Math // Mathematical Thinking

Generalising Patterns

Recognise and use repeated reasoning to generalise: spot calculation patterns, describe rules for sequences, and predict results using known mathematical facts

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

Quick check

When the learner is practising times tables or number patterns, do they spot a shortcut — like "if 6 × 7 = 42, then 6 × 8 must be 48" — and use known facts to work out ones they haven't memorised yet?

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.
  • Use a known doubles fact to derive a near-doubles answer (e.g. 6 + 7 = 6 + 6 + 1 = 13)
  • Notice that subtracting 10 from any two-digit number always reduces the tens digit by 1
  • Describe a rule for a pattern and use it to extend or predict (e.g. 'each time we add 5, the ones digit alternates between 0 and 5')

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
  • Finding efficient methodsRequired

    Mathematical Thinking // Ages 5 to 6

    Age 6-7 generalising from repeated reasoning builds on age 5-6 noticing repeated patterns

  • Addition and subtraction within 20Helpful

    Addition & Subtraction // Ages 6 to 7

    Deriving facts within 20 from known facts (near doubles, make ten) exercises generalising from patterns

  • 10 More or 10 LessHelpful

    Number Representation & Place Value // Ages 6 to 7

    Mentally finding 10 more/less uses generalised repeated reasoning about place value

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Generalising Patterns

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 ages 6-7, this topic builds the habit of looking for and expressing regularity in repeated reasoning (MP8): deriving a near-double from a known double (6+7 = 6+6+1 = 13), noticing that subtracting 10 from a two-digit number drops the tens digit by one, and stating a sequence rule ('+5 each time starting from 5, so the ones digit alternates 5, 0') — then TESTING that rule on a fresh case rather than trusting it from one or two examples. Core teaching moves: place a known fact adjacent to the derived one so the adjustment is seen not recomputed, press for the rule in the child's own words, and always test the generalisation on a new case.

Common misconceptions

  • Recomputes 6+7 from one, not seeing it as the known double 6+6 plus 1
  • Applies the near-double adjustment wrongly (adds 2, or doubles the larger number)
  • States a rule from one or two cases and assumes it holds without testing
  • Subtracts 10 by changing the ones digit or counting back in ones instead of dropping the tens digit

Useful representations

  • Doubles and near-doubles on ten-frames / cubes (double + 1)
  • Hundred square and place-value cards for ±10
  • Number line marked with equal repeated steps (+5)
  • Bead strings for repeated addition patterns

Teaching practices

  • Put a doubles fact next to its near-double (6+6 and 6+7) so the '+1' adjustment is seen, not computed from scratch
  • Ask the child to state the rule ('what's happening each time?') and then test it on the next case to guard against a rule taken from too few examples
  • Use a hundred square / place-value cards so −10 visibly drops the tens digit each time
  • Record a sequence and its step so the repeating structure and its rule are both on show ('+5, +5, +5')

Activities to try

  1. 1

    Near-doubles builder: the double plus one

    Why it works: A near-double can be derived from an adjacent known double by adjusting, instead of counting up from one.

    1. Set out two rows of 6 cubes (or two ten-frames of 6) and confirm the double: 6+6=12.
    2. Add exactly ONE more cube to one row and say the new total: 6+7=13. State it as 'the double I know, and one more'.
    3. Repeat for 6+8 ('two more than the double, so 14') and for another double you know, e.g. 7+7=14 so 7+8=15.
    4. Say aloud which double you used and how many you adjusted by, before giving the total.
  2. 2

    State a rule, then test it

    Why it works: A rule guessed from a couple of cases should be tested on fresh cases before we rely on it: a passing test makes us more confident (it does not prove the rule), while one failing case shows the rule needs revising — this is the MP8 discipline of checking a generalisation.

    1. Lay out the first few terms of a sequence (e.g. 5, 10, 15, 20) and say the repeating step out loud: '+5, +5, +5'. Optionally show the equal +5 jumps on a number line or bead string so the repeating structure is visible.
    2. State the rule in your own words: 'each time we add 5'. Predict the NEXT term before extending.
    3. Predict the ones digit of the next three terms (25, 30, 35) and say why ('they alternate 5, 0, 5').
    4. TEST it: have the adult reveal the next term of the given sequence (e.g. uncover a hidden card or the next group on the bead string) and compare it with your prediction; if it does not match, revise the rule. Extending the sequence yourself with your own rule is not a test.
    5. Then say how you could be sure a rule is right before trusting it.
  3. 3

    Subtracting 10 on a hundred square — and bounding the rule

    Why it works: Subtracting 10 from a two-digit number lowers the tens digit by one and leaves the ones digit unchanged — and a good rule is stated with the conditions under which it holds.

    1. On a hundred square (or with place-value cards), start at a two-digit number such as 45 and subtract 10; land on 35 and notice the tens digit dropped by one while the ones digit stayed the same.
    2. Repeat from two or three other starting numbers to see the same pattern each time; state the rule: 'take away 10 drops the tens digit by one'.
    3. Test the boundary: does taking away ONE do the same thing? Try 45−1 (which gives 44) and see it does not — so the rule is only for −10.
    4. Say the rule in your own words AND what it does NOT cover.

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