Algebra
KS2MA-Y6-D005
Using simple formulae expressed in words and symbols; generating and describing linear number sequences; expressing missing number problems algebraically; finding pairs of numbers that satisfy an equation with two unknowns; enumerating possibilities of combinations of two variables.
National Curriculum context
Algebra is introduced explicitly in Year 6 as a named domain, representing the formal beginning of symbolic mathematical reasoning that pupils will develop extensively throughout KS3 and beyond. Prior to Year 6, algebraic thinking has been present implicitly — in missing number problems from Year 1 onwards, in generalising pattern rules, and in the use of formulae for area and perimeter — but Year 6 formalises this into symbolic notation using letters to represent unknown or variable quantities. The transition from arithmetic to algebraic thinking is one of the most significant cognitive steps in mathematics education, and the Year 6 programme provides a carefully managed entry point through familiar contexts such as simple formulae and linear sequences. Work on equations with two unknowns introduces systematic enumeration of solutions, developing logical reasoning and laying groundwork for simultaneous equations at KS3. Teachers should emphasise the continuity between familiar arithmetic operations and their algebraic generalisations, helping pupils see algebra as a powerful tool for expressing and solving problems rather than a mysterious new subject.
3
Concepts
2
Clusters
0
Prerequisites
3
With difficulty levels
Lesson Clusters
Use algebraic notation and formulae with letters for unknowns
introduction CuratedAlgebraic notation and formulae is the gateway concept for Year 6 algebra, introducing the idea of using letters to represent unknowns. Must precede sequence and equation work.
Generate and describe linear sequences and solve equations in two variables
practice CuratedLinear sequences, two-variable equations and mathematical reasoning/justification are co-taught (C014 and C030 mutually co-teach). Together they represent the Year 6 algebra application and reasoning cluster.
Teaching Suggestions (1)
Study units and activities that deliver concepts in this domain.
Algebra
Mathematics Worked Example SetDomain Vocabulary
32 terms across 3 concepts (32 domain-specific)(2 shared)
algebra(noun)
A branch of mathematics that uses letters and symbols to represent unknown numbers in expressions and equations.
always(adverb)
True in every case without exception; used when stating mathematical rules or generalisations.
argument(noun)
A logical chain of reasoning used to justify or prove a mathematical statement.
coefficient(noun)
The number placed in front of a variable in an algebraic term that shows how many of that variable there are.
common difference(noun)
The fixed amount added or subtracted to get from one term to the next in a linear sequence.
conjecture(noun)
A mathematical statement believed to be true based on observations, but not yet formally proven.
coordinate(noun)
An ordered pair of numbers that describes a precise position on a grid, written as (x, y).
counter-example(noun)
A single example that disproves a general statement or conjecture.
enumerate(verb)
To count or list items one by one in an organised way.
equation(noun)
A mathematical sentence with an equals sign showing that two sides have the same value.
Shared by 2 concepts
evaluate(verb)
To find the numerical value of an expression by substituting known values for variables and calculating.
explain(verb)
To give mathematical reasons and justifications for an answer or method, showing understanding.
expression(noun)
A combination of numbers, variables, and operations that represents a value, but does not contain an equals sign.
formula(noun)
A mathematical rule expressed using letters and symbols that shows the relationship between quantities.
generalise(verb)
To identify and express a pattern or rule that works for all cases, not just specific examples.
justify(verb)
To provide mathematical evidence and reasoning to support an answer or conclusion.
letter symbol(noun)
A letter used in algebra to represent an unknown or variable quantity.
linear sequence(noun)
A number sequence where the same amount is added or subtracted each time, producing a straight line when graphed.
logical(adjective)
Following a clear, step-by-step reasoning process based on mathematical rules.
never(adverb)
Not true in any case; used when a mathematical statement is always false.
prove(verb)
To demonstrate that a mathematical statement is always true by using logical reasoning, not just examples.
reason(verb)
To think logically and make deductions using known mathematical facts and rules.
rule(noun)
A mathematical instruction or pattern that describes how numbers relate to each other.
sequence(noun)
An ordered list of numbers that follows a rule or pattern.
solution(noun)
The value or set of values that make an equation or problem true.
sometimes(adverb)
True in some cases but not all; used when a mathematical statement applies only under certain conditions.
substitute(verb)
To replace a variable in an expression or equation with a specific numerical value.
systematically(adverb)
Working through a problem in an organised, methodical way to ensure all possibilities are considered.
term(noun)
A number in a sequence or pattern, identified by its position (e.g. 1st term, 2nd term).
Shared by 2 concepts
two unknowns(noun)
A problem or equation that contains two variables whose values need to be found.
unknown(noun)
A number that has not been found yet; a missing value in a number sentence.
variable(noun)
A letter or symbol that represents a quantity which can change or take different values.
