Math // Multiplication & Division

Mental multiplication and division

Use place value, known and derived facts to multiply and divide mentally, including multiplying by 0 and 1, dividing by 1, and multiplying together three numbers

Ages
8 to 9
Type
Method
Builds on
2 topics
Leads to
1 topic
Standards
1 code

Quick check

Can the learner work out things like '6 × 0', '1 × 35', or '100 ÷ 100' instantly in their head — using what they know rather than working everything out from scratch?

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.
  • Calculate 40 × 6 = 240 mentally using place value
  • Explain why any number × 0 = 0 and any number × 1 = the number itself
  • Multiply three numbers mentally by choosing a useful pair first (e.g. 2 × 7 × 5 = 2 × 5 × 7 = 70)

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

2 topics
  • All times tables to 12×12Required

    Multiplication & Division // Ages 8 to 9

    Mental multiplication requires known table facts

  • Place value of each digitHelpful

    Number Representation & Place Value // Ages 8 to 9

    Place-value knowledge supports mental strategies like 40 × 6

You are here

Mental multiplication and division

Ages 8 to 9 // Method

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 ages 8–9: multiply and divide mentally using place value, known and derived facts, including the special cases ×0, ×1 and ÷1, and multiplying three numbers by choosing a useful pair first. The goal is flexible strategy choice — looking for the easy route before calculating — with the identity/zero cases understood through the equal-groups meaning rather than memorised as rules. Note: ÷0 is out of scope and is undefined; keep to ×0, ×1, ÷1.

Common misconceptions

  • Thinking ×0 gives the other number, or ×1 doubles/changes it, when treated as rote rules.
  • Believing multiplication always makes a number bigger (so ×1 or later × a fraction surprises).
  • Not reordering three factors, so multiplying left-to-right the hard way.
  • Confusing ÷1 with ÷0 or thinking ÷1 halves.
  • Miscounting place value in 40×6 (getting 24 instead of 240).

Useful representations

  • Equal-groups images for the identity/zero cases (1 group of n; n groups of 1; 0 groups)
  • Rearrangeable factor tiles for three-number products (regroup to make a ten)
  • Place-value counters for × multiples of ten
  • Number line / known-fact chains for derived facts

Teaching practices

  • Teach ×0, ×1, ÷1 through the equal-groups meaning (1 group of n = n; n groups of 1 = n; 0 groups = 0) so they're understood, not memorised as rules.
  • Build the habit of scanning three-factor products for the easiest pair to combine first (make a ten, or a known fact), justified by commutativity/associativity.
  • Use place-value reasoning for mental ×÷ with multiples of ten (40×6 = 24 tens = 240).
  • Verbalise the chosen strategy each time ('I'll pair the 2 and 5 to make 10') to make flexible method-choice explicit.
  • Keep ÷0 out of scope and, if raised, address it meaningfully (you can't make groups that share nothing / it's undefined).

Activities to try

  1. 1

    Reorder three factors to find the easy pair

    Why it works: Reordering factors is always allowed (commutativity/associativity) and the total stays the same — but some orderings turn a fiddly three-number product into a single known fact, so it pays to look for the friendly pair before calculating.

    1. Work out 2 × 7 × 5 in your head, saying aloud which two numbers you multiply first and why.
    2. Try pairing 2 and 5 first to make 10, then multiply by 7 to get 70; compare this with going left to right (2 × 7 = 14, then × 5 = 70).
    3. Check that different orders give the same total: is 2 × 7 × 5 the same as 5 × 2 × 7? Explain why you are allowed to reorder.
    4. Repeat with 2 × 9 × 5 and 4 × 3 × 5, each time scanning for a pair that makes a ten or a known fact first.
  2. 2

    Understand ×1, ×0 and ÷1 through groups

    Why it works: ×1 and ÷1 leave a number unchanged because you have one whole group (or share into groups of one), and ×0 gives 0 because there are no groups (or every group is empty) — these are meanings to be seen, not rules to memorise.

    1. Lay out counters to show 8 × 1 as 'one group of 8' and '8 groups of 1'; confirm both give 8.
    2. Show 8 × 0 as '8 groups of nothing' — every group is empty, so the total is 0.
    3. Show 8 ÷ 1 as one whole group of 8 (or sharing 8 into groups of one) and confirm the result is still 8.
    4. Say in your own words why n × 1 = n, why n × 0 = 0, and why n ÷ 1 = n.
    5. Contrast 8 × 0 with 8 + 0: adding 0 leaves 8, but 0 groups of 8 (or 8 groups of 0) is nothing — notice the roles are different.
  3. 3

    Scale with place value for multiples of ten

    Why it works: A known fact can be scaled by place value: 40 × 6 is ten times 4 × 6, so the answer is 240, not 24 — keeping the 'tens' label attached prevents losing the place value.

    1. Work out 40 × 6 mentally by reasoning '4 tens × 6 = 24 tens = 240'; say the 'tens' name aloud as you go.
    2. Explain how knowing 4 × 6 = 24 helps you get 40 × 6.
    3. Try 30 × 7 and 50 × 8 the same way, keeping the place-value name attached to each digit.
    4. Check your answer is sensible: 40 × 6 is exactly ten times 4 × 6, so 240, not 24.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
Ma/KS2/Y4/MD/2

Math // Multiplication & Division

More in Multiplication & Division

All 57 topics