Curriculum Hub → Maths Lessons → Algebra → Expressions → Introducing Algebra Through Shape
Last Updated: May 2026 | Originally Published: March 2016
Algebraic notation does not have to be taught as a list of rules to be memorised. Taught well, it is a natural extension of what students already know about arithmetic. By drawing on students’ intuition through visual area puzzles made from familiar shapes, this KS2 to KS3 transition maths lesson helps Year 7 students build a genuine understanding of how to collect like terms and simplify expressions. No rote learning required.
Too often, introducing algebraic notation means handing students a set of abstract conventions and asking them to trust that the rules make sense. It is an approach that leaves many behind, and is not much fun to teach.
The area approach is different. Students work through visual puzzles, discuss which shapes can be combined, and discover the notation themselves. That is a better experience for them and, in my experience, a considerably more enjoyable one for the teacher.
When Year 7 students first encounter algebra, the most common difficulty is not with the notation itself – it is with the idea that letters represent quantities. Having spent primary school working with concrete numbers, they arrive at secondary school and are asked to work with unknowns before they have any intuitive sense of what those unknowns mean.
The result is surface-level manipulation. Students learn to collect like terms as a sorting procedure – group the xs, group the ys – without any understanding of why unlike terms cannot be combined. This is the gap the area approach closes.
The key insight: when each letter represents the area of a visually distinct shape, the rules students apply are the same rules of arithmetic they have always used. You cannot add squares to quarter circles for the same reason you cannot add apples to oranges. The visual makes the constraint obvious without a single convention having to be stated.
A strong KS2 to KS3 transition maths lesson should not simply repeat primary content or rush students into secondary notation. It should create a bridge between what students already understand and the more abstract ideas they are about to meet. This lesson does that by connecting familiar arithmetic and area reasoning to early algebraic notation.
Students already understand that repeated quantities can be counted, combined and compared. They also understand that different units cannot always be combined directly. This lesson turns those existing ideas into algebraic thinking. A square with area x cm² and a quarter circle with area y cm² become concrete representations of algebraic terms.
That makes the lesson particularly useful for Year 7 transition units, mixed-attainment classes and departments looking for a more conceptual start to algebra. Students do not begin by memorising rules for collecting like terms. They begin by seeing why the rules are necessary.
This transition lesson builds directly on knowledge students bring from Key Stage 2. Before students write any algebraic expressions, they draw on familiar ideas from arithmetic and geometry:
The transition into algebra happens when those familiar ideas are represented with letters. The notation is not the obstacle — the idea is. Once students understand the idea, the algebraic expression becomes a natural way of recording what they can already see.
The lesson begins with just two shapes. A square has area x cm² and a quarter circle has area y cm². From these two building blocks, every composite shape on the board has an area that students can express algebraically — not because they have been taught a rule, but because they can see the components directly.
The first shape consists of four squares and four quarter circles. A student who writes the area as x + x + x + x + y + y + y + y is immediately asked: “Is there a shorter way?” Without any formal instruction, the majority write 4x + 4y or even 4(x + y). The shorthand is intuitive because it mirrors how they already write repeated addition — four copies of the same thing become four times that thing.
Crucially, no student writes 4yx. They understand intuitively that a square and a quarter circle are different shapes and therefore cannot be combined. The visual prevents the most common algebraic misconception – combining unlike terms – before it can take hold.
This short tutorial video walks students through six worked examples using exactly the same square and quarter-circle model. It includes deliberate pause points where students are challenged to find the expression before the answer is revealed — ideal as a flipped learning task, a transition starter or a Year 7 algebra introduction.
One misconception consistently surfaces as we work through the examples: students write the area of four quarter circles as y4 rather than 4y. This is a useful discussion point. We explore why the coefficient is written before the letter, and students accept this immediately once they see it follows the same left-to-right convention they already use when writing 4 × 7.
A more satisfying moment comes when students are challenged to write the same area in a different way. The majority independently produce y + 3x as an alternative to 3x + y — discovering commutativity in algebraic addition by analogy with arithmetic, not by rule.
In the final teaching example students are presented with a shape made from four copies of the x − y piece and asked to write the total area in two different ways.