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Concepts (3)
Algebraic Notation and Formulae
knowledge AI DirectMA-Y6-C013
Mastery of algebraic notation means pupils can write, interpret and use simple algebraic expressions and formulae in which letters represent unknown or variable quantities, substituting values to evaluate expressions and constructing expressions from word descriptions. A fully secure pupil understands that a letter represents a number (which may be unknown, or may vary), that 3n means '3 times n', and that an expression like 2l + 2w is a general formula applicable to any rectangle — not just a specific case.
Teaching guidance
Draw explicitly on prior work with missing number problems to establish continuity: the box in □ + 5 = 12 is the same idea as n in n + 5 = 12. Introduce letter notation as a more efficient and general representation than boxes. Practise writing expressions from descriptions ('5 more than n' → n + 5; 'three times m, subtract 4' → 3m - 4). Substitution exercises — evaluating an expression for given values — should precede formula derivation. Connect to known formulae (area of rectangle: A = lw; perimeter of rectangle: P = 2(l + w)) as familiar examples. Avoid ambiguity: be clear about the difference between an expression (3n + 2) and an equation (3n + 2 = 17).
Vocabulary (11 terms)
Common misconceptions
Pupils often treat a letter in an expression as a label or abbreviation (e.g., a = apples) rather than as a number. Concatenation notation (3n meaning 3 × n) is unfamiliar and pupils may read it as 'thirty-n' or misinterpret it. Some pupils believe that different letters in an expression must represent different values (e.g., in a + b, a ≠ b). Pupils also confuse 'evaluate an expression' with 'solve an equation' — these are importantly different tasks.
Difficulty levels
Understanding that a letter represents an unknown number and evaluating simple expressions by substitution.
Example task
If n = 5, what is 3n + 2?
Model response: 3 × 5 + 2 = 15 + 2 = 17.
Writing algebraic expressions from word descriptions and evaluating expressions with two operations.
Example task
Write an expression for 'twice a number, subtract 7'. If the number is 12, evaluate.
Model response: 2n – 7. When n = 12: 2 × 12 – 7 = 24 – 7 = 17.
Using formulae in context, substituting values into multi-variable expressions, and forming expressions from problems.
Example task
The perimeter of a rectangle is P = 2(l + w). Find P when l = 8 and w = 5. A pizza costs £p. Write an expression for the cost of 3 pizzas and a £2 delivery charge.
Model response: P = 2(8 + 5) = 2 × 13 = 26. Pizza cost: 3p + 2.
CPA Stages
concrete
Using physical 'function machines' (boxes with input/output slots), mystery number envelopes, and concrete bar models where a letter card represents an unknown number of counters
Transition: Child writes algebraic expressions from word descriptions and solves simple equations by reasoning about the unknown
pictorial
Drawing bar models to represent algebraic expressions and equations, using substitution tables to evaluate expressions for different values
Transition: Child writes and evaluates algebraic expressions and solves one-step equations using bar models or inverse operations on paper
abstract
Writing, evaluating and solving algebraic expressions and simple equations mentally, using formulae in context, and explaining that a letter represents a number
Transition: Child uses algebraic notation fluently, evaluates expressions by substitution, and solves simple equations using inverse operations
Delivery rationale
Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Linear Sequences and Two-Variable Equations
process AI DirectMA-Y6-C014
Mastery means pupils can generate any term of a linear sequence given its first term and common difference, describe a sequence using the language of 'common difference' and 'term', and represent simple sequences and relationships using algebraic expressions. For two-variable equations, a fully secure pupil can enumerate all pairs of positive integers (or values within a given range) that satisfy an equation like a + b = 10 or 2x + y = 8 by working systematically.
Teaching guidance
Use real-world contexts for sequences: tile patterns, fence posts and panels, stair-step shapes. Ask pupils to describe the pattern in words before introducing algebraic notation. Progress from 'add n each time' descriptions to the nth term rule for linear sequences (though full nth term work is a KS3 objective). For two-variable equations, teach systematic enumeration: start with the smallest possible value for one variable, find the corresponding value for the other, and work through all possibilities in order, recording in a table. Connect to coordinate graph work by plotting solution pairs on a grid — visually, the solutions lie on a straight line.
Vocabulary (11 terms)
Common misconceptions
Pupils confuse the term-to-term rule (add 4 each time) with the position-to-term rule (nth term = 4n - 1). When enumerating solutions to two-variable equations, pupils work unsystematically and miss solutions or count duplicates. Some pupils do not consider that variables can take the value 0. Explicit use of tables and a systematic left-to-right approach prevents most enumeration errors.
Difficulty levels
Continuing a linear sequence given the first few terms and describing the rule as 'add n each time'.
