Place Value in Decimals — Preventing "0.7 + 0.5 = 0.12"
"0.7 + 0.5 = 0.12." Plenty of parents have seen this one. The two sides of the decimal point get added as separate whole numbers.
I assumed at first that it was a slip. It isn't, at all. Almost every decimal error traces back to place value — to the fact that the base-ten structure continues to the right of the decimal point.
Where it goes wrong
Cause 1: 0.1 is a symbol, not a number
Japan's Ministry of Education, in its commentary on the national curriculum guidelines for elementary mathematics (2017), describes decimals like this. With whole numbers, "ten of a given unit make the next unit up"; with decimals, "conversely, the size of a unit (1) is divided into ten to make a new unit (0.1), and size is expressed as so many of that unit" (Grade 3, Numbers and Calculation).
So 0.1 is the same machinery as "ten 1s make 10", run one step in the opposite direction. Without that structure, 0.7 + 0.5 becomes symbol pushing: 7 + 5 = 12, so 0.12.
Cause 2: the false rule "×10 adds a zero"
With whole numbers, "×10 means add a zero on the right" happens to work. Carried into decimals it produces "2.5 × 10 = 2.50". What actually happens is that every digit moves up one place. Teach "the places go up" rather than "add a zero" from the whole-number stage and this never breaks.
Cause 3: lining up the right-hand ends in column arithmetic
Writing "2.5 + 0.34" flush right. Whole-number column arithmetic lined up on the right, so this transfer is natural too. With decimals the rule is to line up the decimal points — that is, line up matching places.
How to teach it at home
Step 1: see 0.1 with a litre jug
Grade 3 decimals start from "one tenth of a litre is 0.1 L". Pour it with a measuring cup and confirm with the eye that ten lots of 0.1 L make 1 L. The Decimal Units (Capacity and Length) topic sits exactly at that join between felt quantity and notation.
Step 2: place the digits in a table before calculating
Draw a table — ones, tenths, hundredths — and put 2.5 and 0.34 into their places before adding. Once "line up the decimal points" is visibly a shorthand for that table, it stops being a rule to memorise.
Step 3: put it on a number line
The commentary says that "placing decimals on a number line, positioning them within the same number line as whole numbers, is important for deepening understanding", using 3.6 as its example: it lies between 3 and 4, and it is the sixth mark when the gap from 3 to 4 is divided into ten. A single line drawn on paper does the job.
Step 4: call ×10 and ÷10 "moving house"
"2.5 × 10: the 2 moves to the tens place and the 5 to the ones, so 25." "Every digit moves up one place" carries further than "add a zero" or "shift the point" — it still works for decimals × whole numbers in grade 4 and decimals × decimals in grade 5.
How to practise
- Decimals secures how many 0.1s, the number line, and same-place addition and subtraction
- Adding and Subtracting Decimals is mostly column arithmetic; when it goes wrong, go back to placing the digits in the table. Multiplying and Dividing Decimals is where "moving house" earns its keep
- In grade 5, Multiplication of Decimals and Division of Decimals make placing the point the main challenge. Saying an estimate first — "will 2.5 × 1.3 be more or less than 2.5?" — acts as a detector for misplaced points
Frequently asked questions
Q. "0.5 and 0.50 are the same" hasn't clicked
A. Put them in the place table and 0.50 is simply "a zero in the hundredths place" with the same quantity. The measurement context (why you'd write 0.50 L) is a separate matter, so in grades 3 and 4 "the same size" is enough.
Q. Is it all right to teach with money?
A. Comparing 0.1 to a ten-cent coin fits the intuition for place value and works well. But money has no thousandths, so run it alongside real capacity (L, dL, mL) and length (m, cm, mm) to root it more deeply.
Decimal errors look alarming, but the causes gather into one place. Once your child can draw the place table, several different mistakes disappear at the same time.
This article is written by a parent who builds diglabo. I am not a teacher or an education specialist. The general guidance here is based on how the topic is taught in Japanese primary schools, checked against Japan's national curriculum guidelines.
diglabo's worksheets follow the Japanese curriculum. What that means if you are outside Japan