Inverse Operations and Fact Families

The big idea

Every addition has a corresponding subtraction, and every multiplication has a corresponding division. Operations come in inverse pairs, and understanding this pairing, that one operation ‘undoes’ the other, is the structural foundation for solving equations, checking calculations, and reasoning about mathematical relationships in reverse.

Why this matters for secondary maths

Solving an equation is, at its core, the process of undoing operations to isolate an unknown. When pupils solve $3 x + 5 = 20$ , they subtract 5 (the inverse of adding 5) and then divide by 3 (the inverse of multiplying by 3). If they do not understand why these inverse steps work, if they are following a memorised procedure rather than reasoning about undoing, they will struggle with multi-step equations, equations with the unknown on both sides, and rearranging formulae.

The concept of inverse operations connects several ideas already established in the framework:

The additive identity (Concept Section 4): adding and then subtracting the same value returns to the original number because the net effect is adding zero. So $a + b - b = a + 0 = a$ .

The multiplicative identity (Concept Section 4): multiplying and then dividing by the same value returns to the original number because the net effect is multiplying by one. So $\frac{a \times b}{b} = a \times 1 = a$ (provided b ≠ 0).

Division as the inverse of multiplication (Concept Section 8): the area model shows this relationship visually. If you know two of the three quantities (both sides and the area), you can find the third.

Directed number (Concept Section 5): subtracting a negative is the same as adding the positive, because subtraction ‘undoes’ the negative direction.

Fact families make the inverse relationships concrete. The four related facts 3 + 5 = 8, 5 + 3 = 8, 8 − 3 = 5, and 8 − 5 = 3 are not four separate pieces of knowledge, they are four expressions of a single additive relationship. Similarly, 4 × 7 = 28, 7 × 4 = 28, 28 ÷ 4 = 7, and 28 ÷ 7 = 4 are four expressions of a single multiplicative relationship. Pupils who see fact families as a connected structure, rather than as four things to memorise separately, are better prepared to reason about equations, rearranging, and checking.

Beyond simple one-step equations, inverse operations support the idea that a sequence of operations can be reversed by applying the inverse operations in reverse order. This is the logical basis for function machines (and later, inverse functions), and it connects to the broader algebraic principle that operations are processes that can be undone.

Key representations

Fact family triangles (or triads)

A triangle (or triad diagram) with three numbers, for example, 3, 5, and 8 in an additive triad, or 4, 7, and 28 in a multiplicative triad, shows that these three numbers are connected by a single relationship. From the triangle, pupils can generate all four related facts. The product (or sum) is typically placed at the top, with the two factors (or addends) at the base.

Strengths: Makes the connected nature of fact families visible. Supports the transition from ‘I know one fact’ to ‘I know four facts’. Emphasises that knowing one relationship gives access to all related facts.

Limitation: Can become procedural (‘cover the number you want to find’) if not connected to the structural reasoning. Teachers should ensure pupils understand why the triad works, not just how to use it.

Function machines: operations as reversible processes

A function machine takes an input, applies one or more operations, and produces an output. The key insight is that the machine can be ‘run backwards’: given the output, applying the inverse operations in reverse order recovers the input. For example, a machine that does ‘×3, then +5’ is reversed by ‘−5, then ÷3’.

Strengths: Provides a visual model for multi-step equation solving. Makes the ‘reverse order’ principle concrete. Prepares pupils for function notation and inverse functions in later years.

Limitation: Can become a crutch if pupils rely on drawing machines rather than developing algebraic fluency. The machine is a model for reasoning, not a permanent calculation tool.

The bar model and balance model: equations as relationships

A bar model can show an additive relationship (a bar of length 8 split into parts of 3 and 5) or a multiplicative relationship (a bar of length 28 made of 4 equal parts of 7). From either model, the inverse relationships are visible: if the total is 8 and one part is 3, the other must be 5.

A balance model (two sides of an equation as a balanced scale) shows that performing the same operation on both sides preserves equality. Subtracting 5 from both sides of $3 x + 5 = 20$ is performing an inverse operation while maintaining the balance.

