Using Order of Operations - Virginia Department of Education

ARI Curriculum Companion ? Using Order of Operations and Exploring Properties

Introduction

The lessons in this section focus on order of operations and evaluating numerical expressions. Students will also investigate and recognize properties of real numbers used to solve an equation. These lessons form an outline for your ARI classes, but you are expected to add other lessons as needed to address the concepts and provide practice of the skills introduced in the ARI Curriculum Companion. Some of the lessons cross grade levels, as indicated by the SOL numbers shown below. This is one method to help students connect the content from grade to grade and to accelerate.

Standards of Learning

Order of Operations 5.7 The student will evaluate whole number numerical expressions, using the order of operations

limited to parentheses, addition, subtraction, multiplication, and division. 6.8 The student will evaluate whole number numerical expressions, using the order of operations. 8.1 The student will

a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers; and

8.4 The student will apply the order of operations to evaluate algebraic expressions for given replacement values of the variables.

Properties of Real Numbers 5.19 The student will investigate and recognize the distributive property of multiplication over addition. 6.19 The student will investigate and recognize

a) the identity properties for addition and multiplication; b) the multiplicative property of zero; and c) the inverse property for multiplication. 7.16 The student will apply the following properties of operations with real numbers: a) the commutative and associative properties for addition and multiplication; b) the distributive property; c) the additive and multiplicative identity properties; d) the additive and multiplicative inverse properties; and e) the multiplicative property of zero. 8.1 The student will a) simplify numerical expressions involving positive exponents, using rational numbers, order of

operations, and properties of operations with real numbers; and 8.15 The student will

c) identify properties of operations used to solve an equation.

Table of Contents

Lesson plans pertaining to the following Standards of Learning are found in this section. Click (or CTRL+click) on each to jump to that lesson.

SOL 5.7, 6.8 ....................................................................................................................... 2 SOL 5.7, 6.8 ....................................................................................................................... 6 SOL 5.19, 7.16b ............................................................................................................... 10 SOL 6.19, 7.16c,d,e ......................................................................................................... 14 SOL 7.16 .......................................................................................................................... 17 SOL 8.1a .........................................................................................................Coming soon SOL 8.4 ............................................................................................................................ 24 SOL 8.15c .......................................................................................................Coming soon

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ARI Curriculum Companion ? Using Order of Operations and Exploring Properties

SOL 5.7, 6.8

Lesson Summary

Students discover the order of operations, using the areas of rectangles. (45 minutes)

Materials

Overhead transparency of three rectangles Copies of the attached worksheet

Warm-up

Review with students the process of finding the area of a rectangle. Use an overhead transparency to display three rectangles with the dimensions (9" x 4", 9" x 5", and 9" x 8") of each rectangle shown. Ask students to explain the most direct way to find the total area of any two of these rectangles. They should recognize that the most direct way is by finding the area of each rectangle and then adding the two areas, e.g., 9 ? 4 = 36; 9 ? 5 = 45; 36 + 45 = 81. Ask them to explain the most direct way to find the total area of all three rectangles. (Find the area of each, and then add the three areas.)

Lesson

1. Put the following problem on the board: 4 ? 3 + 2 ? 5 = x. Have students solve the problem and then share their answer with a partner. Ask the students whether anyone got an answer of 70. If so, ask how them how they got that answer. They will probably say that they multiplied 4 and 3 to get 12, then added 2 to 12 to get 14, and finally multiplied 14 by 5 to get 70. Ask the students if anyone got an answer of 22. If so, ask how they got that answer. The students who got this answer should explain that they multiplied 4 by 3 to get 12, then multiplied 2 by 5 to get 10, and finally added 12 and 10 to get 22. Discuss with the students why using these two different procedures would give such different answers.

2. Have students write an expression to model the process they used in the warm-up to find the total area of two rectangles. Students should write one of the following expressions: 9 ? 4 + 9 ? 5, 9 ? 4 + 9 ? 8, or 9 ? 5 + 9 ? 8. Select one of these expressions, and use it to model on the board the two procedures used in the warm-up: (1) doing all operations left to right and (2) doing all multiplications first and then doing addition.

