Objectives

Students will use the order of operations to evaluate expressions that contain grouping symbols. They will  - correctly evaluate numerical expressions with numerous grouping symbols (parentheses, brackets, or braces) using the order of operations.  - create numerical expressions having specified values by grouping symbols as needed.

Core Questions

- How can mathematics help to quantify, compare, depict, and model numbers? - How can mathematics help us communicate more effectively? - How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?

Vocabulary

- Braces: Symbols used to group certain parts of a mathematical expression, { }.  - Brackets: Symbols used to group certain parts of a mathematical expression, [ ].  - Numerical Expressions: A mathematical combination of numbers, operations, and grouping symbols.  - Order of Operations: The steps used to evaluate a numerical expression: 1) Simplify the expressions inside grouping symbols. 2) Evaluate all powers. 3) Do all multiplications and/or divisions from left to right. 4) Do all additions and/or subtractions from left to right.  - Parentheses: Symbols used to group certain parts of a mathematical expression, ( ).

Materials

- Let’s Go Golfing practice worksheet (M-5-6-2_Let's Go Golfing Practice Worksheet and KEY)  - index cards

Assignment

- Use the practice-hole example from the Let's Go Golfing practice worksheet to determine whether students require extra instruction.  - The exit slip can be used to assess if students understand how to apply the order of operations to calculate the value of an expression.

Supports

Scaffolding, Active Engagement, Modeling, and Explicit Instruction  W: The lesson shows how to use the order of operations to evaluate expressions with grouping symbols.  H: Ask students to evaluate an expression containing grouping symbols. Explain to students that without a specific order of operations, they will get many different values for the same expression.  E: Encourage students to evaluate expressions and describe their processes to the class. Peer teaching is an extremely strong technique. The more exposure students have to various explanation of the same concept, the more likely they are to grasp and remember it.  R: Students will write expressions with specific values. Students will use the order of operations to ensure that the expressions they wrote include the specified values. This is an excellent method to review what they've learnt. Have students practice creating and evaluating expressions in pairs.  E: Students will evaluate and create expressions during the the lesson. Make them record their work in their math notebook. Monitor student responses and clarify any misconceptions. Use the exit slip to assess students' progress.  T: To tailor the lesson to your students' requirements, refer to the Extension section for options. The Routine section provides strategies for keeping the concepts relevant throughout the year. The Small Group section offers extra practice opportunities for students who could benefit from more time and practice. The Expansion section presents additional difficulties to students who have mastered the concepts.  O: The lesson emphasizes the need of understanding the order of operations. The lesson begins with evaluating expressions that use grouping symbols. This activity allows  students to understand how to apply the order of operations when an expression contains grouping symbols. Students then create their own expressions with specified values and evaluate their work based on the order of operations. This activity shows students to translate word phrases and mathematical expressions in Lesson 3.

