Math // Addition & Subtraction

Representing Addition and Subtraction

Represent addition and subtraction using objects, drawings, and mental images

Ages
4 to 6
Type
Representation
Builds on
2 topics
Leads to
2 topics
Standards
2 codes

Quick check

If the learner has 3 apples and you give them 2 more, can they show you how many there are altogether — using their fingers, some small objects, or a drawing?

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.
  • Use cubes or counters to show 3 + 2
  • Draw a picture to represent a subtraction situation
  • Use fingers to model an addition problem

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

You are here

Representing Addition and Subtraction

Ages 4 to 6 // Representation

Leads to

2 topics
  • Addition and subtraction word problemsRequired

    Addition & Subtraction // Ages 4 to 6

    Solving word problems within 10 requires ability to represent the operations with objects/drawings

  • Real-World to Maths ConnectionsHelpful

    Mathematical Thinking // Ages 5 to 6

    Representing addition/subtraction with objects and drawings is the core exercise of early quantitative reasoning

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), help children represent addition and subtraction using objects, drawings, fingers, mental images, spoken language, and number sentences — and, crucially, move fluidly between these so they see that they all show the SAME calculation. Deliberately cycle one calculation through multiple representations and link them explicitly ('show me where the 2 is in your drawing'), progressing from concrete toward mental imagery so efficient strategies can replace counting-all.

Common misconceptions

  • Treating each representation as a separate, unrelated task and not seeing that the objects, drawing and number sentence all show the SAME calculation.
  • Drawing elaborate pictures that obscure quantity (detailed apples instead of countable dots), losing the mathematical point.
  • Staying dependent on concrete objects and not progressing toward drawings/mental images, so efficiency never develops.
  • Mis-mapping between representations — e.g. drawing the total when the story described only one part.
  • Using a drawing as decoration after getting an answer another way, rather than as a tool to find or explain the answer.

Useful representations

  • Counters/cubes and fingers (concrete)
  • Ten-frames and five-frames as structured object layouts
  • Part-part-whole ('cherry') diagram
  • Children's own drawings/jottings (dots, marks, crossing-out for removal)
  • Mental images and the spoken number sentence as the bridge to symbols

Teaching practices

  • Cycle ONE calculation through several representations and explicitly link them ('point to the 3 in the cubes, now in your drawing') — the skill is the mapping between representations, not any single one.
  • Move along concrete → pictorial → mental image → symbolic, encouraging children to eventually 'see it in their head' so representations become efficient mental tools rather than crutches.
  • Use consistent structured models (ten-frames, part-whole, fingers) so representations accumulate meaning rather than being ad hoc each time.
  • Have children narrate their representation aloud ('I drew 5 and crossed out 2') to tie drawing to action to language.
  • Value the child's own drawings/jottings while nudging toward clearer, more mathematical ones (dots/marks over detailed pictures) to keep focus on quantity.

Activities to try

  1. 1

    Linked windows onto one calculation

    Why it works: The objects, drawing, ten-frame and number sentence are not separate exercises — they are the same quantity shown different ways; and I can begin to carry the picture in my head.

    1. Place counters onto a mat to show a given sum such as 3 + 2.
    2. Watch linked representations update together: a ten-frame fills, a dot-drawing sketches itself, and the number sentence 3 + 2 = 5 assembles alongside the counters.
    3. Tap the '2' in the number sentence and see exactly two counters, two dots, and two ten-frame cells light up at once across every pane.
    4. Try a 'cover a window' challenge: predict what the drawing will look like before it is revealed, then check.
  2. 2

    Show it with cubes, then with a drawing

    Why it works: I can represent the same calculation two different ways and show how they correspond, using quick countable marks rather than detailed pictures.

    1. Use cubes or counters to show a spoken addition, e.g. build 3 and 2 pushed together to make 5.
    2. Draw a picture (dots or marks) that matches the same calculation.
    3. Narrate aloud what you did, e.g. 'I drew 3 dots and 2 more dots, and that makes 5.'
    4. Point to the '3' and the '2' in the cubes, then point to them in your drawing.
  3. 3

    Enact subtraction as taking away

    Why it works: One way subtraction works is taking away: starting with an amount and removing part of it. That is different from putting two groups together.

    1. Start with 5 cubes and show 5 − 1 by actually removing one cube, leaving four.
    2. Say the number sentence aloud: '5 take away 1 leaves 4, so 5 − 1 = 4.'
    3. Explain which cube you took away and how many are left.
    4. Contrast this with 4 + 1: show how combining to make 5 is different from taking away to leave 4.

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.1
The national curriculum in England: Key stages 1 and 2 framework document
Maths/Y1/AS/4

Math // Addition & Subtraction

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