The shape of the lesson
By the end of the lesson, learners can
- The highest common factor: In the lab choose Common factor and press Next step to see the highest common factor of the numbers, then of each letter, come out in front.
- Quadratics (Extended): To factorise \(x^{2} + bx + c\), find two numbers that multiply to \(c\) and add to \(b\).
- Special patterns (Extended): Learn to recognise these shapes on sight.
- HCF: biggest number, and each letter to its lowest power.
- \(x^{2} + bx + c\): two numbers that multiply to \(c\) and add to \(b\).
- \(ax^{2} + bx + c\): numbers that multiply to \(ac\) and add to \(b\); split and group.
- Difference of two squares: \(p^{2} - q^{2} = (p - q)(p + q)\).
- Check: expand your answer and look for anything still common.
One picture every idea returns to
Always take out the highest common factor first, then look inside the bracket for a pattern: a quadratic, a difference of two squares or a perfect square. Factorise fully means nothing left can be factorised again. Check by expanding.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Expanding, run backwards.. Factorising writes an expression as a product. | Whole class, then pairs |
| 5 to 13 min | Teach and check: The highest common factor | In the lab choose Common factor and press Next step to see the highest common factor of the numbers, then of each letter, come out in front. Project tab 3 of the lab (Algebra lab, factorising) and let a student drive it while the class predicts. Model the worked example Two terms on the board, one line at a time. Your turn: Factorise fully \(14a^{2}b^{3} + 21ab^{2}\). | Whole class, then pairs |
| 13 to 21 min | Teach and check: Quadratics (Extended) | To factorise \(x^{2} + bx + c\), find two numbers that multiply to \(c\) and add to \(b\). Model the worked example Leading coefficient 1 (Extended) on the board, one line at a time. Your turn: (Extended) Factorise \(3x^{2} - 10x + 8\). | Pairs |
| 21 to 30 min | Teach and check: Special patterns (Extended) | Learn to recognise these shapes on sight. Model the worked example Grouping (Extended) on the board, one line at a time. Your turn: (Extended) Factorise completely \(2x^{2} - 50\). | Rally Coach |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 18 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Special patterns (Extended), never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Rally Coach
Pairs share one worksheet. One partner solves the first question aloud while the other coaches, checking each line and offering a prompt rather than the answer; they swap roles for the next question.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| \(3(3x^{2} + 5xy)\) is factorised, but not fully: \(x\) is still common inside the bracket. 'Factorise fully' loses a mark for this. | Return to The highest common factor: ask a pair to state the correct version and show it on the lab before moving on. |
| Getting the right numbers with the wrong signs: \((x + 5)(x - 3) = x^{2} + 2x - 15\), not \(x^{2} - 2x - 15\). | Expand to check the middle term every time. |
| \(x^{2} + 25\) does NOT factorise. | The difference of two squares needs a minus sign between two perfect squares. |
Factorising expressions: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
Factorise fully (a) \(8y - 20\) (b) \(5p^{2} + 15pq\).
Factorise fully \(14a^{2}b^{3} + 21ab^{2}\).
(Extended) Factorise (a) \(x^{2} - 3x - 28\) (b) \(3x^{2} - 10x + 8\).
(Extended) Factorise completely (a) \(25x^{2} - 4y^{2}\) (b) \(3x^{3} - 12x\).
(Extended) Factorise \(xy + 3x - 2y - 6\).
(Extended)
(a) Without a calculator, use factorising to work out \(87^{2} - 13^{2}\).
(b) \(n\) is a whole number. Show that \((2n + 1)^{2} - (2n - 1)^{2}\) is always a multiple of 8.
Answer key and marking notes
- (a) \(4(2y - 5)\)
- (b) \(5p(p + 3q)\)
- HCF \(= 7ab^{2}\)
- \(7ab^{2}(2ab + 3)\)
- (a) Product \(-28\), sum \(-3\): \(-7\) and \(4\). \((x - 7)(x + 4)\)
- (b) \(ac = 24\), numbers \(-6\) and \(-4\): \(3x^{2} - 6x - 4x + 8\)
- \(= 3x(x - 2) - 4(x - 2) = (3x - 4)(x - 2)\)
- (a) \((5x)^{2} - (2y)^{2} = (5x - 2y)(5x + 2y)\)
- (b) \(3x(x^{2} - 4) = 3x(x - 2)(x + 2)\)
- \(x(y + 3) - 2(y + 3)\)
- \(= (y + 3)(x - 2)\)
- (a) \(87^{2} - 13^{2} = (87 - 13)(87 + 13) = 74 \times 100 = 7400\)
- (b) Difference of two squares: \([(2n + 1) - (2n - 1)][(2n + 1) + (2n - 1)]\)
- \(= 2 \times 4n = 8n\)
- \(8n\) is 8 times a whole number, so it is a multiple of 8.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Factorise fully \(15a^{2}b - 10ab^{2}\).
