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1 Order of Operations (Time 20 minutes) Math is a language, just like English, Spanish, or any other language. We use nouns, like "bread", "tomato", "mustard" and "cheese" to describe physical objects. Math has values, like the numbers 1, 2 or, to describe quantities. 3 Students should begin to look at math as a language. To help them with this, you can discuss syntax and grammar of math, and its potential for ambiguity in the absence of a fixed order of operations. Help students see that math is a way to communicate calculations among people. Humans also use verbs like "throw", "run", "build" and "jump" to describe operations on these nouns. Mathematics has functions like addition and subtraction, which are operations performed on numbers. Just as you can "slice a piece of bread", a person can also "add four and five". A mathematical expression is like a sentence: it s an instruction for doing something. The expression tells us to add 4 and 5. To evaluate an expression, we follow the instructions in the expression. The expression evaluates to What does evaluate to? What does evaluate to? What does evaluate to? Sometimes, we need multiple expressions to accomplish a task. If you were to write instructions for making a sandwich, it would matter very much which came first: melting the cheese, slicing the bread, spreading the mustard, etc. The order of functions matters in mathematics, too. If someone says "four plus two minus one", they could mean several things: Add four and two, then subtract one: (4 + 2) 1 Add four to the result of subtracting one from two: 4 + (2 1) Write an expression of your own that can mean several things. Page 10 of 21

2 Depending on which way you read the expression, you might have very different results! This is a problem, because we often use math to share calculations between people. For example, you and your cell phone company should agree upfront on how much you will pay for sending text messages and making calls. Different results might mean that your bill looks wrong. We avoid problems by agreeing on the order in which to use the different operations in an expression. There are two ways to do this: 1. We can all agree on an order to use 2. We can add detail to expressions that indicate the order Write down one reason why it is important to have rules about the order of operations. Mathematicians didn t always agree on the order of operations, but now we have a common set of rules for how to evaluate expressions. The pyramid on the right summarizes the order. When evaluating an expression, we begin by applying the operations written at the top of the pyramid (multiplication and division). Only after we have completed all of those operations can we move down to the lower level. If both operations are present (as in ), we read the expression from left to right, applying the operations in the order in which they appear. This item covers PEMDAS (order of operations in ambiguous expressions). You can skip this item if you do not need to cover PEMDAS. In what order should we apply the functions in the following example? One way to indicate the order of operations in an expression is to first draw the expression as a diagram. This diagram is called a Circle of Evaluation. Here you can see an example of a Circle of Evaluation, for the math expression. 4 5 [Video] This section benefits enormously from visual aids, diagrams, etc. Make sure you have plenty of board space to draw examples! Page 11 of 21

3 Circles of Evaluation show the structure that s going on inside an expression. All Circles of Evaluation have two rules: Circle Rule 1: Each circle must have one function, which goes at the top of the circle. Circle Rule 2: The numbers are written below, in order from left to right. Which rule does this Circle of Evaluation break? Try drawing the Circle of Evaluation for each of the following expressions: It s important for students to view the Circles of Evaluation as "just another way of writing arithmetic". Have students discuss whether associativity and commutativity still matter here (they do). Every Circle of Evaluation evaluates to the result of its corresponding expression. For example, the circle on the right evaluates to 1, because the circle says to use subtraction (the function at the top) on the numbers in order ( ). 4 5 Page 12 of 21

4 To use multiple functions in the same expression, we can combine Circles of Evaluation. Look at the Circle of Evaluation you ve written for 351/ 1. We already know that the Circle for 4 5 will evaluate to 1, so we can replace the number with the expression. Does this change what the expression evaluates to? From this point forward, the Circles of Evaluation are your assessment tool for Order of Operations. Quizzing students by asking them to correctly evaluate an expression is fraught with false negatives, as many student might get the order right but may still have problems with basic calculations. This method is also vulnerable to Commutativity, since a student will correctly evaluate even if they get the order of operations wrong! Circles of Evaluation have neither of these flaws, as they put the emphasis where it should be: exercising a student s ability to see the structure inside the arithmetic. Finish writing the Circle of Evaluation shown here, so that it represents (23 + 7) (13 5) Page 13 of 21

5 What does this Circle of Evaluation evaluate to? Let s review how we evaluate a Circle: We know we are multiplying because that s the function at the top of the Circle. The Number 6 is the first number in the multiplication, because it s on the left-hand side. The second number in the multiplication is on the right-hand side. The right-hand side has a separate circle, so we need to evaluate the number for that circle. The second number is therefore the result of adding 4 and (the inner circle) evaluates to 9, and 6 9 (the outer circle) evaluates to. This circle evaluates to. Convert this Circle of Evaluation into an arithmetic expression. Work through several of these examples with students, asking them to come up with arithmetic expressions and then convert them into Circles, or giving them Circles and having them translate them back into arithmetic. When you talk about Circles, be sure to consistently use the term function for what s on top, rather than similar terms like "operation", "symbol", "procedure", "name", etc. The Circles of Evaluation will help students see the similarity between arithmetic functions (like ) and algebraic functions (like f ) if you use terminology carefully Page 14 of 21

6 Match the following Circles of Evaluation with the corresponding arithmetic expressions: 1 9 (4 3) a / (9 4) 3 2 b (9 3) 4 3 c (2 + 4)/(6 3) 4 d e Page 15 of 21