Math // Mathematical Thinking

Extending Table Patterns

Recognise and use repeated reasoning to generalise: extend multiplication table patterns, derive unknown facts from known ones, and describe rules for sequences

Ages
7 to 8
Type
Thinking skill
Builds on
3 topics
Leads to
1 topic

Quick check

When the learner is learning their times tables, do they use known facts to figure out ones they're not sure of — for example, using 5 × 6 = 30 to work out that 6 × 6 = 36?

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.
  • Notice that all multiples of 4 are even and use this to check answers
  • Derive 8 × 7 from 8 × 5 + 8 × 2 by spotting the pattern
  • Describe a rule for a growing pattern (e.g. 'add 50 each time') and use it to predict the next terms

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

You are here

Extending Table Patterns

Ages 7 to 8 // Thinking skill

Leads to

1 topic
  • Times tables (age 8+)Required

    Mathematical Thinking // Ages 8 to 9

    Age 7-8 repeated reasoning is prerequisite to age 8-9 level

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 7-8, turn multiplication tables from a memorised list into a web of derivable facts and checkable patterns. Children derive a harder fact (8×7) from anchor facts they own (8×5 + 8×2 = 40 + 16 = 56) using an array split, use a pattern as a reasonableness check (multiples of 4 and 8 are even — is my answer even?), and describe/test the rule of a growing sequence ('add 50 each time') to predict onward terms. The core habit is deriving AND self-checking, plus probing what a pattern does and does not claim.

Common misconceptions

  • Skip-counts or guesses harder facts instead of deriving them from known ones (8×5 + 8×2)
  • Splits the derivation wrongly (8×5 + 2, forgetting to multiply the 2 by 8)
  • Accepts an unreasonable answer because the pattern check (multiples of 4 are even) isn't applied
  • Over-generalises a pattern from a few cases without testing, or reverses it wrongly (thinks every even number is a multiple of 4)

Useful representations

  • Arrays / area model split for deriving ×7 from ×5 + ×2
  • Multiplication grid highlighting column/row patterns (evens, digit patterns)
  • Number line with equal repeated steps for growing sequences (+50)
  • Hundred square shading multiples to reveal their pattern

Teaching practices

  • Derive harder table facts from anchor facts (×5 and ×2 to reach ×7) using an array split, so the table is a network not a list to memorise
  • Explicitly use pattern as a check: 'a multiple of 4 must be even — is your answer even?' to build reasonableness monitoring
  • Have children describe the rule of a growing sequence and predict several terms ahead, then verify
  • Surface and test near-generalisations ('all multiples of 4 are even — are all even numbers multiples of 4?') to sharpen what the rule does and doesn't claim

Activities to try

  1. 1

    Split an array to derive a hard fact

    Why it works: A 'hard' table fact can be assembled from facts you already own; the array makes the decomposition (distributive idea) visible rather than relying on recall or skip-counting, and the 8×8 − 8 route is a distinct alternative strategy (adjusting from a near fact), not another split of the 7.

    1. Face 8×7 shown as an array 8 rows tall and 7 wide.
    2. Cut the 7 into 5 and 2 (drag a divider, fold a paper array, or draw a line) so the array cleaves into an 8×5 block and an 8×2 block.
    3. Read off 8×5 = 40 and 8×2 = 16, then combine the two blocks to get 56.
    4. Try a different way to reach the same answer: 8×8 − 8 = 64 − 8 = 56 (take one group of 8 away from eight 8s). Compare which way you prefer and why — this uses one fewer group of 8, which is a different idea from cutting the 7 into 5 and 2.
  2. 2

    Use a table pattern as a reasonableness check

    Why it works: A noticed pattern is not just an observation — it becomes a tool you use to catch your own errors and judge whether an answer looks plausible (this check shows an answer is plausible, e.g. even and within range, not that it is exactly right).

    1. Shade the multiples of 4 (and of 8) on a multiplication grid or hundred square and observe that every one is even.
    2. State the pattern as a check: 'a multiple of 4 must be even — is my answer even?'
    3. Apply the check to a proposed product: given a candidate like an odd 'multiple of 4', decide whether it can be right and say why.
    4. Estimate before computing: 'roughly how big should 8×7 be — between which two facts I know?' (between 8×5=40 and 8×8=64).
  3. 3

    Describe and test a sequence rule

    Why it works: A clearly stated rule lets you predict beyond what you can see and then verify, which is the predict-and-check habit in a sequence context.

    1. Given a growing sequence (e.g. steps of +50 along a number line), describe the rule in words: 'add 50 each time'.
    2. Predict several terms ahead — e.g. the term after 350 (400), and the one after that (450) — then verify by stepping along the number line.
    3. Say how sure you are and how you could check a predicted term.

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