The shape of the lesson
By the end of the lesson, learners can
- Bigger, smaller, same shape: In the lab, place the centre of enlargement and set the scale factor to 2, then to \(\frac{1}{2}\).
- Describing an enlargement: To describe an enlargement fully you need the scale factor and the centre.
- Translation by a vector: In the lab, change the top and bottom numbers of the column vector and watch the image slide.
- Enlarge: step from the centre to each vertex, multiply by \(k\), add back to the centre.
- Scale factor: image length \(\div\) object length.
- Centre: where lines through corresponding vertices meet.
- Translate: add the top number to \(x\) and the bottom number to \(y\).
- Describe: enlargement with scale factor and centre; translation with a column vector.
One picture every idea returns to
Enlargement: every distance from the centre is multiplied by the scale factor \(k\); describe it with the scale factor and the centre. Translation: every point moves by the same column vector \(\begin{pmatrix}x\\y\end{pmatrix}\) (\(x\) across, \(y\) up); describe it with that vector.
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, Resize it, slide it.. An enlargement changes the size of a shape from a fixed centre; a translation slides it without turning. 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: Bigger, smaller, same shape | In the lab, place the centre of enlargement and set the scale factor to 2, then to \(\frac{1}{2}\). Project tab 3 of the lab (Transformations lab, enlargement) and let a student drive it while the class predicts. Model the worked example Scale factor 2 from a centre on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Describing an enlargement | To describe an enlargement fully you need the scale factor and the centre. Model the worked example Finding the scale factor and centre on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Translation by a vector | In the lab, change the top and bottom numbers of the column vector and watch the image slide. Project tab 4 of the lab (Transformations lab, translation) and let a student drive it while the class predicts. Model the worked example Translating a point on the board, one line at a time. Your turn: A translation maps \((-2, 5)\) onto \((3, 1)\). Write down the column vector. | 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, 15 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 Translation by a vector, 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 |
|---|---|
| Multiplying the coordinates when the centre is NOT the origin. | Enlarging \((2, 2)\) by scale factor 2 from \((1, 1)\) gives \((3, 3)\), not \((4, 4)\). Always measure from the centre. |
| Writing the scale factor as 3 (or \(-3\)) when the shape gets smaller. | Divide IMAGE length by OBJECT length: going from 3 to 1 gives \(\dfrac{1}{3}\). A negative scale factor means the image is on the other side of the centre, not that it is smaller. |
| Subtracting the wrong way round. | The vector from object to image is image minus object: from \((1, 2)\) to \((5, -1)\) is \(\begin{pmatrix}4\\-3\end{pmatrix}\), and \(\begin{pmatrix}-4\\3\end{pmatrix}\) would slide the shape back again. |
Enlargement and translation: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
A translation maps the point \((2, 5)\) onto \((7, 3)\). Write down the column vector of the translation.
Triangle \(T\) has vertices \((1, 1)\), \((2, 1)\), \((1, 3)\). Find the vertices of its image after an enlargement, scale factor 3, centre \((0, 0)\).
Triangle \(U\) has vertices \((6, 4)\), \((10, 4)\), \((6, 8)\). Find the vertices of its image after an enlargement, scale factor \(\dfrac{1}{2}\), centre \((2, 0)\).
Triangle \(A\) has vertices \((1, 2)\), \((3, 2)\), \((1, 3)\). Triangle \(B\) has vertices \((3, 4)\), \((9, 4)\), \((3, 7)\).
Describe fully the single transformation that maps \(A\) onto \(B\).
(Extended) Triangle \(V\) has vertices \((2, 1)\), \((4, 1)\), \((4, 2)\). Find the vertices of its image after an enlargement, scale factor \(-\dfrac{1}{2}\), centre \((0, 0)\).
A robot moves on a warehouse floor marked with a coordinate grid. It starts at \((4, 1)\). It is translated by \(\begin{pmatrix}3\\-2\end{pmatrix}\) and then by \(\begin{pmatrix}-5\\4\end{pmatrix}\).
(a) Find the robot's final position.
(b) Write down the single column vector that would have moved it there in one step.
(c) Write down the column vector that takes it from its final position back to the start.
Answer key and marking notes
\(7 - 2 = 5\) across, \(3 - 5 = -2\) up: \(\begin{pmatrix}5\\-2\end{pmatrix}\).
- Multiply each coordinate by 3.
- \((3, 3)\), \((6, 3)\), \((3, 9)\).
- Steps from \((2, 0)\): \(\begin{pmatrix}4\\4\end{pmatrix}\), \(\begin{pmatrix}8\\4\end{pmatrix}\), \(\begin{pmatrix}4\\8\end{pmatrix}\).
- Halve them: \(\begin{pmatrix}2\\2\end{pmatrix}\), \(\begin{pmatrix}4\\2\end{pmatrix}\), \(\begin{pmatrix}2\\4\end{pmatrix}\).
- Add to \((2, 0)\): \((4, 2)\), \((6, 2)\), \((4, 4)\).
- Horizontal sides: \(3 - 1 = 2\) and \(9 - 3 = 6\), so the scale factor is 3.
- Line through \((1, 2)\) and \((3, 4)\): \(y = x + 1\), which passes through \((0, 1)\). Line through \((3, 2)\) and \((9, 4)\): gradient \(\dfrac{1}{3}\), and at \(x = 0\) it gives \(y = 2 - 1 = 1\). They meet at \((0, 1)\).
