Directed Number

The big idea

Numbers can have direction as well as size. Directed number extends the number system to include positions and movements in opposite directions relative to zero, giving pupils the conceptual foundation for working with positive and negative terms throughout algebra.

Why this matters for secondary maths

Directed number is the first place most pupils meet the idea that the number system extends beyond counting numbers. If this concept is not secure, pupils will struggle with every algebraic topic that involves negative terms, which is almost all of them.

The critical conceptual challenge is distinguishing between the sign of a number (whether it is positive or negative) and the operation being performed (whether we are adding or subtracting). The expression $-3 - (- 5)$ involves a negative number, a subtraction operation, and another negative number. Three things that pupils must parse separately. When this distinction is not taught explicitly, pupils fall back on memorised sign rules (‘two negatives make a positive’) without understanding when and why those rules apply.

This same distinction reappears throughout algebra. In the expression $3 x - 5 y + 2 x - (- 3 y)$ , pupils must distinguish between the sign attached to each term and the operations that connect them. A pupil who has learned directed number only as a set of sign rules will be fragile here; a pupil who understands the underlying structure will not.

Key representations

Number lines and vectors: directed number as movement and position

A number line helps pupils see where a number is located, the size of a number, the direction of movement, and the difference between numbers. Starting at 3 and adding −5 means moving 5 units left. Starting at −2 and subtracting −4 means removing a movement of 4 units left, which is equivalent to moving 4 units right.

This model is especially powerful for establishing that a negative number tells us something about direction or position relative to zero, not that it is ‘bad’ or ‘small in a vague sense’. Using arrows or directed movements prepares pupils for translation in geometry, vectors, graphs, and the idea of quantities changing in opposite directions. It also supports the algebraic view that operations can be thought of as transformations.

Limitation: The number line model can become difficult to interpret when subtracting a negative, because ‘removing a leftward movement’ is an abstract concept for many pupils. This is where the counter model provides clearer reasoning.

Two coloured counters: directed number as combining opposites

Two-coloured counters represent +1 and −1 as physical objects. A positive and a negative counter together make a zero pair. They cancel to give zero. This model is especially useful for understanding addition and subtraction with negatives.

For example, 3 + (−5): three positive counters and five negative counters can be paired into three zero pairs, leaving two negative counters, so the result is −2. For 4 − (−3): subtracting −3 means removing three negative counters. If there are not enough negatives to remove, we add zero pairs (which does not change the value), then remove three negatives. The total increases by 3, giving 4 − (−3) = 7.

Limitation: Counters work well for integer arithmetic but become impractical for large numbers or decimals. They are a reasoning tool, not a permanent calculation method.

N.B. The unit tiles from a set of algebra tiles can be used instead of separate counters

Why both models are needed

These representations illuminate different meanings. The number line highlights position, distance, order, and movement. The counter model highlights cancellation, opposites, and the logic of subtraction. Using both avoids a shallow dependence on rules and builds more flexible understanding. Neither is a stepping stone to be abandoned; each remains useful for different purposes throughout secondary mathematics.

Worked examples

Example 1: Addition with negatives using counters

Calculate −4 + 7.

Four negative counters and seven positive counters. Four zero pairs cancel, leaving three positive counters. So −4 + 7 = 3.

Example 2: Subtraction of a negative using the number line

Calculate 2 − (−5).

Start at 2. Subtracting −5 means removing a leftward movement of 5. Equivalently, moving 5 to the right. Arriving at 7. So 2 − (−5) = 7.

Example 3: Subtraction of a negative using counters

Calculate −1 − (−4).

Begin with one negative counter. We need to remove four negative counters, but we only have one. Add three zero pairs (three positive, three negative). Now we have four negative counters and three positive counters. Remove four negatives. Three positive counters remain. So −1 − (−4) = 3.

Example 4: Structural reasoning

Why is 5 − (−2) the same as 5 + 2?

Using counters: removing two negatives from a collection increases the total by 2. Using the number line: reversing a leftward movement of 2 is the same as moving 2 to the right (see additive inverse tool. Both models show that subtracting a negative is equivalent to adding the corresponding positive. This is not a rule to memorise but a structural consequence of how opposites and zero pairs work.

The bridge to algebra

Directed number leads directly into algebra because algebraic expressions involve positive and negative terms, combining opposites, and seeing subtraction as adding the opposite.

$x - 3 = x + (- 3)$

This rewriting of subtraction as adding a negative is the same structural move pupils practised with counters and number lines. It allows algebraic expressions to be treated as sums of signed terms, which simplifies collecting like terms.

$3 x - 5 + 2 x + 5$ $= 3 x + 2 x + (- 5) + 5$ $= 5 x + 0$ $= 5 x$

The −5 and +5 form a zero pair, just as a negative counter and a positive counter cancel. A pupil who has internalised zero pairs in directed number will see this cancellation naturally in algebraic expressions.

