Math // Addition & Subtraction

Positive and Negative Numbers

Understand positive and negative numbers as describing quantities with opposite directions or values; use them in context such as temperature, floors in a building, and bank balances

Ages
11 to 13
Type
Representation
Builds on
1 topic
Leads to
3 topics
Standards
9 codes

Quick check

If the learner sees temperatures like −3°C and +5°C on a weather forecast, can they figure out how much warmer one day is than the other — even when the answer crosses zero?

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.
  • Represent positive and negative numbers on a number line and explain what zero means in context
  • Add a positive or negative number to any integer using number line reasoning
  • Subtract a positive or negative number from any integer, understanding that subtracting a negative is equivalent to adding

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
  • Measuring temperatureRequired

    Number Representation & Place Value // Ages 10 to 11

    Negative number arithmetic extends Y6 using negative numbers in context

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Positive and Negative Numbers

Ages 11 to 13 // Representation

Leads to

3 topics
  • Fractions on a number line (age 11+)Required

    Number Representation & Place Value // Ages 11 to 12

    Ordering all number types (integers, decimals, fractions) on a number line extends the negative-number number-line representation

  • Numbers on a number lineRequired

    Number Representation & Place Value // Ages 11 to 12

    Absolute value requires understanding negative numbers on a number line

  • Sign Rules for MultiplicationRequired

    Multiplication & Division // Ages 11 to 13

    Multiplying/dividing negatives requires adding/subtracting negatives first

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

Positive and negative numbers are quantities with opposite DIRECTIONS from a meaningful zero (freezing point, ground floor, break-even), where sign is direction and magnitude is distance. On a number line, adding a positive moves right/up, adding a negative moves left/down, and subtracting a negative reverses direction to move right/up — so 'subtracting a negative is the same as adding' is derived from the direction of travel, not memorised as a sign rule. The central difficulty is distinguishing the two meanings of the minus symbol: the sign OF a number versus the operation of subtraction. Learners should both READ arrows/journeys and PRODUCE their own for new signed calculations, including moves straight through zero.

Common misconceptions

  • Orders negatives by magnitude - thinks -7 > -3 because 7 > 3
  • Conflates the negative sign with the subtraction operation ('two minuses', reads 4 + (-7) as 4 - 7 or 4 + 7 arbitrarily)
  • Believes numbers 'stop' at zero, so cannot count or move through zero
  • Thinks subtracting always makes smaller and adding always makes bigger (fails for negatives)
  • Treats 'subtract a negative' as more negative rather than as reversing direction

Useful representations

  • Horizontal number line spanning zero (arrows for direction and length)
  • Vertical thermometer / floors of a building / sea level
  • Bank balance and debt (money owed as negative)
  • Two-colour counters (zero pairs) for combining opposites

Teaching practices

  • Anchor every idea on a number line / thermometer where zero is a real reference (freezing point, ground floor, break-even), not 'nothing'
  • Keep magnitude (distance from zero) and sign (direction) as separate ideas - counter 'bigger digit = bigger number' for negatives
  • Use directed-journey language: adding a negative is a move in the negative direction; subtracting a negative reverses direction
  • Distinguish the operation minus from the negative sign explicitly (colour, gesture, or bracket: 4 + (-7))
  • Derive 'subtract a negative = add' on the line (removing a debt / turning to face the other way) rather than stating a rule

Activities to try

  1. 1

    Operate a directed number line: fire and flip the arrow

    Why it works: Signed addition and the direction-reversal of subtracting a negative become visible: 'two minuses make a plus' is a direction reversal you can see, not a spell to recall. Numbers do not stop at zero.

    1. Enter 4 + (-7): start a marker at 4 and draw/trace an arrow 7 units LEFT, landing on -3 — state that the negative sign IS the leftward direction and the 7 is how far.
    2. Enter 1 - (-3): show that the arrow subtraction would send LEFT instead FLIPS to point RIGHT, landing on 4; compare with adding 3 (1 + 3 = 4) and confirm you land in the same place.
    3. Say in words why the arrow reversed: removing a -3 (e.g. clearing a debt of 3) is the same as adding 3.
    4. Replay both journeys vertically on a thermometer/lift mode: for 4 + (-7), count aloud straight down through zero (4, 3, 2, 1, 0, -1, -2, -3); for 1 - (-3), count up from 1 (2, 3, 4) and notice that this journey stays above zero.
  2. 2

    Zero in context and the two minus signs

    Why it works: Zero is a meaningful reference point, and the minus symbol carries two distinct meanings that must be read, not guessed.

    1. On a labelled reference line/thermometer, point to a value and state both its sign (direction from zero) and its distance from zero.
    2. Switch contexts (temperature, floors of a building, bank balance) and say what zero means in each (freezing point, ground floor, break-even — not 'nothing').
    3. For a written calculation like 4 + (-7), mark which minus is the negative SIGN of a number and which (if any) is the subtraction OPERATION; use brackets/colour to keep them apart.
  3. 3

    Place, order and interpret signed calculations

    Why it works: Ordering by position (not by digit size) and reading an arrow's direction and length consolidate that magnitude and sign are separate ideas.

    1. Place -7 and -3 on a number line and state which is smaller, explaining via position (the further left, the smaller).
    2. Order a set of negatives from least to greatest.
    3. Interpret a given arrow (e.g. starting at -3, ending at 2): state the calculation it shows (-3 + 5 = 2, or the difference 2 - (-3) = 5, an interval across 0) and what its direction and length tell you.
    4. Decide and justify which way 1 - (-3) moves before drawing it.

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.56.NS.66.NS.6a7.NS.17.NS.1a7.NS.1b7.NS.1c7.NS.1d
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Num.2

Math // Addition & Subtraction

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