Math // Mathematical Thinking

Numbers on a number line

Select and use appropriate tools and representations (number lines, hundred squares, rulers, part-whole models) to support problem-solving

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

Quick check

When the learner is tackling a maths problem, do they choose a useful tool — like a number line for addition or a hundred square to count on — rather than always working in their head?

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.
  • Choose a number line or hundred square to support adding or subtracting two-digit numbers
  • Select a ruler when a problem requires measuring length
  • Explain why one representation (e.g. a number line vs cubes) is more helpful for a particular 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

3 topics
  • Hands-On Problem SolvingRequired

    Mathematical Thinking // Ages 5 to 6

    Age 6-7 strategic tool selection builds on age 5-6 familiarity with basic tools

  • Adding within 100Helpful

    Addition & Subtraction // Ages 6 to 7

    Adding within 100 requires choosing number line vs hundred square strategically

  • Measuring length (age 6+)Helpful

    Measurement // Ages 6 to 7

    Measuring length with units requires selecting appropriate measurement tools

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Numbers on a number line

Ages 6 to 7 // Thinking skill

Leads to

1 topic

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 6-7 year-olds, strategic tool choice shifts from concrete objects to REPRESENTATIONS (number line, hundred square, ruler, part-whole model), and the skill becomes matching the representation to the problem's PURPOSE and to the size of the numbers. As numbers grow past ten, counting single cubes in ones becomes slow and error-prone, so a key lesson is trading the concrete tool for a representation that carries the tens. The central proposed activity gives a child the same chooser facing two problems — a two-digit addition (34+28) where a number line's tens-and-ones jumps are genuinely quicker than counting a cube pile, and a measuring task where a ruler (or a metre stick for a large object), not a number line, fits — then invites them to justify why each tool fits its problem. Alternatives (a convince-a-partner dialogue, a primer comparison board) are intended to serve different pedagogical purposes.

Common misconceptions

  • Persists with counting single cubes in ones for two-digit numbers (34+28), which is slow and error-prone (EEF: over-reliance on concrete)
  • Mismatches the tool to the problem's purpose — e.g. reaches for a number line for a 'how long is it?' measuring task instead of a ruler
  • Draws a number line but counts every mark in ones anyway, using it as a ritual rather than to jump in tens (EEF: answer, not structure)
  • Thinks a representation is 'babyish' or 'proper' rather than fit-for-purpose, so choice is about status not the problem

Useful representations

  • Number line (jump in tens and ones)
  • Hundred square (place-value moves — down a row = +10)
  • Ruler / measuring tools
  • Part-whole model
  • Cubes / base-ten (for smaller numbers)

Teaching practices

  • Sort problems by 'what are we trying to find?' and match the tool to that purpose (measure → ruler; jump-add → number line) before choosing numbers (EEF: clear rationale)
  • Show number lines explicitly — EEF finds them a particularly effective representation across KS2/3 — modelling jumps of ten rather than counting marks in ones
  • Compare two representations for the same problem and discuss 'which helped more and why', which EEF finds develops conceptual understanding
  • Make the inefficiency of counting big numbers in ones felt, so children choose a representation for efficiency, not just because it is allowed
  • Keep naming each tool's limitation (a number line doesn't measure a real object; cubes are unwieldy past twenty) so choice stays a fit judgement

Activities to try

  1. 1

    Pick the tool, work the problem, justify the fit

    Why it works: The right tool follows the PURPOSE of the problem and the size of the numbers or object, not habit: a representation that carries the tens (number line, hundred square) spares you counting ones, while a length problem needs a measuring tool matched to the object's size. Cubes remain a good fit for small numbers — this is a fit judgement, not a ranking of tools as 'babyish' or 'proper'.

    1. Solve 34 + 28 using a palette of tools: cubes/base-ten, a number line, a hundred square, a ruler.
    2. If you reach for single cubes, build and count 62 in ones — notice it is slow and easy to miscount. (Base-ten rods group the tens, so they avoid most of this counting.)
    3. On the number line, jump 34 → +20 → 54 → +8 → 62 in two jumps.
    4. On the hundred square (numbered with 1 at the top), add the 20 by moving two rows down to 54, then add 8 by moving forward eight squares (crossing into the next row) to reach 62.
    5. After solving, answer: 'Why did that tool help?' and 'When would cubes actually be the better choice?'
  2. 2

    Convince a partner who defaults to cubes

    Why it works: Being able to explain why one representation beats another for a given problem — and when the rejected tool would in fact be right — is what separates a real fit judgement from a blanket ranking of tools.

    1. Work with a partner (or in-role character) who always reaches for cubes, even for 34 + 28.
    2. Argue why a number line or hundred square is more efficient for big numbers — point to jumping the tens and avoiding miscounts.
    3. Concede honestly where cubes still win: give a problem (e.g. a small total like 3 + 4) where cubes really would be the best choice.
    4. Answer a probe such as 'What can a hundred square show you that a number line doesn't when you add 28?'
  3. 3

    Primer: what each tool is for

    Why it works: Each representation has a purpose and a limitation — a number line jumps along a count but does not measure a real object; cubes become unwieldy past twenty — so a warm-up can prime 'what each is for'.

    1. Explore a comparison board for number line vs hundred square vs ruler, revealing 'best for…' and 'not for…' notes for each.
    2. Before the main task, predict which tool you would reach for in an addition and in a measuring problem.

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