Mathematics · Year 10

Active learning ideas

Factorising Quadratics (a=1)

Factorising quadratics is an abstract skill that benefits greatly from concrete and collaborative experiences. Active learning allows students to build connections between numerical properties and algebraic representations, moving beyond rote memorization. These activities encourage students to grapple with the concepts, leading to deeper understanding and retention.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Algebra

• 20–30 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Think-Pair-Share
• 20 min · Pairs

Factor Pairs Puzzle

Provide students with a set of cards, some with quadratic expressions (x² + bx + c) and others with pairs of numbers. Students must match the expression to the pair of numbers that multiply to 'c' and add to 'b'. This can be done individually or in pairs.

Explain the relationship between expanding and factorising quadratic expressions.

Facilitation Tip

During the Think-Pair-Share, give students ample time for individual reflection on the 'Factor Pairs Puzzle' cards before they discuss their matches with a partner.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 02

• Think-Pair-Share
• 30 min · Small Groups

Algebra Tiles Exploration

Using algebra tiles, students can physically represent quadratic expressions. They can arrange the tiles to form a rectangle, then determine the dimensions (the factors) of that rectangle. This visual and tactile approach aids understanding.

Predict the factors of a quadratic expression based on its constant term and coefficient of x.

Facilitation Tip

In Collaborative Problem-Solving, assign roles like 'Materials Manager' or 'Checker' to ensure all students engage with the 'Algebra Tiles Exploration' and the process of building the quadratic.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 03

• Think-Pair-Share
• 25 min · Small Groups

Factorisation Race

Present a series of quadratic expressions on the board. Students work in teams to factorise them as quickly and accurately as possible. The first team to correctly factorise a set number of expressions wins. This encourages rapid recall and application.

Construct a quadratic expression that can be factorised into two linear factors.

Facilitation Tip

During the 'Factorisation Race', circulate to observe teams and provide immediate feedback on their strategy for finding factor pairs and constructing the binomials.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

A few notes on teaching this unit

This topic moves from concrete representations to abstract algebraic manipulation. Start with visual or hands-on methods like algebra tiles or area models to build intuition. Emphasize the inverse relationship between expanding and factorising, and consistently link the numbers p and q to both the product (c) and the sum (b) in ax² + bx + c.

Students will confidently identify the two numbers needed to factorise a quadratic expression into its binomial form. They will be able to explain the relationship between the constant term, the coefficient of the x term, and the two factors. Success looks like students accurately factorising a range of expressions and articulating their reasoning.

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Watch Out for These Misconceptions

• During the 'Factor Pairs Puzzle', watch for students who incorrectly match factor pairs, confusing which number needs to add to 'b' and which needs to multiply to 'c'.

  Redirect students to use the visual clues on the cards or to draw an area model to confirm that the two numbers multiply to the constant term and add to the coefficient of x.

• During the 'Algebra Tiles Exploration', students may struggle to correctly interpret the tiles representing negative constants or coefficients.

  Guide students to use specific 'negative' tiles or to draw representations of negative areas and guide them to see how these combine to form the correct product and sum.

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Methods used in this brief

• Think-Pair-Share