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Introductory algebra formulae and linear sequences - Mathematics Year 6

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Introduction

A letter can help describe a whole family of calculations. Use formulae, missing-number equations, repeating growth and pairs of unknowns to explain what stays fixed and what can change.

Success criteria. Use simple formulae; express and solve missing-number problems; generate a linear sequence; find pairs satisfying a two-unknown equation; and justify an unfamiliar solution with notation, units and a check.

Three chapters combine original explanations, worked examples, unchanged Commons diagrams, an original substitution video and independent native practice. Price and object-count examples are fictional models. A letter's meaning is declared within its own problem; it is not carried unchanged into every other context.

Read a formula and substitute declared values

A letter represents a number in its context. In a fictional activity, the total cost in pounds is given by C = 4n + 3, where n is the number of participants. Four pounds per participant is variable cost; three pounds is a fixed charge.

Multiplication can be written without a cross. Four n means four multiplied by n, not four plus n and not a two-digit number made by putting four beside another digit. For n = 5:

$4 \times 5 + 3 = 20 + 3 = 23$

The model gives £23. It is not £12 from four plus five plus three. The fixed charge is paid once, not once for every participant.

A formula links quantities. If n changes, the model's C changes. For this participant context, n is a non-negative whole number. Although a formula may accept other numerical inputs algebraically, half a participant does not fit the stated count model without a new interpretation.

Use operation order. Multiply before adding in 4n + 3. Brackets can change a rule: 4(n + 3) would give four copies of both n and three. At n = 5:

$4 \times (5 + 3) = 32$

That differs from 23. Read structure before calculating.

A familiar geometric formula. A rectangle's perimeter can be written P = 2(L + W), where L and W are its side lengths in the same unit. With L = 7 cm and W = 4 cm:

$2 \times (7 + 4) = 22$

The answer is 22 cm, a length. Adding all four sides also gives 7 + 4 + 7 + 4 = 22, an independent geometric interpretation.

Use the cuboid's actual labels. Its source letters are uppercase A, B and C. For this separate problem, they stand for three perpendicular dimensions, and volume is V = A × B × C. Assign A = 4 cm, B = 3 cm and C = 2 cm:

$4 \times 3 \times 2 = 24$

The volume is 24 cubic centimetres, not 24 centimetres. Here C is a dimension; it is not the £23 cost symbol in the earlier, different context. No source dimension is inferred from the perspective picture.

Corbettmaths' original Substitution - Primary explains replacing letters with values. Its historical fictional hire example uses a £40 fixed cost and £30 for each day. For three days, forty plus thirty times three gives £130. Treat that as a teaching model, not a current commercial price.

Independent substitution. Attempt F1–F4 in the final chapter. Write the rule, declare the inputs, substitute with brackets where needed, follow operation order, and state the output unit or count. A worked answer should show what each letter meant, not only a final number.

Express missing numbers and describe a linear sequence

An equation says two expressions have equal value. A missing-number task can be written 7 + x = 19. Removing seven from both sides preserves equality, giving x = 12. Check by substituting twelve into the original equation:

$7 + 12 = 19$

The equal sign does not mean merely the answer comes next. Both sides name the same value.

Reverse a two-step rule. In 4n + 3 = 23, the rule multiplies n by four and then adds three. To recover n, undo the addition first, then undo multiplication:

$(23 - 3) \div 4 = 5$

The original check four times five plus three equals twenty-three verifies the recovered input. Guessing may locate five, but the inverse route explains why it works.

Keep balance. Subtracting three from only one side would change the equation. Applying the same reversible step to both sides preserves its solutions. Explain the operation rather than use a phrase such as move it across without understanding the change.

A specified linear sequence. Start at three and add four at every step. The first five terms are:

Position n	Term value
1	3
2	7
3	11
4	15
5	19

The term-to-term rule is add four. The position is not the term value: the third term is eleven, not three.

A position rule for the same sequence. Each term is one less than four times its position, giving term = 4n − 1 for positions n = 1, 2, 3 and so on. At position ten:

$4 \times 10 - 1 = 39$

The rule makes a distant term accessible without repeatedly adding four nine times.

Why does that rule fit every stated position? Begin with three tiles at position one. Each later position adds four tiles. After n minus one additions, the count is 3 + 4(n − 1), equivalent to 4n − 1. This is a reason from the defined growth process, not merely a guess from a few displayed values.

An unshaded grid of one hundred equal cells. — Use a sketch of cells to model three initial tiles and groups of four added at later stages. No sequence regions are already shaded in this unchanged source.

Finite clues do not force a unique continuation. If you are given only the numbers three, seven and eleven, other invented rules could eventually diverge. Here adding four every time is part of the question, so the linear rule is justified. State an assumed rule when the task supplies only a short prefix.

Work backwards from a term. Does thirty-nine belong to this specified sequence? Add one and divide by four, giving position ten. A proposed value forty would give a non-whole position ten and one quarter, so it is not a term at a positive whole-number position.

Independent transfer. Write the first five terms of a sequence that starts at two and adds three every time. Explain its position rule, find the tenth term, and check the first and fifth terms. Distinguish a rule for the next term from a rule using its position.