Without prompting, the vast majority produce both 4(x − y) and 4x − 4y. More importantly, they recognise them as equivalent — not because they have been shown the distributive law, but because they can see it in the shape. Four copies of a region that is x − y is self-evidently the same as four squares with four quarter circles removed.
Lesson design insight: when a mixed-attainment Year 7 class independently recognises the equivalence between 4(x − y) and 4x − 4y, the algebraic notation is doing real cognitive work. Expanding brackets is no longer a procedure to be practised — it is a relationship students have already seen and verified visually.
The same approach extends naturally to more complex activities. Later problems in the lesson use rectangles, triangles, semicircles and octagons, each assigned a different algebraic letter and require careful reasoning to decompose.
2x + 2y or 2(x + y)
4m – 3n
4a + 2b – 2c
4q + 4p or 4(p + q)
The plenary flips the task. Rather than finding the area of a given shape, students are presented with an algebraic expression and asked to draw a diagram with that area. Three component shapes are defined at the start – a rectangle with area w cm², a triangle with area h cm², and a semicircle with area t cm² – and students are asked to produce diagrams for expressions including w + h + t, 2w − 2t + h and 3(w + t).
Students working with mini-whiteboards produce a variety of diagrams – and in doing so, they encounter a fundamental truth about algebra: the same expression can represent infinitely many different physical arrangements. This is the beginning of abstract algebraic thinking, arrived at through concrete visual reasoning.
Students place the letter before the coefficient. Address this by revisiting the count: “We have 4 of these y shapes. We write the number first, just as we write 4 cm rather than cm 4.”
Students write 1y when there is exactly one quarter circle. Ask: “If you see one square, do you write 1x or x?” The analogy with 1 × 7 = 7 usually resolves this.
The visual prevents this error because students can see that the two shapes are different. This is the strongest single argument for the area approach at the KS2–KS3 transition: the misconception is addressed before it becomes embedded.
When a region is removed, some students count all shapes present rather than subtracting. Colour-coding the removed piece differently and asking “what has been taken away?” redirects thinking without lengthy explanation.
The complete lesson — including all differentiated activities, the plenary task, the video tutorial and printable worksheets — is available to Mr Mathematics members. The resource is designed for a 60-minute Year 7 lesson and includes a differentiated worksheet a retrieval starter and full worked solutions.
This lesson is the opening part of a carefully sequenced Year 7 algebra pathway designed to move students from arithmetic reasoning to symbolic manipulation without relying on memorised rules.
Members can access this lesson and the full Writing with Algebraic Notation scheme of work at mr-mathematics.com.
A good KS2 to KS3 transition maths lesson builds on knowledge students already have from primary school while preparing them for the more abstract thinking required in secondary maths. It should connect familiar arithmetic, geometry or number ideas to new notation and methods rather than beginning with rules to memorise.
Algebra should be introduced in Year 7 by giving students a clear meaning for the symbols they are using. Visual models, area representations and simple contexts help students understand that letters represent quantities before they begin manipulating expressions formally.
Students often struggle with algebraic notation because the symbols are introduced before the underlying idea is secure. If students do not understand what a letter represents, collecting like terms becomes a surface-level sorting procedure rather than meaningful mathematical reasoning.
Visual models help students collect like terms because each term represents a distinct object or area. Students can see why three squares can be combined with another square, but a square cannot be combined with a quarter circle. This makes the rule for collecting like terms visible rather than arbitrary.
Students need to be able to count repeated objects, recognise simple shapes, understand area as a measure of surface and use repeated addition. These familiar KS2 ideas are extended into algebraic notation by assigning letters to different component shapes.
Yes. The visual entry point makes the lesson accessible for students who are not yet confident with formal algebra, while the later tasks involving subtraction, brackets and multiple variables provide challenge for higher-attaining students.
This is the first lesson in a series introducing algebra to Year 7. Future lessons in the same scheme extend the approach to writing expressions from words, simplifying more complex expressions, basic substitution and using function machines to solve equations.
The area model does not need to be abandoned as students progress. Whenever a student later struggles with expanding brackets or factorising, returning to the shape representation re-anchors the concept more effectively than repeating the algebraic procedure. The image of four copies of (x − y) spreading into 4x − 4y is one that, once seen, is very hard to forget.
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