Adding and Subtracting Decimals
One instruction carries this whole topic: line up the decimal points, not the ends of the numbers. Everything else is ordinary column arithmetic.
The one instruction
Line up the decimal points, one directly under the other.
That is the whole method. Once the points are aligned, every column contains digits of the same value, and you add or subtract exactly as you always have.
The rule
Write the numbers with their decimal points in a vertical line. Fill any empty places with zeros. Add or subtract column by column, starting from the right.
Put the decimal point in the answer straight below the others, before you write a single digit.
Filling the gaps
Numbers rarely arrive with the same number of decimal places. 12.4 and 3.75 do not match, so make them match.
Write 12.4 as 12.40. It is the same amount, because a zero on the right-hand end changes nothing, and now every column is occupied.
This matters more for subtraction than addition. Trying to work out 6 − 2.35 with nothing under the 6 leads straight to a mess. Write it as 6.00 and it becomes ordinary.
Estimate first
Ten seconds of estimating catches almost every wrong answer, including the ones caused by a misplaced point.
- 12.4 + 3.75 is roughly 12 + 4 = 16. An answer of 1.615 or 161.5 is instantly wrong.
- 6 − 2.35 is roughly 6 − 2 = 4. An answer of 3.65 passes. An answer of 4.35 does not.
Teach the estimate as part of the method, not as a check to do if there is time left. Children who estimate first almost never hand in an impossible answer.
Worked examples
Different lengths
- Estimate: about 12 + 4 = 16.
- Write 12.4 as 12.40 so both have two decimal places.
- Line up the points and add from the right: 0 + 5 = 5, then 4 + 7 = 11, so write 1 and carry 1.
- Carry on into the whole numbers: 2 + 3 + 1 = 6, then 1. The answer is 16.15, close to the estimate.
Subtracting from a whole number
- Estimate: about 6 − 2 = 4, so expect something a bit under 4.
- Write 6 as 6.00. Every whole number has a decimal point, it is just usually invisible.
- Line up the points and subtract from the right, exchanging where needed.
- The result is 3.65, which sits just under 4 as expected.
A money problem
- Two steps here, so do the total first. 8.50 + 1.75 = 10.25.
- Estimate the change: about 20 − 10 = 10 dollars.
- Write 20 as 20.00 and subtract 10.25.
- 20.00 − 10.25 = 9.75, which sits just under the estimate.
Common mistakes
1. Lining up the right-hand ends instead of the points
Writing 12.4 above 3.75 so the 4 sits above the 5. That puts tenths under hundredths, and the answer is nonsense. It happens because with whole numbers, lining up the ends IS lining up the columns. Once a point appears, only the point can be trusted.
2. Losing the decimal point in the answer
Doing all the arithmetic correctly and writing 1615 instead of 16.15. Put the point in the answer FIRST, directly under the others, then fill in the digits around it. It takes one second and it cannot be forgotten later.
3. Leaving a column empty
Subtracting 2.35 from 6 with nothing written under the 6. Fill the gaps with zeros before starting. A blank column invites guessing, and a zero does not.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
Check yourself
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For parents
Give them squared paper. One digit per square forces the columns to line up without anyone having to nag about it, and most of the errors on this topic disappear on their own. It is a cheaper fix than any amount of explaining.
What to learn next
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