The shape of the lesson
By the end of the lesson, learners can
- Adding the pieces: In the lab, build a compound shape and watch which edges the perimeter follows: only the outside ones.
- Parts of shapes and shaded regions: For a shaded region, ask: what is the whole shape, and what has been taken away?
- Compound solids: In the lab, join the solids and watch the surface area: the faces that meet disappear inside the solid, so they are left out.
- Parts of solids: the frustum: (Extended) A frustum is a cone with its top cut off parallel to the base.
- Sketch the pieces and label every length you know.
- Area and volume: add the pieces, subtract the holes.
- Perimeter: outside edges only, never the joins.
- Surface area: outside faces only; joined faces disappear.
- Frustum (Extended): big cone minus small cone.
- Accuracy: keep \(\pi\) and unrounded values until the end.
One picture every idea returns to
Area and volume: add the pieces, subtract the holes. Perimeter and surface area: include only the OUTSIDE edges and faces. A line or face where two pieces join is inside the shape, so it does not count.
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, Break it into pieces.. Exam shapes are rarely a single rectangle or circle. 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: Adding the pieces | In the lab, build a compound shape and watch which edges the perimeter follows: only the outside ones. Project tab 5 of the lab (Area and perimeter lab, compound shapes) and let a student drive it while the class predicts. Model the worked example An L-shape on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Parts of shapes and shaded regions | For a shaded region, ask: what is the whole shape, and what has been taken away? Model the worked example Circle in a square on the board, one line at a time. Your turn: A circular pond of radius 3 m has a path 1 m wide all the way round it. Find the area of the path in terms of \(\pi\). | Pairs |
| 17 to 23 min | Teach and check: Compound solids | In the lab, join the solids and watch the surface area: the faces that meet disappear inside the solid, so they are left out. Project tab 5 of the lab (Surface area and volume lab, compound solids) and let a student drive it while the class predicts. Model the worked example Cone on a hemisphere on the board, one line at a time. | Pairs |
| 23 to 30 min | Teach and check: Parts of solids: the frustum | (Extended) A frustum is a cone with its top cut off parallel to the base. Model the worked example Volume of a frustum (Extended) on the board, one line at a time. | 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, 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 Parts of solids: the frustum, 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 |
|---|---|
| Do not include the join. | In Fig. 1 the 8 cm line under the semicircle is inside the window, so adding it (40.6 cm) is wrong. |
| For a ring, subtract the AREAS, not the radii: \(\pi(7 - 4)^2 = 9\pi\) is wrong. | Square each radius first: \(\pi(7^2 - 4^2) = 33\pi\). |
| For surface area, leave out every face where two parts are joined. | Counting the circle between the cylinder and the hemisphere adds an extra \(16\pi\). |
| You cannot use the frustum's own height in a single cone formula. | Always work with the two complete cones and subtract. |
Compound shapes and parts of shapes: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
A rectangle 12 cm by 9 cm has a rectangle 5 cm by 4 cm cut from one corner. Find (a) the area (b) the perimeter of the shape that is left.
A metal washer is a ring. The outer diameter is 20 mm and the hole has radius 6 mm. Calculate the area of metal.
A shape is a rectangle 10 cm long and 4 cm wide with a semicircle on each of its two 4 cm ends.
Calculate (a) its area (b) its perimeter.
A solid is a cylinder of radius 5 cm and height 8 cm with a cone of radius 5 cm and vertical height 6 cm on top. Calculate its volume.
Calculate the total surface area of the solid in the previous question. The base of the cylinder is flat.
(Extended) A drinking glass is a frustum. Its top has radius 4 cm, its base has radius 3 cm and its height is 6 cm.
The frustum is a cone of height 24 cm and radius 4 cm with a cone of radius 3 cm removed from the bottom.
(a) Show that the volume of the glass is \(74\pi\text{ cm}^3\).
(b) How many glasses can be filled completely from 2 litres of juice?
Answer key and marking notes
- (a) \(12 \times 9 - 5 \times 4 = 108 - 20 = 88\text{ cm}^2\)
- (b) The cut corner does not change the perimeter: \(2(12 + 9) = 42\) cm
- Outer radius \(= 10\) mm.
- Area \(= \pi \times 10^2 - \pi \times 6^2 = 64\pi = 201.0... = 201\text{ mm}^2\)
- Two semicircles of radius 2 make one full circle.
- (a) \(10 \times 4 + \pi \times 2^2 = 40 + 4\pi = 52.56... = 52.6\text{ cm}^2\)
- (b) Two straight sides \(2 \times 10 = 20\), plus one full circumference \(4\pi\): \(20 + 4\pi = 32.6\) cm
- Cylinder \(= \pi \times 25 \times 8 = 200\pi\); cone \(= \tfrac{1}{3}\pi \times 25 \times 6 = 50\pi\).
- Total \(= 250\pi = 785.3... = 785\text{ cm}^3\)
- Cone slant height \(= \sqrt{5^2 + 6^2} = \sqrt{61} = 7.810...\); cone curved \(= \pi \times 5 \times 7.810... = 122.6...\)
- Cylinder curved \(= 2\pi \times 5 \times 8 = 80\pi = 251.3...\)
- Base \(= 25\pi = 78.53...\) (the top of the cylinder is covered by the cone).