Strengths: The balance model connects inverse operations to the meaning of the equals sign (Concept Section 3) and to equation solving. The bar model connects to Concept Section 6 (Additive and Multiplicative Reasoning).

The area model: multiplication and division as a connected pair

As established in Concept Sections 7 and 8, a rectangle with sides $a$ and $b$ has area $a \times b$ . Given any two of the three quantities (side $a$ , side $b$ , area), the third can be found. This makes the inverse relationship between multiplication and division visible and concrete: multiplication finds the area from the sides; division finds a side from the area and the other side.

Worked examples

Example 1: An additive fact family

Start with the fact 7 + 9 = 16. The complete additive fact family is:

7 + 9 = 16 9 + 7 = 16 16 − 7 = 9 16 − 9 = 7

These are not four separate facts. They are four ways of expressing the same additive relationship among the numbers 7, 9, and 16. Knowing any one of these facts gives access to all four, because addition and subtraction are inverse operations, and addition is commutative (Concept Section 2).

Example 2: A multiplicative fact family

Start with the fact 6 × 8 = 48. The complete multiplicative fact family is:

6 × 8 = 48 8 × 6 = 48 48 ÷ 6 = 8 48 ÷ 8 = 6

Using the area model: a rectangle with sides 6 and 8 has area 48. Given the area (48) and one side (6), the other side (8) is found by division. This is the inverse of multiplication in action.

Example 3: Using the inverse to check a calculation

A pupil calculates 347 − 189 = 158. To check, they add: 158 + 189 = 347. The addition confirms the subtraction because the operations are inverses. If the check fails, the pupil knows to look for an error.

This is not just a ‘tip’, it is a structural principle. The inverse operation returns us to the starting point because the net effect of an operation followed by its inverse is doing nothing (the identity), as established in Concept Section 4.

Example 4: A function machine and its inverse

A function machine applies: input → ×4 → +3 → output.

If the input is 5: 5 → 20 → 23. The output is 23.

If the output is 23 and we want the input: we reverse the machine. The inverse operations, in reverse order, are: 23 → −3 → ÷4 → 5.

Crucially, the order of the inverse operations is reversed. This is because the last operation applied (‘+3’) must be the first to be undone. This principle is essential for multi-step equation solving.

Example 5: One-step equation solving using the inverse

Solve $x + 7 = 15$ .

The operation applied to $x$ is ‘add 7’. The inverse of adding 7 is subtracting 7. So: $x = 15 - 7 = 8$ .

Solve $3 x = 18$ .

The operation applied to x is ‘multiply by 3’. The inverse of multiplying by 3 is dividing by 3. So: $x = \frac{18}{3} = 6$ .

In both cases, the logic is the same: identify the operation, apply its inverse. The balance model confirms that performing the same inverse operation on both sides preserves equality.

Example 6: Multi-step equation solving using inverse operations in reverse order

Solve $4 x + 5 = 29$ .

Think of this as a function machine: $x$ → ×4 → +5 → 29.

Reverse: 29 → −5 → ÷4 → $x$ .

Step 1: 29 − 5 = 24 (inverse of +5).

Step 2: 24 ÷ 4 = 6 (inverse of ×4).

So $x = 6$ . Check: 4(6) + 5 = 24 + 5 = 29. ✔

The check itself is a forward pass through the function machine, confirming the answer by using the original operations.

Component 5: The bridge to algebra

Inverse operations are the logical engine behind every algebraic solving process. The arithmetic examples above transfer directly into algebraic form.

Fact families and equation solving:

Arithmetic: 7 + 9 = 16, so 16 − 9 = 7.

Algebra: $x + 9 = 16$ , so $x = 16 - 9 = 7$ .

The same inverse relationship is at work. The only difference is that one of the three values in the fact family is unknown and represented by a letter.

Multiplicative inverses and equation solving:

Arithmetic: 6 × 8 = 48, so 48 ÷ 6 = 8.

Algebra: $6 x = 48$ , so $x = \frac{48}{6} = 8$ .

Again, the same reasoning: division undoes multiplication. The area model makes this visible, if the area is 48 and one side is 6, the other side is $x = 8$ .