$x - (- 3) = x + 3$

Subtracting a negative term works the same way it did with numbers: removing a negative is equivalent to adding the positive. This is essential for simplifying expressions and solving equations throughout KS3 and KS4.

Key vocabulary

Term	Definition
Directed number	A number with both size (magnitude) and direction (positive or negative).
Positive	Greater than zero; to the right of zero on a horizontal number line.
Negative	Less than zero; to the left of zero on a horizontal number line.
Zero pair	A pair consisting of +1 and −1 (or any number and its opposite) that sum to zero.
Additive inverse	The number that, when added to a given number, gives zero. The additive inverse of +5 is −5.
Magnitude / absolute value	The size of a number without regard to its sign. The magnitude of −7 is 7.
Integer	A whole number, which may be positive, negative, or zero.

What we don’t say

Avoid	Why	Say instead
‘Two negatives make a positive’	Only true for subtraction of a negative or multiplication/division of two negatives. Pupils over apply it: they may think −3 + (−5) = 8.	‘Subtracting a negative is the same as adding the positive (or the additive inverse)’ and explain why, using counters or the number line.
‘A minus and a minus make a plus’	Same problem. Conflates the sign of a number with the operation. Obscures the distinction the framework depends on.	Distinguish clearly between the sign of the number and the operation being performed. ‘The number is negative; the operation is subtraction.’
‘Negative numbers are less than nothing’	Mathematically inaccurate and philosophically confusing. Negative numbers are real quantities representing positions, debts, temperatures, etc.	‘Negative numbers are less than zero. They are positioned to the left of zero on the number line.’
‘Just follow the sign rules’	Sign rules without understanding produce fragile recall and prevent pupils from reasoning about unfamiliar situations.	Teach the sign rules as consequences of the zero pair structure and the number line, not as starting points.

Common misconceptions and how to surface them

Misconception 1: The minus sign always means ‘subtract’.

In the expression 5 + (−3), the minus sign on the 3 indicates that the number is negative, not that we are subtracting. Pupils who confuse sign and operation will misread expressions throughout algebra. Surface this by asking: ‘In the expression 8 + (−2), what does the minus sign in front of the 2 tell us?’ A pupil who says ‘it means subtract’ has not yet made the distinction.

Misconception 2: Negative numbers are ‘smaller in every sense’.

Pupils may think −7 is somehow ‘smaller’ than 2 in terms of magnitude. They need to understand that −7 is further from zero than 2, and that ‘smaller’ on the number line means ‘further to the left’, not ‘closer to zero’. A good diagnostic: ‘Which is further from zero: −9 or 4?’

Misconception 3: Memorised sign rules without understanding.

Pupils who can state rules but cannot explain them will fail on unfamiliar problems. Ask: ‘Can you show me why 3 − (−2) = 5 using counters?’ A pupil relying on rote memory will not be able to demonstrate the reasoning.

Misconception 4: Subtraction is only ‘take away’.

Subtraction also means ‘difference’ and ‘adding the opposite’. If pupils only see subtraction as removal, they will struggle to interpret what ‘subtracting a negative’ means. Surface this by asking: ‘What is the difference between 3 and −4?’ and seeing whether pupils can reason about the distance on the number line.

Diagnostic questions

Question 1: Place these in order from smallest to largest: 3, −7, 0, −1, 5, −12.

What this reveals: Pupils who order by magnitude (−1, 3, 5, −7, −12) rather than by position on the number line have not internalised the meaning of negative numbers as positions. Correct: −12, −7, −1, 0, 3, 5.

Question 2: True or false: −4 − (−3) = −7. Explain your reasoning.

What this reveals: A pupil who says ‘true, because a minus and a minus make a minus’ is applying a garbled sign rule. The correct answer is false: −4 − (−3) = −4 + 3 = −1. A pupil who can explain this using counters (removing three negatives) or the number line (reversing a leftward movement) demonstrates structural understanding.

Progression spine

Stage	Key ideas	Notes
Primary (Y5–Y6)	Negative numbers in context (temperature, number lines). Counting through zero. Ordering integers.	Pupils typically meet negatives as positions, not as objects to calculate with.
Year 7	Formal addition and subtraction with directed numbers using both models (number line and counters). The distinction between sign and operation. Zero pairs. Equivalence of subtracting a negative and adding the positive.	Both representations should be developed and connected.
Years 8–11	Multiplication and division of directed numbers. Coordinates in four quadrants. Negative coefficients in expressions and equations. Negative gradients. Working with inequalities involving negatives. Negative indices.	The sign versus operation distinction established in Year 7 continues to be essential. Multiplication of negatives is best understood through pattern and structure, not just rules.