The shape of the lesson
By the end of the lesson, learners can
- Collect, then one bracket: In the lab choose One bracket and set the number outside to a negative value, say \(-3\).
- Two brackets, four products: Each term in the first bracket multiplies each term in the second, giving four products (Fig. 1).
- Three brackets (Extended): Expand two of the brackets first, simplify, then multiply the result by the third bracket.
- Like terms: same letters, same powers. Add their numbers only.
- One bracket: the outside term multiplies every inside term, sign included.
- Two brackets: four products, then collect the middle two.
- Squares: write \((a + b)^{2}\) as \((a + b)(a + b)\).
- Three brackets (Extended): expand two, simplify, then multiply by the third.
One picture every idea returns to
To expand, multiply every term in one bracket by every term in the other, keeping each sign with the term after it. Then collect like terms: only terms with exactly the same letters and powers can be added.
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, Multiply everything inside.. Collect like terms, multiply out single brackets, then pairs of brackets, and (Extended) three brackets. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Collect, then one bracket | In the lab choose One bracket and set the number outside to a negative value, say \(-3\). Project tab 2 of the lab (Algebra lab, expanding) and let a student drive it while the class predicts. Model the worked example Simplify (Extended) on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Two brackets, four products | Each term in the first bracket multiplies each term in the second, giving four products (Fig. 1). Model the worked example Two brackets on the board, one line at a time. Your turn: Expand and simplify \((2x - 3)(3x + 2)\). | Pairs |
| 21 to 30 min | Teach and check: Three brackets (Extended) | Expand two of the brackets first, simplify, then multiply the result by the third bracket. Model the worked example Three brackets (Extended) on the board, one line at a time. Your turn: (Extended) Expand \((x + 1)^{3}\). | Round Table |
| 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, 17 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 Three brackets (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.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
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 |
|---|---|
| \(-2(a - 4)\) is \(-2a + 8\), not \(-2a - 8\). | The minus sign outside multiplies the \(-4\) too, and minus times minus is plus. |
| \((x + 3)^{2}\) is NOT \(x^{2} + 9\). | Write it as \((x + 3)(x + 3)\) and you get the two middle terms: \(x^{2} + 6x + 9\). |
| Simplify the first pair before multiplying by the third bracket. | Carrying four unsimplified terms forward doubles the chance of a sign slip. |
Expanding brackets and simplifying: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Simplify (a) \(7p + 3q - 2p - 8q\) (b) \(3x^{2} + 4x - x^{2} - 6x\).
Expand (a) \(5y(3y - 2)\) (b) \(-2a(4a + 3b)\).
Expand and simplify \(4(3x + 2) - 3(2x - 5)\).
Expand and simplify (a) \((x + 7)(x - 3)\) (b) \((4x - 1)^{2}\).
(Extended) Expand and simplify \((x + 1)(x - 3)(2x + 5)\).
A rectangular card measures \((2x + 3)\) cm by \((x + 4)\) cm. A square of side \(x\) cm is cut out of it.
(a) Show that the area of card left is \((x^{2} + 11x + 12)\) cm2.
(b) Find the area left when \(x = 5\).
Answer key and marking notes
- (a) \(5p - 5q\)
- (b) \(2x^{2} - 2x\)
- (a) \(15y^{2} - 10y\)
- (b) \(-8a^{2} - 6ab\)
- \(12x + 8 - 6x + 15\)
- \(= 6x + 23\)
- (a) \(x^{2} - 3x + 7x - 21 = x^{2} + 4x - 21\)
- (b) \((4x - 1)(4x - 1) = 16x^{2} - 4x - 4x + 1 = 16x^{2} - 8x + 1\)
- \((x + 1)(x - 3) = x^{2} - 2x - 3\)
- \((x^{2} - 2x - 3)(2x + 5) = 2x^{3} + 5x^{2} - 4x^{2} - 10x - 6x - 15\)
- \(= 2x^{3} + x^{2} - 16x - 15\)
- (a) Rectangle: \((2x + 3)(x + 4) = 2x^{2} + 8x + 3x + 12 = 2x^{2} + 11x + 12\)
- Subtract the square: \(2x^{2} + 11x + 12 - x^{2} = x^{2} + 11x + 12\)
- (b) \(25 + 55 + 12 = 92\) cm2
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.
