Math // Multiplication & Division

Writing Number Sentences

Write simple numerical expressions that record calculations with numbers, and interpret numerical expressions without evaluating them (e.g. recognise that 3 × (18932 + 921) is three times as large as 18932 + 921)

Ages
10 to 11
Type
Representation
Builds on
1 topic
Leads to
2 topics
Standards
1 code

Quick check

Can the learner look at '4 × (5 + 3)' and say 'that's 4 times the result of adding 5 and 3' — reading the expression as a description of a calculation, without having to work it out first?

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 an expression for 'add 8 and 7, then multiply by 2' using parentheses
  • Explain what 4 × (365 − 12) represents without computing it
  • Compare two expressions and determine which is larger without evaluating

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
  • Brackets in ExpressionsRequired

    Multiplication & Division // Ages 10 to 11

    Writing/interpreting expressions requires understanding grouping symbols

You are here

Writing Number Sentences

Ages 10 to 11 // Representation

Leads to

2 topics
  • Writing Algebraic EquationsRequired

    Algebra // Ages 10 to 11

    Writing algebraic expressions extends writing/interpreting numerical expressions

  • Using Simple FormulaeHelpful

    Algebra // Ages 10 to 11

    Interpreting formulae connects to interpreting numerical expressions

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 10–11: writing numerical expressions that faithfully record a calculation (using brackets to preserve order) and interpreting an expression's structure WITHOUT evaluating it — e.g. seeing that 3 × (18932 + 921) is three times as large as 18932 + 921. The skill is representational: the point is the mapping between a situation/instruction and its symbolic form, and reasoning about relative size from structure, not computing an answer. Work moves in both directions: words/actions → symbols, and symbols → meaning.

Common misconceptions

  • Compulsively evaluating — rushing to a number instead of describing structure, which defeats the purpose and often introduces arithmetic error.
  • Omitting brackets when order matters, writing 8 + 7 × 2 for 'add then double' (which the order-of-operations convention reads as 8 + 14).
  • Reading 3 × (18932 + 921) as '3 times 18932, plus 921' — mis-scoping the multiplier over the bracket.
  • Assuming two expressions with the same numbers must be equal regardless of brackets/structure (5×(n+3) vs 5×n+3).
  • Confusing 'three times as large' (multiplicative, ×3) with 'three more than' (additive, +3) when narrating an expression.

Useful representations

  • Bar model showing an inner quantity (a + b) and an outer bar three copies long, so 3 × (a + b) is literally three of the inner bars — 'three times as large' made visual without numbers.
  • Two-column translation table: verbal instruction | matching expression, built up so brackets appear exactly where 'first' is spoken.
  • Expression 'family' cards that look alike (5×(n+3), 5×n+3, 5×n×3) sorted by what they mean, to foreground structure.
  • A 'relative size' comparison strip: two expressions placed side by side with a factor arrow ('×3') linking one to the other, no values shown.

Teaching practices

  • Separate WRITING an expression from EVALUATING it — make 'don't work it out' an explicit, repeated instruction, because the whole skill is reading structure; reward a good interpretation ('it's three times the sum') over a computed number.
  • Use very large or awkward numbers deliberately (18932 + 921) so computing is unappealing and the child is nudged to reason about STRUCTURE instead — the size difference is visible from the ×3 alone.
  • Move fluently in both directions: words→symbols (translate an instruction) AND symbols→words/meaning (narrate what an expression records), so the mapping is bidirectional.
  • Anchor on brackets-as-grouping (prerequisite mt_8jFSnXxqQD): brackets are what let a written expression capture 'do this part first', so 'add then multiply' MUST be written with brackets to be faithful.
  • Contrast expressions that look similar but scale differently — 5 × (n + 3) vs 5 × n + 3 — to show that structure, not just the numbers, determines meaning and relative size; this seeds early algebraic reasoning (dependents: writing algebraic equations, simple formulae).

Activities to try

  1. 1

    Expression builder with a live structure mirror

    Why it works: Structure — where the brackets sit and what the multiplier scopes over — determines meaning and relative size; you can read 'three times as large' directly from the form without computing.

    1. Hear or read an instruction such as 'add 18932 and 921, then make it three times as big' and assemble it from number, operator and BRACKET tiles — e.g. 3 × (18932 + 921).
    2. As you wrap (18932 + 921) and prefix × 3, watch a bar-model alongside show one inner bar for the sum and then three copies of that bar stacked, so 'three times as large' appears as literal repeated length — with no total ever calculated.
    3. Switch to a compare mode: two near-identical expressions sit side by side; drag the brackets from around one sub-part to another and watch the bars re-shape, noticing whether and how the comparison changes.
    4. State which expression is larger, giving no numerical value; for a pair like 3 × (18932 + 921) and 18932 + 921, name the FACTOR ('three times'), but for look-alike pairs such as 3 × (18932 + 921) and 3 × 18932 + 921, explain the difference from structure, since there is no simple factor.
    5. Deliberately use very large or awkward numbers so computing is unappealing and you reason from the ×3 structure alone.
  2. 2

    Annotated expression: tap to reveal what each part records

    Why it works: Each piece of notation carries meaning; the bracket groups a 'do first' step and the multiplier scopes over the entire bracket.

    1. Study a worked expression such as 3 × (18932 + 921).
    2. Tap or point to each part in turn — the bracket, the multiplier, the operator — and state what it records and why: the bracket means 'do this part first', the × 3 means 'three copies of the whole bracket'.
    3. Explain, with no computation shown, why the whole expression is three times the sum inside the bracket.
  3. 3

    Dictate, symbolise and justify (words ↔ symbols)

    Why it works: The mapping between situation and symbolic form is bidirectional; brackets are what let a written expression stay faithful to a spoken order, and 'three times' (×3) is different from 'three more' (+3).

    1. Listen to a spoken scenario (e.g. a shopping or recipe instruction) and 'write it down without working it out', using brackets where 'first' is spoken.
    2. Narrate back what an existing expression records in words — reading symbols → meaning.
    3. Answer probing questions such as 'is that three times as much, or three more?' and justify the choice.
    4. Practise both translation tasks: 'subtract 12 from 365, then multiply by 4' → 4 × (365 − 12), and reading 4 × (365 − 12) aloud as its meaning.

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
5.OA.2

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