Objectives To review the order of operations; and to evaluate expressions containing parentheses. Assessment Management

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1 Order of Operations Objectives To review the order of operations; and to evaluate expressions containing parentheses. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Evaluate numerical expressions involving parentheses and exponents. [Number and Numeration Goal 4] Add, subtract, multiply, and divide signed numbers. [Operations and Computation Goals and ] Apply the order of operations to evaluate expressions. [Patterns, Functions, and Algebra Goal ] Key Activities Students evaluate expressions according to the order of operations. Ongoing Assessment: Informing Instruction See page 56. Ongoing Assessment: Recognizing Student Achievement Use journal page. [Patterns, Functions, and Algebra Goal ] 4 Playing Name That Number Student Reference Book, p. 9 Math Masters, p. 46 per partnership: complete deck of number cards (the Everything Math Deck, if available) Students practice applying order of operations. Math Boxes 6 6 Math Journal, p. 4 Students practice and maintain skills through Math Box problems. Study Link 6 6 Math Masters, p. 96 Students practice and maintain skills through Study Link activities. ENRICHMENT Exploring a Grouping Symbol Math Masters, p. 97 Students explore the use of the fraction bar as a grouping symbol. EXTRA PRACTICE Evaluating Expressions Student Reference Book, p. 47 (optional) Math Masters, p. 98 Students practice using the order of operations to evaluate expressions. Key Vocabulary order of operations nested parentheses Materials Math Journal, pp. and Student Reference Book, p. 47 Study Link 6 5 calculator slate Advance Preparation Review pages of the Student Reference Book. If students do not use TI-5 or Casio fx-55 calculators, check that their calculators are programmed for order of operations. Teacher s Reference Manual, Grades 4 6 p Unit 6 Number Systems and Algebra Concepts

2 Getting Started Mathematical Practices SMP, SMP, SMP4, SMP5, SMP6 Content Standards 6.NS.5, 6.EE., 6.EE.c Mental Math and Reflexes Students write a number to model each situation. In golf, par is the number of strokes a player should need to complete a hole or round. In Jeff s first round of golf, he shot over par. In Jeff s second round of golf, he shot 5 under par. -5 Karina s math test score was _ points above the class mean. _ Anastasia s math test score was 5.75 points below the class mean On Monday, the stock market dropped 6. points. -6. On Tuesday, the stock market rose 58 _ 7 points. 58 _ 7 Math Message Evaluate the following expressions / / / 7 6. / Study Link 6 5 Follow-Up Ask students to share the patterns they found in the tables. Make sure that the following patterns are mentioned: x + y = y + x y - x is the opposite of x - y. x y = y x y x is the reciprocal of x y. Teaching the Lesson Math Message Follow-Up WHOLE-CLASS Algebraic Thinking Record students answers on the board. If students disagree, record all proposed answers. Then have students use a scientific calculator to resolve discrepancies. If they do not have scientific calculators, skip ahead to the next section. Order of operations was formally introduced in Fifth Grade Everyday Mathematics. Prior to this lesson, Everyday Mathematics has stressed the use of parentheses to evaluate expressions. If students did not use order of operations in this activity, their answers for expressions, 4, and 5 will differ from the answers they obtained with their calculators. Ask students to insert parentheses in each expression to show which operation the calculator did first. Record these expressions on the board. NOTE Scientific calculators have built-in order of operations. Many four-function calculators do not. Please check students calculators to be sure. Expression Number Expression Left to Right Calculator Parentheses. - 8 / (8 / 4). 4 / (4 / ) (6 5) (9 ) / (6 / ) 6. / ( / 8) + For expressions, 4, and 5, the scientific calculator did not evaluate the expressions in the order in which the numbers and operations were entered. In each of these expressions, the calculator carried out the addition or subtraction last. A scientific calculator will follow order of operations, regardless of the order in which an expression is entered. Lesson

