At a glance. One lesson of about 60 minutes. Students explain volume as unit cubes in layers, find the volume of cuboids and prisms with triangular and compound ends, find a missing dimension, and convert cm³ to litres.
Calculator: no calculator. Nothing here needs one. The numbers are friendly, and the structure of layers is the point.
What students will learn
Know
- Volume is the number of unit cubes that fill a solid, in cubic units.
- Volume of a cuboid is length times width times height.
- Volume of a prism is the area of its end times its length.
Understand
- A layer one unit thick holds as many cubic units as the end has square units of area.
- Prisms with the same end area and the same length have the same volume.
- Lengths scale by k, areas by k times k and volumes by k times k times k.
Do
- Find the volume of a cuboid or a prism, with the right unit.
- Find a missing dimension from the volume.
- Change cm³ to litres, using 1000 cm³ = 1 litre.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: volume of cuboids and prisms (questions 1 to 6).
- Criterion B. Investigating patterns: the second figure and the doubling question. Ask students to predict, then record the volumes for ends with the same area, and then for lengths 1, 2, 3, 4, and state the pattern (question 8 and the what-if).
- Criterion C. Communicating: writing the area of the end as its own step before the length, and giving the unit with every answer (questions 4 to 6).
- Criterion D. Applying mathematics in real-life contexts: crates, tanks and litres, where a student chooses which measurements form the end and which the length (questions 4 and 7).
The one thing to do with the figure
Make the layers slider the lesson. Fill one layer, cover the numbers on the right of the figure with a sheet of paper, then ask “how many cubes will there be when the box is full?” before moving the slider. Students who say “double” for two layers have seen it.
In the second figure, cover the volume line and ask for the volume of the triangle before pressing the button. Many predict less than the rectangle. The equal area is the idea: write 12 and 12 on the board.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The crate and the prediction. First figure, filling one layer at a time. |
| 10 min | Key idea. Students build a 3 by 2 by 2 cuboid from linking cubes, count one layer, then the layers, then write the rule. |
| 10 min | Second figure, then the worked examples; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose D, preview lesson 6 and ask what the six face areas count. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 42 cm²; 41 m²; 27. |
|---|---|
| Crate | 28 boxes in a layer, 84 in the crate. |
| Worked examples | Tank 135 m³; triangular prism 180 cm³. |
| Faded example | End 26 cm², volume 104 cm³. |
| 1. Volume and its unit | A. B has the wrong unit, C adds, D is the surface area. If many choose D, preview lesson 6 and ask what the six face areas count. |
| 2. Volume of a cuboid | 240 cm³. 10 × 6 × 4 = 240. |
| 3. Volume of a cube | 216 cm³. 6 × 6 × 6 = 216. |
| 4. Missing depth | 3 m. 60 ÷ 20 = 3. |
| 5. Volume of a triangular prism | 132 cm³. 12 × 11 = 132. |
| 6. Volume of an L-shaped prism | 110 cm³. 22 × 5 = 110. |
| 7. Volume and litres | 80 litres. 80000 ÷ 1000 = 80. |
| 8. Scaling the volume | 8 times. 2 × 2 × 2 = 8. |
| 9. The end of a prism is not always a rectangle | A. B adds, C overgeneralizes the cuboid rule, D confuses the unit. If many choose C, go back to the second figure and switch between Rectangle and Triangle. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1 | Gives the right number with the area unit. | Ask what a cm² and a cm³ each look like. |
| 20 or 248 in question 2 | Adds the three lengths, or finds the surface area. | Build the box from cubes and count one layer. |
| 18 or 36 in question 3 | Multiplies by 3, or stops at one face. | Ask how many layers a cube of side 6 has. |
| 12 or 1200 in question 4 | Divides by one length, or multiplies. | Find one layer first, then ask how many layers make 60. |
| 264, 165 or 21 in question 5 | Forgets the half, uses the sloping side as the height, or adds. | Write the area of the end as its own line, and mark the height with a square corner before multiplying. |
| 140 or 30 in question 6 | Counts the missing piece, or gives only the missing piece. | Shade the L and ask what its area is. |
| 80000 or 8000 in question 7 | Leaves out the conversion, or divides by 10. | Ask how many 1000s are in 80000. |
| 2, 4 or 6 in question 8 | Scales as a length, as an area, or adds. | Tile the larger cube with the smaller one. |
| C in question 9 | Applies the cuboid rule to a triangular end. | Draw the triangle inside its rectangle and ask what fraction it is. |
Questions worth asking
- Why is volume measured in cubes? To fill a space you stack solid pieces, and a cube stacks with no gaps. Listen for “it fits”.
