The shape of the lesson
By the end of the lesson, learners can
- Prisms and cuboids: In the lab, slide the cutting plane along the prism: every slice has the same shape and area.
- Cylinders, unrolled: In the lab, press Unroll: the curved surface opens out into a rectangle.
- Working backwards: If the volume is given, put it into the formula and solve for the missing length.
- Prism: \(V = Al\), \(A\) = cross-section area (given).
- Cylinder: \(V = \pi r^2 h\) and curved surface \(2\pi rh\) (both given).
- Total surface area: add every face; for a cylinder add \(2\pi r^2\) for the ends.
- Open containers: leave out the missing face.
- Backwards: substitute the volume and solve; round only at the end.
- Capacity: \(1000\text{ cm}^3 = 1\) litre.
One picture every idea returns to
Volume of a prism \(= \text{cross-section area} \times \text{length}\) (\(V = Al\)), so a cylinder has \(V = \pi r^2 h\). Its curved surface area is \(2\pi rh\). All three are in the List of formulas; the TOTAL surface area of a cylinder, \(2\pi rh + 2\pi r^2\), is not, so add the two circular ends yourself.
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, Same slice, all the way.. A prism has the same cross-section all along its length, so its volume is that area times the length. 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: Prisms and cuboids | In the lab, slide the cutting plane along the prism: every slice has the same shape and area. Project tab 1 of the lab (Surface area and volume lab, prisms) and let a student drive it while the class predicts. Model the worked example Cuboid on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Cylinders, unrolled | In the lab, press Unroll: the curved surface opens out into a rectangle. Project tab 2 of the lab (Surface area and volume lab, cylinders) and let a student drive it while the class predicts. Model the worked example Volume and surface area on the board, one line at a time. Your turn: Find the total surface area of a cylinder of radius 4 cm and height 10 cm, in terms of \(\pi\). | Pairs |
| 21 to 30 min | Teach and check: Working backwards | If the volume is given, put it into the formula and solve for the missing length. Model the worked example Height of a cylinder on the board, one line at a time. | Numbered Heads Together |
| 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, 16 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 Working backwards, 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.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
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 |
|---|---|
| Surface area means EVERY face. | In a triangular prism it is easy to count the two triangles and forget one of the three rectangles, especially the sloping one. |
| The List of formulas gives the CURVED surface area \(2\pi rh\) only. | For a closed cylinder add \(2\pi r^2\) for the ends; for an open tin (no lid) add just one \(\pi r^2\). |
| Rounding the radius or an area too early changes the third significant figure. | Store values in the calculator and round once at the end. |
Surface area and volume: prisms and cylinders: worksheet
6 questions, 16 marks. Written in the style of the papers; no past paper question is reproduced.
A cuboid measures 7 cm by 4 cm by 2.5 cm. Find (a) its volume (b) its surface area.
A cylinder has radius 3.5 cm and height 9 cm. Calculate (a) its volume (b) its curved surface area.
A prism is 20 cm long. Its cross-section is a right-angled triangle with sides 5 cm, 12 cm and 13 cm.
Find (a) its volume (b) its total surface area.
Without a calculator, find the volume and the total surface area of a closed cylinder with radius 5 cm and height 8 cm. Give your answers in terms of \(\pi\).
A cylindrical jug is 15 cm tall and holds exactly 1 litre when full.
Calculate the radius of the jug.
A water trough is a prism 2 m long. Its cross-section is a semicircle of radius 30 cm.
(a) Calculate the capacity of the trough in litres.
(b) Water flows into the empty trough at 8 litres per minute. How long does it take to fill?
Answer key and marking notes
- (a) \(7 \times 4 \times 2.5 = 70\text{ cm}^3\)
- (b) \(2(7 \times 4 + 7 \times 2.5 + 4 \times 2.5) = 2(28 + 17.5 + 10) = 111\text{ cm}^2\)
- (a) \(\pi \times 3.5^2 \times 9 = 346.3... = 346\text{ cm}^3\)
- (b) \(2\pi \times 3.5 \times 9 = 197.9... = 198\text{ cm}^2\)
- Cross-section \(= \tfrac{1}{2} \times 5 \times 12 = 30\text{ cm}^2\).
- (a) \(V = 30 \times 20 = 600\text{ cm}^3\)
- (b) Ends \(2 \times 30 = 60\); rectangles \((5 + 12 + 13) \times 20 = 600\); total \(660\text{ cm}^2\)
- \(V = \pi \times 25 \times 8 = 200\pi\text{ cm}^3\)
- Surface area \(= 2\pi \times 5 \times 8 + 2 \times 25\pi = 80\pi + 50\pi = 130\pi\text{ cm}^2\)
- 1 litre \(= 1000\text{ cm}^3\).
