At a glance. One lesson of about 60 minutes. Students draw and recognize nets of cuboids and triangular prisms, find surface area from the faces, handle open boxes, and compare boxes of the same volume.
Calculator: no calculator. Nothing here needs one. The products are small, and the net turns the work into areas students already know.
What students will learn
Know
- A net is a flat pattern that folds into a solid.
- Surface area is the total area of all the faces of a solid, in square units.
- A closed cuboid has three pairs of equal faces; a triangular prism has two triangles and three rectangles.
Understand
- The faces you cannot see match the ones you can, so a net gives six faces from three.
- The rectangles of a prism join in one strip as wide as the end's perimeter.
- Two solids with the same volume can have different surface areas.
Do
- Find the surface area of a cube, a cuboid, a triangular prism and an open box.
- Tell which patterns of squares fold into a cube.
- Compare the card needed for boxes of the same volume.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: nets and surface area of cuboids and prisms (questions 1 to 6).
- Criterion B. Investigating patterns: the two figures and the same-volume question. Ask students to record the surface area of every whole-number cuboid with volume 24 or 36, and to say which is the least and why (question 8).
- Criterion C. Communicating: sketching a net with the faces labelled by their dimensions and writing the sum so the faces can be checked one at a time (questions 3, 4 and 5).
- Criterion D. Applying mathematics in real-life contexts: packaging and card, where the decision depends on both the card used and the shape needed (questions 5 and 8).
The one thing to do with the figure
Ask how many faces they can see, then unfold. In the first figure, cover the numbers on the right with a sheet of paper and ask for the surface area of a box that is 4 by 3 by 2 before they look. Most say 3 faces. Then let the six faces lie flat and count.
In the second figure, ask what shape the three rectangles make when they sit side by side (one long rectangle with the triangle's perimeter as its width). That sentence is the formula.
Choose the 5, 5, 6 triangle and ask which length is the height of the end and which lengths are the widths of the rectangles: the end uses the height at a right angle (4), the rectangles use the three sides (5, 6 and 5).
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The tea boxes, the word net, and the first figure with predictions. |
| 10 min | Key idea. Students cut out and fold a net of a small box from squared paper, then write the six face areas on it. |
| 10 min | Second figure, cube nets (students fold patterns P and Q from squared paper before they meet question 6), then the worked examples; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B, go back to the first figure and count faces. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 120 cm³; 45 cm²; 49. |
|---|---|
| Worked examples | Faces 40, 16 and 10, surface area 132 cm². Wedge: ends 12 cm², rectangles 108 cm², total 120 cm². |
| Faded example | Ends 192 cm², rectangles 240 cm², total 432 cm². |
| 1. All six faces | A. B lists only the visible faces, C gives the pair totals, D doubles twice. If many choose B, go back to the first figure and count the faces. |
| 2. Surface area of a cube | 294 cm². 6 × 49 = 294. |
| 3. Surface area of a cuboid | 280 cm². 2 × 140 = 280. |
| 4. Surface area of a triangular prism | 186 cm². 24 + 162 = 186. |
| 5. Open box | 216 cm². 96 + 72 + 48 = 216. |
| 6. Which pattern folds to a cube | B. A is six in a row, C is a 2 by 3 block, D contains a 2 by 2 block. If many choose A, fold a paper strip of six squares round a cube. |
| 7. Finding a height from surface area | 7 cm. 240 − 72 = 168, 168 ÷ 24 = 7. |
| 8. Same volume, different card | 32 cm². 98 − 66 = 32. |
| 9. Surface area is not volume | A. B adds a count of faces to a volume, C confuses surface area and volume, D thinks the net hides half. If many choose C, return to lesson 5 and compare the unit of each. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Lists only the visible faces, lists the pair totals, or doubles twice. | Count the faces on a real box and write each area on it. |
| 49, 196 or 343 in question 2 | Gives one face, four faces, or the volume. | Hold a cube and count the faces aloud, then ask what a unit of area looks like. |
| 140 or 300 in question 3 | Gives half the surface, or the volume. | Colour the faces in pairs and count the colours. |
| 162, 150, 202 or 174 in question 4 | Leaves out the ends, uses the height for the sloping rectangles, uses the sloping side as the triangle's height, or counts one end. | List the five faces by name, then ask which length is at a right angle to the base and which lengths are the widths of the rectangles. |
| 312 or 120 in question 5 | Includes the lid, or leaves out the base. | Ask which face is missing from the closed box, and which are left. |
| A, C or D in question 6 | Chooses a row of six, a 2 by 3 block, or a pattern with a 2 by 2 block. | Fold a paper strip and see the overlap, or fold a 2 by 2 block and see the gap. |
| 168 or 28 in question 7 | Stops at the side faces, or divides by one side. | Draw the four side faces as one long strip, 24 cm wide. |
| 98, 164 or 66 in question 8 | Gives one box, both boxes added, or the other box. | Write both surface areas, then underline what is asked. |
| B, C or D in question 9 | Mixes up surface area and volume, or thinks the net shows half. | Compare the units: cm² for faces, cm³ for cubes inside. |
Questions worth asking
- Can a net be drawn in more than one way? Yes: a cube has eleven different nets. Listen for the idea that each net has the same six faces.