Procedures

"In this lesson, we will learn how to use the order of operations when an expression contains grouping symbols. This will help us grasp the confusion in Mr. Ryte's math lesson.  "Mr. Hu Ryte asked his students to estimate the value of the following expression: 18 - (4 + 2) + (6 × 5) ÷ 3  "He was surprised when he asked five students, and each had a different answer, as shown in the table below." 'Which student has the right answer?' he wondered." <img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_38.png" width="506" height="139"> “Take a few moments to work together and try to determine the value of this expression.” Ask students to discuss their values and describe the processes they followed. Students will most likely receive a variety of values based on the order in which they performed the operations; some students may not know where to begin. Lesson 1 also focused on the order of operations, but the expressions had no grouping symbols. Ask students to explain why Mr. Hu Ryte's students had so many different values for the same expression. Students are likely to remark that different people performed the calculations in different orders, as this was also discussed in Lesson 1. If needed for a class discussion, here are the procedures Julio and Maya took to get their solutions: Julio: 18 – (4 + 2) + (6 × 5) ÷ 3 =          Maya: 18 – (4 + 2) + (6 × 5) ÷ 3 =          18 – 6 + 30 ÷ 3                                        18 – 4 + 2 + (6 × 5) ÷ 3 =          12 + 30 ÷ 3                                              14 + 2 + 30 ÷ 3 =          42 ÷ 3 = 14                                              46 ÷ 3 = 15 1/3 Now, remind students of the order of operations taught in Lesson 1.  "Mathematicians agree on an order of operations. This is a specified order that ensures everyone receives the same value. Today, we'll use the order of operations to determine which of Mr. Hu Ryte's students has the correct answer. Mathematicians say Parentheses (P) or grouping symbols first, followed by Exponents (E), Multiplication and Division (MD) from left to right, and finally Addition and Subtraction (AS) from left to right. Let's use the order of operations to figure out which of Mr. Hu Ryte's students has the correct answer.” Help students evaluate the expression using the order of operations as shown here. 18 – (4 + 2) + (6 × 5) ÷ 3 = [Parentheses first] Notice: There are no Exponents in this expression. 18 – 6 + 30 ÷ 3 =                 [Multiplication/Division left to right next] 18 – 6 + 10 =                        [Addition/Subtraction left to right next] 22 "Using mathematicians' agreed-upon order of operations, we now know Sierra has the correct value of the expression, 22. But how do we remember the order of operations? Remember that the acronym PEMDAS symbolizes the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. Many students remember PEMDAS using the sentence, 'Please Excuse My Dear Aunt Sally.'" It is very helpful to demonstrate the order of operations in the classroom when students are first learning it. Present the following expressions: (4 + 5) × 8 - 3² (6 + 18) ÷ 3 × (5 – 1) Ask students to work in pairs to figure out the value of these expressions. Observe students while they work. Find a pair of students who found the correct values 63 and 32), and ask them to demonstrate and explain the processes they used on the board. (When students work in pairs, they feel less scared about sharing their work, and they may help each other record and explain the process.) Now present the following expressions: 13 + 21 ÷ (9 – 6) – (3 × 4) + 5 × 2 [(7 – 2) × 3] ÷ [4 + (2 ÷ 2)] – 7 × 0 Help students realize that the second expression has many grouping symbols, including braces and parentheses. “When there are multiple grouping symbols, always work on the innermost grouping symbols first.” Ask students to clarify which operations should be done first in the second expression. Students should begin with "7 - 2" and "2 ÷ 2" because these are the operations in the innermost grouping symbols. Now, ask students to work in pairs again to determine the values of both expressions. Request that pairs of students volunteer to demonstrate and describe on the board the processes they used. (The values of the expressions are 18 and 3, respectively.) The Let's Go Golfing worksheet (M-5-6-2_Let's Go Golfing Practice Worksheet and KEY) can be used to practice the order of operations. It is important to first introduce the game of golf, as not all students are familiar with it.  "In golf, par is the maximum number of strokes a player should take to get the ball into the hole. On a golf course, par for each hole is calculated using the distance from the tee to the hole, the location of water and sand hazards, and other hole features. Golfers attempt to get the golf ball into the hole using at most the number of strokes that is par for that hole. If golfer use only the number of strokes that is par for that hole, they say they made par on that hole. Can you complete the golf course's order of operations with no more than the amount of strokes (par) listed for each hole?"  Distribute the Let's Go Golfing worksheet to each student.  "No golf clubs or tees are required for the order-of-operations golf course. Instead, each hole has a target number. To get the golf ball into the hole on this course, use the order of operations and the numbers 2, 3, 4, 5, 6, and 7. For each hole, create an expression with the target number as its value. Each digit you use counts as one stroke, so try hard to make par in each hole!" Ask students to work in pairs to create an expression with a value of 58, which is the target number for the practice hole.  "First, begin with the practice hole. Notice that the target number is 58. Work with a partner to create an expression with a value of 58. Remember to only use the digits 2, 3, 4, 5, 6, and 7, but you can use them several times."  