The shape of the lesson
By the end of the lesson, learners can
- Mirror, mirror: In the lab, drag the mirror line to a vertical position such as \(x = 1\), then to a horizontal one such as \(y = -1\).
- Turning about a centre: In the lab, put the centre of rotation at the origin and turn the shape through \(90^\circ\), \(180^\circ\) and \(270^\circ\).
- Describe it fully: In the lab, read the full description of each transformation.
- Reflection: same perpendicular distance from the mirror, on the other side.
- Mirror lines: \(x = k\) vertical, \(y = k\) horizontal, (Extended) \(y = x\) and \(y = -x\).
- Rotation: use tracing paper, pinned at the centre.
- Describe a reflection: 'Reflection in the line ...' with its equation.
- Describe a rotation: angle, direction (unless \(180^\circ\)) and centre.
- One transformation only when the question says 'single'.
One picture every idea returns to
To describe a reflection fully, give the equation of the mirror line. To describe a rotation fully, give the angle, the direction (clockwise or anticlockwise) and the centre. Always describe ONE single transformation.
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, Flip it, turn it.. A reflection flips a shape over a mirror line; a rotation turns it about a fixed centre. 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: Mirror, mirror | In the lab, drag the mirror line to a vertical position such as \(x = 1\), then to a horizontal one such as \(y = -1\). Project tab 1 of the lab (Transformations lab, reflection) and let a student drive it while the class predicts. Model the worked example Reflecting in a vertical line on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Turning about a centre | In the lab, put the centre of rotation at the origin and turn the shape through \(90^\circ\), \(180^\circ\) and \(270^\circ\). Project tab 2 of the lab (Transformations lab, rotation) and let a student drive it while the class predicts. Model the worked example Rotation about the origin on the board, one line at a time. Your turn: Rotate the point \((3, 1)\) through \(90^\circ\) anticlockwise about the origin. | Pairs |
| 21 to 30 min | Teach and check: Describe it fully | In the lab, read the full description of each transformation. Project tab 5 of the lab (Transformations lab, describe it fully) and let a student drive it while the class predicts. Model the worked example A half-turn on the board, one line at a time. | Corners |
| 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 Describe it fully, 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.
Corners
Put one answer or viewpoint in each corner of the room. Students move to the corner they agree with, justify the choice to a partner there, then hear one voice from each corner before the reveal.
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 |
|---|---|
| Mixing up \(x = k\) and \(y = k\). | The line \(x = 4\) is VERTICAL, so reflecting in it changes the \(x\)-coordinates, not the \(y\)-coordinates. Sketch the mirror line before you reflect anything. |
| Turning the wrong way. | Clockwise is the way clock hands move. A \(90^\circ\) clockwise turn is the same as a \(270^\circ\) anticlockwise turn, not a \(90^\circ\) anticlockwise one. |
| Naming two transformations, such as 'a reflection and then a translation', or leaving out the centre or direction. | The question asks for a SINGLE transformation, and a rotation without its centre scores very little. |
Reflection and rotation: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Write down the image of
(a) \((2, -3)\) after reflection in the \(y\)-axis
(b) \((4, 1)\) after reflection in the line \(y = 2\).
Triangle \(T\) has vertices \((1, 1)\), \((4, 1)\) and \((4, 2)\).
Find the vertices of the image of \(T\) after a rotation of \(90^\circ\) anticlockwise about the origin.
Triangle \(T\) has vertices \((1, 1)\), \((4, 1)\) and \((4, 2)\).
Find the vertices of the image of \(T\) after reflection in the line \(x = -1\).
Shape \(P\) has vertices \((2, 1)\), \((5, 1)\), \((5, 3)\).
(a) Shape \(Q\) has vertices \((2, -1)\), \((5, -1)\), \((5, -3)\). Describe fully the single transformation that maps \(P\) onto \(Q\).
(b) Shape \(R\) has vertices \((-2, -1)\), \((-5, -1)\), \((-5, -3)\). Describe fully the single transformation that maps \(P\) onto \(R\).
(Extended) Triangle \(A\) has vertices \((1, 1)\), \((3, 1)\), \((1, 2)\). Triangle \(B\) has vertices \((-1, -1)\), \((-1, -3)\), \((-2, -1)\).
Describe fully the single transformation that maps \(A\) onto \(B\).
(Extended) A designer draws a flag shape \(F\) with vertices \((2, 1)\), \((4, 1)\), \((4, 2)\). She rotates it \(90^\circ\) clockwise about the point \((1, -1)\) to make shape \(G\).
(a) Find the vertices of \(G\).
(b) Describe fully the single transformation that maps \(G\) back onto \(F\).
Answer key and marking notes
- (a) \((-2, -3)\)
- (b) \((4, 1)\) is 1 unit below \(y = 2\), so its image is 1 unit above: \((4, 3)\).
- \((x, y) \to (-y, x)\).
- \((1, 1) \to (-1, 1)\), \((4, 1) \to (-1, 4)\), \((4, 2) \to (-2, 4)\).
- \((1, 1)\) is 2 right of \(x = -1\), so it goes to \((-3, 1)\).
- \((4, 1)\) and \((4, 2)\) are 5 right, so they go to \((-6, 1)\) and \((-6, 2)\).
- (a) \(x\) unchanged, \(y\) changes sign: reflection in the \(x\)-axis (\(y = 0\)).
