Why order matters
The expression 36 − 4 × 7 is a compact set of instructions. The multiplication describes one quantity: four groups of seven. We find that quantity before subtracting it. First calculate 4 × 7 = 28, then 36 − 28 = 8.
If we simply work from left to right, we change the expression into a different set of instructions. That is why order of operations is more than a classroom convention to memorize: it lets everyone read the same mathematical sentence the same way.
Teach the hierarchy accurately
Parentheses are handled first because they explicitly group part of the expression. Multiplication and division come next and share the same priority, so they are handled from left to right. Addition and subtraction also share a priority and are handled from left to right.
This shared priority matters. The claim that multiplication always comes before division, or addition always before subtraction, creates errors later. In 24 ÷ 6 × 2, work left to right: 24 ÷ 6 = 4, then 4 × 2 = 8.
Use one clean written routine
Ask the child to underline or point to the operation that happens next. Perform only that operation, rewrite the rest unchanged, and repeat. One line should represent one step. This makes the structure visible and gives both child and parent a place to find an error.
- Circle parentheses or other explicit groups.
- Mark multiplication and division; take the leftmost one.
- Rewrite the expression after doing exactly one operation.
- Finish addition and subtraction from left to right.
- Estimate whether the final answer is plausible.
Questions that reveal understanding
Instead of asking only “What is the answer?”, ask “Which part forms one quantity?” or “What must stay unchanged on the next line?” If the child can explain why 4 × 7 stays together in 36 − 4 × 7, the rule has meaning.
Try changing the structure: compare 36 − 4 × 7 with (36 − 4) × 7. The same numbers produce very different results because the parentheses change the instructions. That contrast is often more memorable than another page of nearly identical exercises.
Three examples worth discussing
For 18 − 12 ÷ 3, division creates one quantity first: 12 ÷ 3 = 4, then 18 − 4 = 14. For 24 ÷ 6 × 2, division and multiplication have equal priority, so work from left to right: 24 ÷ 6 = 4, then 4 × 2 = 8. For 5 + 3 × (9 − 7), the parentheses produce 2, multiplication produces 6, and addition produces 11.
Ask the child to rewrite the complete expression after each step. The unchanged parts are as important as the part being calculated. A common error is to drop the 18 in the first example or carry an operation down incorrectly. Rewriting makes that bookkeeping visible.
After solving, change exactly one feature. Add parentheses around 18 − 12. Swap multiplication and addition. Ask the child to predict whether the answer will increase or decrease before calculating. These small variations test structure more effectively than ten copies of the same format.
Mistakes that point to different needs
A left-to-right answer such as 224 for 36 − 4 × 7 shows that the child may not yet see multiplication as a grouped quantity. Draw four groups of seven or put a box around 4 × 7. An answer based on doing multiplication before the parentheses suggests the hierarchy itself needs revisiting. A mistake in 24 ÷ 6 × 2 may come from an inaccurate slogan that multiplication always beats division.
Calculation errors and order errors should not be treated as the same problem. If the child chooses the correct next operation but computes 4 × 7 as 24, keep the structural praise and repair the fact separately. If every arithmetic fact is correct but the order is wrong, more calculation drill will not fix the misconception.
- Ask the child to point to the next operation before calculating.
- Have them explain which operations share priority.
- Check that every untouched number and symbol appears on the next line.
- Use estimation to catch a final answer that does not fit the expression.
A short practice set with answers
Try these one at a time and ask for the next operation before the answer: 20 − 3 × 4; 30 ÷ 5 × 3; 6 + 2 × (11 − 8); and (14 − 6) ÷ 2 + 5. The answers are 8, 18, 12, and 9. If an answer is wrong, return to the first line where the expressions differ rather than restarting the whole page.
Once these are comfortable, ask the child to invent two expressions using the same numbers but different parentheses so the answers differ. Creating an example is a strong test: it requires the learner to control the structure rather than merely react to it.