- \(\pi r^2 \times 15 = 1000\), so \(r^2 = \dfrac{1000}{15\pi} = 21.22...\)
- \(r = 4.606... = 4.61\) cm
- Cross-section \(= \tfrac{1}{2} \times \pi \times 30^2 = 450\pi = 1413.7...\text{ cm}^2\).
- Length 2 m = 200 cm, so \(V = 1413.7... \times 200 = 282\,743...\text{ cm}^3\).
- (a) \(282\,743... \div 1000 = 283\) litres
- (b) \(282.74... \div 8 = 35.3\) minutes
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 closed box is a cuboid measuring 7 cm by 4 cm by 2.5 cm. Find its total surface area.
- A\(111\text{ cm}^2\)
- B\(70\text{ cm}^2\)
- C\(55.5\text{ cm}^2\)
- D\(91\text{ cm}^2\)
A prism is 20 cm long. Its cross-section is a right-angled triangle with sides 5 cm, 12 cm and 13 cm, the right angle being between the 5 cm and 12 cm sides. Find the volume of the prism.
- A\(600\text{ cm}^3\)
- B\(1200\text{ cm}^3\)
- C\(650\text{ cm}^3\)
- D\(30\text{ cm}^3\)
A cylinder has diameter 12 cm and height 10 cm. Find its volume, in terms of \(\pi\).
- A\(360\pi\text{ cm}^3\)
- B\(1440\pi\text{ cm}^3\)
- C\(120\pi\text{ cm}^3\)
- D\(60\pi\text{ cm}^3\)
A cylindrical can has diameter 8 cm and height 15 cm. How many litres does it hold? Give your answer correct to 3 significant figures.
- A0.754 litres
- B3.02 litres
- C7.54 litres
- D0.377 litres
A closed cylinder has radius 3.5 cm and height 9 cm. Find its total surface area, correct to 3 significant figures.
- A\(275\text{ cm}^2\)
- B\(198\text{ cm}^2\)
- C\(236\text{ cm}^2\)
- D\(346\text{ cm}^2\)
A cylinder has volume \(1000\text{ cm}^3\) and height 12 cm. Find its radius, correct to 3 significant figures.
- A5.15 cm
- B26.5 cm
- C13.3 cm
- D9.13 cm
Quiz key
The three different faces are \(7 \times 4 = 28\), \(7 \times 2.5 = 17.5\) and \(4 \times 2.5 = 10\). Surface area \(= 2(28 + 17.5 + 10) = 111\text{ cm}^2\). The tempting 70 is the volume \(7 \times 4 \times 2.5\) (in cm3). 55.5 counts each face only once, and 91 leaves out the pair of \(4 \times 2.5\) end faces.
Cross-section \(= \tfrac{1}{2} \times 5 \times 12 = 30\text{ cm}^2\), so \(V = Al = 30 \times 20 = 600\text{ cm}^3\). The tempting 1200 forgets the half in the triangle. 650 uses the 13 cm side, which is not perpendicular to the 5 cm side, and 30 is only the cross-section area.
\(r = 6\) cm, so \(V = \pi \times 6^2 \times 10 = 360\pi\text{ cm}^3\). The tempting \(1440\pi\) uses the diameter as the radius, \(\pi \times 12^2 \times 10\). \(120\pi\) is the curved surface area \(2\pi \times 6 \times 10\), and \(60\pi\) forgets to square the radius.
\(r = 4\) cm, so \(V = \pi \times 4^2 \times 15 = 240\pi = 753.98...\text{ cm}^3\). Since \(1000\text{ cm}^3 = 1\) litre, the can holds \(0.754\) litres. The tempting 3.02 uses the diameter 8 as the radius. 7.54 divides by 100 instead of 1000, and 0.377 uses the curved surface formula \(2\pi rh\) instead of the volume.
Curved surface \(= 2\pi \times 3.5 \times 9 = 63\pi\). Two ends \(= 2 \times \pi \times 3.5^2 = 24.5\pi\). Total \(= 87.5\pi = 274.8... = 275\text{ cm}^2\). The tempting 198 is the curved surface alone. 236 adds only one end, and 346 is the volume \(\pi \times 3.5^2 \times 9\).
\(\pi r^2 \times 12 = 1000\), so \(r^2 = \dfrac{1000}{12\pi} = 26.52...\) and \(r = \sqrt{26.52...} = 5.15\) cm. The tempting 26.5 cm stops at \(r^2\) and forgets the square root. 13.3 cm uses the curved surface formula \(2\pi rh\), and 9.13 cm leaves out \(\pi\).
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.