2.9. Order of Operations. A Fairly Ordered Operation. Order of Operations. ACTIVITY 2.9 Investigative. Activity Focus

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1 SUGGESTED LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text,, Quickwrite Ayana and Zachary Wilson are excited about going to the Pace County Fair. General admission to the fair is $8.00 per person. It covers visiting exhibits and some entertainment. Tickets for rides and games must be bought separately. Food and drinks are purchased at the concession stands. A ride ticket costs $ Ayana loves to make lists of things to do to prepare for an activity. She made the following list for the morning of the fair. To Do Comb my hair Order 8 ACTIVITY 2.9 Investigative Activity Focus Materials No special materials are needed. Chunking the Activity #1 #8 #12 #2 4 #9 #13 #5 #10 #14 #6 7 #11 #15 Take a shower Brush my teeth Eat breakfast Put my clothes on Buy my tickets for the fair Put on my shoes Put on my socks Ride to the fair Quickwrite After the students give the order of the steps it is very important that they explain why the order of the steps is important. TEACHER TO You could have students TEACHER write each step on a sticky note or an index card so they can easily play around with the order of the steps before entering the order in the student book. 2 Close Reading, Guess and Check, Quickwrite The intent of this question (and the next two) is to show a need for an agreement on the order in which operations should be performed. After completing this first set of questions, students will learn that multiplication should be performed before addition. Get money out of piggy bank a. Order the steps as you think Ayana will complete them. Answers may vary. Sample answer: See table above. b. Explain why the order of steps is important. Answers may vary. Sample answer: Some things have to be done before others, such as putting on socks before shoes. 2. Ayana plans to go on 5 rides. She wrote the numerical expression to represent the cost of her rides and admission to the fair. a. What is the total cost of Ayana s admission to the fair and the rides she wishes to go on? $23 b. Do you think her expression represents this total cost? Answers may vary. Sample answer: Yes, if you multiply first. 7 Unit 2 Operations with Numbers 115 Unit 2 Operations with Numbers 115

2 Continued 3 Close Reading, Guess and Check, Quickwrite 4 Debrief Students should determine that multiplication should come before addition. Students should also determine that the way expressions are written is important. 5 Close Reading, Guess and Check, Quickwrite Students should be able to explain that multiplication is performed before addition. 6 Close Reading, Guess and Check, Quickwrite With this question, students understanding of order of operations for adding and subtracting continues to be developed. 116 SpringBoard Mathematics with Meaning Level 1 SUGGESTED LEARNING STRATEGIES: Close Reading, Guess and Check, Quickwrite, Work Backward 3. Zachary intends on going on 8 rides in the morning and 5 rides in the afternoon. He wrote the expression to represent the total cost of the rides he wishes to go on. a. What is the total cost of Zachary s rides? $39 b. Do you think his expression represents this total cost? Answers may vary. Sample answer: Yes, if you add first. 4. Is it a problem that both Ayana and Zachary used the same mathematical expression to represent two different costs? Answers may vary. Sample answer: Yes, because the expressions have different values and it is not clear which is correct. Mathematicians have agreed that when evaluating an expression containing both addition and multiplication the operation of multiplication should be performed first. 5. Can Ayana use the expression to represent the cost of her admission ticket and rides or can Zachary use it to represent the cost of his rides? Ayana, because multiplication must be performed before addition. The day of the fair both Ayana and Zachary realized that they had not figured food into the cost of going to the fair. 6. Ayana has $100 in her piggy bank. She took $60 out to go to the fair. Then her dad gave her $5 allowance. She does not need this $5 for the fair so she puts it in her piggy bank. Ayana wrote the expression to represent the amount of money that is now left in her bank. a. What is the total amount left in Ayana s piggy bank? $45 b. Do you think her expression represents this total amount? Answers may vary. Sample answer: Yes, you can add or subtract first and the answer is the same. 116 SpringBoard Mathematics with Meaning Level 1

