Adding and Subtracting Decimals: Line Up That Point
Every year I watch the same mistake. A student writes 3.4 plus 12.75, and instead of stacking them by the decimal point, they shove both numbers over to the right like they're adding whole numbers. So they line the 4 up under the 5, get some number in the thousands, and hand it in looking pleased. The digits were all correct. The lineup was wrong, so the answer was garbage.
Adding and subtracting decimals is not hard math. The arithmetic is the same adding and subtracting they already know. The one new rule is where the numbers go on the page, and almost every early mistake comes from getting that one rule wrong. So let's make it stick.
The one rule: stack the decimal points
When you add or subtract decimals, you line up the decimal points, straight down the page, in a single column. Everything else follows from that.
Take 3.4 plus 12.75. Written correctly, it looks like this:
12.75
+ 3.40
-------
Notice two things. The decimal points are in a vertical line. And I added a zero to 3.4 to make it 3.40, so both numbers have the same number of places after the point. That zero doesn't change the value at all. Three and four tenths is exactly the same as three and forty hundredths. But it fills the empty spot so the columns line up cleanly, and it stops kids from panicking about a "missing" digit.
Once the points are stacked, you add exactly like you always do, right to left, carrying when you need to. Then you bring the decimal point straight down into the answer. It lands directly below the other points. It does not move.
Why "line up the right side" fails
Kids line numbers up on the right because that's what worked for years with whole numbers. With whole numbers, the rightmost digit is always the ones place, so lining up on the right lines up the place values automatically. It's a good habit that quietly stops working the second a decimal shows up.
Here's why. In 3.4, the 4 is in the tenths place. In 12.75, the rightmost 5 is in the hundredths place. Line those two up on the right and you're adding tenths to hundredths, which is like adding four dimes to five pennies and calling it nine of something. It isn't nine of anything.
The decimal point is the anchor. It marks where the whole numbers end and the fractions begin. Line up the points and every place value falls into its correct column on its own. That's the whole reason the rule exists.
The money trick
If a kid is fighting this, I stop talking about tenths and hundredths and switch to money. Almost every kid has a rock-solid mental model of dollars and cents, and dollars and cents are just decimals wearing a costume.
Ask them: you have $3.40 and I give you $12.75. How much do you have?
Suddenly they don't stack it wrong. Nobody lines up money by shoving the numbers to the right, because they know intuitively that dollars go with dollars and cents go with cents. They'll tell you $16.15 without a worksheet. Then I show them that the money they just added is the exact same problem as 3.4 plus 12.75, laid out the exact same way. The dollar amount is the whole number. The cents are the decimal places. The decimal point is the little dot between dollars and cents that they've seen a thousand times.
Money is powerful here for a few reasons:
- Kids already know cents always have two places, which reinforces filling in that extra zero.
- They have real intuition for the size of the answer, so a wildly wrong result feels wrong to them.
- It connects a school skill to something they'll use at a store for the rest of their lives.
Subtracting decimals, and the borrowing that trips kids up
Subtraction works the same way. Stack the points, fill in zeros so both numbers have the same number of places, then subtract right to left and borrow when you need to.
The classic wreck is something like 8 minus 3.6. A kid sees the 8 as just "8" with nothing after the point and stalls out. Rewrite the 8 as 8.0, and now it's honest:
8.0
- 3.6
-----
You can't take 6 tenths from 0 tenths, so you borrow one whole, turning the 8 into 7 and the 0 tenths into 10 tenths. Ten minus six is four, seven minus three is four, and the answer is 4.4. The borrowing is identical to whole number borrowing. The only trick was writing 8 as 8.0 so there was actually something to borrow into.
I tell kids: an invisible zero is still a zero. If a number looks like it's "missing" a decimal place, write the zero in. It costs nothing and it saves the problem.
A quick way to catch mistakes
Estimation is the seatbelt. Before a kid does the careful arithmetic, have them guess the ballpark. For 3.4 plus 12.75, round to 3 plus 13 and expect something near 16. When the careful answer comes out to 16.15, it passes the sniff test. When a lined-up-wrong answer comes out to 3,415 or something absurd, the kid catches it themselves.
That habit does more than fix decimals. It builds the instinct that an answer should make sense, not just fall out of a procedure. A child who estimates first and checks after is doing real thinking, not just cranking a handle. That's worth far more than a stack of finished worksheets.
Here's the short version to tape near the homework spot. Line up the decimal points. Fill empty places with zeros. Add or subtract as usual. Bring the point straight down. If you can do money, you can do this, because it's the same thing.