The shape of the lesson
By the end of the lesson, learners can
- Points, lines and angles: Open the lab and drag the arm of the angle slowly from 0° all the way round to 360°.
- Triangles and quadrilaterals: The special quadrilaterals form a family.
- Polygons, solids and nets: A polygon is a flat shape with straight sides; it is regular when all its sides AND all its angles are equal.
- The parts of a circle: In this lab tab, click each label in turn and watch the matching part of the circle light up.
- Angles: acute below 90°, obtuse 90° to 180°, reflex 180° to 360°.
- Quadrilaterals: check parallel sides, equal sides, right angles and diagonals in turn.
- Solids: count faces, then edges, then vertices; check \(F + V - E = 2\).
- Circles: sector = two radii and an arc; segment = chord and arc.
- Use the exact word: the mark is for the correct term, spelt clearly.
One picture every idea returns to
Name a shape by the properties it MUST have, not by how it looks: a trapezium has exactly one pair of parallel sides, a rhombus has four equal sides, and a square is a special rectangle AND a special rhombus.
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, Say it in geometry.. Every geometry question is written in a precise vocabulary: reflex, perpendicular, rhombus, frustum, segment. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: Points, lines and angles | Open the lab and drag the arm of the angle slowly from 0° all the way round to 360°. Project tab 1 of the lab (Angles lab, measuring angles) and let a student drive it while the class predicts. Model the worked example Naming angles on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Triangles and quadrilaterals | The special quadrilaterals form a family. Model the worked example Name the quadrilateral on the board, one line at a time. Your turn: A triangle has angles \(45^\circ\), \(45^\circ\) and \(90^\circ\). Use two words to describe it. | Pairs |
| 17 to 23 min | Teach and check: Polygons, solids and nets | A polygon is a flat shape with straight sides; it is regular when all its sides AND all its angles are equal. Model the worked example Counting faces, edges and vertices on the board, one line at a time. Your turn: How many faces, edges and vertices does a pentagonal pyramid have? | Pairs |
| 23 to 30 min | Teach and check: The parts of a circle | In this lab tab, click each label in turn and watch the matching part of the circle light up. Project tab 1 of the lab (Circle theorems lab, parts of a circle) and let a student drive it while the class predicts. | Jot Thoughts |
| 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 The parts of a circle, 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.
Jot Thoughts
Teams write ideas on slips, one idea per slip, as many as they can in two minutes, saying each aloud as they place it in the middle. Then they sort the slips, which is where the structure of the topic appears.
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 reflex angle is NOT just a big obtuse angle. \(170^\circ\) is obtuse; anything over \(180^\circ\) is reflex. | When asked for a reflex angle, subtract the inside angle from \(360^\circ\), not from \(180^\circ\). |
| A kite's equal sides are next to each other (adjacent); a parallelogram's equal sides are opposite. | And a trapezium has exactly ONE pair of parallel sides: a parallelogram is not a trapezium in IGCSE. |
| Do not count the same edge twice. | Every edge is shared by exactly two faces, so counting the edges of each face and adding gives double the true number. |
| Sector and segment are the classic mix-up. | A sector always reaches the centre (two radii); a segment never does unless the chord is a diameter. |
Geometrical terms: shapes, solids and nets: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Write down the type of each angle: (a) \(89^\circ\) (b) \(91^\circ\) (c) \(181^\circ\) (d) \(90^\circ\).
Name the quadrilateral:
(a) two pairs of equal adjacent sides, and diagonals that cross at right angles but are not equal and do not both get bisected;
(b) opposite sides equal and parallel, four right angles, and two different side lengths.
Write down the word for:
(a) a straight line that touches a circle at exactly one point;
(b) a chord that passes through the centre;
(c) the region enclosed by two radii and an arc.
A pyramid has a hexagon as its base. Find the number of (a) faces (b) edges (c) vertices.
The hands of a clock show exactly 8 o'clock.
(a) Find the obtuse angle between the hands.
(b) Find the reflex angle between the hands.
(c) At what whole hour (other than 8 o'clock) is the angle between the hands the same as at 8 o'clock?
A flower bed PQRS is a quadrilateral. PQ is parallel to SR, PQ = 6 m, SR = 10 m and PS = QR = 5 m.
(a) Name the shape PQRS.
(b) Explain why PQRS is not a parallelogram.
(c) Is angle PSR acute or obtuse? Give a reason.
(d) Diagonal PR splits the bed into two triangles. Are these triangles congruent? Explain.