- Which faces of a prism are always rectangles? The faces joining the two ends. Listen for “the ones that go along the length”.
- Why can two boxes hold the same and need different card? Surface area and volume measure different things: one is the outside, one is the space inside.
- Is a bigger volume always more surface area? Not in general. For the same shape it is (a bigger cube has more of both), but a 1 by 1 by 24 box holds 24 cm³ with 98 cm² of card, while a 3 by 3 by 4 box holds more, 36 cm³, with less, 66 cm².
Differentiation
A way in for anyone stuck
- Give a ready-drawn net on squared paper so students can cut, fold and label before calculating.
- Provide a table with the three face areas and a column for how many of each there are.
- Start with cubes, where every face is the same, then move to a cuboid.
Stretch
- Show that a cube of edge n has surface area 6 × n × n and volume n × n × n, and that doubling the edge multiplies the surface area by 4 and the volume by 8.
- Among whole-number cuboids with volume 36 cm³, which has the least surface area, and what is it? (3 by 3 by 4, 66 cm². 2 by 3 by 6 has 72, 1 by 6 by 6 has 96.)
Students learning in English as an additional language
“Net” has an everyday meaning (a fishing net). Show a flat folded box as the meaning. “Face” is a flat side of a solid, not a person’s face. “Surface” is the outside.
| Word | Plain meaning | Example |
|---|---|---|
| net | a flat pattern that folds up into a solid | A flattened cereal box |
| face | one flat side of a solid | A cube has six faces |
| surface area | the total area of all the faces of a solid | The card needed to make a box |
| edge | where two faces meet | A cube has twelve edges |
| cuboid | a box with six rectangular faces | A shoe box |
| open box | a box with one face missing | A tray or a pencil case without a lid |
Sentence starters for explaining aloud or in writing:
- The faces come in pairs: … and …
- The surface area is … because the faces are …
- This pattern folds into a cube because …
Two things not to say
“Surface area is the three numbers multiplied.” That is the volume. Surface area is a sum of products.
“Count the faces you can see.” A net exists because the hidden faces are as real as the visible ones.
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.
- CubeA cube has edges of 3 cm. Find its surface area.
54 cm². - CuboidA closed box is 4 cm by 2 cm by 1 cm. Find its surface area.
28 cm²: 2 × (8 + 4 + 2).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA gift box is 20 cm by 15 cm by 10 cm and closed. How much wrapping paper covers it exactly, with no overlap?
1300 cm²: 2 × (300 + 200 + 150). - SpacingFrom Unit 7, lesson 5: a prism has a triangular end with base 6 cm and height 5 cm and is 8 cm long. Find its volume.
120 cm³.
3Challenge
Reasoning, generalising and working backwards.
- Working backA closed cube has surface area 150 cm². How long is each edge?
5 cm: 150 ÷ 6 = 25. - ReasoningA closed box is 2 cm by 2 cm by x cm and has surface area 40 cm². Find x.
x = 4: the two ends are 2 × 4 = 8, the four sides are 4 × 2x = 8x, so 8 + 8x = 40 and x = 4.
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 and a few sheets of card do the same job: cut, slide and fold.