3 Order of Operations In many everyday situations, the order in which things are done is important. When you bake a cake, for example, you crack the eggs before adding them to the batter. In mathematics, too, many operations should be done in a certain order. Rules for the Order of Operations. Do the operations inside parentheses. Follow rules 4 when you are working inside parentheses.. Calculate all expressions with exponents.. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. Some people remember the order of operations by memorizing this sentence: P E M D A S P E M D A S Evaluate. 5 * 4 6 *? Multiply first. 5 * 4 6 * Subtract next. 0 8 Then add. 5 * 4 6 * 4 4 Evaluate 5 ( * 4 ) / 5. Clear parentheses first. 5 ( * 4 ) / 5 Calculate exponents next. 5 0 / 5 Divide. 5 0 / 5 Then add ( * 4 ) / / 8. (5 * 0) / 4. 0 * 6 / * ( / 6 4) / Check your answers on page 4. Student Reference Book, p. 47 Algebra Reviewing Order of Operations (Student Reference Book, p. 47) WHOLE-CLASS DISCUSSION Algebraic Thinking Write the following sentence on the board: Nancy fed Tom the big gray cat. Ask students to punctuate the sentence so it is clear that Nancy is not giving Tom a cat to eat. Nancy fed Tom, the big gray cat. Explain that the use of parentheses in an expression and the order of operations make up a mathematical punctuation system. As a class, read and work through the examples on Student Reference Book, page 47. Discuss why, in the absence of parentheses, it is necessary to adhere to the order of operations. Sample answer: Everyone can agree on the same value for an expression. Use the mnemonic device Please Excuse My Dear Aunt Sally to identify the order in which operations should be performed. Point out that in expressions with only addition and subtraction or only multiplication and division, neither operation has priority over the other. They are done in order from left to right. For example, the answer to is, not 0, and the answer to 4 / 4 is, not. Adjusting the Activity ELL Have students create original mnemonic devices to remember the order of operations, for example, Please Exile My Devious Annoying Sibling. Post students devices in the classroom. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Have students evaluate the Check Your Understanding expressions and explain how they got their results. Suggest that they follow the format of the examples in the Student Reference Book and write each step on a separate line. This makes it easy to keep track of the order in which operations are done. Have students use a calculator to check their results. If the calculator is scientific, students can enter the expression from left to right. Pose additional expressions if students need more practice. 56 Unit 6 Number Systems and Algebra Concepts

4 Evaluating Expressions (Math Journal, pp. and ) PROBLEM SOLVING INDEPENDENT ELL Algebraic Thinking Assign journal pages and. Have students evaluate the expressions independently and then compare results with partners. Write disagreements on the board and ask students to justify their thinking. Have students use a scientific calculator to resolve any disputes. Ongoing Assessment: Informing Instruction Watch for students who may be confused by the omission of the multiplication symbol in algebraic expressions. Remind students that expressions such as N and N are equivalent. Similarly, 5(4 + ) and 5 (4 + ) have the same meaning. Date 6 6 Order of Operations Show your work. Then compare your results to those of your partner / ( 5) 4. 4 / Time / Some expressions use two sets of parentheses, one inside the other. The operation in the inner parentheses is done before the operation in the outer parentheses. Double parentheses are often referred to as nested parentheses. To support English language learners, write nested parentheses on the board and include some examples. When most students have finished, bring the class together and discuss expressions that students found difficult. Math Journal, p. Ongoing Assessment: Recognizing Student Achievement Journal Page Problems 8 Use journal page, Problems 8 to assess students ability to apply the order of operations to evaluate numerical expressions. Students are making adequate progress if they are able to evaluate expressions in Problems 8. Some students may be able to evaluate expressions containing exponents (Problems 9 and 0). [Patterns, Functions, and Algebra Goal ] Date 6 6 Time Order of Operations continued (( 4 ) ) 6. (8) (4) 6 () / (8) / 6 ( 5) Try This 5. 5(6 ()) / ( 4 ) ( ( )) 4 7. Evaluate the following expressions for x. 5 a. x x 4 / b. x (6 x) / 9 Math Journal, p. Lesson

