Math // Measurement

Measuring length

Order three objects by length and compare the lengths of two objects indirectly using a third object

Ages
6 to 7
Type
Method
Builds on
1 topic
Leads to
2 topics
Standards
1 code

Quick check

If the learner has three pencils of different lengths but can only compare two at a time, can they figure out which is longest — by using a third pencil as a go-between?

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.
  • Put three ribbons in order from shortest to longest
  • Use a piece of string to compare the heights of two objects that cannot be placed side by side
  • Explain that if A is longer than B and B is longer than C, then A is longer than C

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
  • Comparing Lengths & HeightsRequired

    Measurement // Ages 4 to 6

    Ordering 3 objects by length and indirect comparison extends direct length comparison

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Measuring length

Ages 6 to 7 // Method

Leads to

2 topics

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

Children learn to seriate three or more objects by length and to compare two objects that cannot be placed side by side by using a third object (string or paper strip) as a fixed intermediary length. The underlying reasoning is transitivity: if A is longer than B and B is longer than C, then A is longer than C — and a string kept at one fixed length lets a comparison be carried across a gap. Enactive ordering and carrying an intermediary build the reasoning; recognition formats and static displays can check or summarise it.

Common misconceptions

  • Compares only pairs and never chains the inference — insists on placing A directly against C instead of reasoning through the intermediary.
  • Orders by trial-and-error with no image of an increasing series, and cannot say where a new object should be inserted.
  • Does not keep the intermediary constant — stretches, re-cuts, or re-aligns the string between the two comparisons.
  • Non-conservation: believes the string's length changed when it was bent or moved.

Useful representations

  • A set of three-plus ribbons or cube-towers arranged as an increasing 'staircase'
  • A length of string or paper strip used as an intermediary copy of a length
  • The seriation staircase image (each step the smallest increase in length)

Teaching practices

  • Serial ordering: give three (then more) ribbons or cube-towers and encourage a strategy — pick the shortest, then the next shortest — rather than random trial-and-error.
  • Set a genuine problem where two objects cannot be moved together (a doorway vs a bookshelf) so the child discovers a string/strip as a portable copy of a length.
  • Voice the transitive inference explicitly: 'the door is taller than the string; the shelf is shorter than the string; so the door is taller than the shelf.'
  • Keep the intermediary constant — uncut, unstretched, aligned at the same end — because a re-cut or stretched string breaks the comparison; use it to build conservation of length.
  • Use insert-into-series tasks: 'where does this new ribbon go?' to push beyond trial-and-error toward a mental image of an increasing series.

Activities to try

  1. 1

    Carrying a length across a gap with a string

    Why it works: A string kept at one fixed length can be carried to a place where two objects cannot meet, letting you decide which is longer by comparing each to the same intermediary.

    1. Face two objects (e.g. two cube-towers, or a tall object by the door and a shelf across the room) that are clearly too far apart to bring together and line up.
    2. Hold a fixed length of string against the first object, aligned at the same starting end, and note whether that object is taller or shorter than the string; keep the string's length unchanged for both comparisons (do not change it between the two objects).
    3. Keeping the string uncut and unstretched, carry it across to the second object and see whether it overshoots the top or falls short.
    4. Compare each object to the same string and state the conclusion, voicing the transitive chain — e.g. 'this one is taller than the string; that one is shorter than the string; so this one is taller than that one.'
    5. Try a second far-apart pair to transfer the idea.
  2. 2

    Building and extending a seriation staircase

    Why it works: Transitivity becomes the rule that orders a whole set: when no two lengths are equal, each item is longer than everything before it and shorter than everything after, so a new item of a different length from all the others slots into exactly one place (if it matches an existing length, it can go right next to that item).

    1. Order three ribbons or cube-towers from shortest to longest using a strategy — pick the shortest, then the next shortest — rather than random swapping.
    2. Progress to ordering five or more into an increasing 'staircase' where each step is the smallest available increase in length.
    3. Check the finished line and fix any inversion (a shorter item placed after a longer one).
    4. Take a new ribbon and decide where it belongs: it goes after every shorter ribbon and before every longer one, so it may land in between two ribbons or at either end if it is the shortest or longest of all.
    5. Optional screen version: drag a fourth ribbon into a staircase that only settles when it sits above the shorter and below the longer neighbour, with a nudge when an inversion is left.
  3. 3

    Naming the transitive chain (check and consolidate)

    Why it works: The transitive conclusion can be stated and recognised in words once it has been built through ordering and carrying an intermediary.

    1. Order a given set of ribbons shortest to longest as an ordering check.
    2. Answer a transitive-inference question in words, e.g. 'The bookshelf is shorter than the string and the door is taller than the string — which is taller?'
    3. Review a staircase display of the 'A > B > C, so A > C' chain, naming each term, as a summary reference after doing the hands-on work.

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
1.MD.1

Math // Measurement

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