Use your expression to calculate what the cost of the job would be if it did take 4 hours. 75 + (4 • 25) $175 Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Independent Practice Evaluate the expressions in Exercises 4–7. ÷ 8 – 2 • 5 • (10 – 6 ÷ 3) –4 • 3 7. 3 • (5 – 3) + (27 ÷ 3) 25 47 15 8. Paul buys 5 books priced at $10 and 3 priced at $15. He also has a coupon for $7 off his purchase. Write an expression with parentheses to show the total cost, after using the coupon, and then simplify it to show how much he spent. (5 • 10) + (3 • 15) – 7. He spent $88. Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Independent Practice 9. Insert parentheses into the expression – 6 • 4 to make it equal to 48. ( – 6) • 4 Simplify the expressions in Exercises 10–12 as far as possible. 10. x – 7 • 2 11. y + x • (4 + 3) – y (60 – x • 3) x – 14 7x 66 – 3x Solution follows…

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**The Order of Operations**

Lesson 1.1.3 The Order of Operations Round Up If you evaluate an expression in a different order from everyone else, you won’t get the right answer. That’s why it’s so important to follow the order of operations. This will feature in almost all the math you do from now on, so you need to know it. Don’t worry though — just use the word PEMDAS or GEMA to help you remember it.

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