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.

• Year 10
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Analyze the relationship between the factors of a quadratic expression (ax² + bx + c) and the coefficients a, b, and c.
2. Evaluate the efficiency of using the difference of two squares formula compared to general factorisation methods for specific quadratic forms.
3. Create quadratic expressions that are factorisable by the difference of two squares, specifying the values of the terms.
4. Demonstrate the process of factorising quadratics where a ≠ 1, justifying each step through expansion verification.

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.

Prepare & details

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.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

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.

Prepare & details

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.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

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.

Prepare & details

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.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

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.

Prepare & details

Differentiate between various factorisation methods for different quadratic forms.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Quick Check

After Card Sort: Quadratic Matches, ask students to write a one-sentence explanation for each pair they matched, naming the method used and the key step they applied.

Exit Ticket

After Algebra Tiles: Building Factors, provide the expression 9x² - 16 and ask students to: 1. identify the method, 2. write the factorisation, and 3. explain why this method is efficient here.

Discussion Prompt

During Relay Race: Factorisation Challenges, pause the race to ask, ‘What two numbers did you look for when factorising 6x² + 11x + 4, and why?’ Facilitate a brief group discussion to surface the ‘a×c’ logic before restarting.

Extensions & Scaffolding

• Challenge students to write a quadratic expression of their own that factors using both methods, then create a short ‘how-to’ guide for a peer.
• Scaffolding: Provide partially filled factorisation templates for the relay race, with blanks for the correct pair of numbers and signs.
• Deeper exploration: Ask students to generalise the difference of squares to higher powers, such as x⁴ - y⁴, and factor it using the same structure.

Key Vocabulary

Difference of Two Squares	A binomial of the form a² - b², which factorises to (a - b)(a + b). It applies when a quadratic expression has two perfect square terms separated by a minus sign.
Quadratic Trinomial	A polynomial expression of the form ax² + bx + c, where a, b, and c are constants and a ≠ 0. This topic focuses on cases where a ≠ 1.
Factor Pairs	Two numbers that multiply together to give a specific product. For factorising ax² + bx + c, students find factor pairs of 'ac' that sum to 'b'.
Expansion	The process of multiplying out the terms of a factored expression, typically using the distributive property or FOIL method, to return to the original polynomial form. Used for verification.