Home → Curriculum Hub → Maths Lessons → Schemes of Work → Two-Year IGCSE Higher → Algebraic Expressions
Algebraic Expressions is the first algebra unit in the IGCSE Mathematics Higher scheme of work. It introduces important skills that students will use throughout the course, including expanding and factorising expressions.
Students learn to take out common factors, expand two and three brackets and collect like terms. They then use these skills in reverse to factorise quadratic expressions, including quadratics where the coefficient of (x2) is greater than 1.
By the end of this unit students will be able to:
Students should be secure with the following before beginning this unit.
The factor outside the bracket is the largest number and the lowest power that divides every term. 6x3 − 9x2 is 3x2(2x − 3), not 3x(2x2 − 3x). Check by expanding.
For two brackets, four products appear, then the like terms are collected. A missing inner or outer product leaves the linear term wrong. For three binomials, expand two first, then multiply the quadratic by the third bracket.
For x2 + bx + c, find two numbers that multiply to c and add to b. Expand to check before the brackets are declared. A pair that multiplies to c but does not add to b is not a factorisation.
For ax2 + bx + c, the numbers in the brackets must multiply to ac and still give the linear term when expanded. Factorising by grouping, or by trying pairs that include the factors of a, both work. Leaving a outside and factorising x2 + bx + c as if a were 1 does not.
| Misconception | Teaching focus |
|---|---|
| Taking out a common factor that is not the highest power, so a power of x remains in every term inside. | Ask which power of x divides every term. That lowest power, with the numerical HCF, goes outside. |
| Expanding two brackets as the product of the first terms plus the product of the last terms, and missing the two cross products. | Write a two-by-two grid, or list first, outer, inner, last, then collect the middle pair. |
| When expanding three brackets, multiplying only the end terms of the third bracket by the quadratic. | Expand the first pair fully. Then multiply every term of that quadratic by each term of the third bracket. |
| Choosing a factor pair that multiplies to c but does not add to b. | Expand the proposed brackets before moving on. If the linear term is wrong, the pair is not a factorisation. |
| Factorising ax2 + bx + c by ignoring a, then tacking a on the front. | The constants in the brackets must multiply to ac. Expand to check the coefficient of x2 as well as the constant. |
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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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