Home → Curriculum Hub → Maths Lessons → Schemes of Work → Three-Year GCSE Higher → Algebraic Expressions
This Higher GCSE scheme of work unit covers algebraic expressions in Year 9. It starts at a single bracket, then factorising with powers, then the product of two and three brackets, and finishes by factorising x² + bx + c and ax² + bx + c.
Students have to multiply every term in a bracket, including a minus, and they have to factorise completely. Those two habits are what later quadratic equations, algebraic fractions and the difference of two squares all use.
A student who has secured this unit:
Students should be secure with the following before beginning this unit.
2(x + 5) is 2x + 10. The second term is multiplied as well as the first. A minus outside the bracket multiplies both terms.
18x + 24y is 6(3x + 4y), not 2(9x + 12y). The HCF of the coefficients and of the letters has to come out.
A quadratic product is four terms before they collect. A cubic product is a quadratic times a third bracket, not three letters written side by side.
x² + bx + c needs two numbers that multiply to c and add to b. When the x² coefficient is not 1, those two numbers multiply to ac.
| Misconception | Teaching focus |
|---|---|
| Multiplying only the first term in a bracket, so 3(x + 4) becomes 3x + 4. | Point to the second term and ask for 3 × 4 before the line is accepted. |
| Stopping at a partial factorisation, such as 18x + 24y = 2(9x + 12y). | Ask whether anything still divides both terms inside the bracket. |
| Losing the middle x term when two brackets are expanded, so (x + 3) (x + 1) becomes x² + 3. | Fill all four cells of the grid and add them, including both x terms. |
| Choosing two numbers that add to c and multiply to b, the wrong way round. | Write ‘multiply to c, add to b’ above the first five x² + bx + c questions. |
| Ignoring the a in ax² + bx + c and factorising as if a were 1. | Form ac first, find the pair that adds to b, then split the middle term. |
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Students learn how to expand algebraic expressions from a set of brackets.
As learning progresses, they find the sum and difference of two or more brackets with positive and negative terms.
Students use the highest common factor to factorise algebraic expressions.
As learning progresses, they are challenged to investigate possible perimeters for a rectangle with algebraic lengths.
Factorising algebraic expressions with powers
Students learn how to factorise expressions with powers by taking out the highest common factor.
Later, as learning progresses, the most common factor includes numerical and algebraic terms.
Students learn how to expand the product of two brackets and collect the like terms.
As learning progresses, they apply their learning to calculate the area of rectangles and triangles with algebraic lengths.
Students learn how to write the product of three binomials by expanding a cubic expression.
Learning progresses onto solving cubic identities and using an expansion to calculate a cube number.
Students learn how to factorise quadratics in the form x2 + bx + c into two brackets.
As the learning progresses, students begin to use the difference of two squares to factorise quadratics with a zero x term.
Students learn how to factorise quadratics in the form ax2 + bx + c.
As learning progresses they consider non-prime coefficient of x2.
Students gain additional practice expanding a pair of brackets and simplifying the result.
The problem-solving activity challenges students to write the expression for the area of a composite shape.
Students gain additional practice factorising expressions into a single bracket.
The problem-solving activity challenges students to collect like terms around the perimeter of a hexagon to solve an algebraic identity.
Students gain additional practice expanding the product of two brackets.
The problem-solving activity challenges students to find the average of three quadratic expressions.
Students gain additional practice factorising quadratic expressions in the form ax2 + bx + c.
The problem-solving activity challenges students to find the perimeter of a rectangle when the area is given in the form ax2 + bx + c.
Students revise how to expand a single pair of brackets by a number and term.
Learning progresses to simplifying the sum and difference of multiple brackets while linking to shape and space.
Students revise how to expand and simplify the product of two brackets.
As learning progresses, they consider quadratic identities and link the notation to the surface area of a cuboid.
Students revise how to factorise a quadratic in the form x2 + bx + c.
As learning progresses, they link quadratic expressions to the area and perimeter of rectilinear shapes.
Students revise how to expand a cubic expression from three binomials.
As learning progresses, they link cubic expressions to algebraic identities and the volume of prisms.
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