Math // Addition & Subtraction

Numbers up to 10 into pairs

Decompose numbers up to 10 into pairs in more than one way (part-part-whole)

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

Quick check

If the learner has 8 grapes, can they split them into two groups in different ways — like 3 and 5, or 4 and 4 — and explain that both groups together still make 8?

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.
  • Show that 5 = 1 + 4, 5 = 2 + 3, 5 = 0 + 5 etc.
  • Use objects or drawings to find all pairs that make a given number
  • Record decompositions as equations

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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Numbers up to 10 into pairs

Ages 4 to 6 // Concept

Leads to

2 topics
  • Number bonds to 9Required

    Addition & Subtraction // Ages 4 to 6

    Making 10 is a specific application of decomposing numbers into pairs

  • Showing Your WorkingHelpful

    Mathematical Thinking // Ages 5 to 6

    Explaining part-part-whole decompositions exercises showing and telling

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

For ages 4-6 (Reception/Y1), build flexible decomposition of numbers up to 10 into part-pairs (part-part-whole): 5 = 0+5 = 1+4 = 2+3, and their reversals. The core insight is that one whole is a FAMILY of pairs — the whole stays constant while the two parts trade. Fix the whole, systematically vary the split, and record each as an equation, connecting to number bonds and missing-part subtraction. Several complementary formats each play a role: concrete manipulatives and a screen part-whole bench both let the child generate pairs and feel the whole conserved, while a static anchor diagram and recognition checks consolidate the bond facts. Physical manipulation is emphasised at this age in line with the curriculum.

Common misconceptions

  • Thinking a number has only ONE decomposition (e.g. 5 is 'just' 4 and 1) and not seeing it as a family of pairs.
  • Losing the whole — changing a part but not compensating the other, so the total drifts (5 becomes 2+4).
  • Missing pairs or double-counting when not working systematically, so 'all the ways' is never securely established.
  • Treating 2+3 and 3+2 as entirely unrelated rather than the same pair reordered.
  • Not connecting decomposition to subtraction — failing to see that if 5 = 2 + 3 then 5 − 2 = 3.

Useful representations

  • Part-part-whole ('cherry'/bar) model with a fixed whole and two changeable parts
  • Ten-frame showing a number split by colour into two parts
  • Bead string or two-colour counters partitioned into two groups
  • A systematic list/table of all pairs for a number (0+5, 1+4, …) with equations
  • Rekenrek (arithmetic rack) for showing splits of 5 and 10

Teaching practices

  • Keep the WHOLE fixed and physically trade objects between the two parts, so children see the same total split many ways — the conceptual heart of the topic.
  • Work systematically (0+5, 1+4, 2+3, …) and record each split as an equation, so the set of pairs is seen as complete and patterned, not random.
  • Use the part-part-whole frame consistently so 'whole' and 'part' become stable ideas that carry into missing-part subtraction and bonds.
  • Connect to commutativity gently (2+3 and 3+2 use the same pair) without yet formalising it, and to the emerging idea that knowing one pair gives its partner.
  • Record with drawings AND equations so the abstract bond is tied to a picture of two parts making a whole.

Activities to try

  1. 1

    Part-whole bench: slide the divider, keep the whole

    Why it works: A number is not one fixed pair but a whole family of pairs; the parts trade while the total is conserved, and each known pair also tells a missing-part subtraction fact.

    1. Place a fixed whole of counters (e.g. 5) into a single frame, then move a divider across so the counters fall into two parts.
    2. Slide the divider to a new spot and read the two parts aloud (e.g. 1 and 4; then 2 and 3).
    3. Record each split as an equation (5 = 1 + 4, 5 = 2 + 3, …) next to a drawing of the two groups.
    4. Keep a running list of pairs found and check whether every pair has been discovered.
    5. In a 'make 10' mode, fix the whole at 10 and find pairs that make ten.
  2. 2

    Physical trading with counters and a part-whole frame

    Why it works: Physically trading objects between two parts of a fixed whole makes conservation tangible: the total never changes, only how it is split.

    1. Lay out a fixed whole of two-colour counters or a bead string and physically move one counter from one part to the other.
    2. After each move, say the two parts and write the matching equation.
    3. Build a systematic list of all pairs for 5, then for 10, recording each with a drawing and an equation.
    4. Explain in your own words why the total stays the same when a counter moves across.
    5. Show that 2+3 and 3+2 use the same two groups reordered.
  3. 3

    Anchor diagram and recognition checks

    Why it works: Seeing several decompositions laid out and practising missing-part completion consolidates number-bond facts and helps spot pairs that do not make the target whole.

    1. Study a part-whole (cherry/bar) diagram for one number and reveal each pair in turn (tap or lift-the-flap 'another way?').
    2. Fill in the missing part: 5 = 2 + __, and set a slider or write the number to complete the bond.
    3. From a set of candidate pairs that also contains a distractor pair whose total is off by one (so it does not make 5), multi-select only the pairs that make 5; the distractor should be left unselected.
    4. Match given pairs to the correct whole.

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
K.OA.3
The national curriculum in England: Key stages 1 and 2 framework document
Maths/Y1/AS/2

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