The shape of the lesson
By the end of the lesson, learners can
- Line symmetry: In the lab, place a mirror line on each shape and reflect it.
- Rotational symmetry: Imagine tracing the shape and spinning the tracing about its centre.
- Symmetry of solids (Extended): In the solids tab, slice each solid with a plane and see whether the two halves are mirror images.
- Lines: test every possible mirror: through vertices, through midpoints of sides.
- Order: count the fits in one full turn, including the start; minimum 1.
- Angle of turn: \(360^\circ \div \text{order}\).
- Regular polygons: \(n\) sides, \(n\) lines, order \(n\).
- Solids (Extended): planes along the length from the cross-section's lines, plus any plane across the middle.
One picture every idea returns to
Count lines of symmetry by asking where a mirror could go; count the order of rotational symmetry by asking how many times the shape fits onto itself in one full turn. Every shape has order at least 1, and a regular \(n\)-sided polygon has \(n\) lines and order \(n\).
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, Fold it, turn it.. A shape has line symmetry if it folds onto itself, and rotational symmetry if it fits onto itself more than once in a full turn. 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: Line symmetry | In the lab, place a mirror line on each shape and reflect it. Project tab 5 of the lab (Constructions lab, symmetry) and let a student drive it while the class predicts. Model the worked example Symmetry tells you about the shape on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Rotational symmetry | Imagine tracing the shape and spinning the tracing about its centre. Model the worked example Letters on the board, one line at a time. Your turn: Write down the number of lines of symmetry and the order of rotational symmetry of a regular octagon. What is the smallest angle of turn that fits? | Pairs |
| 21 to 30 min | Teach and check: Symmetry of solids (Extended) | In the solids tab, slice each solid with a plane and see whether the two halves are mirror images. Project tab 6 of the lab (Constructions lab, symmetry of solids) and let a student drive it while the class predicts. Model the worked example Planes of a prism (Extended) on the board, one line at a time. Your turn: (Extended) How many planes of symmetry does a right prism with an equilateral triangle cross-section have? | Quiz-Quiz-Trade |
| 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 Symmetry of solids (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.
Quiz-Quiz-Trade
Each student holds a card with a question on the front and the answer on the back. They pair up, quiz each other, coach if needed, trade cards and find a new partner. Three rounds covers the whole set.
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 |
|---|---|
| A rectangle's diagonals are NOT lines of symmetry: folding along a diagonal does not match the corners. | A rhombus is the shape whose diagonals are lines of symmetry. |
| Order 0 does not exist. | A shape with no rotational symmetry has order 1, because it always fits once, at the full turn. |
| For prisms, remember the extra plane that cuts across the middle of the length. | For a cube, the six diagonal planes (through opposite edges) are easy to miss: \(3 + 6 = 9\). |
Line and rotational symmetry: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
Write down the number of lines of symmetry and the order of rotational symmetry of (a) a kite (b) a rhombus (c) an isosceles trapezium.
For each capital letter, write down the number of lines of symmetry and the order of rotational symmetry: (a) N (b) X (c) E.
A regular polygon has rotational symmetry of order 12. Find (a) the smallest angle of turn that maps it onto itself (b) its number of lines of symmetry.
A regular polygon maps onto itself after a rotation of \(24^\circ\), and after no smaller rotation. How many sides does it have?
(Extended) Write down the number of planes of symmetry of (a) a cube (b) a right pyramid with a square base (c) a cylinder. (d) State the order of rotational symmetry of the square-based pyramid about its vertical axis.
Triangle PQR has exactly one line of symmetry, which passes through P and meets QR at M. QR = 8 cm and angle QPR = \(52^\circ\).
(a) What type of triangle is PQR? (b) Find angle PQR. (c) Find the length QM. (d) Find angle PMQ, giving a reason.
