Math // Addition & Subtraction

Adding and subtracting (age 7+)

Add and subtract numbers with up to three digits using formal written methods of columnar addition and subtraction

Ages
7 to 8
Type
Method
Builds on
2 topics
Leads to
5 topics
Standards
1 code

Quick check

Can the learner set out a sum like '347 + 265' or '512 − 278' in columns and work through it step by step, carrying or borrowing where needed?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Set out a columnar addition correctly with digits aligned by place value
  • Carry out columnar subtraction with exchange (borrowing) when needed
  • Check the answer using the inverse operation or estimation

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics

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Adding and subtracting (age 7+)

Ages 7 to 8 // Method

Leads to

5 topics

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Teach formal columnar addition and subtraction to three digits as a compact written algorithm that keeps its place-value meaning. Introduce it only after children can compose/decompose concretely, and bridge from the expanded method (partial sums) to the compact one so each carried or exchanged digit still stands for a real ten or hundred. The skill has three parts: correct place-value alignment, the carry/exchange mechanic, and checking with estimation or the inverse. Pair written work with place-value counters during the transition, and rehearse the hard across-zero exchange explicitly.

Common misconceptions

  • Place-value misalignment: writing digits ragged so ones and tens are added together
  • Smaller-from-larger: subtracting the smaller digit from the larger in each column regardless of position (83−25→62 pattern, now at three digits)
  • Carry omitted or mis-placed: forgetting to add the carried ten/hundred to the next column
  • Borrow-across-zero errors when the adjacent column is 0 (502−276)

Useful representations

  • Column grid / squared paper with H-T-O headings
  • Place-value counters alongside the algorithm
  • Expanded written method (partial sums) as the stepping stone
  • Base-ten blocks for the exchange step
  • Number line / estimation for the check

Teaching practices

  • Bridge expanded-to-compact: start with partitioned partial sums, then compress, so the compact carries keep their meaning (NCETM; DfE Appendix 1)
  • Insist on place-value alignment — use a column grid or squared paper so ones sit under ones
  • Keep base-ten/place-value counters available alongside the written method during the transition, especially for exchange
  • Teach subtraction exchange as decomposition ('take a ten, make ten ones'), not a memorised 'borrow', and rehearse across-zero cases
  • Always check: estimate first (round) and verify with the inverse (DfE Year 3)

Activities to try

  1. 1

    Carry and exchange with counters beside the columns

    Why it works: A carried or exchanged digit is a real ten or hundred moving between columns, not a meaningless mark; across a zero you must decompose a hundred into tens before you can get a ten, then a ten into ones.

    1. Set out 347 + 185 in columns on squared paper or a headed H-T-O grid, ones under ones.
    2. Add the ones: 7 + 5 = 12. Write the 2 in the ones column and the carried 1 above the tens; alongside, exchange ten of the 12 one-counters for a single ten-counter and place it in the tens column, so the written '1' and the real ten move together.
    3. Add the tens: 4 + 8 + 1 = 13. Write 3, carry 1 hundred, and exchange ten of the 13 ten-counters for a single hundred-counter, placing it in the hundreds column.
    4. Add the hundreds: 3 + 1 + 1 = 5, giving 532.
    5. Now set out 502 − 276. To take 6 from 2 in the ones you need more ones, but the tens column is 0: first break one hundred into ten tens (502 becomes 400 + 90 + 12 after the cascade), then break a ten into ten ones.
  2. 2

    Bridge from expanded partial sums to the compact column

    Why it works: The compact algorithm is a shorthand for the expanded method; every carry corresponds to a partial sum crossing into the next place.

    1. Solve 347 + 185 first in the expanded written layout, writing each partial sum: 7 + 5 = 12, 40 + 80 = 120, 300 + 100 = 400.
    2. Add the partial sums (12 + 120 + 400 = 532) and notice where the extra ten and the extra hundred come from.
    3. Rewrite the same calculation in the compact column, carrying the 1 ten and 1 hundred, and check the answer matches 532.
    4. Say aloud what each carried digit in the compact version stands for, pointing back to the partial sum it came from.
  3. 3

    Spot the error in a worked column

    Why it works: Many errors that look careless are the surface of a systematic wrong procedure (misalignment, smaller-from-larger, borrow-across-zero); naming the bug helps avoid it.

    1. Examine a worked example where the digits are set out ragged (for example the 5 of 185 placed under the 4 of 347) and identify what has gone wrong and why the answer is unreliable.
    2. Examine a worked across-zero subtraction where a 9 or 0 has been written at the empty tens column without decomposing a hundred, and explain the correct cascade.
    3. Rework each flawed example correctly on a headed grid and state what the fixed carry or exchange represents.
    4. Complete a few mixed column calculations for fluency, estimating first and checking with the inverse.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
Ma/KS2/Y3/AS/2

Math // Addition & Subtraction

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