Math // Mathematical Thinking

Connecting Representations

Move between real-world situations, drawings, and number sentences, explaining how each representation connects to the others (quantitative reasoning)

Ages
6 to 7
Type
Thinking skill
Builds on
3 topics
Leads to
1 topic

Quick check

When the learner solves a word problem, can they draw a diagram or write a number sentence to show their thinking — and then explain how the picture and the numbers match the real situation?

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.
  • Write a number sentence (e.g. 14 + 5 = 19) to match a word problem and explain the connection
  • Draw a bar model or part-whole diagram to represent a problem, then solve using the diagram
  • Translate between a concrete model and a symbolic equation, describing what each number represents

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

3 topics
  • Real-World to Maths ConnectionsRequired

    Mathematical Thinking // Ages 5 to 6

    Age 6-7 quantitative reasoning builds on age 5-6 concrete-to-representation skills

  • Reading +, −, and = symbolsHelpful

    Addition & Subtraction // Ages 5 to 6

    Connecting representations to number sentences requires fluency with +, -, = symbols

  • Inverse: addition undoes subtractionHelpful

    Addition & Subtraction // Ages 6 to 7

    Inverse relationship understanding requires moving between addition and subtraction representations

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Connecting Representations

Ages 6 to 7 // Thinking skill

Leads to

1 topic
  • Understanding fractionsRequired

    Mathematical Thinking // Ages 7 to 8

    Age 7-8 quantitative reasoning builds on age 6-7

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 6-7, learners move among three representations of an addition/subtraction situation — a real-world story, a bar or part-whole drawing, and a symbolic number sentence — and explain how each connects to the others, saying what every number stands for. Two priorities drive the design: (1) the bar model is used to expose STRUCTURE (which quantity is the whole, which are the parts, where the unknown sits), not as a computing trick, and a single part-whole bar can be matched by several correct sentences (a=b+c, a=c+b, a−b=c, a−c=b); comparison bars serve the distinct 'how many more' type of situation. (2) The equals sign must be built as RELATIONAL ('the same amount on both sides'), because children very commonly read '=' as 'work out the answer' and will write compute-left-to-right sentences regardless of structure. True/false and open number sentences with the unknown in varied positions are the documented remedy; simply telling children '= means the same as' is reported insufficient on its own.

Common misconceptions

  • Reads '=' as 'work out the answer' (operational), so accepts or writes left-to-right computes like 8 + 4 = □ + 5 → 12 regardless of the structure
  • Grabs the two numbers in a story and adds them without asking which is the whole and which is a part
  • Ignores 'altogether'/total language when deciding what to combine
  • Can produce a matching sentence but cannot say what each number represents in the story

Useful representations

  • Part-whole bar model (whole on top, parts beneath; unknown marked with ?)
  • Comparison bars for 'how many more' situations
  • Number sentence with the unknown in varied positions (□ + 5 = 12, 7 + □ = 6 + 4)
  • True/false number sentences as a relational-equality routine

Teaching practices

  • Teach the bar model to expose structure, not to compute: which is the whole, which are the parts, and where the unknown sits (NCETM) — then map each part to a referent in the story
  • Build a RELATIONAL equals sign with true/false sentences (7 = 3 + 4, 8 + 2 = 10 + 4, 8 = 8) and open sentences with the unknown in varied positions (□ = 9 + 5, 7 + □ = 6 + 4), the Falkner/Levi/Carpenter remedy — telling children '= means the same as' is not enough
  • Require the child to explain the connection ('the 10 in the story is the whole, so it can't be what I add') rather than just producing a sentence
  • Move in all directions among situation, bar, and equation — not only story→sentence — so representations are read for meaning both ways
  • Use the bar to show addition and subtraction as one relationship (part-part-whole) so a matching sentence can be written several correct ways

Activities to try

  1. 1

    Build the bar from the story, then write the sentence

    Why it works: A part-whole bar shows the structure of a situation — which quantity is the whole and which are the parts — so you must identify the whole before deciding what to add, and one bar can be matched by several correct sentences (adding or subtracting). A comparison bar is a different representation that suits 'how many more' situations by lining up two amounts to show the difference between them.

    1. Read a story such as: '6 red apples, and some green apples, 10 apples altogether.'
    2. Build a part-whole bar: first decide and label which quantity is the WHOLE, then place the known part, then mark the missing part with '?'.
    3. Notice the classic error — putting 6 and 10 side by side as two parts — and correct it by recognising that 10 is the whole, 6 is a part, and the green apples are the missing part.
    4. Write a matching number sentence (6 + ? = 10) and then write at least one other correct form for the same bar (e.g. 10 − 6 = ?).
    5. Say aloud what each number in the sentence stands for in the story.
  2. 2

    True/false and balance: '=' means 'the same as'

    Why it works: The equals sign means 'the same amount on both sides', not 'the answer comes next'; so you cannot just copy the left-hand total into the box — you must make both sides equal.

    1. Decide whether sentences are true or false: 8 = 8; 7 = 3 + 4; 8 + 4 = 12 + 5; 8 + 4 = 7 + 5.
    2. Solve open sentences with the unknown in varied positions: □ = 9 + 5, 7 + □ = 6 + 4, 8 + 4 = □ + 5.
    3. For 8 + 4 = □ + 5, try putting 12 in the box, check whether both sides are really equal, and adjust until they match (7).
    4. Explain for each: how do you decide whether the two sides are the same amount?
  3. 3

    Explain-why with a study partner

    Why it works: Explaining the connection out loud — naming the whole, the parts, and what each symbol stands for — is part of understanding, not an extra; a tempting-but-wrong sentence can be rejected with a structural reason.

    1. Consider a partner who says: 'The story has 6 red and 10 altogether, so I wrote 6 + 10 = 16 — good match?'
    2. Decide whether this matches the story and say why or why not, naming what 10 is in the story (the whole).
    3. When you write or accept a sentence, point at each number and state what it represents in the story.
    4. Respond to a follow-up: 'Could you match the same story using take-away instead of add? Why does that work too?'

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