“In today’s lesson, we are going to learn about using the order of operations when there are grouping symbols in an expression. This will help us understand the confusion in Mr. Ryte’s math class.
“Mr. Hu Ryte asked his students to determine the value of this expression:
18 – (4 + 2) + (6 × 5) ÷ 3
“He was very surprised when he asked five students and all had different answers, as listed in the table below. ‘Which student has the correct answer?’ he wondered.”
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Julio
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Maya
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Xui
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Sierra
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Ryan
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14
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15 1/3
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-6
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22
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2
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“Take a few moments to work together and try to determine the value of this expression.” Ask students to share their values and explain the processes they used. Students will likely get many different values depending on the order in which they performed the operations; and some students might not know where to begin. Although Lesson 1 also focused on the order of operations, the expressions in Lesson 1 did not include any grouping symbols.
Ask students to discuss why Mr. Hu Ryte’s students had so many different values for the same expression. Students will likely say that different people did the calculations in different orders, as this was also discussed in Lesson 1. If needed for class discussion, here are the steps Julio and Maya used 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 that was introduced in Lesson 1.
“Mathematicians agree on an order of operations. This is a specific order that is used so everyone is sure to get the same value. Today we will use the order of operations to decide which of Mr. Hu Ryte’s students has the correct answer. Mathematicians say Parentheses (P) or grouping symbols first, Exponents (E) next, then 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 determine 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 the order of operations that mathematicians agree upon, we now know that Sierra has the correct value of the expression, 22. But, how can we remember the order of operations? Remember the acronym PEMDAS represents 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 display the order of operations in the classroom while 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 try to determine the value of these expressions. Observe students as they work. Find a pair of students who found the correct values (63 and 32), and ask the pairs of students to show and explain the processes they used on the board. (Working as a pair, students tend to feel less threatened when sharing their work, and students can support each other in both recording and explaining 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 notice the second expression includes multiple grouping symbols, both braces and parentheses. “When there are multiple grouping symbols, always work on the innermost grouping symbols first.” Ask students to explain which operations should be performed first in the second expression. Students should say to start with “7 – 2” and “2 ÷ 2” as these are the operations in the innermost grouping symbols.
Now, ask students to work in pairs again to find the values of both of these expressions. Ask for pairs of students to volunteer to show and explain on the board the processes they used. (The values of the expressions are 18 and 3, respectively.)
For practice using the order of operations, introduce the Let’s Go Golfing worksheet (M-5-6-2_Let's Go Golfing Practice Worksheet and KEY.docx). It is important to first introduce the game of golf, as all students may not be familiar with the game.
“In golf, par is the maximum number of strokes that a golfer should need to get the golf ball in the hole. On a golf course, par for each hole is based on the distance from the tee to the hole, the location of water and sand hazards, and other features of the hole. Golfers try to get the golf ball in the hole using at most the number of strokes that is par for that hole. If golfers use only the number of strokes that is par for that hole, they say they made par on that hole. Can you finish the order-of-operations golf course using at most the number of strokes (par) listed for each hole?”
Distribute the Let’s Go Golfing worksheet to all students.
“No golf clubs or tees are needed for the order-of-operations golf course. Instead, each hole has a target number. To get the golf ball in the hole on this course, use the order of operations and the digits 2, 3, 4, 5, 6, 7. For each hole, write an expression that has the target number as its value. Each digit you use counts as one stroke, so try hard to make par for each hole!”
Ask students to work in pairs to write an expression with a value of 58, as that is the target number for the practice hole.
“First, start with the practice hole. Notice the target number is 58. Work together with a partner to write an expression that has a value of 58. Remember, only use the digits 2, 3, 4, 5, 6, and 7, but you can use these digits more than once.”
Some students may not know how to begin. For students who are having difficulty, 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, for example decomposing 58 to 50 and 8. Students may then work to attain 50 first and then focus on adding 8, such as (5 × 5 × 2) + (6 + 2).
Ask a few pairs of students to present their expressions on the board. Work together as a class using the order of operations to verify that the values of these expressions are 58. The goal is to have students share their expressions and, most importantly, the strategies they used for writing these expressions.
If needed for discussion, some sample expressions for the practice hole are provided here. 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.
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Hole
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Target Number
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Par
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Write an Expression
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Your Score
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Practice
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58
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5
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(4 + 4) × 9 – (7 × 2)
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5
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Practice
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58
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5
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7 × 6 + 4²
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4
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Practice
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58
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5
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4³ – 6
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3
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Ask all students: “What strategies did you use to write an expression that had a value of 58?” In writing expressions, students must think carefully about the order of operations. For most students, writing expressions with a target value is more challenging than applying the order of operations to expressions provided, as in the class examples used at the beginning of the lesson and the activities in Lesson 1. For this reason, it is important to help all students think about strategies they can use to write these expressions.
Be sure all students understand that par for the Practice Hole is 5, and help students understand how the score for each hole is determined. Help students count the number of digits in one of the expressions provided and understand that this would be the score for that hole. For example, if the expression was (7 × 6) + 4², the score would be 4 as the four digits 7, 6, 4, and 2 were used. Also, remind students that in golf you want the lowest score possible—the least number of strokes to get the ball in the hole.
Notice, the worksheet has been divided into the front 9 holes and the back 9 holes. Ask students to complete the front 9 holes of the Let’s Go Golfing Practice Worksheet in groups of two or three. You may then choose to assign holes 10–18 as homework.
Monitor students’ progress as they work. Provide necessary interventions and support as needed. Use the tips such as beginning with a known fact or decomposing numbers to help students generate an expression equivalent to a target value. Also, students may need support remembering the order of operations and help verifying the value of the expressions they create.
With 5 to 8 minutes left in the class period, distribute an index card to each student. Present the expression 4 + (5² – 8) × (3 + 7), and ask students to work individually to find the value of this expression. Remind students that it is important they write down each step in the process, as this is an opportunity for you to assess both what they know and what they are struggling with.
Collect all of these “exit slips” before students exit the classroom. Review the exit slips before the next class period to identify common errors students are making and specific students who need extra support. (The value of the expression is 174.)
If time permits, review the expression 4 + 5² – 8 × 3 + 7 on the exit slip from Lesson 1 with the exit slip expression 4 + (5² – 8) × (3 + 7) here in Lesson 2. These expressions are very similar, except for the grouping symbols. Comparing the values of these expressions (12 and 174) emphasizes that inserting the grouping symbols often changes the value of an expression.
Extension:
- Routine: Students who need some extra practice using the order of operations should use this site. The site is interactive and asks students to identify the steps of evaluating an expression using the order of operations. The site would be particularly beneficial for students who need more practice. http://www.learnalberta.ca/content/mec/flash/index.html?url=Data/1/A/A1A2.swf
- Small Group: Students can play Order of Operations Bingo to practice applying the order of operations to evaluate expressions. See Order of Operations Bingo at http://illuminations.nctm.org/LessonDetail.aspx?id=L730.
- Expansion: Challenge students to write expressions with the value of 1 to 100 using only the digit 4. Make a poster with these values on it, and be sure to provide space for students to record their expressions. Also, provide students with a similar chart on paper. This is an activity that students can work on whenever there are a few minutes to spare and that can engage students throughout the academic year. Recognize students who contribute expressions to the poster. Occasionally use an expression from the poster as a warm-up activity, as a homework problem, or on a quiz or test to help students review the order of operations.