- Why does the shape of the end not matter? Only its area does: each layer has that many cubes. Listen for “same area, same layer”.
- If I double the length of a prism, what happens to the volume? It doubles: twice as many layers. Compare with doubling every edge.
- Can two boxes hold the same but be different shapes? Yes: 12 by 1 by 1 and 3 by 2 by 2 both hold 12. This leads into lesson 6.
Differentiation
A way in for anyone stuck
- Provide linking cubes or sugar cubes and let students build the box before they multiply.
- Give a three-column table: end area, length, volume, and fill the area column first.
- Start with cuboids only, then show one prism with a triangular end.
Stretch
- A cuboid has volume 72 cm³ and whole-number edges. List all the different cuboids. (There are 12: 1 by 1 by 72, 1 by 2 by 36, 1 by 3 by 24, 1 by 4 by 18, 1 by 6 by 12, 1 by 8 by 9, 2 by 2 by 18, 2 by 3 by 12, 2 by 4 by 9, 2 by 6 by 6, 3 by 3 by 8 and 3 by 4 by 6.)
- A prism has the same volume as a cube of edge 6 cm, and its end is a triangle of area 54 cm². How long is it? (216 ÷ 54 = 4 cm.)
Students learning in English as an additional language
“Volume” also means loudness. Say “volume of a box” with hands showing the space inside. “Cubic” links to “cube”: three directions.
| Word | Plain meaning | Example |
|---|---|---|
| volume | the amount of space inside a solid, counted in unit cubes | A box holds 36 cm³ |
| cubic centimetre (cm³) | the volume of a cube with edges of 1 cm | One sugar cube is about 1 cm³ |
| cuboid | a box shape with six rectangular faces | A brick or a shoe box |
| prism | a solid with the same end shape all along its length | A tent, a Toblerone box, a block of cheese |
| cross-section | the shape you see when you cut straight through a solid | The triangle at the end of a tent |
| layer | one level of unit cubes in a box | The bottom layer of a crate |
Sentence starters for explaining aloud or in writing:
- One layer has … cubes because …
- The area of the end is …, so the volume is …
- These two prisms hold the same because …
Two things not to say
“Volume is length times width times height.” That is only true when the end is a rectangle. Say “area of the end times the length”.
“Volume and area are the same thing in a different unit.” Volume is space in three directions, area in two.
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.
- CuboidA box is 8 cm by 5 cm by 3 cm. Find its volume.
120 cm³. - CubeA cube has edges of 4 cm. Find its volume.
64 cm³.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA swimming pool is 25 m long, 10 m wide and 2 m deep. How many cubic metres of water does it hold?
500 m³. - SpacingFrom Unit 7, lesson 4: the end of a prism is a house shape, a 6 m by 4 m rectangle with a triangle on top 2 m high and 6 m wide. Find the area of the end.
24 + 6 = 30 m².
3Challenge
Reasoning, generalising and working backwards.
- Working backA prism's end has area 18 cm² and the prism holds 117 cm³. How long is it?
6.5 cm. - ReasoningA cuboid is 4 cm by 6 cm by 10 cm. Another cuboid has the same volume and a square end 5 cm by 5 cm. How long is it?
240 ÷ 25 = 9.6 cm.
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.