Find pairs of unknowns and communicate complete solutions

One equation can have several pairs of solutions. Consider 3a + b = 14, where a and b must both be positive whole numbers. Choose a, calculate b, and check the pair in the original equation.

a	b	Check
1	11	3 × 1 + 11 = 14
2	8	3 × 2 + 8 = 14
3	5	3 × 3 + 5 = 14
4	2	3 × 4 + 2 = 14

These are four different ordered pairs. The order matters because a is multiplied by three while b is not.

Explain completeness within a domain. Because a is positive, begin at one. If a were five, three a would already be fifteen, leaving b negative. No larger positive whole a can work with positive b. The table therefore includes all pairs under the stated conditions.

Changing the domain changes the answer set. If zero is allowed, a = 0 and b = 14 is another solution. If negative numbers or fractions are allowed, more pairs become possible. A claim about all solutions must name its allowed values.

A deliberate failed pair. Swapping the valid pair (2, 8) to (8, 2) gives:

$3 \times 8 + 2 = 26$

Twenty-six is not fourteen. Each unknown has a defined role; exchanging values is not automatically harmless.

Four formula tasks. These are independent contexts. Re-declare each letter and its units before substituting.

Code	Rule and inputs
F1	C = 4n + 3; n = 5; C is in pounds.
F2	P = 2(L + W); L = 7 cm, W = 4 cm.
F3	V = A × B × C; A = 4 cm, B = 3 cm, C = 2 cm.
F4	T = 3k − 2; k = 6; T is an abstract tile count.

A combined unfamiliar task. A fictional club rule is C = 4n + 3. The total is £35. Undo the fixed charge and divide by four:

$(35 - 3) \div 4 = 8$

There are eight participants in the model. Substitution gives thirty-two plus three, recovering thirty-five. If the total were £34, the inverse would give 7.75, which does not fit a whole-participant count. That mismatch calls for checking the context or model, not silently rounding and claiming equality.

Compare a table with an inverse. For a small participant range, a table can list allowed n values and their costs, making the count restriction visible. An inverse calculation is faster for a distant target. Both need a final check in the original rule.

A complete algebra response. Define letters, domain and units; express the relationship; choose substitution, inverse reasoning, a growth model or systematic enumeration; show working; and check the original conditions. The number of examples tested determines what has been shown: three valid pairs prove at least three solutions, but completeness needs a bound.

Your independent dossier. Complete all four formula tasks, solve the missing-number examples, generate both specified linear sequences and provide at least three pairs for the two-unknown equation. Then justify the full positive-pair list. Finish by correcting one misleading interpretation of a letter, an equal sign, a term position or a missing-value operation.

Interactive Tasks

Quiz: Test Your Knowledge

What does four n mean in the formula C = 4n + 3? (Four multiplied by n) (!Four plus n) (!A two-digit number beginning with four) (!Four divided by n in every case)

What is C when n is five in C = 4n + 3? (Twenty-three) (!Twelve) (!Thirty-two) (!Twenty)

Why is 4(n + 3) different from 4n + 3? (The brackets multiply the added three by four as well) (!Brackets never affect a calculation) (!The letter always becomes three) (!Multiplication changes into subtraction)

What does an equal sign state? (The expressions on both sides have the same value) (!Only that a new calculation will follow) (!That the left side must be larger) (!That letters can be removed without changing anything)

What solves seven plus x equals nineteen? (x equals twelve) (!x equals twenty-six) (!x equals seven) (!x equals nineteen sevenths)

For the specified sequence starting at three and adding four, what is the third term? (Eleven) (!Three) (!Twelve) (!Seven)

Which position rule matches that specified sequence for positive whole-number positions? (Four times the position minus one) (!Four plus the position) (!Three times the position) (!Four times the position plus three)

Which pair satisfies three a plus b equals fourteen? (a equals two and b equals eight) (!a equals eight and b equals two) (!a equals two and b equals four) (!a equals five and b equals one)

Why must a complete solution-pair list state the domain? (Allowing zero, negatives or fractions can change which pairs qualify) (!The domain changes the meaning of addition) (!Only the colour of a table determines its solutions) (!Every equation has exactly one pair regardless of allowed values)

A cuboid has dimensions four centimetres, three centimetres and two centimetres. What unit belongs to its volume? (Cubic centimetres) (!Centimetres) (!Square centimetres) (!Pounds)

Memory Game

Variable	A symbol whose value may vary in a stated model
Unknown	A value to determine from given conditions
Formula	A rule connecting quantities
Substitution	Replacing a symbol with its declared value
Equation	A statement that two expressions are equal
Term	One value in a sequence
Domain	The allowed values in a particular problem

Drag and Drop

Evaluate a formula	Declare inputs, substitute and follow the operation structure
Recover a missing input	Undo operations in reverse order and check the original rule
Generate a linear sequence	Use its specified start and constant additive step
Find ordered pairs	Choose an allowed first value and determine the second systematically
Claim a complete list	State the domain and explain a bound that excludes further cases

Crossword Puzzle

FORMULA	A rule relating quantities
UNKNOWN	A value to find
EQUATION	A statement of equality
SEQUENCE	An ordered list generated by a rule
SUBSTITUTE	Replace a symbol with a given value
DOMAIN	The values allowed by a problem

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Four formulae: Evaluate F1–F4. Define the letters afresh in each context, show substitution and give the output unit or count.
Missing numbers: Solve 7 + x = 19 and 4n + 3 = 23. Describe inverse operations and substitute back to check.
Specified sequence: Start at three and add four each time. Write the first five terms, identify the third term and explain the difference between position and value.
Three valid pairs: Give at least three positive whole-number pairs satisfying 3a + b = 14. Show a check for each in its ordered roles.

