• The Exit Slip may be used to determine if students understand how to use the order of operations to find the value of an expression.

To help students learn about the need for the order of operations, introduce the A Lesson from Aunt Sally practice worksheet (M-5-6-1_A Lesson from Aunt Sally-Worksheet and KEY.docx). Distribute a copy of the A Lesson from Aunt Sally practice worksheet to all students.

“Jeremiah and his Aunt Sally are working on his math homework. Jeremiah has to find the value of these expressions:

• 18 – 4 + 2
• 24 + 8 ¸ 4
• 15 – 2 × 3
• 36 ÷ 2 × 9

“But, Jeremiah and his Aunt Sally always get different answers. ‘Why does Aunt Sally always get a different answer than I do?’ Jeremiah wondered. In today’s lesson, we will learn about the order of operations. This will help us understand why Jeremiah and Aunt Sally get different answers. First, Jeremiah needs your help. Work together in pairs to find the value of these expressions. Show your work in the ‘Try It’ column. The second column will be used later.”

Observe students as they work in pairs. Do all students evaluate these expressions performing the operations strictly from left to right? This is what we would expect. Students read left to right, so they tend to also want to perform the operations from left to right. Yet, some students may first perform the operation they perceive to be the easiest. The table shows Jeremiah’s and Aunt Sally’s solutions. While you are observing, make note of any students who get these same answers. You may want to call on them to ask about the process they used when the class discussion of these solutions occurs later in the lesson.

After the pairs of students are finished evaluating these expressions, write a table on the board of Jeremiah’s and Aunt Sally’s solutions. Note, only write the values of the expressions Jeremiah and Aunt Sally computed, as noted in boldface. Do not write the processes shown in the table. They are provided for use in class discussion.

“Let’s continue to help Jeremiah. For each expression, work together in pairs again to understand how both Jeremiah and Aunt Sally got their solutions. To do so, follow these steps:

• For each expression, decide if your solution is the same as Jeremiah’s, Aunt Sally’s, or neither.
• Now, let’s try again. Try to understand the solution you did not have. For example, if you got Jeremiah’s solution, try to determine how Aunt Sally got her solution. If you didn’t get either of the solutions, try to determine how both Jeremiah and Aunt Sally got their solutions.
• Record your work in the ‘Try It Again’ column. Be prepared to share your thinking with the class.”

Observe students as they work in pairs. The purpose is to help students understand that performing the computations in different orders results in different solutions. This activity will motivate the need for an order of operations that all agree on.

Write each expression on the board. Ask students to write the processes Jeremiah and Aunt Sally used for each expression on the board. Then ask students to explain the processes used by both Jeremiah and Aunt Sally.

Now ask students, “Why did Jeremiah and Aunt Sally get different solutions for each expression?” Students will likely mention that Jeremiah and Aunt Sally performed the operations in different orders.

Next, introduce students to the order of operations.

“Mathematicians agree on an order of operations. This is a specific order used so everyone is sure to get the same value. Today we will use the order of operations to decide whether Jeremiah or Aunt Sally got the correct solutions for each expression. 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 the correct answers for each of these expressions.”

Help students evaluate the expressions using the order of operations as shown here. [Note: As shown, it may be helpful to have students circle the operation they should perform at each step. Not all students will need to use circles, but it helps some students to focus on a single task during each step.]

Expression:      18 – 4 + 2 =    

Notice: There are no Parentheses or Exponents in this expression. There is also no Multiplication/ Division in this expression.

               [Perform Addition/Subtraction left to right]

Expression:      24 + 8 ÷ 4 =

Notice: There are no Parentheses or Exponents in this expression.

            [Perform Multiplication/Division left to right]

                       [Perform Addition/Subtraction left to right]

Expression:      15 – 2 × 3 =

Notice: There are no Parentheses or Exponents in this expression.

                    [Perform Multiplication/Division left to right]

          [Perform Addition/Subtraction left to right]

Expression:   36 ÷ 2 × 9

Notice: There are no Parentheses or Exponents in this expression.

                [Perform Multiplication/Division left to right]

“Using the order of operations that mathematicians agree upon, we now know that Aunt Sally had the correct values for the expressions. But, how will we remember the order of operations? Notice, 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.’” Encourage students to create their own sentences to remember PEMDAS, if they choose. Also, it is very helpful to display the order of operations in the classroom while students are first learning it.

Write the following expressions on the board:

• 4 + 5 × 8 – 3²
• 2 + 18 ÷ 3 × 4 – 1
• 5 – 2³ ÷ 4 × 2

Ask students to work in pairs to determine the value of these expressions. Encourage them to follow the order of operations carefully so they get the correct value for each expression.

Observe students as they work. Find pairs of students who found the correct values (35, 25, and 1). When students are finished working, ask one pair of students to write and explain the processes they used for each expression 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.) Be sure to have students ask questions about processes they don’t yet understand. It is important that all questions are answered before students begin the practice activity.

For practice using the order of operations, introduce the What Happened to Aunt Sally? worksheet (M-5-6-1_What Happened to Aunt Sally Practice Worksheet and KEY.docx). Distribute the What Happened to Aunt Sally? worksheet to all students.

“There are 16 expressions on the worksheet. Each expression has a corresponding letter. Evaluate the expression. On page 2, find all boxes with the value of the expression and write its corresponding letter. For example, if the value of expression E is 128, you would write E in the boxes above every 128 in the puzzle.”

Monitor students’ progress as they work. Provide necessary interventions and support as needed. Students may need to be reminded of the order of operations. Also, remember to suggest that struggling students circle the operation to be performed in each step. Also you will likely need to remind students that multiplication and division are done from left to right in the same step, as are addition and subtraction. In the acronym PEMDAS, students often think multiplication is done before division, but these operations are performed at the same time. Similarly, this also holds true for addition and subtraction.

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 concepts they have yet to master.

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 may benefit from extra support. (The value of the expression is 12.)

Extension:

Use the following strategies and activities to meet the needs of your students during the lesson and throughout the year.

• Routine: Throughout the year, use this Web site to keep students fresh on the concept of order of operations. Students can practice evaluating numerical expressions without grouping symbols at this site. The site is interactive and records how many expressions students evaluate correctly in 2 minutes. http://cemc2.math.uwaterloo.ca/mathfrog/english/kidz/order.shtml
• Small Group: Students who need additional practice may find this online game, Calculator Chaos, to be beneficial. Students practice using operations to generate a target number given a broken calculator, with select keys and numbers. http://www.mathplayground.com/calculator_chaos.html
• Expansion: Students who are ready for a greater challenge may find it in this online game, One to Ten. Students use the order of operations and select numbers to generate the numbers 1 through 10. Generating an expression is usually more difficult for students than evaluating given expressions. So, the One to Ten Game can be used to challenge students: http://www.theproblemsite.com/games/onetoten2.asp