Order of Operations

We are at the start of a new unit where we will learn about expressions, including expressions with exponents. Today we’re going to talk about exponents for the first time. We’ll talk about what it means to be squared, what it means to be to raise a number to the power of zero and what’s the same as saying 8 x 8 x 8 x 8 x 8 x 8. Then we’re going to talk about the order of operations and how we ought to show work when simplifying expressions that include a few different operations, so that no matter where you go in the world, everyone will have the same answer to the fourth problem.

At the start of today's class students work alone on the TAB problem.  After work time, I ask students to share out their answers for Problem 1 and Problem 2. Then I say: 

“For Problem 3, some of you rewrote this expression as 5+5+5+5+5. Today, we are learning about another way to rewrite this expression using an exponent. Just like writing a repeated addition expression using a simpler form, you can write an expression with repeated multiplication in a simpler form. You would rewrite 5x5 as 5² and we are going to learn why.”

On Problem 4 students give a variety of answers.  I call on students and record the different responses on the board.  I ask, “Is it possible for all of these to be right? No!  How confusing would it be if there were no agreed upon way of simplifying expressions with multiple operations.  Luckily mathematicians have created an agreed upon order for us to follow.”

As we get ready for new material I say, "We are going to practice following the order of operations today as we learn to simplify numerical expressions that have exponents!”

In this lesson, there is a lot of new material, including a number of new vocabulary words.  For this reason, I present the new material in the form of cloze notes before working through problems with my students.  I fill in the key points in the INM on the document camera as students fill in their own pages.  See Key Points Key for the filled-in notes.

After modeling Part a and Part b, by thinking aloud, I ask students to help me with the remaining examples. 

Students then complete the quick practice on their own.  I cold call students to read each problem (ex – “four cubed”), identify the base, identify the exponent, express the problem as repeated multiplication (4*4*4), and evaluate.

Next, we move on to the order of operations.  I do not teach students PEMDAS or ‘Please Excuse My Dear Aunt Sally,’ because I feel that students get confused later in math when suddenly P/Please means braces, brackets, radicals, etc.  Instead, I teach that grouping symbols are the first step.

We read the steps chorally together, and then work through the first example.  This first problem includes five different operations. The purpose of this model will be to show the order and the logical steps we use to show the work as well as how to neatly organize one’s work.

Example 1:  5 x 2² - 8 ÷ 2

1. First, we look for any grouping symbols.  Parentheses, absolute value, brackets.  Do we have any here?
2. Next, we look for exponents.  Underline the part of the expression with an exponent.
3. What is 2²?
4. Rewrite 5 x 4 – 8/2
5. So far, we’ve looked for grouping symbols, and exponents.  Now, we look for multiplication and division, and evaluate from left to right. What do we see first?
6. Underline 5 x 4. What is 5 x 4
7. Rewrite 20 – 8/2.
8. What would we do next? Vote.  Division.  Why division and not subtraction?  Because we do division before we subtract according to the order of operations. 
9. What is our final step? Subtraction.

 Example 2: 10 - 4 x 2 + 2 x 5²

1. Are there any grouping symbols?
2. What do we do next? Exponents.  Underline and evaluate.  Rewrite.
3. What do we do next? Multiplication.  How many instances of multiplication do we have?  Which do we do first?
4. What do we do next? Subtraction and addition.  Which do we do first? Subtraction.  Why?  Because it comes first from left to right. 

For the third example, students are posed with students’ answers. For one answer, someone does it the correct way, and someone does it the incorrect way.

Students Turn-and-Talk to discuss which answer they think is correct and explain why. I will Cold Call students for answer as a final share out.

For this lesson, I made the decision to not have students work in partners.  I want to be sure that they are carefully following the order of operations and also organizing their work as expected.  We work through the Guided Practice problems together, much like in the intro to new material section.  As students get more comfortable with the process, they take on more of the work with me asking fewer guiding questions.  

Students complete the Final_CFU problem on their own.  I pull a popsicle stick for student work to be displayed on the document camera.  The lucky student talks through the steps (s)he took to evaluate the expression.  I then open it up for feedback from the class on how the problem can be improved.  Feedback here is usually focused on neatness and organization, if the student has completed all of the steps correctly.   

Then, I will ask my students to work on Independent Practice for 15 minutes.

As they work, I circulate and look to see:

• Are scholars accurately following the order of operations?
• Are students organizing their work correctly by writing new expressions after each operation is performed?
• Are students correctly evaluating exponents by multiplying the base by itself the number of times the exponent indicates?

I also check in with students and ask them:

• How did you know which operation in the expression to simplify first??
• What would happen if you started calculating out of order?
• When do you subtract before you add? Divide before you multiply?
• What is the base?  What does it indicate?  What is the exponent?  What does it indicate?

To make the most of the conversation in today's class, I begin the lesson closing by asking students to analyze the work of our (imaginary) friend BoBo (see Error For Student Analysis).  I give them 45 seconds of silent think time to determine what mistake BoBo made.  Students then have 30 seconds to turn and talk with their neighbor about the mistake.  I ask one student to share out the error (BoBo evaluated three squared as 6, and not 9).  Finally, I ask for someone to share the correct answer, and display their work on the document camera.

Finally, students work on their Exit Ticket to end the lesson.