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Order of Operations

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Children should be led to make their own investigations, and to draw upon their own inferences. They should be told as little as possible which can produce unlimited learning potential. Herbert Spencer Intellectual Moral and Physical 1864

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ORDER OF OPERATIONS How to do a math problem with more than one operation in the correct order.

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Order of Operations Problem:Evaluate the following arithmetic expression: 3 + 4 x 2 Solution: Student 1 3 + 4 x 2 = 7 x 2 = 14 Student 2 3 + 4 x 2 3 + 8 11

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Order of Operations It seems that each student interpreted the problem differently, resulting in two different answers. Student 1 performed the operation of addition first, then multiplication Student 2 performed multiplication first, then addition.

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Order of Operations When performing arithmetic operations there can be only one correct answer. We need a set of rules in order to avoid this kind of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation.

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Order of Operations Rule 1:First perform any calculations inside parentheses. Rule 2:Next perform all multiplications and divisions, working from left to right. Rule 3:Lastly, perform all additions and subtractions, working from left to right.

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Example Expression Evaluation Operation 6 + 7 x 8 = 6 + 7 x 8 Multiplication = 6 + 56 Addition = 62 16 ÷ 8 - 2 = 16 ÷ 8 - 2 Division = 2 - 2 Subtraction = 0 (25 - 11) x 3 = (25 - 11) x 3 Parentheses = 14 x 3 Multiplication = 42

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Time to do some computing! Evaluate using the order of operations. 3 + 6 x (5 + 4) ÷ 3 - 7 Solution: Step 1: = 3 + 6 x 9 ÷ 3 - 7 Parentheses Step 2: = 3 + 54 ÷ 3 - 7 Multiplication Step 3: = 3 + 18 - 7 Division Step 4: = 21 - 7 Addition Step 5: = 14 Subtraction

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1) 5 + (12 – 3) 5 + 9 14 2) 8 – 3 2 + 7 8 - 6 + 7 2 + 7 9 3) 39 ÷ (9 + 4) 39 ÷ 13 3 Examples

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Fractions Evaluate the arithmetic expression below: This problem includes a fraction bar, which means we must divide the numerator by the denominator. However, we must first perform all calculations above and below the fraction bar BEFORE dividing. The fraction bar can act as a grouping symbol

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Thus Evaluating this expression, we get: Fractions

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Mr. Smith charged Jill $32 for parts and $15 per hour for labor to repair her bicycle. If he spent 3 hours repairing her bike, how much does Jill owe him? Solution: 32 + 3 x 15= 32 + 3 x 15= 32 + 45 = 77 Jill owes Mr. Smith $77. Write an arithmetic expression

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Add Parentheses to Obtain Result 7 − 1 + 6 = 0 7 − ( 1 + 6 ) = 0 1 + 2 × 5 + 6 = 21 ( 1 + 2 ) × 5 + 6 = 21 2 + 2 × 5 ÷ 3 − 1 = 10 ( 2 + 2 ) × 5 ÷ ( 3 − 1 ) = 10 3 + 8 ÷ 2 = 7 3 + ( 8 ÷ 2 ) = 7 8 + 3 − 7 = 4

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When evaluating arithmetic expressions, the order of operations is: Simplify all operations inside parentheses. Perform all multiplications and divisions, working from left to right. Perform all additions and subtractions, working from left to right. If a problem includes a fraction bar, perform all calculations above and below the fraction bar before dividing the numerator by the denominator. Summary

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