In this lesson, students will evaluate expressions based on their order of operations. They will:
- learn the order of operations.
- appreciate the importance of the order of operations.
- correctly evaluate numerical expressions (with no grouping symbols) by following the order of operations.
- convert simple word problems into numerical expressions.
- 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?
- 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.
- A Lesson from Aunt Sally practice worksheet (M-5-6-1_A Lesson from Aunt Sally-Worksheet and KEY)
- What Happened to Aunt Sally? practice worksheet (M-5-6-1_What Happened to Aunt Sally Practice Worksheet and KEY)
- index cards
- 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.
Scaffolding, Active Engagement, Modeling, and Explicit Instruction
W: Students will learn how to apply the order of operations to evaluate expressions.
H: Introduce expressions and ask students to analyze them. Encourage students to grasp that without a specific order for the operations, they will obtain a variety of 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 get to various explanations of the same idea, the more likely they are to grasp and remember it.
R: Students will practice and review their understanding of evaluating expressions using order of operations. Allow students to work in pairs so that they can verify each other's processes and solutions.
E: Students will be assessing expressions throughout the lesson. Make them record their work in their math notebook. Monitor and assess student responses, and correct any misconceptions.
T: The lesson can be tailored to the needs of your students by following the suggestions in the Extension section. Specific suggestions are provided to assist students who may require further practice in assessing expressions, and the Expansion section gives additional difficulties for students who are ready for a challenge that exceeds the requirements of the standard.
O: The lesson teaches students the importance of following the correct order of operations. The lesson focuses on evaluating expressions with no grouping symbols. This teaches students to use order of operations to evaluate expressions containing multiple grouping symbols in Lesson 2.
Introduce the A Lesson from Aunt Sally practice worksheet (M-5-6-1_A Lesson from Aunt Sally-Worksheet and KEY) to help students understand the importance of the order of operations. Distribute a copy of the A Lesson from Aunt Sally practice worksheet to each student.
"Jeremiah and his Aunt Sally are working on his math homework. Jeremiah must find the value of these expressions:
18 – 4 + 2
24 + 8 ÷ 4
15 – 2 × 3
36 ÷ 2 × 9
"However, Jeremiah and his Aunt Sally always get different answers. 'Why does Aunt Sally always get different answers than I do?' Jeremiah wondered. In today's lesson, we will discuss the order of operations. This will help us understand why Jeremiah and Aunt Sally receive different answers. First, Jeremiah needs your help. Work in pairs to determine the value of these expressions. Display your work in the 'Try It' column. The second column will be used later."
Observe students working in pairs. Do all students evaluate these expressions performing the operations strictly from left to right? This is what we should expect. Students read from left to right, thus they also prefer to do operations from left to right. However, some students may first perform the operation that they believe is the easiest. The table shows Jeremiah and Aunt Sally's suggestions. While you're observing, take note of any students who receive the same answers. You might want to ask them about the process they used when the class discusses these solutions occurs later in the lesson.
After the pairs of students have completed assessing these expressions, draw a table on the board with Jeremiah's and Aunt Sally's results. Only write the values of the expressions Jeremiah and Aunt Sally computed, as indicated in boldface. Do not write the processes shown in the table. They are offered for use during class discussions.
"Let's continue to help Jeremiah. For each expression, work in pairs again to understand how Jeremiah and Aunt Sally got their solutions. To do so, take the following steps:
For each expression, determine whether your solution matches Jeremiah's, Aunt Sally's, or neither.
Now let's try again. Try to understand the solution you didn't have. For example, if you have Jeremiah's solution, try to figure out how Aunt Sally got hers. If you didn't get either of the solutions, try to figure out how Jeremiah and Aunt Sally got their solutions.
Record your work in the 'Try It Again' column. Be prepared to share your ideas with the class."
Observe students working in pairs. The goal is to help students understand that doing the computations in different orders results in different results. This activity will highlight the importance for an order of operations that all agree on.
Write each expression on the board. Ask students to write down the steps Jeremiah and Aunt Sally took for each expression on the board. Then have students discuss 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 are likely to mention that Jeremiah and Aunt Sally completed the operations in different order.
Next, introduce students to the order of operations.
"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 whether Jeremiah or Aunt Sally have the correct answers for each equation. 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 find the correct answers for each of these expressions."
Help students evaluate expressions using the order of operations given above. [Note: As shown students can circle the operation they need to perform at each step. Circles are not necessary for all students, but they may help some students focus on a single task during each step.]
Expression: 18 – 4 + 2 =
Notice: There is no Parentheses or Exponents in this expression. There is 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 can now conclude that Aunt Sally had the correct values for the expressions. But how will we remember the order of operations? The acronym PEMDAS refers to 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 make their own sentences to recall PEMDAS, if they choose. It is also very helpful to demonstrate the order of operations in the classroom when 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 figure out the value of these expressions. Encourage them to carefully follow the order of operations to ensure that each expression gets the correct value.
Observe students while they work. Find pairs of students who got the correct answers (35, 25, and 1). When students have finished their work, assign one pair of students to write and explain the processes they used for each expression 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.) Make sure that students ask questions about processes that they do not yet understand. Before beginning the practice activity, students should have all of their questions answered.
Introduce the What Happened to Aunt Sally? worksheet (M-5-6-1_What Happened to Aunt Sally Practice Worksheet and KEY) to help students practice using the order of operations. Distribute the What Happened to Aunt Sally? worksheet to all students.
"There are 16 expressions on this worksheet. Each expression has a corresponding letter. Evaluate the expression. On page 2, find all boxes containing the value of the expression and write the corresponding letter. For example, if the value of expression E is 128, you would write letter E in the boxes above each 128 in the puzzle."
Keep track of students' progress while they work. Provide interventions and support as needed. Students may need to be reminded of the order of operations. Also, recommend that struggling students circle the operation to be done at each step. You'll also want to remind students that multiplication and division, like addition and subtraction, are done from left to right in the same step.
Students often think that multiplication comes before division when they see the acronym PEMDAS, but these operations are performed at the same time. Similarly, this applies to addition and subtraction.
With 5 to 8 minutes remaining in the class time, hand out an index card to each student. Present the expression 4 + 5² - 8 × 3 + 7 and ask students to solve it on their own. Remind students that writing down each step in the process is important because it allows you to assess both what they know and what concepts they still need to grasp.
Collect all of the "exit slips" before the students leave the classroom. Review the exit slips before the following class period to discover typical mistakes students make and specific students who may benefit from extra help. (The value of the expression is 12).
Extension:
Use the strategies and activities below to satisfy your students' needs during the lesson and throughout the year.
Routine: Use this website throughout the year to refresh students' understanding of the concept of order of operations. Students can practice evaluating numerical expressions without grouping symbols on this website. The site is interactive, and it records how many expressions students correctly evaluate in two minutes.
http://cemc2.math.uwaterloo.ca/mathfrog/english/kidz/order.shtml
Small Group: Students that need more practice may benefit from playing Calculator Chaos, an online game. Students learn applying operations to get a target number from a broken calculator, using select keys and numbers.
http://www.mathplayground.com/calculator_chaos.html
Expansion: Students who are ready for a more difficult challenge may find it in the online game One to Ten. Students produce the numbers 1 through 10 by following the order of operations and selecting numbers. Creating an expression is typically more challenging 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