Answer key and marking notes
- (a) 1 line, order 1
- (b) 2 lines, order 2
- (c) 1 line, order 1
- (a) N: 0 lines, order 2
- (b) X: 2 lines, order 2
- (c) E: 1 line (horizontal), order 1
- (a) \(360^\circ \div 12 = 30^\circ\)
- (b) 12
- Order \(= 360^\circ \div 24^\circ = 15\).
- A regular polygon's order equals its number of sides: 15 sides.
- (a) 9
- (b) 4
- (c) infinitely many
- (d) order 4
- (a) Isosceles, with PQ = PR.
- (b) \((180^\circ - 52^\circ) \div 2 = 64^\circ\).
- (c) The mirror line cuts QR in half: QM \(= 8 \div 2 = 4\) cm.
- (d) The reflection maps angle PMQ onto angle PMR, so they are equal, and together they make a straight line: each is \(90^\circ\).
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.
Which capital letter has exactly 2 lines of symmetry and rotational symmetry of order 2?
- AH
- BN
- CA
- DT
A regular polygon maps onto itself after a rotation of \(40^\circ\), and after no smaller rotation. How many sides does it have?
- A9
- B4.5
- C8
- D18
The diagram shows a pattern of shaded squares on a 4 by 4 grid. Describe the symmetry of the whole pattern.
- A0 lines of symmetry, rotational symmetry of order 2
- B2 lines of symmetry, rotational symmetry of order 2
- C0 lines of symmetry, rotational symmetry of order 0
- D1 line of symmetry, rotational symmetry of order 2
A quadrilateral has exactly 2 lines of symmetry, rotational symmetry of order 2, and its diagonals ARE its lines of symmetry. What is it?
- ARhombus
- BRectangle
- CSquare
- DParallelogram
How many planes of symmetry does a cuboid measuring 4 cm by 4 cm by 10 cm have?
- A5
- B3
- C9
- D4
A right prism has a regular hexagon as its cross-section. How many planes of symmetry does it have?
- A7
- B6
- C12
- D8
Quiz key
H has a vertical and a horizontal line of symmetry and looks the same after a half turn, so it has 2 lines and order 2. The tempting N has order 2 but no lines of symmetry: a mirror turns it back to front. A and T each have one vertical line and order 1.
The order of rotational symmetry is \(360^\circ \div 40^\circ = 9\), and a regular polygon has order equal to its number of sides, so it has 9 sides. The tempting 4.5 divides \(180^\circ\) by \(40^\circ\); a full turn is \(360^\circ\). 8 forgets that the full turn itself counts as a position, and 18 divides \(720^\circ\).
A half turn about the centre maps each shaded square onto another shaded square, but a quarter turn does not, so the order is 2. Folding along the vertical, horizontal or either diagonal line sends a shaded square onto an unshaded one, so there are no lines of symmetry. The tempting "2 lines" assumes that a shape with a half-turn symmetry must also have mirror lines; a parallelogram shows it need not. Order 0 does not exist: every shape has order at least 1.
A rhombus folds onto itself along each diagonal and has no other lines, so it has exactly 2 lines (the diagonals) and order 2. The tempting rectangle also has 2 lines and order 2, but its lines join the midpoints of opposite sides; folding a rectangle along a diagonal does not match the corners. A square has 4 lines, and a parallelogram has none.
Every cuboid has 3 planes through the midpoints of opposite faces. Because the ends are squares, the 2 planes through the diagonals of the square ends (running along the 10 cm length) are also planes of symmetry: \(3 + 2 = 5\). The tempting 3 is the answer for a cuboid with three different lengths. 9 is the cube's total, which needs all faces square, and 4 counts only the planes containing the long axis.
The hexagon has 6 lines of symmetry, and each gives a plane running along the length of the prism. One more plane cuts across the middle of the length, parallel to the ends: \(6 + 1 = 7\). The tempting 6 misses that cross-wise plane. 12 doubles the hexagon's lines, and 8 is the number of faces.
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.