- (b) Both coordinates change sign: rotation, \(180^\circ\), about the origin \((0, 0)\).
- Each point \((a, b)\) maps to \((-b, -a)\): for example \((3, 1) \to (-1, -3)\).
- The midpoint of \((3, 1)\) and \((-1, -3)\) is \((1, -1)\), which lies on \(y = -x\); \((1, 1)\) and \((-1, -1)\) give \((0, 0)\), also on it.
- Reflection in the line \(y = -x\).
- (a) From \((1, -1)\): \((2, 1)\) is 1 right, 2 up. Clockwise quarter-turn: 2 right, 1 down, giving \((3, -2)\).
- \((4, 1)\) is 3 right, 2 up \(\to\) 2 right, 3 down: \((3, -4)\). \((4, 2)\) is 3 right, 3 up \(\to\) 3 right, 3 down: \((4, -4)\).
- (b) Undo the turn: rotation, \(90^\circ\) anticlockwise, about \((1, -1)\).
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.
Describe fully the single transformation that maps triangle \(A\) onto triangle \(B\).
- AReflection in the line \(x = 4\)
- BReflection in the line \(y = 4\)
- CReflection in the \(y\)-axis
- DTranslation by \(\begin{pmatrix}-6\\0\end{pmatrix}\)
The point \((3, -2)\) is reflected in the line \(y = 1\). What are the coordinates of its image?
- A\((3, 4)\)
- B\((-1, -2)\)
- C\((3, 2)\)
- D\((-3, -2)\)
The point \((2, 5)\) is rotated \(90^\circ\) anticlockwise about the origin. What are the coordinates of its image?
- A\((-5, 2)\)
- B\((5, -2)\)
- C\((-2, -5)\)
- D\((5, 2)\)
To describe a rotation fully, which facts must you give?
- AThe angle, the direction and the centre
- BThe angle and the direction only
- CThe centre and the mirror line
- DThe angle and a column vector
The point \((-3, 5)\) is reflected in the line \(y = x\). What are the coordinates of its image?
- A\((5, -3)\)
- B\((-5, 3)\)
- C\((3, 5)\)
- D\((3, -5)\)
Describe fully the single transformation that maps triangle \(P\) onto triangle \(Q\).
- ARotation, \(90^\circ\) anticlockwise, centre \((1, 2)\)
- BRotation, \(90^\circ\) clockwise, centre \((1, 2)\)
- CRotation, \(90^\circ\) anticlockwise, centre \((0, 0)\)
- DReflection in the line \(y = x\)
Quiz key
\(B\) is a mirror image of \(A\) (the sloping side faces the other way), so it is a reflection. Matching vertices \((5, 1)\) and \((3, 1)\) are equally far from \(x = 4\), as are \((7, 1)\) and \((1, 1)\), so the mirror line is the VERTICAL line \(x = 4\). The tempting \(y = 4\) is a horizontal line, which would flip the triangle upside down. The \(y\)-axis is \(x = 0\), which would put \(B\) at negative \(x\), and a translation cannot turn the triangle into its mirror image.
The point is \(1 - (-2) = 3\) units below the line \(y = 1\), so the image is 3 units above it: \(y = 1 + 3 = 4\), and \(x\) stays 3. Image \((3, 4)\). The tempting \((-1, -2)\) reflects in the vertical line \(x = 1\) instead. \((3, 2)\) reflects in the \(x\)-axis (\(y = 0\)), and \((-3, -2)\) reflects in the \(y\)-axis.
A \(90^\circ\) anticlockwise turn about the origin sends \((x, y)\) to \((-y, x)\), so \((2, 5)\) goes to \((-5, 2)\), in the second quadrant. The tempting \((5, -2)\) is the \(90^\circ\) CLOCKWISE image. \((-2, -5)\) is a half-turn (\(180^\circ\)), and \((5, 2)\) just swaps the coordinates, which is a reflection in \(y = x\), not a rotation.
A rotation is fixed by its angle (such as \(90^\circ\)), its direction (clockwise or anticlockwise) and its centre. The tempting "angle and direction only" misses the centre, and a rotation without its centre scores very little: turning about a different point puts the image somewhere else. A mirror line describes a reflection, and a column vector describes a translation.
In \(y = x\), \((x, y)\) maps to \((y, x)\), so \((-3, 5)\) maps to \((5, -3)\). The tempting \((-5, 3)\) is the reflection in \(y = -x\), which swaps AND changes both signs. \((3, 5)\) is the reflection in the \(y\)-axis, and \((3, -5)\) is a half-turn about the origin.
\(P\) has vertices \((2, 1)\), \((4, 1)\), \((4, 2)\) and \(Q\) has \((2, 3)\), \((2, 5)\), \((1, 5)\). The side along \(y = 1\) pointing right now points up, so the turn is \(90^\circ\) anticlockwise. About \((1, 2)\): \((2, 1)\) is 1 right and 1 down of the centre; a quarter turn anticlockwise makes it 1 right and 1 up, which is \((2, 3)\). Check \((4, 1)\): 3 right, 1 down becomes 1 right, 3 up, which is \((2, 5)\). The tempting clockwise turn about \((1, 2)\) would send \((2, 1)\) to \((0, 1)\). About the origin, \((2, 1)\) would go to \((-1, 2)\), and a reflection in \(y = x\) would send it to \((1, 2)\).
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.