Reverse order of operations:

Arithmetic function machine: 5 → ×4 → +3 → 23. Reverse: 23 → −3 → ÷4 → 5.

Algebra: Solve $4 x + 3 = 23$ . Subtract 3 first (undoing the last operation): $4 x = 20$ . Then divide by 4 (undoing the first operation): $x = 5$ .

The ‘reverse order’ principle is the same in both cases, and it extends naturally to more complex equations.

Rearranging formulae:

Given the formula $v = u + a t$ , make $t$ the subject.

Step 1: $v - u = a t$ (subtract $u$ , the inverse of adding $u$ ).

Step 2: $\frac{v - u}{a} = t$ (divide by a, the inverse of multiplying by a).

Rearranging a formula is structurally identical to solving an equation: identify the operations applied to the subject, and undo them in reverse order. Pupils who have practised this reasoning with function machines and numerical fact families are well prepared for the algebraic version.

Inverse functions:

In later KS3/KS4 work, pupils meet the idea that if $f (x) = 3 x + 2$ , then the inverse function $f^{- 1} (x) = \frac{x - 2}{3}$ is found by applying the inverse operations in reverse order. This is a direct extension of the function machine reasoning developed in this section.

Key vocabulary

Term	Definition
Inverse operation	An operation that ‘undoes’ another. Addition and subtraction are inverses; multiplication and division are inverses.
Fact family	A set of related addition/subtraction (or multiplication/division) facts linking the same three numbers. For example: 3, 5, 8 generate 3 + 5 = 8, 5 + 3 = 8, 8 − 3 = 5, 8 − 5 = 3.
Function machine	A model that takes an input, applies one or more operations in sequence, and produces an output. The machine can be reversed by applying inverse operations in reverse order.
Solving (an equation)	Finding the value of the unknown by applying inverse operations to isolate it. For example, solving $x + 7 = 15$ by subtracting 7 from both sides.
Checking (by inverse)	Verifying a calculation by applying the inverse operation. If 347 − 189 = 158, then 158 + 189 should equal 347.
Rearranging (a formula)	Changing the subject of a formula by applying inverse operations to both sides. Structurally identical to solving an equation.

What we don’t say

Avoid	Why	Say instead
‘Move it to the other side and change the sign’	This is a description of what happens on paper, not an explanation of why it works. Pupils who ‘move terms’ without understanding inverse operations make errors with multi-step equations and with sign changes. The term is not moving anywhere, we are performing an inverse operation on both sides to maintain equality.	‘We subtract 7 from both sides because subtraction is the inverse of addition. This keeps the equation balanced and isolates the unknown.’
‘Whatever you do to one side, do to the other’ (as the sole explanation)	While true, this statement tells pupils what to do but not why to do it or which operation to choose. It is a procedural instruction that does not build understanding of why the inverse is the right operation to apply.	‘We apply the inverse operation to both sides. The inverse ‘undoes’ the operation that was applied to the unknown, leaving the unknown on its own. We do it to both sides to keep the equation balanced.’
‘The answer is the number on its own’	This encourages pupils to look for the answer visually (‘the number by itself’) rather than to reason about why inverse operations work. It also fails for equations where the unknown appears on both sides.	‘We use inverse operations to isolate the unknown. When the unknown is alone on one side, we have found its value.’
‘Swap the side, swap the sign’	Another description of surface appearance, not structure. It leads to errors when the operation involved is multiplication or division (where there is no ‘sign’ to swap). It also fails to generalise beyond simple linear equations.	Describe each step explicitly: ‘I am dividing both sides by 3, because the inverse of multiplying by 3 is dividing by 3.’

Common misconceptions and how to surface them

Misconception 1: ‘Moving’ terms rather than using inverse operations.

Pupils who have learned to ‘move terms to the other side’ often apply this mechanically, forgetting to change the operation or changing the wrong one. For example, from $x + 5 = 12$ , they may write $x = 12 + 5 = 17$ . Ask: ‘What operation was done to $x$ ? What undoes that operation?’ If the pupil can answer (‘add 5; subtract 5’), they are reasoning structurally. If they say ‘you move the 5 over’, they are following a surface rule.