Some students may be unsure how to begin. To help struggling students, suggest they begin with a fact such as 7 × 6 = 42 and then try to find a way to add 16 more to that total, such as (7 × 6) + 4². Students may also find it helpful to decompose a number, such as decomposing 58 to 50 and 8. Students can start by attaining 50 and then focus on adding 8, for example (5 × 5 × 2) + (6 + 2).  Ask a few pairs of students to show their expressions on the board. Working as a class, use the order of operations to check that the values of these expressions are 58. Students are encouraged to share their expressions, as well as the strategies that they used to write them. If needed for discussion, here are some sample expressions for the practice hole. If no students write an expression using exponents, suggest 7 × 6 + 4² as a possible expression for the practice hole. Remind students that exponents are next in the order of operations after parentheses and grouping symbols. <img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_39.png" width="605" height="290"> Ask all students: “What strategies did you use to write an expression that had a value of 58?” When composing expressions, students must carefully consider the order of operations. For most students, creating expressions with a target value is more difficult than applying the order of operations to expressions presented, as in the class examples at the start of the lesson and the exercises in session 1. As a result, it is critical that all students think about strategies they can use to write these expressions. Make sure that all students understand that the par for the Practice Hole is 5, and explain how the score for each hole is calculated. Help students count the number of digits in one of the given expressions and understand that this represents the score for that hole. For example, if the expression was (7 × 6) + 4², the score would be 4 since the four digits 7, 6, 4, and 2 were used. Also, remind students that the goal in golf is to have the lowest possible score—the fewest number of strokes required to get the ball into the hole.  Notice, the worksheet has been divided into the front 9 holes and the back 9 holes. Students should complete the first 9 holes of the Let's Go Golfing Practice Worksheet in groups of two or three. You may then assign holes 10-18 as homework.  Keep track of students' progress while they work. Provide interventions and support as needed. Use techniques like starting with a known fact or decomposing numbers to help students in creating an expression similar to a target value. Additionally, students may require support remembering the order of operations and checking the value of the expressions they create. With 5 to 8 minutes remaining in the class time, hand out index cards to each student. Give students the expression 4 + (5² - 8) × (3 + 7) and ask them to work individually to find the value of this expression. Remind students that writing down each step in the process allows you to assess both what they know as well as what they are struggling with.  Collect all of the "exit slips" before the students leave the classroom. Review the exit slips before the next class period to discover common mistakes students are making and specific students who require additional support. (The value of the expression is 174.)If time allows, compare the exit slip equation 4 + 5² - 8 × 3 + 7 from Lesson 1 to the exit slip expression 4 + (5² – 8) × (3 + 7) in Lesson 2. These expressions are quite similar, with the exception of the grouping symbols. Comparing the values of these expressions (12 and 174) demonstrates that inserting the grouping symbols frequently changes the value of an expression. Extension: Routine: Students who need additional practice with the order of operations should use this website. The site is interactive, and students are asked to identify the steps of evaluating an expression based on order of operations. The website would be very useful for students who require additional practice. <a href="http://www.learnalberta.ca/content/mec/flash/index.html?url=Data/1/A/A1A2.swf">http://www.learnalberta.ca/content/mec/flash/index.html?url=Data/1/A/A1A2.swf</a>  Small Group: Students can play Order of Operations Bingo to practice using the order of operations to evaluate expressions. Refer to the Order of Operations Bingo at <a href="http://illuminations.nctm.org/LessonDetail.aspx?id=L730">http://illuminations.nctm.org/LessonDetail.aspx?id=L730</a>. Expansion: Challenge students to write expressions from 1 to 100 using only the digit 4. Create a poster with these values on it, and provide space for students to record their expressions. Also, give students a similar chart on paper. This is an activity that students can complete anytime they have a few minutes to spare and that will keep them engaged throughout the academic year. Recognize students who have contributed expressions to the poster. Occasionally use one expression from the poster as a warm-up activity, homework problem, or on a quiz or test to help students remember the order of operations.

Layla Nguyen

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Use Order of Operations to Evaluate Expressions with Grouping Symbols (M-5-6-2)

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Layla Nguyen

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Description

Students will use the order of operations to evaluate expressions that contain grouping symbols. They will
- correctly evaluate numerical expressions with numerous grouping symbols (parentheses, brackets, or braces) using the order of operations.
- create numerical expressions having specified values by grouping symbols as needed.

Grade

Grade 5