Math // Algebra

Linear number sequences

Generate and describe linear number sequences, including those with negative and decimal steps; identify the term-to-term rule

Ages
10 to 11
Type
Method
Builds on
2 topics
Leads to
2 topics
Standards
1 code

Quick check

If the learner sees a number sequence like 3, 7, 11, 15… can they describe the rule, continue it, and predict what the 10th number will be?

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.
  • Continue the sequence 2.5, 4.0, 5.5, ... and state the rule as 'add 1.5'
  • Generate terms of a sequence that crosses zero (e.g. 3, 1, −1, −3, ...)
  • Describe the term-to-term rule for a given linear sequence and predict the 10th term

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
  • Shape patternsRequired

    Multiplication & Division // Ages 9 to 10

    Linear sequences with negative/decimal steps extends Y5 pattern generation

  • Measuring temperatureHelpful

    Number Representation & Place Value // Ages 10 to 11

    Sequences crossing zero require negative number understanding

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Linear number sequences

Ages 10 to 11 // 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

Pupils (Year 6, ages 10–11) generate and describe linear (arithmetic) number sequences, stating the term-to-term rule precisely as a signed amount and giving it with the starting number so the sequence can be rebuilt ('start at 2.5, add 1.5'; 'start at 5, subtract 2'), not a vague 'goes up/down'. The work widens to decimal and fractional steps and to decreasing sequences that stride past zero into negative numbers. A decreasing sequence is treated as 'adding a negative each time' — the same single structure as an increasing one — laying the concrete foundation for the KS3 position-to-term (nth-term) rule.

Common misconceptions

  • States the rule vaguely ('it goes down') without the amount, or gives the amount but not the starting number
  • Treats a decreasing sequence as a different phenomenon rather than as adding a negative each time; notices only increasing sequences
  • Believes the sequence 'stops' at zero — doubts that negative numbers exist or that you can go below zero
  • Reads or writes a negative term wrongly, e.g. −1 written or read as 1 (the raised minus sign not recognised)
  • Fixates on the last term and imposes a familiar pattern (4, 7, 10, 13, 16 → 18 as 'even numbers') instead of the constant step

Useful representations

  • Horizontal number line with zero central (positives right, negatives left)
  • The term-to-term '+d / −d' arrow chain drawn under the numbers
  • A temperature scale / number track for below-zero contexts
  • A position-vs-term table

Teaching practices

  • Insist the rule names BOTH the starting number and the signed amount, refining 'it goes down' into 'start at 5, subtract 2 each time'
  • Treat a decreasing sequence as 'adding a negative' each time, the same structure as an increasing one — not a separate kind
  • Anchor on a horizontal number line with zero in the MIDDLE so continuing past zero is a continued equal step, not a bounce or a stop
  • Use real below-zero contexts (temperature, sea level, floors below ground) to make negative terms concrete
  • Include decimal and fractional steps so 'add 1.5' feels as ordinary as 'add 2'
  • Ask pupils to predict a far term (e.g. the 10th) to motivate looking for structure rather than only counting on
  • When a wrong term appears, distinguish a slip (miscount) from a misconception (misread rule) by checking the step between every pair

Activities to try

  1. 1

    Number-line step machine: lay equal hops that stride past zero

    Why it works: A linear sequence has one constant signed step; because the step never changes at zero, the sequence does not stop or bounce — it continues into the negatives. A rule that names BOTH a start and a signed amount uniquely rebuilds the whole sequence, and the same structure covers increasing, decreasing, whole-number and decimal cases.

    1. On a horizontal number line with zero in the MIDDLE (positives to the right, negatives to the left), set a starting number and a signed step.
    2. Lay each term as an equal-sized hop along the line, one tap/hop at a time: e.g. start 5, step −2, so 5 → 3 → 1.
    3. At the next hop, watch the marker stride the SAME distance straight past zero to −1, then −3, −5, −7 — the step never changed, so the sequence simply continues.
    4. Commit to a rule in a 'state the rule' box in the form 'start at ___, add/subtract ___ each time'; if you type only 'goes down', answer the prompt 'by how much, and from where?'.
    5. Repeat with a decimal preset (start 2.5, add 1.5) and confirm the identical equal hop works for tenths: 2.5, 4.0, 5.5, 7.0, 8.5, 10.0.
  2. 2

    State a rule precise enough to rebuild the sequence

    Why it works: The term-to-term rule gives the exact signed step; together with the starting number, it pins down the sequence so anyone can reconstruct it from the words alone.

    1. Continue 5, 3, 1, … and give the next four numbers (−1, −3, −5, −7), then state the rule as 'start at 5, subtract 2 each time'.
    2. Continue 2.5, 4.0, 5.5, … and give the next three numbers (7.0, 8.5, 10.0), then state the rule as 'start at 2.5, add 1.5 each time'.
    3. Phrase each rule so clearly that a partner could rebuild the whole sequence from the words alone, without seeing the numbers.
    4. Test your own rule: use only your stated start and signed amount to regenerate the given terms and check they match.
  3. 3

    Check every step, not just the last term

    Why it works: A linear sequence is defined by the same step between every pair of terms; the next term follows the step, not a surface pattern noticed in the last number alone.

    1. Consider 4, 7, 10, 13, 16 continued by someone as 18 'because they look even-ish'. Decide what the next term should be and explain why.
    2. Find the step between EVERY adjacent pair, not just the last two, to confirm the constant step (+3) and the correct next term (19).
    3. Explain to the person why fixating on the surface feature of the last number (its being even/odd) is not the rule.
    4. Decide whether a wrong term is a slip (a miscount) or a misconception (a misread rule) by re-checking the step between each pair.

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
Ma6/2.5b

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