simplifying expressions using the order … · · 2013-08-07267 activity 6.3 — simplifying...

265

6.3 SIMPLIFYING EXPRESSIONS USING THE ORDER OF OPERATIONS

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Bringit on!

• how to apply the Order of Operations to simplify an expression with multiple operations

• the purpose for an agreed upon Order of Operations

• validation techniques for Order of Operations

• Simplifying expressions– accuracy– documentation of steps

Brian has decided to open a savings account, where he will earn 4% interest, compounded semi-annually. He will deposit $500 to open the account. He plans to leave the account alone for 1 year, making no additional deposits and no withdrawals. At the end of that year, he will withdraw only the interest he has earned.

That amount can be determined by evaluating the following expression:

$500 × (1 + 0.04 ÷ 2)2 – $500

Complete thewithdrawal slip

at right, fi lling inthe amount Brian

will withdrawfrom his account

in one year.

20.20

266

Chapter 6 — Signed Numbers, Exponents, and Order of Operations

Example 1: Simplify: 75 ÷ 5 × 2 + 23 − 4 (11 − 8)2

Example 2: Simplify: (7 + 2)2 + 5 × 9 − 42 + 12 ÷ 4 Try It!

Steps in the Methodology Example 1 Example 2

Step 1

Identify the terms.

Use brackets, [ ] or { }, to identify the terms of the expression. Recall that addition and subtraction signs separate the terms of an expression.

[75÷5×2]+[23]–[4(11–8)2] [ ] [ ]4 12 42 + ÷

[( ) ] [ ]7 2 5 92+ + × −

Step 2

Simplify operations in parentheses.

Simplify the operation(s) within Parentheses, if there are any, for each term.

As each term is simplifi ed to one number, you may drop the brackets surrounding it.

To ensure that you are doing the steps in the correct order of operations, it may be helpful to label each step as you compute it.

[75÷5×2]+[23]–[4(11–8)2]

=[75÷5×2]+[23]–[4(3)2] P

[( ) ] [ ]9 5 92 + × −P[ ] [ ]4 12 42 + ÷

The Expression is Written as a Fraction with an Expression in Either or Both the Numerator and Denominator (see Model 4)

Special Case:

Order of Operations

To simplify an expression, the process must follow the Order of Operations:First, simplify all operations within Parentheses.Then, simplify all factors and terms with Exponents.Then, compute Multiplication and Division, left to right as they occur in each term.Finally, compute Addition and Subtraction, left to right as they occur in the expression.

While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem. Space has been left for you to do precisely that.

267

Activity 6.3 — Simplifying Expressions Using the Order of Operations

Steps in the Methodology Example 1 Example 2

Step 3

Simplify numbers with exponents.

Simplify the numbers with Exponents, if there are any, in each term.

=[75÷5×2]+[23]–[4(3)2]

=[75÷5×2]+ 8 –[4×9] E2×2×2 3×3

E [ ] [ ]81 5 9+ × −

[ ] [ ]16 12 4+ ÷

Step 4

Multiply and Divide left to right.

Compute Multiplication and Division, left to right as they are situated in each term.

=[75÷5×2]+ 8 –[4×9]

=[15×2]+ 8 –[4×9]

=[30]+ 8 –[36] M & D

M&D [ ] [ ]81 45+ −

[ ] [ ]16 3+

Step 5

Add and Subtract left to right.

Compute Addition and Subtraction of the simplifi ed terms, left to right as they are situated in the expression.

= 30+ 8 – 36

= 38 – 36

= 2 A & S

A&S 81 45 16 3

126 16 3110 3

113

+ − +

− +

+

Step 6

Present the answer.

Present your fi nal answer.

Note: If the answer is in fraction form, reduce it.

