# BM1.3 – The Order of Operations Agreement

Chapter 1, Lesson 3

In this lesson students will:

• understand the need for an Order of Operations Agreement
• use the PEMA acronym for simplifying arithmetic expressions
• ensure that the business calculator and Google Sheets are set-up properly to coincide with our agreement

## The Need for Agreement

Often there are mathematical posts floating around social media asking you to simplify a string of arithmetic operations. Take a few moments to think about how to simplify $6 \div 2(1+2)$.

Most folks come up with one of two answers, 1 or 9, and defending your answer on social media has become quite common and a bit comical. Now, as it turns out, neither answer is technically incorrect – it’s all about how you are interpreting the mathematical symbols. For instance, if we interpret the brackets as needing to be evaluated first along with all values to the right of $\div$ being divided, we might simplify to

\begin{aligned} 6 \div 2(1+2) &= 6 \div 2(3) \\ &= 6 \div 6 \\ &= 1. \end{aligned}

However, those who believe that multiplication and division should be done in the order that they appear might evaluate this expression as

\begin{aligned} 6 \div 2(1+2) &= 6 \div 2(3) \\ &= 6 \div 2 \times 3 \\ &= 3 \times 3 \\ &= 9. \end{aligned}

What all of this unwarranted drama boils down to is the fact that we need some kind of agreed upon interpretation of mathematical symbols. This is important not only for simplifying calculations by hand, but ensuring that our calculator is simplifying the way that we want it to as well. In our course, we call our agreed upon interpretation the Order of Operations Agreement.

## The Order of Operations Agreement

Many of you likely know the order of operations agreement by an acronym such as PEMDAS or BEDMAS. In our course, we will use PEMA. The letters stand for the following:

P – operations that are equivalent to simplifying parentheses are done first
E – operations that are equivalent to simplifying exponential expressions are done second
M – operations that are equivalent to simplifying multiplication are done third
A – operations that are equivalent to simplifying addition are done last

As you may already know, division is a part of the M-step and subtraction will be a part of the A-step. What is a bit more unclear is where operations such as square roots, fraction bars, or logarithms might go. In general, we will group the following operations together:

P – parentheses, complex expressions
E – exponentiation, logarithms, square roots
M – multiplication, division, the fraction bar
A – addition, subtraction

This should cover all operations that we will use in our course. We will also add the following rules to our agreement:

• Complex expressions can be broken up into smaller and more manageable chunks by inputting parentheses. In these cases we will make them the highest priority at the P-step and try to simplify them first.
• Since logarithms are the inverse of exponentiation, they will be placed in the E-step. More information about logarithms will be given later in our course.
• Since square roots are equivalent to exponentiation by a fraction, they will be placed in the E-step.
• The division symbol ($\div$) will apply only to the number that comes directly after it.
• The fraction bar can be interpreted as a division symbol, if needed.
• All operations in the same tier will be evaluated on a left-to-right basis.

## The TI BAII Plus, Google Sheets & Order of Operations Agreement

Let’s take a few moments to practice using the order of operations agreement on our business calculator. In order to do this, we need to make sure that it is set to AOS mode and not to Chn mode. To do this, turn the calculator on, press the button, then hit the sub-menu above the decimal point. This will bring up the formatting options, and you can use the up and down arrows to scroll through the options. Scroll until you find the DEC option. Hit and press the key. Now your calculator will display nine decimal places for all calculations. Keep scrolling until you see Chn. Hit the button followed by . This should change Chn to AOS. If you see AOS already, no action has to be done. Your calculator is now set for order of operations.

Next, let’s highlight a few buttons that will be helpful dealing with the order of operations agreement:

• – this is the plus/minus button. This button can be used to convert a positive number to a negative number, or vice versa. To get a negative number, type the number first, then hit the plus/minus button.
• – this is the square root button. To find the square root of a number, input the number you would like to square root, then press the square root button.
• – this is the square button. To square a number, input the number you would like to square, then press this button.
• – these are the bracket buttons.
• – this is the exponentiation button. This button can calculate any power of any base number. To use, you must input the base number first, then hit the button, then input the number representing the power. After hitting you will get your answer.

Give the following questions a shot on your calculator, and watch the video below to see if you obtained the correct answer.

1. $(-3+2) + 8 \div 2 \times 4$
2. $\displaystyle \frac{5+7}{4} - 1 + 3^{2} \times 2$
3. $(-2)^{3} + 15 - (-2)^{2} \div 2 - \sqrt{16}$
4. $\displaystyle \frac{5 \times 2}{4 - 2} - 8 + (-1)^{2} - 4$

Now that we are more comfortable with the order of operations agreement on our calculator, let’s turn our attention to working on Google Sheets. Recall that you can input numbers into Google Sheets in the typical way. If we want to do any simplification, we can use the equal sign command. If Google Sheets notices a cell beginning with an equal sign, it understands to compute any information that comes afterward. This, and other basic arithmetic operations are given in the following list.

• Equal Sign: if you want Google Sheets to simplify something, begin the cell with an = sign.
• Addition & Subtraction: to perform an addition or subtraction, use the standard + and – keys.
• Multiplication: to perform a multiplication, use the * key (Shift 8).
• Division: to perform a division, use the / key. Be mindful that this operation only works with the two numbers that are immediately beside the division symbol. If you require more numbers as part of your division, use brackets.
• Brackets: brackets can be used in the standard way using Shift 9 for “(” or Shift 0 for “)”.
• Exponents: to input an exponential expression, use the ^ key (Shift 6).
• Square Roots: to find the square root of a number, use the SQRT( ) command. For example, to find the square root of four, input =SQRT(4) into a cell.

Give the following questions another shot, but this time try to use Google Sheets. The video below will guide you through the commands if you are having trouble getting the correct answer.

1. $(-3+2) + 8 \div 2 \times 4$
2. $\displaystyle \frac{5+7}{4} - 1 + 3^{2} \times 2$
3. $(-2)^{3} + 15 - (-2)^{2} \div 2 - \sqrt{16}$
4. $\displaystyle \frac{5 \times 2}{4 - 2} - 8 + (-1)^{2} - 4$