Order of Operations: The Mistakes Behind Wrong Answers

Two students can work the same expression, use the same numbers, and get different answers. Usually neither made an arithmetic slip. One of them did the steps in a different order. The order of operations exists so that every expression has exactly one correct value, and this guide covers the specific places where students most often go wrong.

1. Why there is an agreed order

The Common Core grade 6 standards ask students to perform arithmetic operations, including whole-number exponents, "in the conventional order when there are no parentheses to specify a particular order." That phrase matters. The order is a convention, an agreement everyone uses so a written expression means the same thing to every reader.

The grade 5 standards cover the other half: using parentheses, brackets, or braces in numerical expressions, and evaluating expressions with these symbols. Grouping symbols are how a writer overrides the default order.

2. The conventional order

1. Grouping symbols: parentheses, brackets, braces, and anything above or below a fraction bar.
2. Exponents.
3. Multiplication and division, worked from left to right.
4. Addition and subtraction, worked from left to right.

Many students learn this as PEMDAS. The acronym is helpful, but it hides the detail that causes the most mistakes, which comes next.

3. Mistake 1: treating M before D as a rule

PEMDAS lists M before D, so it is tempting to always multiply first. But multiplication and division share the same level. You work them in the order they appear, from left to right.

Example: 20 / 4 x 5

• Correct: 20 / 4 = 5, then 5 x 5 = 25.
• Wrong: 4 x 5 = 20, then 20 / 20 = 1.

The correct answer is 25.

4. Mistake 2: treating A before S as a rule

The same thing happens with addition and subtraction. They are also on one level, worked left to right.

Example: 10 - 4 + 3

• Correct: 10 - 4 = 6, then 6 + 3 = 9.
• Wrong: 4 + 3 = 7, then 10 - 7 = 3.

The correct answer is 9.

5. Mistake 3: adding before multiplying

Example: 8 + 2 x 5

Reading left to right, it feels natural to add first: 8 + 2 = 10, then 10 x 5 = 50. But multiplication comes before addition, so the correct steps are 2 x 5 = 10, then 8 + 10 = 18.

If the writer really meant to add first, they would write (8 + 2) x 5, which equals 50. The parentheses change the meaning, which is exactly what the grade 5 standard is about.

6. Mistake 4: squaring the wrong thing

An exponent applies only to the number or group directly in front of it.

Example: 2 x 3²

• Correct: square first, 3² = 9, then 2 x 9 = 18.
• Wrong: multiply first, 2 x 3 = 6, then 6² = 36.

The value 36 belongs to a different expression, (2 x 3)². This mistake shows up constantly in formulas. The grade 6 standard gives a good example: the surface area of a cube is A = 6s². For a cube with sides of length 1/2:

• Correct: s² = (1/2)² = 1/4, then 6 x 1/4 = 3/2, or 1.5 square units.
• Wrong: 6 x 1/2 = 3, then 3² = 9.

The wrong answer is six times too large. The same standard pairs it with the volume formula V = s³, which gives (1/2)³ = 1/8 cubic units.

7. A longer example, step by step

Evaluate 6 + 18 / (5 - 2) x 2.

1. Parentheses: 5 - 2 = 3. The expression becomes 6 + 18 / 3 x 2.
2. Division and multiplication, left to right: 18 / 3 = 6, then 6 x 2 = 12. The expression becomes 6 + 12.
3. Addition: 6 + 12 = 18.

The answer is 18. Notice that in step 2, the division came first only because it appeared first.

8. Nested grouping symbols

When grouping symbols sit inside each other, start with the innermost one and work outward.

Evaluate 2 x [3 + (8 - 2) / 3].

1. Innermost parentheses: 8 - 2 = 6.
2. Inside the brackets, divide before adding: 6 / 3 = 2, then 3 + 2 = 5.
3. Outside: 2 x 5 = 10.

The answer is 10.

9. How to check your work

• Rewrite after every step. Copy the whole expression with one operation done. It is slower, but it prevents skipping or merging steps.
• Underline the next operation before doing it. If two operations share a level, underline the one farther left.
• Read the expression as a sentence. The grade 5 standards ask students to interpret expressions without evaluating them, for example seeing 2 x (8 + 7) as "add 8 and 7, then multiply by 2." If your steps do not match that sentence, recheck them.
• Be careful with calculators. Different calculators can handle long expressions differently. Enter parentheses where you need them, or work in pieces.

10. Key takeaways

• The order of operations is a shared convention that gives each expression one correct value.
• Work grouping symbols first, then exponents, then multiplication and division, then addition and subtraction.
• Multiplication and division share one level, worked left to right; so do addition and subtraction.
• An exponent applies only to what sits directly in front of it, so 2 x 3² is 18, not 36.
• Rewriting the expression after each step is the most reliable way to avoid mistakes.