If possible, obtain at least one calculator that does not use the conventional order of operations. Simple four-function calculators often do not follow the conventional order, while the more advanced scientific calculators usually do. You can test whether a calculator follows the order of operations with the following keystrokes:
If the calculator gives an answer of 0, it uses the conventional order of operations. Calculators that give an answer of 6 do not use the conventional order of operations.
Order of Operations. At first glance, an expression such as 5 + 4 × 3 is open to various interpretations. One approach takes 5 + 4 × 3 to mean, “First add 5 and 4 to get 9, and then multiply by 3 to obtain 27.” This simple left-to-right interpretation is the way many four-function calculators evaluate the expression, but it is incorrect, according to mathematical convention accepted the world over.
The standard mathematical meaning of 5 + 4 × 3 is, “First multiply 4 times 3 to get 12, and then add 5 and 12 to get 17.” This interpretation is based on a convention known as the algebraic order of operations. This convention specifies the order in which operations in mathematical expressions are to be carried out:
- Do calculations in parentheses first.
- Do all multiplications and divisions in order from left to right.
- Then do all additions and subtractions in order from left to right.
For example, 48 ÷ 6 − 3 × 2 = 8 − 6 = 2.
Parentheses change the order, so 48 ÷ (6 − 3) × 2 = 48 ÷ 3 × 2 = 16 × 2 = 32.
Avoiding Misconceptions. Various mnemonic devices are sometimes offered to help students remember the order of operations. While mnemonics are helpful in remembering some things, we strongly advise against their use in teaching this topic. The mnemonics that are commonly used to remember order of operations often lead to incorrect computations.
The phrase “My Dear Aunt Sally” is a mnemonic that reminds students to do the operations in this order: multiplication, division, addition, subtraction. This is an incorrect order—it suggests that multiplication is done before division and addition is done before subtraction. This would cause a student to incorrectly evaluate the expression 8 − 4 + 1 as 3. The correct answer, gotten by evaluating left to right because addition and subtraction have the same priority, is 8 − 4 + 1 = 5. Similarly, since multiplication and division have the same priority, 8 ÷ 4 × 2 = 4. A student who follows My Dear Aunt Sally's rule and does the multiplication first would incorrectly calculate 1 to be the answer.
To use Aunt Sally correctly, you need to tell students the additional condition that M and D are at the same level, and A and S are at the same level. But our experience shows that, as memory fades, many students and adults remember only the order given by the Aunt Sally mnemonic (MDAS), but not the extra condition.
Instead of offering a mnemonic, encourage students to think why the conventional order makes sense: multiplication and division are more complicated so we do them first to get them out of the way. They have the same priority level because they “go together”—one is the inverse of the other—so neither is higher than the other. Addition and subtraction “go together” too, so they have the same priority level. They are the final step—they put all the pieces together.
