The shape of the lesson
By the end of the lesson, learners can
- Measuring and drawing: Every construction starts with accurate measuring.
- A triangle from three sides: In the lab, set three side lengths and build the triangle step by step: draw the base, swing an arc from each end, and see where they cross.
- Nets, surface area and volume: Fold and unfold solids in the lab.
- Check first: the two shorter sides must add to more than the longest.
- Base: draw one side accurately with a ruler.
- Arcs: compasses set to each other length, centred on the matching end.
- Join and keep: join to the crossing point; leave the arcs showing.
- Nets: surface area = total area of every face in the net.
One picture every idea returns to
To construct a triangle from three sides: draw one side with a ruler, then draw an arc from each end with the compasses set to the other two lengths. The arcs cross at the third vertex. Leave the arcs showing.
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, Ruler, compasses, arcs.. Given three sides, there is exactly one triangle, and a pair of compasses finds its third corner for you. 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: Measuring and drawing | Every construction starts with accurate measuring. Model the worked example Drawing a reflex angle on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: A triangle from three sides | In the lab, set three side lengths and build the triangle step by step: draw the base, swing an arc from each end, and see where they cross. Project tab 1 of the lab (Constructions lab, construct a triangle) and let a student drive it while the class predicts. Model the worked example Construct a triangle on the board, one line at a time. Your turn: Can a triangle be constructed with sides 4 cm, 5 cm and 10 cm? Explain. | Pairs |
| 21 to 30 min | Teach and check: Nets, surface area and volume | Fold and unfold solids in the lab. Project tab 2 of the lab (Constructions lab, nets) and let a student drive it while the class predicts. Model the worked example From a net to surface area and volume on the board, one line at a time. Your turn: A cuboid measures 6 cm by 4 cm by 3 cm. Find the total area of its net. | 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, 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 Nets, surface area and volume, 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 |
|---|---|
| Reading the wrong scale on the protractor is the most common slip: \(50^\circ\) read as \(130^\circ\). | Always decide acute or obtuse by eye before you read the number. |
| Rubbing out the construction arcs loses marks even when the triangle is perfect. | Cambridge says construction arcs must be shown. Also, set each arc's radius from the correct end: the 6 cm arc goes round A because AC = 6 cm. |
| Do not forget the factor of 2 for the pairs of faces in a cuboid, and do not include the slanted 5 cm side in the triangle's area: the area of a right-angled triangle uses the two sides that meet at the right angle. | Return to Nets, surface area and volume: ask a pair to state the correct version and show it on the lab before moving on. |
Ruler and compasses constructions: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Describe, step by step, how to construct triangle PQR with PQ = 7 cm, PR = 5 cm and QR = 4 cm using a ruler and compasses only.
For each set of lengths, say whether a triangle can be constructed. Give a reason.
(a) 2 cm, 5 cm, 8 cm (b) 5 cm, 6 cm, 10 cm
Kofi wants to draw an angle of \(290^\circ\) using a protractor that goes up to \(180^\circ\). What angle should he measure?
The net of a cube has a total area of 150 cm2. Find the volume of the cube.
A triangular prism is 15 cm long. Its cross-section is a right-angled triangle with sides 6 cm, 8 cm and 10 cm.
(a) Sketch the net and list the faces. (b) Find the surface area. (c) Find the volume.
A box with no lid is made from a net. The box is a cuboid 20 cm long, 12 cm wide and 8 cm high.
(a) How many faces does the net have? (b) Find the area of card in the net. (c) Will the box hold 2 litres of rice? (1 litre = 1000 cm3)
Answer key and marking notes
- Draw PQ = 7 cm with a ruler.
- Compasses set to 5 cm, point on P: draw an arc above PQ.
- Compasses set to 4 cm, point on Q: draw an arc crossing the first arc at R.
- Join PR and QR with a ruler; leave the arcs showing.
- (a) No: \(2 + 5 = 7 < 8\), so the arcs do not meet.
- (b) Yes: \(5 + 6 = 11 > 10\).
\(360^\circ - 290^\circ = 70^\circ\). He draws \(70^\circ\) and marks the reflex angle outside it.
- A cube's net has 6 equal squares: each has area \(150 \div 6 = 25\) cm2.
- Edge length \(= \sqrt{25} = 5\) cm.
- Volume \(= 5^3 = 125\) cm3.
- (a) Two right-angled triangles (6, 8, 10) and three rectangles: \(6 \times 15\), \(8 \times 15\), \(10 \times 15\).
- (b) Triangles: \(2 \times \tfrac{1}{2} \times 6 \times 8 = 48\). Rectangles: \((6 + 8 + 10) \times 15 = 360\). Total 408 cm2.
- (c) \(24 \times 15 = 360\) cm3.
- (a) A cuboid has 6 faces; with no lid the net has 5.