3 SUGGESTED LEARNING STRATEGIES: Close Reading, Guess and Check, Work Backward, Quickwrite, Think Aloud, Marking the Text, 7. Zachary has $100 in his piggy bank. He took $60 out for rides, food, and admission to the fair, and he took $5 out for a souvenir. Zachary wrote the expression to represent the amount of money now left in his bank. a. What is the total amount left in Zachary s piggy bank? $35 b. Do you think his expression represents this total amount? Answers may vary. Sample answer: No, because you should not add when you take something away. Think about the order in which operations should be performed. When evaluating a numerical expression with addition and subtraction, the operations of addition and subtraction should be performed in the order in which they appear from left to right. 8. Does the expression represent the amount left in Ayana s bank or the amount left in Zachary s bank? It represents the amount in Ayana s bank: = 45. After that Zachary decided to take only $60 to the fair. Ayana and Zachary s mom decided they needed to take some snacks to the fair. She took a bag of 6 granola bars and divided the bars evenly into three sacks. She did the same thing with three more bags of 6 granola bars, sharing them evenly into the same three sacks. She then gave Ayana and Zachary each a sack. 9. When Ayana asked Zachary how many granola bars were in each sack, he said the expression represented the number of granola bars. a. Ayana evaluated the expression and said the bag contained less than 1 whole granola bar. Explain how she arrived at this answer. Answers may vary. Sample answer: = 6 12 =.5. Continued 7 Guess and Check, Quickwrite Some students may see that enclosing in parentheses and then subtracting from 100 makes the expression true. TEACHER TO After Question 7 would TEACHER be a good time to have students connect mathematical properties and order of operations to the rules by which they play a game, such as soccer, chess, or volleyball. Paragraph Close Reading 8 Guess and Check, Quickwrite Students will use what they just learned about the order in which to do addition and subtraction to do this problem. 9 This question gives students practice in deciding the order in which to perform multiplication and division. b. How many granola bars were actually in each bag? Explain how Zachary derived his expression. Explanations may vary. Sample answer: 8 bars; 6 bars 3 = 2; 2 4 bags of bars = 8. Unit 2 Operations with Numbers 117 Unit 2 Operations with Numbers 117

4 Continued SUGGESTED LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text,, Guess and Check, Create Representations 0 Guess and Check, Create Representations This is a summative question that asks students to use their knowledge of the order of operations for addition, subtraction, multiplication and division to play the game. Point out that the row of three can be horizontal, vertical, or diagonal. To extend this activity, you might have students make their own game cards on 3 3 grids, putting the numbers from 1 through 9 in any order they choose. Students may want to try using numbers other than 2 with the number sentences. Suggested Assignment CHECK YOUR UNDERSTANDING p. 122, #1 6 UNIT 2 PRACTICE p. 137, #59 63 ACADEMIC VOCABULARY The order of operations is a set of rules for evaluating expressions with more than one operation. The order is: Do calculations inside parentheses first. Evaluate expressions with exponents. Multiply or divide from left to right. Add or subtract from left to right. Tic-tack-two game card Answers may vary. Sample answers shown Numerical expressions can contain the operations of addition, subtraction, multiplication, and division, so it is important to follow an order of operations to avoid confusion. The operations of multiplication and division are evaluated from left to right. The operations of addition and subtraction are also evaluated from left to right. 10. At the fair Ayana used one ticket to play Tic-Tack-Two. She was handed a Tic-Tack-Two game card and this score sheet. To play, each person should fill in the spaces between the 2s with any operation sign: +, -,, or to equal the number at the end of the row. You can use an operation more than once in a row. When you get a number, put an X over that number on your Tic-Tack-Two card. The first player to get three in a row wins. An example, which is not a value on the game card, has been done for you. Play this game with the members of your group. Remember to use the order of operations = = = = = = = = = = SpringBoard Mathematics with Meaning Level SpringBoard Mathematics with Meaning Level 1

5 SUGGESTED LEARNING STRATEGIES: Close Reading, Summarize/Paraphrase/ Retell, Guess and Check, Work Backward, Quickwrite 11. Zachary decided to join Ayana in a game of darts. They found the following target to be very unusual. Continued a Close Reading, Think Aloud, Marking the Text, Summarize/ Paraphrase/Retell, Guess and Check, Work Backward, Quickwrite (b, c) The intent of this question is to show that exponents must be evaluated before multiplication Each player gets to throw two darts at the target. The numbers on the target are the points a player gets for each dart that lands in that region of the target. Both of Zachary s darts landed in the gray region of the target. a. Ayana expressed his score as What did Ayana get for his Zachary s score? 18 b. Zachary expressed his score as Explain how he would have to evaluate this expression so that he would get the same score as Ayana did? Answers may vary. Sample answer: He would have to evaluate 3 2 before he multiplies. c. What do you think is the order that mathematicians have agreed upon when evaluating an expression with exponents and multiplication? Exponents must be evaluated before multiplication. Unit 2 Operations with Numbers 119 Unit 2 Operations with Numbers 119