3. Have students work in pairs to construct an explanation of why they must use the second procedure when finding the area of two rectangles. Allow plenty of time for discussion.

4. Allow pairs time to present their explanations. At the end, summarize the discussion to explain why the second procedure is correct.

5. Have the students write a rule, based on the discussion, for simplifying expressions that involve multiplication and addition. The rule should state that all multiplication must be done before any addition.

6. Have the students write an expression to find the total area of all three rectangles from the warm-up (e.g., 9 ? 4 + 9 ? 5 + 9 ? 8), and have students use their rule to simplify the expression.

7. Have the students work in pairs to see if there is another way to solve their original warm-up problem. Since the rectangles have one dimension the same, they could be put next to each other to create one big rectangle. For example, the 9 x 4 and 9 x 5 rectangles together create a 9 x 9 rectangle with an area of 81. Have students write an expression that models the above approach. One expression could be 9 ? (4 + 5). To simplify this expression, explain that 4 and 5 need to be added first and then the sum multiplied by 9 to get 81. Have students write a rule for simplifying expressions that involve parentheses and multiplication. The rule should state that all expressions inside of parentheses must be simplified before any multiplication.

8. Ask the students to simplify 32 and then add 5 to the answer. Have the students write an expression that models the problem. Students will most likely write 32 + 5. Explain that the commutative property allows the expression also to be written 5 + 32. Have the students write a rule for simplifying

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ARI Curriculum Companion ? Using Order of Operations and Exploring Properties expressions that involve exponents and addition. The rule should state that all exponents must be simplified before any addition.

9. Have students write a new rule that combines the three rules they have written. The rule should state that when simplifying, expressions inside parentheses must be simplified first, then all exponents must be simplified, then all multiplication is done, and, finally, all addition is done. Explain that this "order of operations" rule includes division with multiplication and subtraction with addition. Multiplication and division must be done from left to right first, and then addition and subtraction is done from left to right.

10. Help the class create a complete "order of operations" rule similar to the following: When simplifying, do all expressions inside parentheses first, then all exponents, then all multiplication and division operations from left to right, and finally all addition and subtraction operations from left to right.

It might be helpful for students to see how to organize the order of operations in a numbered list of steps, as shown below:

1. Do any work within parentheses ( ) or other grouping symbols [ ] first. 2. Do any work with exponents (powers) or roots. 3. Do any multiplication and division in order from left to right. 4. Do any addition and subtraction in order from left to right. 11. Have students create and write two expressions, each containing parentheses, exponents, and all operations. On a separate sheet of paper, have them simplify their expressions, showing each step in the order of operations and the final answer. Have them exchange their problems with a partner, simplify each other's expressions, and discuss the problems until agreement is reached on the correct order of operations and final answer.

Reflection

Have students complete the "Simplifying Expressions" worksheet.

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ARI Curriculum Companion ? Using Order of Operations and Exploring Properties

Name:

Simplifying Expressions

Simplify each expression, showing each step in the order of operations. To the right of each step, identify the step as parentheses, exponents, multiplication, division, addition, or subtraction.

Example (4 + 5) ? 4 - 32 + 9(2)

9 ? 4 - 32 + 9(2)

parentheses--addition

9 ? 4 - 9 + 9(2)

exponents

36 - 9 + 18

multiplication, left to right

27 + 18

subtraction, left to right

45

addition

1. 9 - 23

2. 72 - (7 + 8) ? 4

3. 64 - 4 ? 23 + 7

4. 3 + 7(23 - 6)2

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ARI Curriculum Companion ? Using Order of Operations and Exploring Properties

Name: ANSWER KEY

Simplifying Expressions

Simplify each expression, showing each step in the order of operations. To the right of each step, identify the step as parentheses, exponents, multiplication, division, addition, or subtraction.