Simplify \(4p^{2} - 3pq + 2 - p^{2} + 5pq - 7\).
- A\(3p^{2} + 2pq - 5\)
- B\(5p^{2} + 2pq - 5\)
- C\(3p^{2} - 8pq - 5\)
- D\(3p^{2} + 2pq + 9\)
Expand and simplify \(4(3x - 2) - 3(x - 5)\).
- A\(9x + 7\)
- B\(9x - 23\)
- C\(9x - 13\)
- D\(15x + 7\)
Expand and simplify \((x - 6)(x + 2)\).
- A\(x^{2} - 4x - 12\)
- B\(x^{2} + 4x - 12\)
- C\(x^{2} - 12\)
- D\(x^{2} - 8x - 12\)
Expand \((x - 4)^{2}\).
- A\(x^{2} - 8x + 16\)
- B\(x^{2} + 16\)
- C\(x^{2} - 16\)
- D\(x^{2} - 8x - 16\)
Expand and simplify \((2x + 3)(x - 1)(x + 4)\).
- A\(2x^{3} + 9x^{2} + x - 12\)
- B\(2x^{3} + 7x^{2} - 7x - 12\)
- C\(2x^{3} - 12\)
- D\(2x^{3} + 9x^{2} + x + 12\)
Expand and simplify \((2x - 3)^{2} - (x + 2)(x - 5)\).
- A\(3x^{2} - 9x + 19\)
- B\(3x^{2} - 15x - 1\)
- C\(3x^{2} + 3x + 19\)
- D\(3x^{2} - 9x - 1\)
Quiz key
\(p^{2}\): \(4 - 1 = 3\). \(pq\): \(-3 + 5 = 2\). Numbers: \(2 - 7 = -5\). Answer \(3p^{2} + 2pq - 5\). \(3p^{2} - 8pq - 5\) treats \(+5pq\) as \(-5pq\); \(5p^{2}\) ignores the minus in front of \(p^{2}\).
\(4(3x - 2) = 12x - 8\) and \(-3(x - 5) = -3x + 15\). Total: \(12x - 3x - 8 + 15 = 9x + 7\). \(9x - 23\) writes \(-3 \times -5\) as \(-15\); minus times minus is plus. \(9x - 13\) multiplies only the \(x\) by \(-3\).
\(x \times x = x^{2}\), \(x \times 2 = 2x\), \(-6 \times x = -6x\), \(-6 \times 2 = -12\). Collect: \(x^{2} + 2x - 6x - 12 = x^{2} - 4x - 12\). \(x^{2} - 12\) forgets the two middle terms; \(x^{2} + 4x - 12\) gets the sign of \(2x - 6x\) wrong.
\((x - 4)(x - 4) = x^{2} - 4x - 4x + 16 = x^{2} - 8x + 16\). \(x^{2} + 16\) squares each term separately and loses the two middle terms. \(x^{2} - 8x - 16\) forgets that \(-4 \times -4 = +16\).
\((2x + 3)(x - 1) = 2x^{2} - 2x + 3x - 3 = 2x^{2} + x - 3\). Then \((2x^{2} + x - 3)(x + 4) = 2x^{3} + 8x^{2} + x^{2} + 4x - 3x - 12 = 2x^{3} + 9x^{2} + x - 12\). \(2x^{3} + 7x^{2} - 7x - 12\) comes from a sign slip in the first pair (\(-x\) instead of \(+x\)). \(2x^{3} - 12\) multiplies only the first and last terms.
\((2x - 3)^{2} = 4x^{2} - 12x + 9\) and \((x + 2)(x - 5) = x^{2} - 3x - 10\). Subtract: \(4x^{2} - 12x + 9 - x^{2} + 3x + 10 = 3x^{2} - 9x + 19\). \(3x^{2} - 15x - 1\) applies the minus only to \(x^{2}\), not to \(-3x\) and \(-10\). \(3x^{2} + 3x + 19\) writes \((2x - 3)^{2}\) as \(4x^{2} + 9\).
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.