Math // Addition & Subtraction

Inverse: addition undoes subtraction

Recognise and use the inverse relationship between addition and subtraction to check calculations and solve missing-number problems

Ages
6 to 7
Type
Concept
Builds on
1 topic
Leads to
3 topics
Standards
2 codes

Quick check

If the learner works out 13 + 5 = 18, can they immediately use that to say '18 − 5 = 13' — without having to work the subtraction out all over again?

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.
  • Check 15 + 7 = 22 by calculating 22 − 7 = 15
  • Use the inverse to solve: □ + 9 = 14, so □ = 14 − 9 = 5
  • Explain that addition and subtraction 'undo' each other

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

1 topic

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Inverse: addition undoes subtraction

Ages 6 to 7 // Concept

Leads to

3 topics
  • Estimating by roundingRequired

    Addition & Subtraction // Ages 7 to 9

    Using inverse to check answers requires understanding the inverse relationship

  • Unknown in Addition & SubtractionHelpful

    Addition & Subtraction // Ages 6 to 7

    Solving missing-number problems benefits from knowing the inverse relationship

  • Connecting RepresentationsHelpful

    Mathematical Thinking // Ages 6 to 7

    Inverse relationship understanding requires moving between addition and subtraction representations

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

Help 6–7 year-olds see that addition and subtraction undo each other, so one known sum (e.g. 15 + 7 = 22) generates all four related facts, lets you check a calculation by reversing it, and lets you solve a missing-number equation (□ + 9 = 14) by subtracting. The core aim is to treat these as ONE relationship between operations, not four separate facts to memorise. Builds on fact families and part-whole/unknown-addend ideas.

Common misconceptions

  • Treating the four fact-family facts as unrelated things to memorise separately.
  • Believing subtraction facts can also be reversed freely (10 − 6 = 6 − 10), over-extending the 'undo' idea.
  • Checking a sum by repeating the SAME operation rather than using the inverse.
  • Solving □ + 9 = 14 by adding 9 and 14, not recognising subtraction as the inverse move.
  • Thinking 'undoing' requires a different, brand-new procedure rather than the operation they already know.

Useful representations

  • Fact-family / part-whole triangle (one whole, two parts, four facts)
  • Bar model with the whole and two parts, read as add or subtract
  • Empty number line showing a forward jump and its reversing backward jump
  • Balance model where adding then subtracting the same amount returns to start
  • Cuisenaire rods showing a+b and the reverse whole−a

Teaching practices

  • Anchor on the fact-family triangle / part-whole model so all four related facts come from one whole and two parts.
  • Make 'checking by the inverse' a routine habit after every calculation, not an add-on.
  • Use inverse explicitly to convert a missing-number equation into a calculation the child can do (□ + 9 = 14 → 14 − 9).
  • Keep the SAME three numbers while flipping the operation, so the child sees one relationship rather than four separate facts.
  • Contrast with situations where inverse reasoning is misapplied, e.g. reminding that order matters in the subtraction facts (10 − 6 vs 6 − 10).

Activities to try

  1. 1

    Do and undo it on a number line (with a fact-family triangle)

    Why it works: Subtraction reverses addition over the SAME three numbers, so taking one part from the whole returns the other part; one sum gives four related facts and a built-in way to check.

    1. Start at 15 on a number line and make a forward jump of +7, landing on 22.
    2. Before reversing, predict out loud where a backward jump of 7 will land and why.
    3. Make the backward jump of −7 and confirm you return exactly to 15.
    4. Say in your own words why the subtraction lands back on the start: it is the same jump run in reverse.
    5. Fill in a fact-family triangle with the whole (22) and the two parts (15 and 7), then write all four facts: 15 + 7 = 22, 7 + 15 = 22, 22 − 7 = 15, 22 − 15 = 7.
  2. 2

    Solve a missing number by reversing

    Why it works: A missing-addend equation becomes a calculation you can already do by using the inverse operation, and the answer can be confirmed by reversing again.

    1. Look at □ + 9 = 14 and decide what operation would find the missing part.
    2. Explain why subtracting (14 − 9) finds the box: addition and subtraction undo each other.
    3. Compute □ = 14 − 9 = 5 and check by putting it back: 5 + 9 = 14.
    4. Try one or two more missing-number equations, each solved by the inverse and then checked by adding back.
  3. 3

    Explain WHY adding back proves a subtraction

    Why it works: Checking works because addition reverses subtraction over the same whole and parts; the 'undo' idea applies to taking a part from the whole, not to any rearrangement of the three numbers.

    1. Given 22 − 7 = 15, explain how you could be sure it is right WITHOUT redoing the subtraction.
    2. Discuss the difference between reversing (adding the parts back) and simply repeating the same subtraction.
    3. Respond to the claim that 'subtraction undoes too, so 15 − 7 = 22' and explain why that is wrong using which number is the whole.
    4. State a DIFFERENT true subtraction from the same three numbers (22 − 15 = 7) and justify it.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
1.OA.4
The national curriculum in England: Key stages 1 and 2 framework document
Maths/Y2/AS/9

Math // Addition & Subtraction

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