Example (4 + 5) ? 4 - 32 + 9(2)

9 ? 4 - 32 + 9(2)

parentheses--addition

9 ? 4 - 9 + 9(2)

exponents

36 - 9 + 18

multiplication, left to right

27 + 18

subtraction, left to right

45

addition

1. 9 - 23 9 - 8 1

exponents subtraction

2. 72 - (7 + 8) ? 4 72 - 15 ? 4 72 - 60 12

parentheses--addition multiplication subtraction

3. 64 - 4 ? 23 + 7 64 - 4 ? 8 + 7 64 - 32 + 7 32 + 7 39

exponents multiplication subtraction, left to right addition

4. 3 + 7(23 - 6)2 3 + 7(8 - 6)2 3 + 7(2)2 3 + 7(4) 3 + 28 31

parentheses--exponents parentheses--subtraction exponents multiplication addition

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ARI Curriculum Companion ? Using Order of Operations and Exploring Properties

SOL 5.7, 6.8

Lesson Summary

Students review and practice simplifying expressions, using the order of operations. (45 minutes)

Materials

Copies of the attached worksheets Scissors

Warm-up

Review the Order of Operations handout with students. Talk with students about the different kinds of grouping symbols. Remind students that multiplication and division must be simplified from left to right before addition and subtraction are simplified from left to right.

Lesson

1. Distribute a copy of "Order of Operations 4 x 4 Square" and a pair of scissors to each student. Have students cut the squares apart. (Alternatively, pre-cut the squares and make sets of 16 squares inserted into envelopes for distribution to students.)

2. Have students simplify each expression on a separate piece of paper and record the answer below the printed expression.

3. Have students then match up each answer to one that is already printed on another square. Once all expressions are matched to their correct answers, a new 4 x 4 square will be formed.

Reflection

Hand out a copy of the blank "Order of Operations 3 x 3 Square" worksheet to each student. Have the students create and write an expression on each of two sides of each square (in the shaded boxes), simplify each expression, and write each answer on the matching side of the adjacent square. Have students exchange worksheets with a partner to check and verify their work. (You may wish to use these squares in another class for additional practice.)

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ARI Curriculum Companion ? Using Order of Operations and Exploring Properties

Order of Operations

Grouping Symbols Exponents

( ) { } [ ] |abs| Fraction bar

Multiplication Left to Division right

Addition Left to

Subtraction right

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ARI Curriculum Companion ? Using Order of Operations and Exploring Properties

Order of Operations 4 x 4 Square

1. Cut the squares apart along the heavy lines to get 16 small squares. 2. Simplify each expression on a separate sheet of paper, and write each answer below the

printed expression. 3. Lay the 16 small squares side by side so that each answer matches up to the same

number printed on another square. You should form a new 4 x 4 square.

24 ? 5 + 3

44 ? 62 + 1

(36 + 4) ? 12

52 ? 2 + 4

1 2

5

4 3

(5 ? 2) + 3

2 (22 + 6) ? 5

5 ? 2 + 4 ? 2

16 - 2 ? 6 + 1

15 (6 - 2)2 - 1

8 7 + 15 ? 3 - 4

11 21 - 5 ? 2

2 24 ? (6 ? 2)

1 4

1 1

5

5 2

(4 ? 7) - 6

(24 ? 6) + (2 ? 5)

(2 ? 3)2 - 25

25 ? (4 + 1)

8 30 ? (1 + 4) + 2

16 4 ? 3 + 8 ? 2

8 24 ? 6 ? 2

37 (8 + 4) ? (1 + 2) + 1

2 2

1 0

1 4

2 8

2 ? 92 ? 13

4 + 62 ? 2

(6 - 2) ? 1 + 6

(52 + 3) ? 2

3 6 - (22 - 1)

36 (30 ? 1) + (4 + 2)

9 8 + 4 ? (1 + 2 + 1)

14 36 ? 6 + 2 ? 4

2 4

1

1 3

3 1

17 ? 6 + 3

6 + 5 ? 4 - 2

7 - (2 ? 6) ? 2

14 + 1 - 6 ? 3

64

69

72

12

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