Misconception 2: Applying inverse operations in the wrong order.

For $4 x + 5 = 29$ , some pupils divide by 4 first, getting $x + 5 = 7.25$ , and then subtract 5 to get $x = 2.25$ (incorrect). The error is undoing the operations in the wrong order: the +5 was applied last, so it must be undone first. Use the function machine to make the order visible: $x$ → ×4 → +5 → 29. Reversing: the +5 is undone before the ×4. Ask: ‘If you put on your socks and then your shoes, which do you take off first?’ The everyday analogy helps.

Misconception 3: Not seeing the connection between fact families.

Some pupils know that 6 × 7 = 42 but cannot immediately produce 42 ÷ 7 = 6 without a separate calculation. Present a fact family triangle with 6, 7, and 42, and ask the pupil to generate all four facts. If they can produce the multiplication facts but hesitate on the division facts, they have not internalised the inverse relationship. This connects to the relational fluency work in Concept Section 9 (Scaled Multiplication Tables).

Misconception 4: Treating checking as a separate, optional activity.

Pupils often skip checking because they see it as an add-on rather than as a structural tool. Reframe checking as ‘using the inverse operation to verify that the forward operation was correct’, the same structural idea that underpins equation solving. Ask: ‘You calculated 23 × 15 = 345. How can you use division to check this?’ A pupil who says ‘345 ÷ 15 should give 23’ is using the inverse relationship.

Misconception 5: Struggling to reverse operations when they involve fractions or negatives.

The inverse of ‘multiply by $\frac{1}{2}$ ’ is ‘divide by $\frac{1}{2}$ ’ (which is the same as multiplying by 2). The inverse of ‘subtract −3’ is ‘add −3’ (which is the same as subtracting 3). These are structurally consistent with the same inverse principle, but pupils often find them counterintuitive. Using directed number reasoning (Concept Section 5) and the identity properties (Concept Section 4) helps clarify: the inverse operation is always the one that brings you back to where you started.

Diagnostic questions

Question 1: I know that 8 × 12 = 96. Without calculating, write three other facts that must be true.

What this reveals: A pupil who immediately writes 12 × 8 = 96, 96 ÷ 8 = 12, and 96 ÷ 12 = 8 sees the fact family as a connected structure. A pupil who can produce the commutative fact (12 × 8) but hesitates on the division facts has not fully connected multiplication and division as inverses. A pupil who cannot produce any related facts is treating multiplication facts as isolated.

Question 2: Explain, step by step, how you would solve $5 x - 3 = 27$ . For each step, name the operation you are using and explain why you chose it.

What this reveals: A pupil who says ‘add 3 to both sides because adding is the inverse of subtracting, then divide both sides by 5 because dividing is the inverse of multiplying’ demonstrates structural understanding. A pupil who says ‘move the 3 over and change the sign, then move the 5 over and put it underneath’ is following a surface procedure. The key diagnostic is whether the pupil can name the inverse operation and explain why it is chosen.

Progression spine

Stage	Key ideas	Notes
Primary (Y1–6)	Addition and subtraction as inverse operations. Multiplication and division as inverse operations. Using the inverse to check calculations. Number bonds and fact families. Simple missing-number problems (? + 5 = 12).	Pupils develop an intuitive understanding of inverse operations through fact families and checking. The concept is often taught procedurally rather than as a structural principle.
Year 7	Explicit naming of inverse operations as structural relationships. Fact families as connected structures (not four separate facts). Function machines and their reversal. Solving one-step and two-step equations using inverse operations. The reverse-order principle for multi-step problems. Connection to identity properties (Concept Section 4).	The emphasis is on understanding why inverse operations work, not just on executing steps. Function machines provide the bridge between arithmetic checking and algebraic equation solving.
Years 8–11	Solving multi-step linear equations. Solving equations with the unknown on both sides. Rearranging formulae. Inverse functions. Checking algebraic work by substitution (a form of using the inverse). Working with more complex inverses (e.g. squaring and square-rooting).	The principles established in Year 7 extend to increasingly complex situations. Pupils who understand inverse operations as a structural principle can adapt to new equation types; pupils who learned procedures must be re-taught for each new form.