6 Continued b Guess and Check, Work Backward The intent of this question is to show how parentheses can change the order in which operations are performed. c Guess and Check, Work Backward This question provides practice evaluating an expression with parentheses. SUGGESTED LEARNING STRATEGIES: Close Reading, Summarize/Paraphrase/ Retell, Guess and Check, Work Backward As they drove home from the fair, Zachary wondered if there was a way to change the order in which an expression is evaluated. T heir father explained that parentheses can be placed around the numbers and operations to be performed first. 12. Add parentheses to the expression so that it can be evaluated in such a way as to give Zachary the cost of buying tickets for 8 rides in the morning and 5 rides in the afternoon at $3.00 per ride. (8 + 5) 3 Mathematicians have agreed that in evaluating an expression involving grouping symbols, such as parentheses or brackets, the operations inside the grouping symbols should be completed before completing those operations outside the grouping symbols. 13. T he Wilsons decide to stop at a restaurant on the way home from the fair. Part of the menu is shown below. SNACKS VALUE MEALS The Wilsons order one hamburger value meal, four apples, two juices, and two pizza value meals. a. Write an expression to find the total cost of the family s meal. 8 + (4 + 2) b. Find the total cost of the family s meal. Show your work. $ SpringBoard Mathematics with Meaning Level SpringBoard Mathematics with Meaning Level 1

7 Continued SUGGESTED LEARNING STRATEGIES: Summarize/ Paraphrase/Retell, Create Representations 14. Since every numerical expression should have only one value, mathematicians have agreed upon an order of operations for evaluating expressions. Describe the steps involved in simplifying a numerical expression using order of operations. Step 1: Evaluate all expressions in parentheses. Step 2: Evaluate all expressions involving exponents. Step 3: Complete all multiplication and division in order from left to right. Step 4: Complete all addition and subtraction in order from left to right. When Ayana arrived home she wanted to f igure out how much she and Zachary had left of the money they took to the fair. Both she and Zachary paid $8 for an admission ticket. She went on a total of 5 rides and Zachary went on 8 rides in the morning and 5 rides in the af ternoon at $3 per ride. She played 2 games and he played 1 at $3 per game. Zachary bought a $5 souvenir. The following expression represents the total amount of money she and Zachary have lef t of the $60 that they each took with them to the fair [2 8 + ((8 + 5) + 5) 3 + (1 + 2) 3 + 5] d This is a summative question that asks students to put all of the facts they have learned together and come up with a set of steps describing the order in which a numerical expression can be evaluated. Notice the parentheses within parentheses. T hese are called nested parentheses. You must always work the innermost parentheses first. Notice too that brackets have been used. Brackets can be used in place of parentheses, usually on the outside of parentheses. Unit 2 Operations with Numbers 121 Unit 2 Operations with Numbers 121

8 Continued e Summarize/Paraphrase/ Retell, Create Representations This question provides practice using nested parentheses. Suggested Assignment CHECK YOUR UNDERSTANDING p. 122, #7 12 UNIT 2 PRACTICE p. 137, # No parentheses are needed (2 + 3) = (4-4) 4 = Answers may vary. Sample answer: 7 + (3-1) 1 = Answers may vary. Sample answer: 10 + (5-5) Answers may vary. Sample answer: Situations such as those in this lesson can result in confusion when writing an expression unless there is understanding about how to write expressions and how to interpret them. CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work Simplify using order of operations (8 + 2) 4 4. (1 + 3) (56 8 3) SpringBoard Mathematics with Meaning Level 1 SUGGESTED LEARNING STRATEGIES: Close Reading, Summarize/Paraphrase/ Retell, Guess and Check, Work Backward 15. Ayana started to evaluate the expression. Finish her work. Notice that parentheses around a single number can be dropped without changing any values [2 8 + ((8 + 5) + 5) 3 + (1 + 2) 3 + 5] [2 8 + ((13) + 5) 3 + (1 + 2) 3 + 5] [2 8 + (18) 3 + (3) 3 + 5] [ ] [ ] [ ] Insert parentheses when needed to make the number sentence true = = = 0 Use parentheses and the symbols +, -,, and to make each number sentence true = = MATHEMATICAL Give some reasons why REFLECTION mathematicians had to agree on an order of operations to be used to simplify a numerical expression. 122 SpringBoard Mathematics with Meaning Level 1