- A\(5ab(3a - 2b)\)
- B\(5(3a^{2}b - 2ab^{2})\)
- C\(ab(15a - 10b)\)
- D\(5a(3ab - 2b^{2})\)
Which of these is \(8x^{2} + 12x\) factorised fully?
- A\(4x(2x + 3)\)
- B\(2x(4x + 6)\)
- C\(4(2x^{2} + 3x)\)
- D\(x(8x + 12)\)
Factorise fully \(6x^{2}y - 9xy^{2} + 3xy\).
- A\(3xy(2x - 3y + 1)\)
- B\(3xy(2x - 3y)\)
- C\(3(2x^{2}y - 3xy^{2} + xy)\)
- D\(xy(6x - 9y + 3)\)
Factorise \(x^{2} - 3x - 28\).
- A\((x - 7)(x + 4)\)
- B\((x + 7)(x - 4)\)
- C\((x - 14)(x + 2)\)
- D\((x - 7)(x - 4)\)
Factorise \(3x^{2} + 5x - 2\).
- A\((3x - 1)(x + 2)\)
- B\((3x + 1)(x - 2)\)
- C\((3x - 2)(x + 1)\)
- D\((3x + 2)(x - 1)\)
Factorise completely \(3x^{3} - 12x\).
- A\(3x(x - 2)(x + 2)\)
- B\(3x(x^{2} - 4)\)
- C\(3x(x - 2)^{2}\)
- D\(3x(x - 4)(x + 4)\)
Quiz key
HCF of the numbers is 5, both terms contain \(a\) and both contain \(b\), so take out \(5ab\): \(15a^{2}b \div 5ab = 3a\) and \(10ab^{2} \div 5ab = 2b\), giving \(5ab(3a - 2b)\). The other answers are correct but not full: \(5a(3ab - 2b^{2})\) still has \(b\) common inside the bracket.
The HCF of \(8x^{2}\) and \(12x\) is \(4x\), so \(8x^{2} + 12x = 4x(2x + 3)\). In \(2x(4x + 6)\) a 2 is still common inside; in \(4(2x^{2} + 3x)\) an \(x\) is still common; in \(x(8x + 12)\) a 4 is still common.
The HCF is \(3xy\). Dividing each term: \(2x\), \(-3y\) and \(3xy \div 3xy = 1\). So \(3xy(2x - 3y + 1)\). \(3xy(2x - 3y)\) loses the last term: expanding it gives only two terms. The other two are not fully factorised.
\(-7 \times 4 = -28\) and \(-7 + 4 = -3\), so \(x^{2} - 3x - 28 = (x - 7)(x + 4)\). \((x + 7)(x - 4)\) has the right numbers but the wrong signs: it expands to \(x^{2} + 3x - 28\). \((x - 7)(x - 4)\) gives \(+28\).
\((3x - 1)(x + 2) = 3x^{2} + 6x - x - 2 = 3x^{2} + 5x - 2\). The others give the wrong middle term: \((3x + 1)(x - 2) = 3x^{2} - 5x - 2\), \((3x - 2)(x + 1) = 3x^{2} + x - 2\) and \((3x + 2)(x - 1) = 3x^{2} - x - 2\).
\(3x^{3} - 12x = 3x(x^{2} - 4)\), and \(x^{2} - 4\) is a difference of two squares: \((x - 2)(x + 2)\). So \(3x(x - 2)(x + 2)\). Stopping at \(3x(x^{2} - 4)\) is not complete. \((x - 2)^{2}\) is \(x^{2} - 4x + 4\), and \((x - 4)(x + 4)\) is \(x^{2} - 16\).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.