At a glance. One lesson of about 60 minutes. Students draw or read a net of a prism, find the area of every face, and explain why the surface area is two ends plus perimeter times length.
Calculator: no calculator. The sides are integers and the faces are small, and the structure of the net is the learning.
What students will learn
Know
- A prism has two identical ends and rectangular sides.
- The surface area is the total area of all the faces, in square units.
- Surface area = 2 × end area + perimeter of the end × length.
Understand
- The sides unwrap into one rectangle as wide as the perimeter of the end.
- Changing the length changes only the sides, not the ends.
- A net shows every face once.
Do
- Find the surface area of a triangular or trapezoidal prism.
- Find a missing length from the surface area.
- Decide which faces a real object has.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: faces, nets, perimeter and area of the end (questions 1 to 9).
- Criterion B. Investigating patterns: change the length with the end fixed and describe what changes in the surface area, then generalize (the second figure and question 7).
- Criterion C. Communicating: draw a net with its measurements and label each face with its area (questions 2 and 5).
- Criterion D. Applying mathematics in real-life contexts: a tent, with and without a floor, and a gift box, where the real object decides which faces to count (questions 6 and 8).
The one thing to do with the figure
Ask for the number of faces before any are added. Choose each shape in the first figure and ask how many faces its net has and what shapes they are. Then add them one at a time.
In the second figure, set the length to 6, then 3, and ask: “I halved the length. Did the surface area halve?” Most say yes. The ends do not change.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The tent and the prediction. First figure: add the faces one at a time. |
| 10 min | Key idea. Students draw the net of a small prism on squared paper, cut it out and fold it. |
| 10 min | Second figure, worked examples, then the faded one in pairs. |
| 5 min | Question 1, the check. If many choose D, go back to the second figure. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 20 cm²; 18 cm²; 54 cm². |
|---|---|
| Worked examples | 468 cm²; 184 cm²; 104 m². |
| Faded example | 360 cm². |
| 1. What surface area is | A. B leaves out the ends, C has one end, D is the volume. If many choose D, go back to the second figure and ask which numbers are in square centimetres. |
| 2. Surface area of a triangular prism | 336 cm². 48 + 288 = 336. |
| 3. Isosceles triangle end | 444 cm². 120 + 324 = 444. |
| 4. Trapezoidal prism | 416 cm². 56 + 360 = 416. |
| 5. Missing length | 15 cm. 192 − 12 = 180, and 180 ÷ 12 = 15. |
| 6. A tent without a floor | 21 m². 6 + 15 = 21. |
| 7. How much the area grows | 120 cm². 252 − 132 = 120. |
| 8. A hexagonal prism | 324 cm². 84 + 240 = 324. |
| 9. Surface area, not volume | A. B is the volume, C leaves out the ends, D has one end. Dina has found the volume without noticing. This is the bridge to lesson 6. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| D in question 1 | Takes the volume as the surface area. | Ask which units each one is measured in. |
| 288 or 312 in question 2 | Leaves out the ends, or counts one. | Ask how many faces a triangular prism has. |
| 324 or 384 in question 3 | Leaves out the ends, or counts one. | Count five faces on the net. |
| 360, 388 or 266 in question 4 | Leaves out the ends, counts one, or leaves the slopes out of the perimeter. | Go round the end with a finger and count the sides. |
| 16 or 15.5 in question 5 | Divides without taking off the ends, or takes off one. | Ask what the ends add to before dividing. |
| 33 or 27 in question 6 | 33 includes the floor. 27 leaves out the ends but adds the floor. | List the faces the tent really has. |
| 132 or 252 in question 7 | Doubles the whole area, or gives one area. | Colour the faces that change. |
| 282 or 240 in question 8 | Counts one hexagon, or leaves out the ends. | How many hexagons are there? |
| B, C or D in question 9 | Takes the volume, leaves out the ends, or counts one end. | Ask which faces Dina found. |
Questions worth asking
- How many faces does a triangular prism have? A trapezoidal prism? 5 and 6. Listen for the count of sides plus two.
- Why do the sides unwrap into one rectangle? Each side is as tall as the length and as wide as one edge of the end, and they are joined edge to edge. Listen for “all the way round the end”.
- If the length doubles, why does the surface area not double? The ends do not change. Listen for “only the sides double”.
- Where would the area of a trapezium from lesson 1 appear in this lesson? In the end, when the end is a trapezium. Listen for “the ends”.
Differentiation
A way in for anyone stuck
- Give a prism made from card or a printed net to cut out and fold.
- Ask for a list of faces with their shapes before any calculation.
- Colour the ends one colour and the sides another on every diagram.
Stretch
- A prism has a square end of side 5 cm and a length of 20 cm. Compare its surface area with that of a cube of side 5 cm. (Prism 450 cm², cube 150 cm²: three times, not four, since the ends stay the same.)
- A triangular prism has a 3, 4, 5 end. For what length is the area of the sides equal to the area of the two ends? (12 × L = 12, so L = 1 cm.)
Students learning in English as an additional language
“Net” is also a fishing word and a computer word; here it is a flat pattern that folds into a solid. “Face” is a flat surface of a solid, not a person's. “Prism” has the same ending in many languages: show a triangular prism of glass or card.
| Word | Plain meaning | Example |
|---|---|---|
| prism | a solid with two identical ends joined by flat rectangular sides | A tent, a ramp |
| net | a flat pattern that folds up into a solid | A box before it is folded |
| face | one flat surface of a solid | The floor of a tent |
| surface area | the total area of all the faces of a solid | The fabric needed for a tent |
| cross-section | the shape you see when you cut straight through a solid | The triangle at each end of a tent |
| perimeter | the distance all the way round a shape | The edge of the tent's end |
Sentence starters for explaining aloud or in writing:
- The end is a … with area … and perimeter …
- The net has … faces: … rectangles and … triangles.
- The surface area is two ends plus …
Two things not to say
“Surface area is the volume with squares.” They are different quantities with different units, and the exit ticket tests that.
“Add up all the sides.” Sides of what? The perimeter of the end and the sides of the prism are different things.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FluencyA triangular prism has a 6, 8, 10 end and a length of 5 cm. Find its surface area.
168 cm²: 2 × 24 + 24 × 5 = 48 + 120. - FluencyA box has a rectangular end 3 cm by 4 cm and a length of 5 cm. Find its surface area.
94 cm²: 2 × 12 + 14 × 5 = 24 + 70.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA wooden ramp in Lagos is a triangular prism 1 m long. Its end is a right-angled triangle with sides of 0.6 m, 0.8 m and 1 m. How many square metres of paint cover all its faces?
2.88 m²: 2 × 0.24 + 2.4 × 1 = 0.48 + 2.4. - SpacingFrom lesson 3: a circle has a radius of 10 cm. What is its circumference? Use 3.14.
62.8 cm.
3Challenge
Reasoning, generalizing and working backwards.
- MissingA prism has a 5, 12, 13 end and a surface area of 540 cm². How long is it?
16 cm: 540 − 60 = 480 and 480 ÷ 30 = 16. - ReasoningPrism A has a 3, 4, 5 end and length 5. Prism B has all lengths doubled. Is the surface area of B double that of A? What is it?
No. A is 72 cm² and B is 288 cm², four times as much, because every length doubles and area depends on two lengths.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Every figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper, scissors, string and a few sheets of card do the same job: cut, turn, wrap and fold.