Post on 29-Mar-2015
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Lesson 2.1.4Lesson 2.1.4
Order of OperationsOrder of Operations
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Lesson
2.1.4Order of OperationsOrder of Operations
California Standards:Algebra and Functions 1.3Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
Algebra and Functions 1.4Solve problems manually by using the correct order of operations or by using a scientific calculator.
What it means for you:You’ll see that it matters in which order you evaluate an expression, and you’ll learn the correct order.
Key Words:• parentheses• exponent • multiply and divide• add and subtract• expression• evaluate
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Sometimes you’ll meet expressions containing different combinations of operations (such as ×, ÷, +, and –).
Lesson
2.1.4Order of OperationsOrder of Operations
When you do, you need to be sure to do all of them in the right order.
2 × 4 – 6 4 – 10 + 7
2 – x ÷ 3 a × 4 + b × 3
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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
2.1.4Order of OperationsOrder 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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Example 1
Evaluate 2 + 3 × 7, and (2 + 3) × 7.
Solution follows…
Lesson
2.1.4Order of OperationsOrder 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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Example 2
Evaluate: (i) 32 – 3 × 8(ii) 56 – 8 ÷ 2
Solution follows…
Lesson
2.1.4Order of OperationsOrder 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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You do multiplications and divisions working from left to right.
Lesson
2.1.4Order of OperationsOrder of Operations
You also do additions and subtractions from left to right — after you’ve 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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= 47 – 6
Then + and –, from left to right
45 + 34 ÷ 17 – 2 × 3
Example 3
Evaluate: 45 + 34 ÷ 17 – 2 × 3.
Solution follows…
Lesson
2.1.4Order of OperationsOrder of Operations
Solution
= 45 + 2 – 6
= 45 + 2 – 2 × 3
× and ÷ first, from left to right
= 41
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Guided Practice
Solution follows…
Lesson
2.1.4Order of OperationsOrder 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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Guided Practice
Solution follows…
Lesson
2.1.4Order of OperationsOrder 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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Remember... multiplications and divisions before additions and subtractions — unless parentheses tell you otherwise.
Lesson
2.1.4Order of OperationsOrder of Operations
Order of Operations:
( ) [ ] 1. Evaluate anything inside parentheses (or brackets)
× ÷ 3. Multiply and divide from left to right
y2 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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An easier way to remember the order of operations is to remember the word PEMDAS.
Lesson
2.1.4Order of OperationsOrder of Operations
( ) [ ] 1. Parentheses
× ÷ 3. Multiplication / Division
y2 2. Exponents
+ – 4. Addition / Subtraction
This word stands for:
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Example 4
Evaluate: 32 × 5 – (20 – 2) ÷ 32.
Solution follows…
Lesson
2.1.4Order of OperationsOrder of Operations
Solution
32 × 5 – (20 – 2) ÷ 32 = 32 × 5 – 18 ÷ 32
= 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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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…
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2.1.4Order of OperationsOrder of Operations
Alan: 24 ÷ 3 – 22 = 8 – 22
= 62
= 36
Jessica: 24 ÷ 3 – 22 = 24 ÷ 3 – 4= 8 – 4= 4
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Example 5
Lesson
2.1.4Order of OperationsOrder 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 – 22 = 8 – 22
= 62
= 36
Jessica: 24 ÷ 3 – 22 = 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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Guided Practice
Solution follows…
Lesson
2.1.4Order of OperationsOrder 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. 32 + 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
32 + 4 × 9 = 9 + 4 × 9 = 9 + 36 = 45
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Guided Practice
Solution follows…
Lesson
2.1.4Order of OperationsOrder of Operations
Evaluate the expressions shown in Exercises 9–18.
14. 23 – 9 ÷ 3
15. (4 + 1)2 – 12 ÷ 4 – 1
16. 3 × (42 – 5) + 11
17. (62 – 32) ÷ (6 – 3)2
18. (15 ÷ 3 – 2) × (33 – 5 × 4)
23 – 9 ÷ 3 = 8 – 9 ÷ 3 = 8 – 3 = 5
(4 + 1)2 – 12 ÷ 4 – 1 = 52 – 12 ÷ 4 – 1 = 25 – 12 ÷ 4 – 1 = 25 – 3 – 1 = 21
3 × (42 – 5) + 11 = 3 × (16 – 5) + 11= 3 × 11 + 11 = 33 + 11 = 44
(62 – 32) ÷ (6 – 3)2 = (36 – 9) ÷ (6 – 3)2 =27 ÷ 32 = 27 ÷ 9 = 3
(15 ÷ 3 – 2) × (33 – 5 × 4) = (5 – 2) × (27 – 20)= 3 × 7 = 21
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Order Rules Also Apply to Expressions with Variables
The order rules apply to all types of expressions, including those with variables.
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2.1.4Order of OperationsOrder of Operations
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Example 6
Evaluate x + yz when x = –3, y = 9, and z = 10.
Solution follows…
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2.1.4Order of OperationsOrder 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 + 9 × 10
–3 + 90
87
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Example 7
Evaluate 19 – f 2 × (w + q) when f = 3, w = 5, and q = 2.
Solution follows…
Lesson
2.1.4Order of OperationsOrder 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 – 32 × (5 + 2)
19 – 32 × 7
19 – 9 × 7
19 – 63
–44
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Guided Practice
Solution follows…
Lesson
2.1.4Order of OperationsOrder 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
(32 – 22) ÷ 5 = (9 – 4) ÷ 5 = 5 ÷ 5 = 1
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Guided Practice
Solution follows…
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2.1.4Order of OperationsOrder 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) ÷ 4q
24. q – sk 3
10 × 3 + 1 × 10 + 23 = 10 × 3 + 10 + 23= 30 + 10 + 23 = 63
(32 – 1) ÷ 4 × 1 = (9 – 1) ÷ 4 × 1 = 8 ÷ 4 = 2
1 – 10 × 33 = 1 – 10 × 27 = 1 – 270 = –269
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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 OperationsOrder of Operations
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15
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1
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Independent Practice
Solution follows…
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2.1.4Order of OperationsOrder 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
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?Sylvia’s mistake was to do subtraction before multiplication.
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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 × 52 = (6 × 5)2
Order of OperationsOrder 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, 52 is first — equals 150. But on right, × is first — equals 900.
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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 OperationsOrder of Operations
50 – (1.5 × 8) – (2 × 4) = 30
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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 OperationsOrder of Operations
12 × (7 + c)
12 × (7 + 8.5) = $186
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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. 18y – 2j = (18 – 2) × (y – j)
Order of OperationsOrder 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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Lesson
2.1.4Order of OperationsOrder of Operations
Round UpRound Up
Hopefully you can now see how important the order of operations is when evaluating expressions.
You’ll need to be able to get the order right from now on — so make sure you memorize it.