Standard

Position versus next-term rule: Explain both rules for the sequence beginning three with constant step four. Find the tenth term and check positions one and five.
A different linear sequence: Start at two and add three every time. Give five terms, a position rule and the tenth term; explain the initial offset.
Complete positive-pair list: Find every positive whole-number pair for 3a + b = 14. Explain why the first unknown cannot be five or larger.
Club target: Use C = 4n + 3 to find the participant count for total £35. Compare inverse reasoning with a small table of allowed inputs.

Advanced

Changed domain: Repeat the pair problem allowing non-negative whole numbers. Identify the additional pair and explain why negative or fractional domains would require a different answer set.
Sequence membership: Decide whether thirty-nine and forty are terms of the specified start-three add-four sequence. Find candidate positions and interpret whole-number restrictions.
Model mismatch: Explain what a total of £34 implies under the club cost rule and why rounding the resulting participant count would fail the original equality.
Method and evidence dossier: Create a new formula or sequence problem. Declare its domain, justify an efficient method, include units if appropriate and a check; distinguish evidence from a few cases from a general reason.

Learning Check

Four formula outputs: Check F1–F4.
Missing inputs and club model: Check inverse reasoning.
Two linear sequences: Check terms, rules and distant values.
Pairs and completeness: Check the full positive whole-number set.
Membership and units: Check the unfamiliar conditions.
Communication and justification: Assess a complete original solution.

Worked checks and assessment guidance

F1 = £23. F2 = 22 cm. F3 = 24 cm³. F4 = 16 tiles. Show the declared substitutions: 4 × 5 + 3; 2 × (7 + 4); 4 × 3 × 2; 3 × 6 − 2. The uppercase C in the separate volume context is a dimension, not the earlier cost.
7 + x = 19 gives x = 12. 4n + 3 = 23 gives n = 5. A club total of £35 gives n = 8, checked by 4 × 8 + 3 = 35. A total of £34 gives 7.75 mathematically, outside the whole-participant model; rounding to 8 would give £35, not £34.
Start 3, step 4: first five terms 3, 7, 11, 15, 19; position rule 4n − 1; tenth 39. Start 2, step 3: first five terms 2, 5, 8, 11, 14; position rule 3n − 1; tenth 29. Positions start at one. The defined constant growth, not a short prefix alone, justifies each rule.
For 3a + b = 14, positive ordered pairs are (1,11), (2,8), (3,5), (4,2). At a = 5, b would be −1, so no larger positive a works with positive b. If non-negative whole numbers are allowed, add (0,14). Swapping (2,8) gives 3 × 8 + 2 = 26, which fails.
Thirty-nine gives position (39 + 1)/4 = 10 and is a term. Forty gives 41/4 = 10.25, outside positive whole-number positions, so is not. Perimeter uses length units; volume uses cubic units. A letter or diagram alone supplies no physical measurement.
Require declared letters, domain and units; correct relation and operation structure; suitable substitution, inverse, sequence or enumeration method; and a check against the original conditions. Distinguish 4n from 4 + n, term position from value, equal expressions from an instruction to continue, and a few valid examples from a bounded complete solution list.

Learning evidence

Keep four formula calculations, two missing-input solutions, both specified sequences, the complete positive pair table and an unfamiliar checked problem. State domains, units and why a method fits. Annotated or oral reasoning can show understanding; no complete curriculum qualification or observed physical measuring competence is claimed.

OER and Wikipedia

Checked 11 November 2026. Department for Education: mathematics programmes of study, Year 6 algebra, informs simple formulae, missing-number relationships, linear sequences and pairs satisfying a two-unknown equation. Systematic bounds support complete enumeration.

All rules, original explanations, fictional cost contexts, tasks and feedback are newly authored. Domains and units are stated; an inferred non-whole count is not silently rounded to claim equality.

Embedded original Substitution - Primary, corbettmaths. Its price example is historical fictional teaching material, not an offer. The native course separately teaches sequences, equations, domains and complete pair lists.

Commons assets are unchanged. The cuboid uses uppercase A/B/C and no stated numerical dimensions. The hundred grid is unshaded; proposed growing-tile regions are learner representations, not existing marks in the source.

Cuboid.abc.png, Cuboid.png: en:User:Tomruen, CC BY-SA 4.0. Unchanged.

100 grid.PNG, The original uploader was Leightwing at English Wikiversity., Public domain. Unchanged.