Order of operations: learn and practice

Who this lesson is for

This order of operations lesson is for grades 5–6. Grade 5 meets several operations in one expression. Grade 6 uses the same order on longer expressions, including percents on the grade 6 hub. Both grades link here instead of to a copied page.

One expression, one agreed order

An expression such as 3 + 4 × 2 can be read in more than one way if you invent your own order. Mathematics uses one order so that every reader gets the same value. Parentheses are done first, because they are a written instruction to treat what is inside as a single quantity. After that, multiplication and division are done from left to right. Addition and subtraction are done last, also from left to right. Multiplication does not outrank division, and addition does not outrank subtraction.

Parentheses change the groups

Without parentheses, 3 + 4 × 2 means 3 plus the product 4 × 2. With parentheses, (3 + 4) × 2 means the sum 3 + 4, and then that sum times 2. The symbols are almost the same and the values are different: 11 and 14. When you copy an expression onto the board, copy the parentheses too. Dropping them changes the problem. Nested parentheses are done from the inside out. In 2 × (3 + (4 − 1)) you first do 4 − 1, then add 3, then multiply by 2.

Worked example 1: multiply before you add

Evaluate 3 + 4 × 2. There are no parentheses. The multiplication comes before the addition. 4 × 2 = 8. The expression is now 3 + 8. 3 + 8 = 11. So the value is 11. The left-to-right reading 3 + 4 = 7 and then 7 × 2 = 14 is a different problem, the one with parentheses around 3 + 4. If your answer is 14, you added first. Circle the multiplication, do it, and only then add the 3 that was waiting.

Worked example 2: parentheses first

Evaluate (3 + 4) × 2. The parentheses say to add before multiplying. 3 + 4 = 7. The expression is now 7 × 2. 7 × 2 = 14. So the value is 14. Compare this with example 1. The only change is the parentheses, and the value changes from 11 to 14. That is why a grouping symbol is stronger than the usual multiply-before-add rule. The parentheses replace that rule for the numbers they enclose.

Worked example 3: left to right inside a rank

Evaluate 20 − 8 ÷ 2 + 1. Division is in the higher group, so do it before the subtraction and the addition. 8 ÷ 2 = 4. The expression is now 20 − 4 + 1. Subtraction and addition share a rank, so go left to right. 20 − 4 = 16. Then 16 + 1 = 17. The value is 17. Doing the addition 4 + 1 before the subtraction would give 20 − 5 = 15, which is wrong. Left to right is the tie-breaker inside one rank. It is not a reason to add before you divide.

A line for each operation

Rewrite the expression after every operation and keep the numbers you have not used yet. If a line changes two operations at once, split it. When you finish, you should be able to point at each line and name the single operation it did. Then practice a new expression on the board. Start with one multiplication beside an addition, then add parentheses, then mix division with subtraction so the left-to-right rule has work to do.

Common questions about the order of operations

Does multiplication always come before division?

No. Multiplication and division are done in the same pass, from left to right. 8 ÷ 2 × 4 is 4 × 4, which is 16, not 8 ÷ 8.

Do parentheses beat multiplication?

Yes. What is inside parentheses is treated as one quantity before you multiply or divide by something outside. (3 + 4) × 2 is 14, while 3 + 4 × 2 is 11.

What about addition and subtraction?

They are partners too. After the higher operations are finished, do addition and subtraction from left to right. Do not do every addition before every subtraction.