Skills / Ages 11 to 14 / Math
Order of operations
The agreed order for working out math with more than one operation: parentheses, then powers, then multiplication and division left to right, then addition and subtraction left to right.
Why it matters
The book's algebra lessons put numbers in for letters and work out expressions like 5 + 4n. With n = 6 that is 5 + 4 x 6, which is 29. Adding first gives 54, a different answer. Everyone uses one order so an expression means the same thing to everyone who reads it.
What it looks like when your child can do it
- Looking at 4 + 6 x 3, your child multiplies first and gets 22, not 30.
- Your child works the inside of parentheses before anything else, so (4 + 6) x 3 comes out 30.
- Your child rewrites the problem one step at a time, a line per step, instead of doing it all in their head.
How to teach it at the kitchen table
- 01
Show why an agreed order is needed
At the store you buy one 3 dollar loaf of bread and 2 bags of apples at 4 dollars each. That cost is 3 + 2 x 4. Your child knows the bill is 11 dollars, so the multiplying has to happen before the adding.
- 02
Learn the four steps in order
First, work inside parentheses. Second, powers, like 2 to the power of 3. Third, multiplication and division, from left to right. Fourth, addition and subtraction, from left to right.
- 03
Write one line per step
Copy the problem, do one operation, and write what is left on the next line. For 25 - 3 x 5, the first line after it is 25 - 15, and the last line is 10.
- 04
Parentheses change the order on purpose
12 - 4 ÷ 2 is 10, because the division comes first. (12 - 4) ÷ 2 is 4, because the parentheses say subtract first. Doing both side by side shows your child that parentheses are an instruction.
You will need: Paper and a pencil · A grocery receipt
Try it together
Try 1
Work out 4 + 6 x 3.
Listen for 22. Multiply first, 6 x 3 = 18, then add 4. An answer of 30 means your child went straight left to right.
Try 2
Work out (4 + 6) x 3.
Listen for 30. The parentheses come first, so 4 + 6 = 10, then 10 x 3 = 30. Put it beside the last one so your child sees the parentheses change the answer.
Try 3
Work out 25 - 3 x 5.
Listen for 10, because 3 x 5 = 15 and 25 - 15 = 10. An answer of 110 comes from subtracting first: 22 x 5.
Try 4
Work out 3 x 2³, where the small raised 3 belongs to the 2 only.
Listen for 24. The power comes first: 2 x 2 x 2 = 8, and 3 x 8 = 24. Multiplying first would cube 6 instead, which is 216.
Try 5
Work out 18 ÷ 3 + 6 x 2.
Listen for 18. The division and the multiplication both come before the adding: 18 ÷ 3 = 6 and 6 x 2 = 12, then 6 + 12 = 18.
Watch for
- Working straight left to right like reading a sentence. It usually gives a wrong answer when an addition or subtraction comes before a multiplication or division.
- Treating a power as multiplying by the small raised number, so 2 to the power of 3 becomes 6 instead of 8.
If it is not there yet
- Go back to two-operation problems with no parentheses, like 5 + 2 x 3, until multiplying first is automatic. Then bring in parentheses.
- Use real receipts. Write a grocery total as a math sentence and check it against the actual receipt, so the order has a reason behind it.
A few days of this is normal, and it is time well spent. The lesson will still be there, and it will go faster.
Where this skill comes up
Grade 6 Math Practice Workbook · Lesson 17
When a Letter Stands for a Number
Needs it because evaluating an expression correctly depends on it.