- Total \(= 452.5... = 453\text{ cm}^2\)
- Small cone height \(= 24 - 6 = 18\) cm (check: \(\tfrac{3}{4}\) of 24).
- (a) \(\tfrac{1}{3}\pi \times 16 \times 24 - \tfrac{1}{3}\pi \times 9 \times 18 = 128\pi - 54\pi = 74\pi\text{ cm}^3\)
- \(74\pi = 232.4...\text{ cm}^3\), and 2 litres \(= 2000\text{ cm}^3\).
- (b) \(2000 \div 232.4... = 8.60...\), so 8 glasses can be filled completely.
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 diagram shows a lawn made from a rectangle 20 m by 10 m with a semicircle on each short end. Find the area of the lawn, correct to 3 significant figures.
- A\(279\text{ m}^2\)
- B\(357\text{ m}^2\)
- C\(239\text{ m}^2\)
- D\(514\text{ m}^2\)
The diagram shows a lawn made from a rectangle 20 m by 10 m with a semicircle on each short end. Find the perimeter of the lawn, correct to 3 significant figures.
- A71.4 m
- B91.4 m
- C103 m
- D55.7 m
A ring is the region between two circles with the same centre, of radii 9 cm and 6 cm. Find the area of the ring, in terms of \(\pi\).
- A\(45\pi\text{ cm}^2\)
- B\(9\pi\text{ cm}^2\)
- C\(6\pi\text{ cm}^2\)
- D\(117\pi\text{ cm}^2\)
A solid is a cylinder of radius 5 cm and height 12 cm with a cone of radius 5 cm and vertical height 9 cm on top. Find the volume of the solid, correct to 3 significant figures.
- A\(1180\text{ cm}^3\)
- B\(942\text{ cm}^3\)
- C\(1650\text{ cm}^3\)
- D\(550\text{ cm}^3\)
A solid is a cylinder of radius 5 cm and height 12 cm with a cone of radius 5 cm and vertical height 12 cm joined to its top. The bottom of the cylinder is flat. Find the total surface area of the solid, correct to 3 significant figures.
- A\(660\text{ cm}^2\)
- B\(738\text{ cm}^2\)
- C\(644\text{ cm}^2\)
- D\(581\text{ cm}^2\)
A cone of radius 8 cm and vertical height 18 cm has a small cone of radius 4 cm and height 9 cm cut from its top, leaving a frustum. Find the volume of the frustum, correct to 3 significant figures.
- A\(1060\text{ cm}^3\)
- B\(1210\text{ cm}^3\)
- C\(3170\text{ cm}^3\)
- D\(452\text{ cm}^3\)
Quiz key
Rectangle \(= 20 \times 10 = 200\). The two semicircles (radius 5 m) make one whole circle: \(\pi \times 5^2 = 25\pi\). Area \(= 200 + 25\pi = 278.5... = 279\text{ m}^2\). The tempting 357 counts each semicircle as a full circle. 239 includes only one semicircle, and 514 uses the diameter 10 m as the radius.
The outside edge is the two straight 20 m sides and two semicircular arcs, which together make one full circumference \(\pi \times 10 = 10\pi\). Perimeter \(= 40 + 10\pi = 71.4\) m. The tempting 91.4 m also adds the two dashed 10 m joins, which are inside the lawn. 103 m uses 10 m as the radius, and 55.7 m includes only one arc.
Big circle \(= \pi \times 9^2 = 81\pi\); small circle \(= \pi \times 6^2 = 36\pi\). Ring \(= 81\pi - 36\pi = 45\pi\text{ cm}^2\). The tempting \(9\pi\) subtracts the radii first, \(\pi(9 - 6)^2\). \(6\pi\) is the difference of the circumferences, and \(117\pi\) adds the two circles instead of subtracting.
Cylinder \(= \pi \times 5^2 \times 12 = 300\pi\). Cone \(= \tfrac{1}{3}\pi \times 5^2 \times 9 = 75\pi\). Total \(= 375\pi = 1178.0... = 1180\text{ cm}^3\). The tempting 942 is the cylinder alone. 1650 forgets the \(\tfrac{1}{3}\) for the cone (the same as a cylinder of height 21 cm), and 550 applies \(\tfrac{1}{3}\) to the whole 21 cm height.
Cone slant height \(= \sqrt{5^2 + 12^2} = 13\). Outside surfaces: cylinder curved \(2\pi \times 5 \times 12 = 120\pi\), flat base \(25\pi\), cone curved \(\pi \times 5 \times 13 = 65\pi\). Total \(= 210\pi = 659.7... = 660\text{ cm}^2\). The tempting 738 also counts the circle where the cone meets the cylinder, which is hidden inside. 644 uses the vertical height 12 in \(\pi rl\), and 581 leaves out the flat base.
Big cone \(= \tfrac{1}{3}\pi \times 8^2 \times 18 = 384\pi\). Small cone \(= \tfrac{1}{3}\pi \times 4^2 \times 9 = 48\pi\). Frustum \(= 336\pi = 1055.5... = 1060\text{ cm}^3\). The tempting 1210 is the big cone with nothing removed. 3170 forgets the \(\tfrac{1}{3}\) in both cones, and 452 wrongly uses the frustum's own height 9 cm with the difference of the base areas, \(\tfrac{1}{3}\pi(64 - 16) \times 9\).
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.