Project Objective To introduce the rules for order of operations. www.everydaymathonline.com etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Doing the Project Recommended Use During or after Unit 7 Key Concepts and Skills Use basic facts to solve numeric expressions. [Operations and Computation Goals 1 and 3] Recognize that numeric expressions can have different values depending on the order in which the operations are carried out. [Patterns, Functions, and Algebra Goal 3] Key Activities To explore order of operations, children use two calculators, one that follows the conventional, algebraic order of operations and one that follows a simple left-toright order of operations. Key Vocabulary order of operations Extending the Project Mathematical Practices SMP1, SMP5, SMP6, SMP8 Content Standards 3.OA.5, 3.OA.8 Materials Student Reference Book, p. 74H Math Masters, p. 394A Class Data Pad slate Per group: 2 calculators one that follows the conventional order of operations (such as a scientific calculator) and one that follows a left-to-right order of operations (such as most four-function calculators, including the TI-108 and Casio SL-450). See Advance Preparation. Using a calculator that follows a left-to-right order of operations to solve number sentences, children record the key sequences that should be used to follow the conventional, algebraic order of operations. Children practice skills through Home Link activities. Materials Math Masters, p. 394A Per group: 2 calculators. See Materials list in Part 1. Student Reference Book, p. 74H Additional Information Once children begin to encounter expressions such as 5 + 3 2, they have to decide the order in which the indicated operations should be carried out. The simplest order is to begin at the left and carry out the operations in the order in which they appear. This simple left-to-right order of operations is built into many four-function calculators. But it is neither the order of operations commonly used in higher mathematics nor the one built into more advanced calculators. When people refer to, by convention they mean the algebraic order of operations in which multiplication and division take precedence over addition and subtraction and operations of equal precedence are performed from left to right. There are other, more specialized orders of operation that are used in technical fields but are not pertinent to school mathematics. In Everyday Mathematics, we follow common usage and refer to the conventional order of operations as the. See Section 10.1.2 of the Teacher s Reference Manual for further discussion. Advance Preparation For Part 1, gather enough calculators to provide two for each small group of 3 or 4 children. You might consider using calculators on cell phones and on the Internet. For each group, one calculator should follow the conventional order of operations (such as a scientific calculator); the other should follow a simple left-to-right order of operations (such as most four-function calculators including the TI-108 and Casio SL-450). To check whether a calculator follows the conventional order of operations, enter 6 5 3. Calculators that follow the conventional order of operations will show 21 in the display. Calculators that follow the left-to-right order will show 33 in the display. Copy the list of rules for order of operations from Teacher s Lesson Guide, page 502B onto the Class Data Pad. 936 Project 7
1 Doing the Project Exploring SMALL-GROUP Tell the class that you have a mystery for them. Explain that you wanted to solve 6 + 5 3. But when you used two different calculators, you got two different answers. One calculator gave the result as 21; the other gave 33. Write the following number sentences on the board: 6 + 5 3 = 21 6 + 5 3 = 33 Ask children to figure out how each calculator solved the problem. To get 21, one calculator multiplied 5 3 first and then added the result to 6; 5 3 = 15; 6 + 15 = 21. To get 33, the other calculator added 6 + 5 first and then multiplied the result by 3; 6 + 5 = 11; 11 3 = 33. Distribute two calculators to each small group one that follows the conventional order of operations (such as a scientific calculator) and one that follows a left-to-right order of operations (such as most four-function calculators, including the TI-108 and Casio SL-450). Have each group use each calculator to solve 15-7 2 =? Have them share their answers. 16 (left-to-right) and 1 (scientific) Ask children to explain how each of their calculators solved the problem. Sample answers: One calculator (scientific) multiplied first and then subtracted. The other calculator (left-to-right) subtracted first and then multiplied. Explain that some calculators carry out operations in the order they are entered, or left-to-right, in the number sentence. Other calculators follow a special set of rules that tell which operation to do first and which operation to do next. Project 7 936A
Student Page Operations and Computation In many situations, the order in which things are done is important. For example, you always put on your socks before your shoes. In solving number models, certain things must be done in a certain order. For example, to solve 8 + 4 3 =, you must know whether to add first or to multiply first. There are rules that tell you what to do first and what to do next. 74H 1. Do operations inside parentheses first. 2. Multiply and divide in order, from left to right. 3. Add and subtract in order, from left to right. Solve. 8 + 4 3 = Multiply first. 8 + 4 3 Then add. 8 + 12 The answer is 20. 20 8 + 4 3 = 20 Solve. 10 - (9-6 2) = Start inside parentheses. Divide first. 10 (9 6 2) Then subtract. 10 (9 3) Subtract again. 10 6 The answer is 4. 4 10 (9 6 2) = 4 Student Reference Book, p. 74H Scientific calculators always follow the rules for the order of operations. Introducing Order of Operations (Student Reference Book, p. 74H) WHOLE-CLASS Tell children that to avoid confusion when solving number sentences, mathematicians have agreed to a set of rules, called the order of operations. These rules tell you what to do first and what to do next. Refer children to the list of rules you prepared on the Class Data Pad. 1. If there are parentheses, do the operations inside the parentheses first. Follow rules 2 and 3 when computing inside parentheses. NOTE It is important to emphasize that multiplication and division are of equal priority. This means that you perform whichever of these two operations comes first, from left to right. Likewise, addition and subtraction are of equal priority. As a class, read about the order of operations on Student Reference Book, page 74H. Practicing (Math Masters, p. 394A, Student Reference Book, p. 74H) WHOLE-CLASS Name PROJECT 7 Project Master Date Use order of operations to solve each number sentence below. Show your work. To check your work, use a calculator that follows order of operations. 1. If there are parentheses, do the operations inside the parentheses first. 1. 2 = 11 3 3 2. 15 3 + 2 = 7 3. 8 = 20 5 2 4. 6 + 4 2 = 8 5. (22 + 8) 2 = 60 6. 16 = (7 2) + 14 7 Write number sentences on the board and have children copy them onto their slates. Referring to the Rules for the Order of Operations on the Class Data Pad or on Student Reference Book, page 74H, children underline the operation that should be performed first and then solve. Have volunteers share their solution strategies. Suggestions: 8-2 3 = 2 20 = 0 20 + 20 12 + 6 2 = 15 13 = 15-8 + 6 10 5 (2 + 5) = 14 6 = (15 6) 9 + 5 When children are ready, have them work in small groups, following the rules for the order of operations to solve the number sentences on Math Masters, page 394A. They use scientific calculators (those that follow the conventional order of operations) to check their work. Bring the class together to share answers. Math Masters, p. 394A 936B Project 7
2 Extending the Project Extension Suggestions (Math Masters, p. 394A) PARTNER For each problem on Math Masters, page 394A, have children record the key sequence that should be used to correctly follow the conventional order of operations with a calculator that follows a left-to-right order. Children use a left-to-right calculator to check their work. For example, the key sequence to solve 11 3 3 is: 3 3 9 11 9 2. Home Link Suggestions (Student Reference Book, p. 74H) INDEPENDENT Children discuss the order of operations and share Student Reference Book, page 74H with family members. Children and family members determine whether their home calculators follow the conventional order of operations. Project 7 936C
Name Date PROJECT 7 Use order of operations to solve each number sentence below. Show your work. To check your work, use a calculator that follows order of operations. 1. If there are parentheses, do the operations inside the parentheses first. 1. = 11 3 3 2. 15 3 + 2 = 3. = 20 5 2 4. 6 + 4 2 = Copyright Wright Group/McGraw-Hill 5. (22 + 8) 2 = 6. = (7 2) + 14 7 394A