Home → Curriculum Hub → Maths Lessons → Schemes of Work → Two-Year IGCSE Foundation → Algebraic Manipulation
Algebraic Manipulation builds on earlier work with number and the order of operations by introducing the language and methods of algebra. This unit develops the skill of simplifying and rewriting expressions.
Students learn to substitute using the order of operations, collect like terms and simplify products. They then expand and factorise linear expressions, before expanding two brackets, leaving solving equations and factorising quadratics for later units.
By the end of this unit students will be able to:
Students should be secure with the following before beginning this unit.
The multiplication sign is dropped, so ab means a × b. Division is written as a fraction. A coefficient of 1 is not written. 2a means 2 × a, not 2 + a.
Only terms with the same letter and the same power can be collected. 3x and 3x² are not like terms. ab and ba are like terms.
Every term inside the bracket is multiplied. 3(a + b) is 3a + 3b and −2(x + 5) is −2x − 10. Two brackets need four products, then the linear terms are collected.
Factorising is not finished until the highest common factor of the numbers and the letters has been taken out. 18x + 24y is 6(3x + 4y), not 2(9x + 12y).
| Misconception | Teaching focus |
|---|---|
| Treating 2a as 2 + a, or writing a + b as ab. | Substitute a = 3 and b = 4 into both forms so students can see they are equal or not. |
| Collecting 3x and 3x², or treating ab and ba as unlike terms. | Write the power next to each letter before any collecting starts. |
| Multiplying only the first term in a bracket, especially when the multiplier is negative. | Draw two arrows from the term outside onto every term inside, including the sign. |
| Stopping factorising at a common factor that is not the highest, so 4x + 8 becomes 2(2x + 4). | Check the bracket: if the numbers inside still share a factor, the work is not finished. |
| Expanding (x + 2)(x + 3) as x² + 6, missing the two middle products. | Use an area grid so all four products are visible before like terms are collected. |
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Students learn how to substitute values into algebraic expressions using the order of operations.
As learning progresses, they develop calculator and non-calculator techniques.
Students learn to simplify algebraic expressions by collecting like terms using basic notation.
As learning progresses, they evaluate the perimeter around shapes with algebraic lengths.
Students learn how to simplify algebraic products using the rules of indices.
As learning progresses, students begin to apply their learning to evaluate the area of a rectangle with algebraic lengths.
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.
Fluency questions provide extra practice writing algebraic expressions from written descriptions.
Mastery and Problem Solving questions require students to form expressions with multiple operations and apply to real-life situations.
The lesson builds from substitution to reasoning and application. In fluency, students practise substituting values and simplifying accurately.
IProblem solving applies formulae in real contexts, interpreting and rearranging simple expressions to model relationships.
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 simplifying expressions by collecting like terms.
The problem-solving activity challenges students to write the perimeter of a composite shape as a simplified expression.
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 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 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 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 simplify expressions by collecting like terms.
As learning progresses they simplify algebraic products and solve problems involving the perimeter of shapes.
Students revise how to factorise algebraic expressions by taking out the highest common factor.
As learning progresses, they factorise expressions involving powers and solve algebraic identities.
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.
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