Example task
Continue this sequence: 3, 7, 11, 15, ?, ?. What is the rule?
Model response: 19, 23. The rule is add 4 each time.
Generating terms of a sequence from a rule and finding pairs of values that satisfy a two-variable equation.
Example task
Find all pairs of positive whole numbers where a + b = 8.
Model response: 1+7, 2+6, 3+5, 4+4, 5+3, 6+2, 7+1.
Describing linear sequences algebraically, enumerating solutions to two-variable equations, and solving problems involving sequences.
Example task
A sequence starts at 5 and adds 3 each time. Is 50 in the sequence? Explain.
Model response: Terms: 5, 8, 11, 14... Each term is 3n + 2 (where n starts at 1). For 50: 3n + 2 = 50, 3n = 48, n = 16. Yes, 50 is the 16th term.
CPA Stages
concrete
Building linear patterns with physical objects (tiles, matchsticks, cubes) and recording the number in each term, then finding all integer pairs that satisfy two-variable equations using counters
Transition: Child describes the common difference, predicts any term, and enumerates all positive integer solutions of a two-variable equation systematically
pictorial
Drawing pattern sequences and recording in tables, writing the term-to-term rule, and plotting solution pairs of two-variable equations on a coordinate grid
Transition: Child generates sequences from rules and enumerates solutions on a coordinate grid, noticing the linear pattern
abstract
Working with sequences and two-variable equations abstractly: finding the nth term rule (informally), predicting terms, and finding all solutions within a range
Transition: Child finds any term of a linear sequence and systematically enumerates solutions to two-variable equations
Delivery rationale
Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Mathematical Reasoning and Justification
process AI DirectMA-Y6-C030
Mastery of mathematical reasoning at Year 6 means pupils can construct and communicate a logical mathematical argument, explain why a procedure works (not just how), and use examples and counter-examples to investigate conjectures. A fully secure pupil can distinguish between a demonstration (showing that something is true in a specific case) and a proof (showing that it must always be true), and uses correct mathematical language and notation to communicate their reasoning clearly and precisely.
Teaching guidance
Embed reasoning tasks throughout all mathematical content rather than treating reasoning as a separate topic. Use prompts such as 'Always, Sometimes, Never' (e.g., is it always true that multiplying two numbers gives a larger answer?), 'Prove it' challenges, and 'Find the error in this reasoning' activities. Teach pupils to use language precisely: 'The answer is always even because...' rather than 'I think it's even'. Connect to the algebra domain: algebraic notation is a tool for expressing general mathematical truths. Require pupils to write mathematical explanations as complete sentences that would be understood by someone who has not seen the working.
Vocabulary (12 terms)
Common misconceptions
Pupils often confuse explanation ('this is what I did') with justification ('this is why it must work'). Checking a rule with one or two examples is often mistaken for proof. Some pupils communicate mathematical reasoning in informal or ambiguous language that does not precisely convey their thinking. The distinction between 'showing' and 'proving' needs explicit teaching with examples from familiar mathematical content.
Difficulty levels
Explaining what they did and why in a single-step calculation using mathematical vocabulary.
Example task
Explain why you added 37 and 28 to solve the problem about apples.
Model response: I added because the problem said 'altogether', which means combining two groups. 37 + 28 = 65 apples altogether.
Using examples and counter-examples to investigate whether a mathematical statement is always, sometimes or never true.
Example task
Is this always, sometimes or never true: 'Multiplying two numbers gives a bigger answer'?
Model response: Sometimes true. True: 3 × 4 = 12 (bigger than both). False: 0.5 × 6 = 3 (smaller than 6). Also false: 0 × 5 = 0 (not bigger than 5). So it depends on the numbers.
Constructing a chain of logical reasoning to prove or disprove a conjecture, distinguishing proof from demonstration.
Example task
Prove that the sum of three consecutive numbers is always a multiple of 3.
Model response: Let the three consecutive numbers be n, n+1, n+2. Their sum = n + (n+1) + (n+2) = 3n + 3 = 3(n+1). Since 3(n+1) is 3 times a whole number, it is always a multiple of 3.
CPA Stages
concrete
Using physical examples and counter-examples to test conjectures: building arrays, sorting shapes, or testing number properties with real objects
Transition: Child tests conjectures systematically using examples and counter-examples and refines statements based on findings
pictorial
Recording mathematical arguments on paper: writing 'if...then' statements, drawing diagrams to support reasoning, and using algebraic notation to express general rules
Transition: Child writes mathematical justifications that explain WHY something must be true, not just that it IS true in specific cases
abstract
Constructing and communicating mathematical arguments using precise language, generalising from patterns, and distinguishing between examples and proof
Transition: Child constructs logical arguments using algebraic reasoning and clearly distinguishes between demonstration and proof
Delivery rationale
Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.