Order of Operations

BEDMAS is not six steps. It is four, because two of the letters are a pair and so are two more, and that is exactly where most marks are lost.

Whole numbers · topic 10 of 10Grades 5 to 7Number6 min read7 practice questions

Why an order exists

Without an agreed order, 2 + 3 x 4 has two answers, and both of them are defensible.

Work left to right and you get 20. Do the multiplication first and you get 14. Neither is more natural than the other, and mathematicians did not discover which was correct; they agreed on one so that everybody writing a calculation means the same thing by it. Telling children that it is an agreement rather than a law of nature usually helps, because they stop looking for a reason inside the numbers.

The four levels

Four, not six

1. Brackets

2. Exponents, the small raised numbers

3. Multiply and divide, together, left to right

4. Add and subtract, together, left to right

BEDMAS has six letters and that is what causes the trouble. Children read it as a queue of six and conclude that division waits for multiplication, and that addition waits for subtraction. Neither is true. They are two pairs, and inside a pair you simply work left to right.

Same level, left to right

Take 20 divided by 4 x 5. If division genuinely came before multiplication you would do 4 x 5 first, get 20, and end with 1. Working left to right instead gives 5 x 5 = 25, and 25 is the correct answer.

The same happens with the other pair. In 10 - 3 + 2, doing the addition first gives 10 - 5 = 5, while left to right gives 7 + 2 = 9. Again, 9 is correct.

Write BEDMAS in two columns

Put DM on one line and AS on the next, side by side rather than in a queue. It is a small change to how it is written down and it removes the most persistent error in the whole topic.

Worked examples

Work out 2 + 3 x 4.

  1. No brackets, no exponents.
  2. Multiply and divide level: 3 x 4 = 12.
  3. The calculation is now 2 + 12.
  4. Add and subtract level: 14.

14

Work out 6 + 2 x (9 - 4).

  1. Brackets first: 9 - 4 = 5.
  2. The calculation is now 6 + 2 x 5.
  3. Multiply: 2 x 5 = 10.
  4. Add: 6 + 10.

16

Work out 36 divided by 6 x 2.

  1. No brackets, no exponents.
  2. Divide and multiply are the same level, so work left to right.
  3. 36 divided by 6 = 6.
  4. 6 x 2 = 12.

12, not 3

Common mistakes

1. Doing all the multiplying before any dividing

Reading BEDMAS as a queue of six. In 36 divided by 6 x 2 that gives 3 instead of 12. Multiply and divide are one level and you take them in the order they appear on the page.

2. Adding before subtracting

The same error in the other pair. In 10 - 3 + 2 it gives 5 rather than 9. It is less commonly taught and just as costly, and it is worth testing separately, because a child can have the multiply and divide pair right and still miss this one.

3. Working left to right and ignoring the order entirely

Answering 20 to 2 + 3 x 4. This one is not really a mistake about BEDMAS, it is a sign the child has not met it yet, or has forgotten it applies. If it appears, go back to why the agreement exists rather than drilling the letters.

Practice questions

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What to learn next

The order of operations is the last piece of whole-number arithmetic, and it is also the first piece of algebra: an expression like 3n + 2 only means one thing because of these rules.

For parents

Try 10 - 3 + 2 on your child, not 2 + 3 x 4. Almost every child is drilled on the multiply-before-add case and almost none on the subtract-then-add one, so the second question tells you far more about whether they have understood the levels or memorised one example.

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