- (b) Base \(20 \times 12 = 240\); two long sides \(2 \times 20 \times 8 = 320\); two short sides \(2 \times 12 \times 8 = 192\).
- Total \(240 + 320 + 192 = 752\) cm2.
- (c) Volume \(= 20 \times 12 \times 8 = 1920\) cm3 = 1.92 litres, which is less than 2 litres, so no.
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.
Triangle ABC has AB = 8 cm, AC = 6 cm and BC = 5 cm. AB has been drawn with a ruler. What is the correct next step?
- ADraw an arc of radius 6 cm centred on A and an arc of radius 5 cm centred on B; C is where they cross
- BDraw an arc of radius 5 cm centred on A and an arc of radius 6 cm centred on B; C is where they cross
- CDraw arcs of radius 6 cm and 5 cm, both centred on A
- DMeasure an angle of \(60^\circ\) at A and draw a 6 cm line
Two sides of a triangle are 4 cm and 9 cm. Which of these could be the length of the third side?
- A7 cm
- B5 cm
- C3 cm
- D14 cm
Sam measures an obtuse angle with a protractor and writes down \(38^\circ\). He has read the wrong scale. What is the size of the angle?
- A\(142^\circ\)
- B\(322^\circ\)
- C\(52^\circ\)
- D\(218^\circ\)
A triangular prism is 10 cm long. Its cross-section is a right-angled triangle with sides 3 cm, 4 cm and 5 cm. Find the total area of its net.
- A\(132\text{ cm}^2\)
- B\(126\text{ cm}^2\)
- C\(140\text{ cm}^2\)
- D\(60\text{ cm}^2\)
The net of a closed cuboid has a total area of \(94\text{ cm}^2\). The cuboid is 5 cm long and 4 cm wide. Find its height.
- A3 cm
- B8.22 cm
- C6 cm
- D4.7 cm
The net of a pyramid is a square of side 8 cm with an isosceles triangle on each edge. The sloping edges of each triangle are 5 cm. Find the total surface area of the pyramid.
- A\(112\text{ cm}^2\)
- B\(144\text{ cm}^2\)
- C\(160\text{ cm}^2\)
- D\(48\text{ cm}^2\)
Quiz key
Every point on an arc of radius 6 cm centred on A is 6 cm from A, and every point on an arc of radius 5 cm centred on B is 5 cm from B. Where the arcs cross is the one point that is both, so it is C. The tempting option swaps the radii, which makes AC = 5 cm and BC = 6 cm, a different triangle. Two arcs from A never cross, and a guessed \(60^\circ\) angle is not part of a ruler and compasses construction.
The two shorter sides must add to MORE than the longest side. With 7 cm: \(4 + 7 = 11 > 9\), so the arcs cross and a triangle exists. The tempting 5 cm gives \(4 + 5 = 9\), exactly equal, so the arcs only touch on the line and the triangle is flat. 3 cm gives \(4 + 3 = 7 < 9\), and 14 cm gives \(4 + 9 = 13 < 14\): the arcs never meet.
The inner and outer scales give readings that add to \(180^\circ\), so the true angle is \(180^\circ - 38^\circ = 142^\circ\), which is obtuse as expected. The tempting \(322^\circ\) subtracts from \(360^\circ\), which would make a reflex angle. \(52^\circ\) subtracts from \(90^\circ\), and \(218^\circ\) adds \(180^\circ\); neither is obtuse.
Each triangle has area \(\tfrac{1}{2} \times 3 \times 4 = 6\text{ cm}^2\), so two give \(12\text{ cm}^2\). The rectangles are \(3 \times 10\), \(4 \times 10\) and \(5 \times 10\), total \(120\text{ cm}^2\). Net area \(= 12 + 120 = 132\text{ cm}^2\). The tempting 126 includes only one triangle. 140 uses the 5 cm side in the triangle's area (\(\tfrac{1}{2} \times 4 \times 5\)), and 60 is the volume, \(6 \times 10\), not an area.
\(2(5 \times 4 + 5h + 4h) = 94\), so \(20 + 9h = 47\), \(9h = 27\) and \(h = 3\) cm. Check: \(2(20 + 15 + 12) = 94\). The tempting 8.22 cm forgets the factor 2 (\(20 + 9h = 94\)). 6 cm doubles only the top and bottom (\(40 + 9h = 94\)), and 4.7 cm divides the area by \(5 \times 4\) as if it were a volume.
Each triangle has base 8 cm, so its height is \(\sqrt{5^2 - 4^2} = 3\) cm and its area is \(\tfrac{1}{2} \times 8 \times 3 = 12\text{ cm}^2\). Total \(= 64 + 4 \times 12 = 112\text{ cm}^2\). The tempting 144 uses 5 cm as the height (\(4 \times 20 + 64\)). 160 forgets to halve (\(4 \times 24 + 64\)), and 48 leaves out the square base.
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.