Math // Multiplication & Division

Factors, multiples, and primes (age 11+)

Use the concepts and vocabulary of prime numbers, factors, multiples, common factors, common multiples, highest common factor (HCF), lowest common multiple (LCM), and prime factorisation including product notation and the unique factorisation property

Ages
11 to 12
Type
Concept
Builds on
1 topic
Leads to
0 topics
Standards
2 codes

Quick check

Can the learner find the highest common factor of two numbers like 24 and 36 — working out the biggest number that divides exactly into both — and explain how they found it?

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.
  • Express any integer as a product of its prime factors using index notation
  • Find the HCF and LCM of two numbers using prime factorisation
  • Apply the unique factorisation theorem to explain why every number has exactly one set of prime factors

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
  • Factors, multiples, and primesRequired

    Multiplication & Division // Ages 9 to 11

    HCF/LCM/prime factorisation extends Y5-6 factor pairs and common factors/multiples

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Factors, multiples, and primes (age 11+)

Ages 11 to 12 // Concept

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Multiplication & Division

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 11–12: build a number's prime factorisation with a factor tree, record it in index form (e.g. 360 = 2³ × 3² × 5), and use prime factorisations to find HCF (lowest power of each shared prime) and LCM (highest power of each prime present). The conceptual anchor is the unique-factorisation property — however the tree is built, the multiset of primes is identical. A Venn diagram of two numbers' prime factors organises HCF (intersection product) and LCM (whole-diagram product). Includes production, reference/anchor, consolidation, and Socratic-discussion activities, with scaffold limits noted.

Common misconceptions

  • Thinking a number can have different prime factorisations depending on how the tree is drawn (missing the uniqueness property).
  • Treating 1 as prime, or including 1 in a factorisation.
  • Confusing HCF with LCM (taking the largest common multiple or smallest common factor), or muddling which is 'highest' vs 'lowest'.
  • Taking the highest index for HCF / lowest for LCM (rule reversed).
  • Stopping a factor tree at composite leaves (e.g. leaving 4 or 6 unfactored) so the factorisation isn't fully prime.

Useful representations

  • Factor tree branching a number down to its prime leaves, then collected into index form.
  • Venn diagram of two numbers' prime factors: intersection → HCF, union (whole diagram) → LCM.
  • Index-notation product (360 = 2³ × 3² × 5) as the number's unique 'prime fingerprint'.
  • Prime-factor tiles/counters that can be regrouped to show different trees yield the same multiset of primes.

Teaching practices

  • Anchor on the UNIQUE FACTORISATION property by having children build the SAME number with different first splits (360 = 4×90 or 8×45) and confirm the prime fingerprint is identical — the tree branches differ, the primes don't.
  • Teach HCF and LCM THROUGH prime factorisation rather than as separate list-and-compare tricks: a Venn diagram of the two numbers' prime factors gives HCF (intersection product) and LCM (whole-diagram product) at a glance.
  • Insist on index notation as compact, meaningful shorthand (2³ not 2×2×2) and connect it to the powers/indices work (dependent mt_hCVPYlF-7Y).
  • Ground HCF/LCM in problems that MOTIVATE them — HCF for simplifying/sharing into equal groups, LCM for repeating-event and fraction-denominator problems — so the concepts have purpose, not just procedure.
  • Reinforce that 1 is neither prime nor composite and that primes are the multiplicative 'atoms', linking back to the Y5-6 primes and factor-pairs work (prerequisite mt_FHIAv6dfhU).

Activities to try

  1. 1

    Prime factory: build the tree, discover uniqueness

    Why it works: Every whole number greater than 1 has exactly one prime 'fingerprint': the order and shape of the splits do not change which primes (and how many of each) you end up with. Composite leaves must be fully broken down before the factorisation is complete.

    1. Take a number such as 360 and repeatedly split it into a factor pair (e.g. 4 × 90), then keep splitting each branch until every leaf is a prime.
    2. Collect the prime leaves and write them in index form: 360 = 2³ × 3² × 5.
    3. Now build 360 a second time starting from a different first split (e.g. 8 × 45 instead of 4 × 90). Record the prime leaves again.
    4. Lay the two sets of prime leaves side by side and check: the trees branched differently, but the collected primes are identical (2³ × 3² × 5 both times).
    5. Check each tree for composite leaves left unsplit (e.g. a 4 or a 9 that should become 2×2 or 3×3) and finish splitting them.
  2. 2

    Venn organiser for HCF and LCM

    Why it works: A common factor can only be built from primes BOTH numbers have, taken at the lowest shared power — that is the HCF. The LCM must contain enough of every prime to be a multiple of both, so each prime appears at its highest power. The Venn regions are intended to make lowest-for-HCF / highest-for-LCM visible rather than a rule to memorise.

    1. Write the prime factorisations of two numbers, e.g. 24 = 2³ × 3 and 36 = 2² × 3².
    2. Place each number's prime factors (as individual copies) in a two-circle Venn diagram, putting the factor copies both numbers share in the overlap and the leftover copies in the outer regions.
    3. Read the HCF as the product of the overlap (the shared primes, each taken at the lowest power both numbers have): 2² × 3 = 12.
    4. Read the LCM as the product of the whole diagram (each prime at its highest power): 2³ × 3² = 72.
    5. Check the relationship HCF × LCM = product of the two numbers: 12 × 72 = 864 = 24 × 36.
  3. 3

    Reference anchor: reveal the index form and the Venn result

    Why it works: A worked tree plus Venn serves as a clear reference for the layout of index form and the intersection/union logic of HCF and LCM.

    1. Study a completed factor tree for a number; before revealing, predict each leaf's prime value and the full index form.
    2. Reveal the index form (e.g. 360 = 2³ × 3² × 5) and compare with your prediction.
    3. Study a two-number Venn of prime factors; predict the HCF (overlap) and LCM (whole diagram) before revealing them.
    4. Reveal the HCF and LCM and check your reasoning about which powers went where.

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
6.NS.4
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Num.3

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