Montessori Math

The Lesson AlbumDecimal Fractions · lesson 2ages 9–12grades 4–6

Decimal Operations

With the places to the right of the unit secure, the decimal board now compares, adds, and subtracts decimal quantities. The rules are the ones your child already knows from the golden beads — put together and trade ten up, take away and break one down — carried straight across the decimal point.

Materials

  • The decimal board and colored counters from the introduction lesson (paper columns and buttons or beads in four colors)
  • A second paper board, or the same sheet folded in half, so two quantities can sit side by side for comparing
  • Three small cards written <, =, and > for comparison work
  • Slips of paper with problems: comparison pairs like 0.3 ? 0.25, sums like 1.23 + 0.45 and 1.4 + 0.75, and differences like 2.87 − 1.34 and 2.1 − 0.35
  • Pencil and paper for recording each problem and its answer

No materials at home? Use the virtual Decimal Board.

Before this lesson

Aims

Direct

  • To compare decimal quantities place by place, starting from the largest place
  • To add decimals concretely, trading ten of a place for one of the place to its left
  • To subtract decimals concretely, breaking one of a place into ten of the place to its right — including one unit into ten tenths, across the decimal point

Indirect

  • Preparation for the written algorithms for decimal addition and subtraction, with the decimal points lined up
  • Guarding against the common misreading that 0.25 is more than 0.3 because twenty-five looks bigger than three
  • Remote preparation for decimal multiplication and division, percent, and measurement work

Presentation

  1. 1 Begin with a comparison. Have your child build 0.3 on one board and 0.25 on the other, checking each against its slip.Which is bigger? Let us not guess — the board will show us.
  2. 2 Compare column by column from the left: units first (none on either side), then tenths — three against two.Three tenths against two tenths. The tenths decide it: no matter what comes after, 0.3 is greater.
  3. 3 Let your child test the trap. If twenty-five still “looks bigger,” trade the 3 tenths for 30 hundredths and count 30 hundredths against 25.Thirty hundredths against twenty-five hundredths — now you can see which is more.
  4. 4 Place the correct card between the boards and record the sentence: 0.3 > 0.25. Work two or three more comparison pairs the same way.We always compare the biggest places first.
  5. 5 Move to addition. Write 1.23 + 0.45 on a slip. Have your child build 1.23 across the top of the board and 0.45 below it, then slide the two quantities together column by column and read the answer: 1.68.Adding is just putting together — the same as it was with the golden beads.
  6. 6 Record 1.23 + 0.45 = 1.68 in pencil. Now write 1.4 + 0.75 and have your child build and combine the two quantities the same way.This time, keep an eye on the tenths column.
  7. 7 Count the tenths column: eleven. Trade ten of them for one unit, carrying it left across the decimal point, and let your child read the answer.Ten tenths — trade! — one unit crosses the point. Now read it: two point one five.
  8. 8 Record 1.4 + 0.75 = 2.15 in pencil. Now write 2.87 − 1.34 and have your child build only 2.87.In taking away, we build just the starting number. The answer will be whatever is left on the board.
  9. 9 Take away 1.34 place by place — 4 hundredths, then 3 tenths, then 1 unit — with no trades needed. Read what remains and record 2.87 − 1.34 = 1.53.Two point eight seven, take away one point three four — one point five three is left.
  10. 10 Now write 2.1 − 0.35 and have your child build only 2.1, then ask them to take away 5 hundredths. The hundredths column is empty — so trade 1 tenth for 10 hundredths, then take the 5.Nothing to take! So we break one tenth into ten hundredths — and now we can.
  11. 11 Ask for 3 tenths next. No tenths remain — so break 1 unit into 10 tenths, straight across the decimal point, and take the 3.Even a whole unit will break into ten tenths when we need it to.
  12. 12 Read what remains — 1 unit, 7 tenths, 5 hundredths — and record 2.1 − 0.35 = 1.75. Have your child check by putting the taken-away pieces back and confirming the board reads 2.1 again.One point seven five. Put back what we took — do we land exactly on two point one?

Points of interest

  • The trap pairs — 0.3 versus 0.25, 0.4 versus 0.399 — where the shorter numeral wins
  • Carries and borrows sail straight across the decimal point as if it were not there
  • Eleven tenths sitting in one column, visibly waiting to be traded before the answer can be read
  • Checking subtraction by adding back what was taken away

Control of error

  • A column holding ten or more cannot be read — the material itself demands the trade before an answer can be written
  • Taking from an empty column is impossible — the board forces the exchange at exactly the spot where a written algorithm would borrow
  • Putting the taken-away pieces back must restore the starting number exactly
  • On the virtual board, comparisons are judged exactly and the Check button marks every column ✓ or ✗ without giving the answer away

Vocabulary

  • greater than
  • less than
  • equal to
  • sum
  • difference
  • exchange
  • carry
  • borrow

Variations

  • Comparison pairs where the shorter numeral is larger (0.4 vs 0.399, 0.5 vs 0.487) until place-by-place comparing is automatic
  • Addition with a full cascade of trades, like 0.999 + 0.001, where every column trades up in turn
  • Subtraction with a double borrow across the point, like 2.1 − 0.35 — then a full cascade of borrows, like 3 − 0.001

Extensions

  • Estimate before building: for 1.4 + 0.75 ask, “Will it be more or less than 2?” — then let the board settle it
  • Work each problem twice, on the board and in columns on paper with the decimal points lined up, until your child prefers the paper
  • Money problems with real coins: $2.10 − $0.35 makes the same borrow with dimes and pennies

Follow-up work (pencil & paper)

  • Print a mixed sheet of decimal comparison, addition, and subtraction problems and have your child work it in pencil at the table, building on the board only the problems they doubt.Printable: Decimal Fractions
  • Trap-pair journal: your child writes five comparison pairs like 0.3 ? 0.25 in pencil, circles the greater number in each, and writes one sentence explaining which place decided it.
  • Check by adding back: for every subtraction worked on paper, your child adds the answer to the number taken away and confirms in writing that it restores the starting number.

What comes next

Fluent comparing, adding, and subtracting on the board lead naturally to the written algorithms: the same problems worked in pencil columns with the decimal points aligned. From there your child is ready for decimal multiplication and division, percent, and the measurement work of upper elementary.