Why do we need an Order of Operations?

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Objective

SWBAT: • Explain the Order of Operations • Use the OOO graphic organizer to simplify numerical expressions

Big Idea

Why do we need an Order of Operations? What is 5 + 3 x 4? What is 3 x 4 + 5? Students work through examples to get at these questions and work with the order of operations to simplify numerical expressions.

Do Now

10 minutes

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Here, I want students to review what they learned about exponents in the previous lesson.  I am looking for students to write out the repeated multiplication to show which base and exponent is greater.  A common mistake is for students to think 6^2 as 6 times 2.  If students make this mistake, I remind students that exponents indicate repeated multiplication.  I also ask students to share an efficient way to multiply powers of 10. 

Problems

10 minutes

Students work on the two problems and questions independently.  I walk around to monitor student progress.  When most students have completed the problems and questions we come back together as a class.  I have students Think Pair Share about their answers.  Then I call on students to share their observations with the class.  Some students may say that both problems result in 17.  Some students may say that the problems have different answers.  Possible answers could be 32 or 27. 

One option is to add context to the problems.  You have a bunch of coins.  You have a stack of 5 coins and 3 stacks of 4 coins.  You can represent this number of coins with these two expressions.  How many coins do you have?  What if you have 3 stacks of 4 coins and one stack of 5 coins?  Have students draw the pictures to compare and contrast the problems.

The Order of Operations

7 minutes

A common misconception for my students is to think about the order of operations as a six step process, and forget that you evaluate multiplication and division problems (or addition and subtraction) from left to right.  For this reason I do not use the mnemonic PEMDAS in my class, I refer to them as the order of operations.  I’ve found students latch on to PEMDAS and then assume multiplication always comes before division and addition before subtraction.  I use this horizontal graphic organizer with my students.  In each box I put the symbol for the step (or a number with an exponent).  I regularly switch where I place the multiplication and division (and addition and subtraction) symbols as a constant reminder that students must look for these symbols from left to right.

Then I ask these questions:

  • What would happen if we did not agree and we approached evaluating expressions in our own ways?
  • What problems might arise?

I am looking for students to recognize that the order of operations is a set of rules that have been decided upon by mathematicians.  Without these rules, different people would solve the same problems and get vastly different answers.  I am interested to hear what problems students imagine would occur without agreement on the order of operations.

Practice

23 minutes

We complete the 4 questions on the “Practice” page together.  With each problem I ask what we need to identify and work with first, next, etc.  I stress to students that it is essential that they only complete one step of the expression at a time and that they copy the rest of the expression, until they are left with an answer.  Many students will be tempted to do it in their heads and write down their answers.  I explain that I will not check any work that does not show a step-by-step progression.  I explain that in order for me to understand what they did (correct or incorrect) I need to see their work.  Here it is crucial that students use MP6: Attend to precision.

For students who are doing well with the problems, they can move on to Practice 1.  I explain that the practice pages get more difficult.  If students are feeling confident, they can complete the last 3 problems on Practice 1.  Once they finish, I quickly check their work.  If all three are correct they can move on to Practice 2.  If there are any mistakes, I direct them to try the incorrect problem again.  Posting A Key can make checking student work much more efficient for me and the students. Unit 1.10 Posted Key 1.jpg Unit 1.10 Posted Key 2.jpeg

I am looking to see if students are using the graphic organizer to help them.  Do they perform multiplication and division (as well as addition and subtraction) from left to right?  Are they working correctly with exponents?  Are they making silly math mistakes?  What are the most common mistakes?  I will use these observations to inform the Closure of the lesson.

For students who are struggling, there are a few ways I may intervene:

  • Pull a small group to the back of the class to work on Practice 1 problems together.  Once students are more comfortable, they return to their seats to work independently.
  • I have them work with their partners together and I regularly check in with them.
  • If there are enough students struggling, I’ll pass out whiteboards and markers.  I have students show the step-by-step work on the first few practice problems.
  • Give students a multiplication chart or calculator to help them access the problems.

For students who successfully complete the classwork, there are a few choices I give them:

  • Serve as a tutor to a student I have identified who needs extra help
  • Complete the Challenge problems
  • Pair up with another student and take turns creating and evaluating numerical expressions using the order of operations
  • Playing “Show me the Money!” from Show what you know Factors and Multiples + Introduction to Exponents (students can use the 1-10 spinner for added difficulty).

Closure and Ticket to Go

10 minutes

 For Closure I have students turn to problem 6 in Practice 3.  What do we do first?  Some students will struggle since there are two operations inside the parentheses.  Should we subtract first or simplify 2^3?  I have students explain the step by step process.  I ask again, why do we have an order of operations?  What might happen if we didn’t?

With the last few minutes of class I give a Ticket to Go for students to independently complete.  I pass out the HW Why do we need an order of operations at the end of class.