2 113

Model 1

Simplify 4.6 ÷ 2 – (0.5)2 + 2 (8 – 1.5) Validation:

Step 1: Identify the terms: [4.6 ÷ 2] – [(0.5)2] + [2 (8 – 1.5)]

Step 2: P = [4.6 ÷ 2] – [(0.5)2]0 5 0 5. .×

+ [2 (6.5)] of Step 26.5 + 1.5 = 8

Step 3: E = [4.6 ÷ 2] – [0.25] + [2 (6.5)]

Step 4: M & D = 2.3 – 0.25 + 13 of Step 42.3 × 2 = 4.6 13.0 ÷ 6.5 = 2

Step 5: A & S

left to rightas they occur

= 2.05 + 13 = 15.05

of Step 515.05 – 13 + 0.25 = 2.05 + 0.25 = 2.30 Order: P, E, M&D, A&S

Step 6: Answer: 15.05

268


Model 2

Simplify 5 (–4) – (6 – 1)3 – 7 (–2) Validation:

Step 1: Identify the terms: [5(–4)] – [(6 – 1)3] – [7 (–2)]

Step 2: P = [5(–4)] – [(5)3] 5 5 5× ×

– [7 (–2)] of Step 25 + 1 = 6

Step 3: E = [5(–4)] – 125 – [7 (–2)]

Step 4: M = –20 – 125 – (–14) of Step 4–20 ÷ (–4) = +5 –14 ÷ (–2) = +7

Step 5: S

change all subtraction to

addition

= –20 + (–125) + (+14)= –145 + (+14)= –131

of Step 5–131 + (–14) + 125 = –145 + 125 = –20 Order: P, E, M, S

Step 6: Answer: –131

Model 3

Simplify 12

23

35

102⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟ + − ×

Validation:

Step 1: 12

23

35

102⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

+⎡

⎣⎢⎢

⎤

⎦⎥⎥

− ×⎡

⎣⎢⎢

⎤

⎦⎥⎥

Step 2: skip this step—no operations inside Parentheses

Step 3: E

12

12

×

3

5

101

61

61

2

× = == +⎡

⎣⎢⎢

⎤

⎦⎥⎥

− ×⎡

⎣⎢⎢

⎤

⎦⎥⎥

14

23

35

10 6 10610

35

÷ = =

of Step 4

of Step 5

Order: E, M, A & S

1112

23

1112

812

312

14

− = −

= =

51

12+ = +6

1112

–Step 4: M = +⎡

⎣⎢⎢

⎤

⎦⎥⎥

− 614

23

Step 5: A & S, left to right as they occur

14

23

312

812

1112

+ = + =1112

6 5112

− = −1112

7212

1112

7212

11 7212

6112

51

12

− = + −⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟⎟

=+ −

=− =−( )

=

Step 6: Answer: 51

12 –

269


Model 4 Special Case: The Expression is Written as a Fraction with an Expression in Either or Both the Numerator and Denominator

The fraction bar indicates that the numerator and denominator are to be treated as two separate expressions. Simplify each expression separately, following the Order of Operations procedure; then reduce the resulting fraction, paying careful attention to the correct sign of the answer.

Simplify 4 2 7 2 5 2

10 2 1−( )− −( )− − −

Step 1: 4 2

10 2 1

2 7 5 2

( )⎡⎣

⎤⎦ − ( )⎡

⎣⎤⎦

− − −

––Validation:

of Step 2

–5 + 7 = 23 + 2 = 5

of Step 4

–20 ÷ (–5) = +46 ÷ 3 = 2

of Step 5

–26 + 6 = –20–13 + 1 + 2= –12 + 2 = –10

Order: P, M, S

Step 2: P =( )⎡

⎣⎤⎦ − ( )⎡

⎣⎤⎦

− − −

4 2

10 2 1

5 3–

Step 3: no Exponents, skip this step

Step 4: M =− −

− − −20 6

10 2 1

Step 5: S

change to addition

=− −( )

− + −( ) + −( )=

−−

20 6

10 2 12613

+

Step 6: Reduce:−−

= +2613

2

Answer: 2

Validation and the Order of Operations

When simplifying an expression with the Order of Operations, you can be fully confi dent in your answer only when you apply the correct order as well as do each computation accurately.

Models 1 through 4 demonstrate that keeping track of steps by labeling each one as you go is an effective way to ensure that the order is correct. Validating the accuracy of each computation as you work through the problem (as demonstrated in the models) can further ensure the accuracy of your fi nal answer.

270


1. What is the order of operations to follow when simplifying an expression?

2. What is the purpose of the Order of Operations?

Make Your Own ModelEither individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of.