5 Date 6 6 Math Boxes. Solve. Simplify your answers. a b c Triangles JKL and MNO are congruent. J K Which MO side corresponds with JL? L M O N 9 9 Time. Multiply or divide. a. 0(4.5) b. 4 (5) 70 c Label the axes of this mystery graph and describe a situation it might represent. Sample answers: x-axis Time (seconds) y-axis Speed (mph) Situation A car begins to move after the traffic light turns green Ongoing Learning & Practice Playing Name That Number (Student Reference Book, p. 9; Math Masters, p. 46) PARTNER Students use the order of operations to match a target number. For each round, students write their answers as a single number sentence on the game record sheet. Ask students to organize their number sentences to avoid using parentheses whenever possible. 5. Two dice are tossed. Some possible outcomes appear in the table below. Complete the table. (,) (,) (,) (,4) (,5) (,6) (,) (,) (,) (,4) (,5) (,6) (,) (,) (,) (,4) (,5) (,6) (4,) (4,) (4,) (4,4) (4,5) (4,6) (5,) (5,) (5,) (5,4) (5,5) (5,6) (6,) (6,) (6,) (6,4) (6,5) (6,6) 6 a. How many equally likely outcomes are there? b. What is the probability of tossing a multiple of on both dice? c. What is the probability of tossing a composite number on the first die and a prime number on the second die? Math Journal, p. 4 9, 6 or 4 6, 6 or Math Boxes 6 6 (Math Journal, p. 4) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-8. The skill in Problem 5 previews Unit 7 content. Writing/Reasoning Have students write a response to the following: Explain how you know that L and O in Problem are congruent. Sample answer: The two triangles are congruent and L and O are corresponding, so L and O are congruent. Study Link 6 6 (Math Masters, p. 96) INDEPENDENT Home Connection Students use order of operations to evaluate expressions. Study Link Master STUDY LINK 6 6 Using Order of Operations Game Master Name That Number Record Sheet 9 4 Please Excuse My Dear Aunt Sally Parentheses Exponents Multiplication Division Addition Subtraction 47 Round. 5 6 º. 4 º 9 / ( 4 6) 5 8 º 4. (. 7.8) º ( ( 5)) 6. (( º 4) ) º 6 / Round Evaluate the following expressions for m. 7. m m ((4 ) º ) / 9 º ( m) 8 º m m ( (m )) 8 0. Practice Find each missing number gal 7 qt 4 gal qt. 5 gal qt qt. pt qt pt 4. 0 c qt pt 5. 8 qt gal pt Math Masters, p Units of Capacity cups (c) pint (pt) pints quart (qt) 4 quarts gallon (gal) -5 Name That Number Record Sheet Round Round Math Masters, p Unit 6 Number Systems and Algebra Concepts

6 Differentiation Options 6 6 Teaching Master Another Grouping Symbol A fraction bar (sometimes referred to as the vinculum) indicates division. An expression such as (8 4) (5 ) can be written as ( 8 4). ( 5 ) ENRICHMENT INDEPENDENT Exploring a Grouping Symbol (Math Masters, p. 97) 5 5 Min To further explore order of operations, students use the fraction bar as a grouping symbol. They evaluate expressions that involve fractions and nested parentheses. EXTRA PRACTICE PARTNER Evaluating Expressions (Student Reference Book, p. 47; Math Masters, p. 98) 5 0 Min Copyright Wright Group/McGraw-Hill Copyright Wright Group/McGraw-Hill A fraction bar also acts as a grouping symbol. Both the numerator and the denominator can be treated as if there were parentheses around them. Any operations in the numerator or the denominator must be performed before the division. Example: Evaluate º º 7 ( 9 7) (4 º ) ( 5 7) 6( 6 4) 7(8 ) (4 ) [4 (7 )] (0 7) Another Grouping Symbol A fraction bar (sometimes referred to as the vinculum) indicates division. An expression such as (8 4) (5 ) can be written as ( 8 4). ( 5 ) A fraction bar also acts as a grouping symbol. Both the numerator and the denominator can be treated as if there were parentheses around them. Any operations in the numerator or the denominator must be performed before the division. Example: Evaluate º º 7 ( 9 7) (4 º ) Students practice evaluating numerical expressions involving the order of operations. This page provides practice with order of operations without the distraction of difficult computations ( 5 7) 6( 6 4) 7(8 ) (4 ). 4. 6[4 (7 )] (0 7) 7 9 Math Masters, p Teaching Master 6 6 Order of Operations Please Excuse My Dear Aunt Sally Parentheses Exponents Multiplication Division Addition Subtraction Compare your result to a partner s. If you don t agree, discuss how you evaluated the expression to decide which result is correct º / / / (7 ) / º 4 / Math Masters, p. 98 Lesson