- Enlargement, scale factor 3, centre \((0, 1)\).
- Multiply each coordinate by \(-\dfrac{1}{2}\).
- \((-1, -0.5)\), \((-2, -0.5)\), \((-2, -1)\): half the size, on the other side of the origin, upside down.
- (a) \((4 + 3, 1 - 2) = (7, -1)\), then \((7 - 5, -1 + 4) = (2, 3)\).
- (b) \(3 + (-5) = -2\) and \(-2 + 4 = 2\): \(\begin{pmatrix}-2\\2\end{pmatrix}\).
- (c) The reverse move: \(\begin{pmatrix}2\\-2\end{pmatrix}\).
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.
The point \((6, 4)\) is enlarged by scale factor 3, centre \((2, 1)\). Where is its image?
- A\((14, 10)\)
- B\((18, 12)\)
- C\((12, 9)\)
- D\((16, 11)\)
A translation maps the point \((-2, 5)\) onto the point \((3, 1)\). Find the column vector of the translation.
- A\(\begin{pmatrix}5\\-4\end{pmatrix}\)
- B\(\begin{pmatrix}-5\\4\end{pmatrix}\)
- C\(\begin{pmatrix}1\\6\end{pmatrix}\)
- D\(\begin{pmatrix}-4\\5\end{pmatrix}\)
Describe fully the single transformation that maps triangle \(P\) onto triangle \(Q\).
- AEnlargement, scale factor 2, centre \((0, 1)\)
- BEnlargement, scale factor 2, centre \((0, 0)\)
- CEnlargement, scale factor \(\dfrac{1}{2}\), centre \((0, 1)\)
- DEnlargement, scale factor 2, centre \((1, 2)\)
The point \((8, 6)\) is enlarged by scale factor \(\dfrac{1}{2}\), centre \((2, 0)\). Where is its image?
- A\((5, 3)\)
- B\((4, 3)\)
- C\((3, 3)\)
- D\((14, 12)\)
The point \((4, 1)\) is enlarged by scale factor \(-2\), centre \((2, 3)\). Where is its image?
- A\((-2, 7)\)
- B\((6, -1)\)
- C\((-8, -2)\)
- D\((0, 5)\)
The point \((1, 4)\) is translated by the vector \(\begin{pmatrix}3\\-6\end{pmatrix}\), and then its image is reflected in the \(y\)-axis. Find the final point.
- A\((-4, -2)\)
- B\((2, -2)\)
- C\((4, 2)\)
- D\((4, -2)\)
Quiz key
From \((2, 1)\) to \((6, 4)\) is 4 across and 3 up. Multiply by 3: 12 across and 9 up. From the centre: \((2 + 12, 1 + 9) = (14, 10)\). The tempting \((18, 12)\) multiplies the coordinates by 3, which only works when the centre is the origin. \((12, 9)\) forgets to add the step on to the centre, and \((16, 11)\) is \((18, 12)\) minus the centre.
Across: \(3 - (-2) = 5\). Up: \(1 - 5 = -4\). The vector is \(\begin{pmatrix}5\\-4\end{pmatrix}\). The tempting \(\begin{pmatrix}-5\\4\end{pmatrix}\) subtracts the wrong way round (object minus image), which would slide the point back again. \(\begin{pmatrix}1\\6\end{pmatrix}\) adds the coordinates, and \(\begin{pmatrix}-4\\5\end{pmatrix}\) puts the components in the wrong order with the signs also wrong.
The bottom side of \(P\) is 2 units and the bottom side of \(Q\) is 4 units, so the scale factor is \(4 \div 2 = 2\). Lines through matching vertices, \((1, 2)\) to \((2, 3)\) and \((3, 2)\) to \((6, 3)\), both meet at \((0, 1)\). Check: from \((0, 1)\), \((3, 2)\) is 3 across, 1 up; doubled is 6 across, 2 up, which is \((6, 3)\). The tempting centre \((0, 0)\) would send \((3, 2)\) to \((6, 4)\), one unit too high. Scale factor \(\dfrac{1}{2}\) maps \(Q\) onto \(P\), the wrong way, and centre \((1, 2)\) would leave the vertex \((1, 2)\) where it is.
From \((2, 0)\) to \((8, 6)\) is 6 across and 6 up. Half of that is 3 across and 3 up, so the image is \((2 + 3, 0 + 3) = (5, 3)\). The tempting \((4, 3)\) halves the coordinates, which treats the origin as the centre. \((3, 3)\) is only the halved step, not added to the centre, and \((14, 12)\) uses scale factor 2 instead of \(\dfrac{1}{2}\).
From \((2, 3)\) to \((4, 1)\) is \(\begin{pmatrix}2\\-2\end{pmatrix}\). Multiply by \(-2\): \(\begin{pmatrix}-4\\4\end{pmatrix}\). Image: \((2 - 4, 3 + 4) = (-2, 7)\). The tempting \((6, -1)\) uses scale factor \(+2\), so the image stays on the same side of the centre. \((-8, -2)\) multiplies the coordinates by \(-2\) as if the centre were the origin, and \((0, 5)\) uses scale factor \(-1\).
Translate: \((1 + 3, 4 - 6) = (4, -2)\). Reflect in the \(y\)-axis: \((-4, -2)\). The tempting \((2, -2)\) does the reflection first, \((-1, 4)\), and then the translation; the order matters. \((4, 2)\) reflects in the \(x\)-axis instead, and \((4, -2)\) stops after the translation.
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.