Problem: _________________________________________________________________________ Answers will vary.

PEMDAS: Perform operations in parenthesis fi rst; then do all exponents; then do all multiplications or divisions from left to right; then do all additions or subtractions, from left to right.

It serves as a standard to evaluate an expression and to simplify arriving at an answer.

271


3. How do you identify the terms of an expression? Give an example of an expression with four terms.

4. When computing a series of multiplication and division operations within a single term, in what order must they be done?

5. In the fi nal step of the Order of Operations, why must addition and subtraction be computed from left to right as they occur in the expression?

6. What is a strategy you can use to validate an order of operations problem?

7. Why do you think the Exponents must be computed before the Multiplication and Division step is computed?

8. What aspect of the model you created is the most diffi cult to explain to someone else? Explain why.

Multiplications and divisions must be done in order from left to right as they come in order. Divisions are done fi rst if they are the fi rst operations beginning at the left of the expression to be evaluated.

The terms of an expression are separated by addition (+) and subtraction (–) signs. Every expression has at least one term.

The operation which comes fi rst when working from left to right is what is to be done fi rst. It could be the subtraction that is done fi rst. This is done this way because that is the way it is stated in the Order of Operations.

It is easier to validate each step as you do it.

This is universally accepted as the Order of Operations. Probably parentheses and exponents are done fi rst because they are more complicated to work out and more than one process is usually involved.

Answers will vary.

272


Expression Validation (optional)

1) 16 – 23 ÷ 4 (2)

2) 32 × 5 ÷ 9 + 82 – 7

3) 100 ÷ 4 × 5 + 10

4) –2 (5.2 – 1.3) + (2.2)2 – 1

5) (7.1)2 – (19.1 + 25.9) + 2 (0.2) – 12.3

Simplify each of the following expressions:

273


Expression Validation (optional)

6) 512

38

14

23

2

÷ +⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟−

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

7) 15 3

14 2 1

2−− − −( )

8) 16 3 7

8 3 4

− ( )− + −( )

9) 14 5 2

3 1 10

− −( )−( )−

Simplify each of the following expressions:

1. 15 – 10 ÷ 5 × 2 11

2. –3 (5 – 9) – 5 (3 – 6) 27

3. (0.2)3 + 0.5 × (0.3 + 6.5) 3.408

4. (2.1)2 – 7.5 + 3 (0.63 + 0.27) –4

5. − −( )− −( )

− − −3 4 7 5 7 2

5 2 1 2

6. 5 2

53

16 5

2⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟−

− −( ) − 1

3

7. 26 14 1 3

50 5 4

2− −( )+ −( ) –1

8. 78

12

13

12

33

+⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ × − ÷

34

274


Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the second column. If the worked solution is incorrect, solve the problem correctly in the third column. You can validate your work in the fourth column.

Worked SolutionWhat is Wrong Here? Identify the Errors Correct Process Validation

(optional)

1) Simplify:

2 – 3 (5 + 4)

Did not follow order of operationsP, E, M&D, A&S.Added 2 and –3 before multiplying.The two terms are2 and 3(5+4).

Answer: –25

2 - 3(5 + 4)= 2 - [3(5 + 4)]= 2 - [3(9)] P= 2 - 27 M= 2 + (-27) A= -25

Order P, M, A

9 - 4 = 5 27 ÷ 9 = 3

-25 + 27 = +2

2) Simplify:

–24 ÷ (–3) (–4)

Work all multiplications OR divisions working from left to right.

3) Simplify: − + − −( )

÷ −( ) +3 5 2

16 4 2

2 Perform order of operations PEMDAS.

Exponents should be done fi rst.

275


Worked SolutionWhat is Wrong Here?

Identify the Errors Correct Process Validation(optional)

4) Simplify:

(0.04)2 + 6.3 – (15 – 4.7)

The decimal point is placed incorrectly in the product of (.04)2.

Line up the decimal points and trailing zeros when subtracting 4.7 from 15.

5) Simplify:

− −( )− ( )−( ) + −( )

2 7 3 5 1

2 3 5 63

There needs to be a single sign in the fi nal answer.

simplifying expressions using the order … · · 2013-08-07267 activity 6.3 — simplifying...

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