Answer key and marking notes
(a) acute (b) obtuse (c) reflex (d) right angle
- (a) kite
- (b) rectangle
- (a) tangent
- (b) diameter
- (c) sector
- (a) 1 hexagon + 6 triangles = 7 faces.
- (b) 6 base edges + 6 slant edges = 12 edges.
- (c) 6 base vertices + 1 apex = 7 vertices.
- Check: \(7 + 7 - 12 = 2\).
- (a) Each hour gap is \(30^\circ\). From 8 round to 12 is 4 gaps: \(4 \times 30 = 120^\circ\).
- (b) \(360 - 120 = 240^\circ\).
- (c) The hour hand must be 4 gaps from 12 on the other side: 4 o'clock.
- (a) Exactly one pair of parallel sides: a trapezium (an isosceles trapezium, since PS = QR).
- (b) In a parallelogram both pairs of opposite sides are equal, but PQ = 6 m and SR = 10 m are not equal.
- (c) SR is the longer parallel side, so S and R are the base corners and PS slopes inwards: the horizontal step is \((10 - 6) \div 2 = 2\) m, so angle PSR is acute (its co-interior partner SPQ is obtuse).
- (d) No. Triangle PQR has sides 6, 5 and PR, but triangle PSR has sides 5, 10 and PR. They share PR but the other sides differ, so they are not identical.
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.
A quadrilateral has diagonals that are equal in length and bisect each other, but they do NOT cross at right angles. What is the quadrilateral?
- ARectangle
- BSquare
- CRhombus
- DKite
Angle PQR \(= 128^\circ\). Find the reflex angle PQR.
- A\(232^\circ\)
- B\(52^\circ\)
- C\(308^\circ\)
- D\(128^\circ\)
How many edges does a pentagonal-based pyramid have?
- A10
- B15
- C6
- D20
The diagram shows a circle with centre O and a chord AB. What is the name of the shaded region?
- AA segment
- BA sector
- CAn arc
- DA semicircle
A prism has 12 vertices and 18 edges. What shape is its cross-section?
- AHexagon
- BDodecagon (12 sides)
- CNonagon (9 sides)
- DOctagon
A cube is cut by a plane that passes through two opposite edges (one top edge and the bottom edge diagonally opposite it). What shape is the flat face made by the cut?
- AA rectangle that is not a square
- BA square
- CA rhombus that is not a square
- DAn equilateral triangle
Quiz key
Only the square and the rectangle have equal diagonals that bisect each other. A square's diagonals also cross at right angles, so a shape whose diagonals do not cross at \(90^\circ\) must be a rectangle (that is not a square). The tempting square fails the right-angle condition. A rhombus has perpendicular diagonals that are not equal, and a kite's diagonals cross at right angles and only one is bisected.
The reflex angle and the angle of \(128^\circ\) together make a full turn, so reflex angle PQR \(= 360^\circ - 128^\circ = 232^\circ\). The tempting \(52^\circ\) subtracts from \(180^\circ\), which gives an acute angle, not a reflex one. \(308^\circ\) adds \(180^\circ\), and \(128^\circ\) is the obtuse angle itself.
The pentagon base has 5 edges, and one edge runs from each of the 5 base vertices up to the apex: \(5 + 5 = 10\) edges. The tempting 15 is the number of edges of a pentagonal PRISM. 6 is the number of faces (or vertices), and 20 counts the edges of every face separately (\(5 + 5 \times 3\)), so each edge is counted twice.
The shaded region is bounded by the chord AB and the arc AB, and it does not reach the centre, so it is a segment (the minor segment). The tempting sector is bounded by two RADII and an arc, so it always has a corner at O. An arc is only part of the circumference (a curve, not a region), and a semicircle would need AB to be a diameter through O.
A prism with an \(n\)-sided cross-section has \(2n\) vertices and \(3n\) edges (\(n\) on each end plus \(n\) along the length). \(2n = 12\) gives \(n = 6\), and \(3n = 18\) checks, so the cross-section is a hexagon. The tempting dodecagon uses all 12 vertices on one end. Nonagon halves the edges (\(18 \div 2\)), and octagon comes from the number of faces, \(n + 2 = 8\).
The cut has four right angles: two of its sides are cube edges of length \(a\), and the other two are face diagonals of length \(a\sqrt{2}\), so it is a rectangle with sides \(a\) and \(a\sqrt{2}\), not a square. The tempting square assumes all four sides are edges of the cube. A rhombus would need four equal sides, and a triangle comes from a plane through three corners, not through two opposite edges.
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.