Mathematics · Year 10

Active learning ideas

Factorising Quadratics (a≠1) and Difference of Two Squares

Active learning works because factorising quadratics demands systematic testing and pattern recognition, both of which are strengthened through movement, visual models and collaborative reasoning. These methods help students move beyond rote procedures to flexible problem-solving as they manipulate expressions, justify steps and correct errors in real time.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Algebra

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Stations Rotation
• 25 min · Pairs

Card Sort: Quadratic Matches

Prepare cards with unfactorised quadratics, factor pairs, and expanded forms. Pairs sort them into matching sets, then verify by multiplying factors. Extend by creating their own cards for classmates to solve.

Differentiate between various factorisation methods for different quadratic forms.

Facilitation Tip

During Card Sort: Quadratic Matches, circulate and listen for students verbalising the ‘a×c’ rule aloud before pairing cards, reinforcing the method’s purpose.

What to look for

Present students with three quadratic expressions: one simple trinomial (a=1), one trinomial where a≠1, and one difference of two squares. Ask them to write the factorisation for each and briefly state which method they used and why.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 02

• Stations Rotation
• 35 min · Small Groups

Algebra Tiles: Building Factors

Provide algebra tiles for expressions like 3x² + 10x + 8. Small groups arrange tiles into rectangles, identify factors from dimensions, and record steps. Compare models across groups.

Justify the steps involved in factorising a quadratic where a ≠ 1.

Facilitation Tip

During Algebra Tiles: Building Factors, ask each group to demonstrate one completed factorisation by placing tiles on a mini-whiteboard so peers can see the model.

What to look for

Provide students with the expression 4x² - 25. Ask them to: 1. Identify the type of factorisation needed. 2. Show the steps to factorise it. 3. Write one sentence explaining why this method is efficient here.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 03

• Stations Rotation
• 30 min · Whole Class

Relay Race: Factorisation Challenges

Divide class into teams. One student solves a quadratic at the board, tags next teammate. Include a≠1 and difference of squares. Whole class cheers and checks answers together.

Construct a quadratic expression that can be factorised using the difference of two squares.

Facilitation Tip

During Relay Race: Factorisation Challenges, stand at the finish line to observe how students record and justify their final factorisations, catching systematic errors early.

What to look for

Pose the question: 'When factorising 6x² + 11x + 4, what are the two numbers you are looking for, and why?'. Facilitate a discussion where students explain the 'ac' method and the role of the sum 'b'.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 04

• Stations Rotation
• 20 min · Small Groups

Squares Spotter: Pattern Hunt

List expressions around room; small groups identify difference of two squares, factorise, and justify. Create new ones from measurements like room length minus width squared.

Differentiate between various factorisation methods for different quadratic forms.

What to look for

Present students with three quadratic expressions: one simple trinomial (a=1), one trinomial where a≠1, and one difference of two squares. Ask them to write the factorisation for each and briefly state which method they used and why.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

A few notes on teaching this unit

Teach factorising quadratics by modelling the ‘a×c’ process on the board with think-alouds, then transition to structured pair work to reduce cognitive load. Focus on correcting sign errors immediately using tile models, as these persist across all methods. Use exit tickets to gauge whether students can distinguish between methods before moving on.

Students will move from guessing to methodical testing, explaining each step clearly and verifying results through expansion. They will recognise when to apply each method, articulate why it fits, and adjust their approach when errors arise during hands-on work.

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Watch Out for These Misconceptions

• During Card Sort: Quadratic Matches, watch for students treating all quadratics as if a=1 and ignoring multiplication by a.

  Direct students to the ‘a×c’ card in each set and ask them to list the product before sorting. Peers can verify by checking that their paired factors multiply back to the original expression.

• During Relay Race: Factorisation Challenges, watch for students applying difference of squares only to numerical constants, not to variables.

  Prompt teams to include at least one mixed expression like 9x² - 25y² in their rotation. Ask them to model it with tiles to see the variable symmetry before expanding.

• During Algebra Tiles: Building Factors, watch for sign errors in the final factors, especially when the middle term is negative.

  Have students lay out the tiles and then verbalise the sign pattern of their factors before writing them. A peer checks by flipping the tile arrangement to confirm the original expression is reproduced.

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Methods used in this brief

• Stations Rotation