1 Lesson 2.1.4 Order of Operations. 2 Lesson 2.1.4 Order of Operations California Standards: Algebra...

1

Lesson 2.1.4Lesson 2.1.4

Order of OperationsOrder of Operations

2

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

3

Sometimes you’ll meet expressions containing different combinations of operations (such as ×, ÷, +, and –).

Lesson


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

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


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.

5

Example 1

Evaluate 2 + 3 × 7, and (2 + 3) × 7.

Solution follows…

Lesson


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).

6

Example 2

Evaluate: (i) 32 – 3 × 8(ii) 56 – 8 ÷ 2

Solution follows…

Lesson


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

7

You do multiplications and divisions working from left to right.

Lesson


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

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


Solution

= 45 + 2 – 6

= 45 + 2 – 2 × 3

× and ÷ first, from left to right

= 41

9

Guided Practice

Solution follows…

Lesson


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

10

Guided Practice

Solution follows…

Lesson



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

11

Remember... multiplications and divisions before additions and subtractions — unless parentheses tell you otherwise.

Lesson


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:

12

An easier way to remember the order of operations is to remember the word PEMDAS.

Lesson


( ) [ ] 1. Parentheses

× ÷ 3. Multiplication / Division

y2 2. Exponents

+ – 4. Addition / Subtraction

This word stands for:

13

Example 4

Evaluate: 32 × 5 – (20 – 2) ÷ 32.

Solution follows…

Lesson


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

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


Alan: 24 ÷ 3 – 22 = 8 – 22

= 62

= 36

Jessica: 24 ÷ 3 – 22 = 24 ÷ 3 – 4= 8 – 4= 4

15

Example 5

Lesson


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.

16

Guided Practice

Solution follows…

Lesson



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

17

Guided Practice

Solution follows…

Lesson



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

18

Order Rules Also Apply to Expressions with Variables

The order rules apply to all types of expressions, including those with variables.

Lesson


19

Example 6

Evaluate x + yz when x = –3, y = 9, and z = 10.

Solution follows…

Lesson


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

20

Example 7

Evaluate 19 – f 2 × (w + q) when f = 3, w = 5, and q = 2.

Solution follows…

Lesson


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

21

Guided Practice

Solution follows…

Lesson


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

22

Guided Practice

Solution follows…

Lesson


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

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)


20

15

16

1

24


Solution follows…

Lesson


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.

25


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


False. On left, + is first — equals 30.But on right, × is first — equals 20.

True — both sides equal 9.


False. On left, ÷ is first — equals 25. But on right, × is first — equals 1.


False. On left, 52 is first — equals 150. But on right, × is first — equals 900.

26


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.


50 – (1.5 × 8) – (2 × 4) = 30

27


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.


12 × (7 + c)

12 × (7 + 8.5) = $186

28


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)


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.

29

Lesson


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.

1 Lesson 2.1.4 Order of Operations. 2 Lesson 2.1.4 Order of Operations California Standards: Algebra...

Documents

Transcript of 1 Lesson 2.1.4 Order of Operations. 2 Lesson 2.1.4 Order of Operations California Standards: Algebra...