What is this skill?
When an expression has more than one operation, mathematicians have agreed on an order so everyone gets the same answer. Work inside parentheses first. Then multiplication and division, from left to right. Then addition and subtraction, from left to right. So 3 + 4 × 5 is 3 + 20 = 23, because the multiplication comes first, while (3 + 4) × 5 is 7 × 5 = 35, because the parentheses come first.
Fifth graders also see brackets and nested grouping, as in [8 + (2 × 3)] ÷ 2. Work from the innermost group outward: 2 × 3 = 6, then 8 + 6 = 14, then 14 ÷ 2 = 7. The common memory aid PEMDAS can mislead because multiplication and division share the same rank. In 24 ÷ 4 × 2, go left to right: 24 ÷ 4 = 6, then 6 × 2 = 12. Doing the multiplication first would give 3, which is wrong.
Why it matters
Order of operations is the grammar of math. Without it, an expression like 3 + 4 × 5 would have two different answers, and calculators, spreadsheets and computer programs could not agree with one another. It is also the first step toward algebra, where expressions with variables follow exactly the same rules.
It rewards careful, step-by-step thinking too. Children who write each step on its own line, instead of trying to do it all in their head, make fewer mistakes in every kind of multi-step problem.
Common mistakes
- Working strictly left to right, so 3 + 4 × 5 becomes 7 × 5 = 35. This is the most common error.
- Treating multiplication as always before division, or addition as always before subtraction, when each pair is equal in rank and goes left to right.
- Ignoring the parentheses, or stopping after the first step inside them.
- Losing part of the expression when rewriting it. Copy the unchanged parts carefully on every line.
- Making a small arithmetic slip on a step that is otherwise correct. Writing each step down makes the slip visible.
How to help at home
- Write each step on its own line, with the work underlined. Ask your child to say which operation comes first and why.
- Make up expressions together, for example “a pack of 6 pencils at 2 dollars plus a 3-dollar eraser” and write it as 6 × 2 + 3.
- Show how parentheses change the meaning: (2 + 3) × 4 versus 2 + 3 × 4. Same numbers, different stories and different answers.
- Skip the acronym if it causes confusion, and teach it as three levels: groups, then multiply and divide, then add and subtract.
- Try the same problem on a calculator. Many calculators follow the order, and some basic ones do not. Seeing the difference is a good conversation.
Worked examples
These are real questions from the Tutorsaurus order of operations practice, with the answers worked out.
Example 1
4 + 7 × 2 = ?
Answer: 18. Multiply first: 7 × 2 = 14. Then 4 + 14 = 18.
Example 2
24 ÷ 8 + 8 = ?
Answer: 11. Divide first: 24 ÷ 8 = 3. Then 3 + 8 = 11.
Example 3
9 × (6 + 9) − 6 = ?
Answer: 129. Parentheses: 6 + 9 = 15. Multiply: 9 × 15 = 135. Subtract 6: 129.