At a glance. One lesson of about 60 minutes. Students explain why the volume of a prism is the area of the cross-section times the length, apply it to triangles, trapeziums, polygons and L shapes, convert cubic units to litres, and describe how volume and area scale.
Calculator: no calculator. The products are friendly and the key idea is the layer. A calculator would only matter for the hexagonal candle, which uses a decimal given in the question.
What students will learn
Know
- V = area of the cross-section × length.
- A volume is in cubic units; 1000 cm³ is 1 litre.
- A length is in cm, an area in cm², a volume in cm³.
Understand
- Every slice of a prism is the same, so the volume is one layer repeated.
- The cross-section need not be a rectangle.
- Multiplying every length by k multiplies the surface area by k² and the volume by k³.
Do
- Find the volume of a prism whose end is a triangle, trapezium, polygon or L shape.
- Find a missing length from a volume.
- Convert between cubic centimetres and litres.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: volume as area times length, units and conversions (questions 1 to 9).
- Criterion B. Investigating patterns: multiply every length by 2, 3 and 4 and describe how surface area and volume change, and then generalize (the second figure and question 8).
- Criterion C. Communicating: draw the cross-section on its own and write its area before multiplying by the length (questions 3 and 6).
- Criterion D. Applying mathematics in real-life contexts: channels, troughs and pools, where the cross-section is the side view (questions 3 and 7 and the channel).
The one thing to do with the figure
Ask for the volume of a layer. In the first figure, choose the L shape with a length of 1 cm and ask what the volume is. Students see it equals the area of the end in cubic units. Then ask for 5 layers.
In the second figure, hide the readouts and ask for the volume when every length is multiplied by 2. Many say double. Press the slider, and let them find 8 times as big, then ask why.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The channel and the prediction. First figure: show the slices. |
| 10 min | Key idea. Students stack cut-out paper shapes into a prism, and count layers. |
| 10 min | Second figure, worked examples, then the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B, ask which numbers are lengths, areas, volumes. |
| 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; 20 cm²; 90 cm². |
|---|---|
| Worked examples | 450 m³; 126 cm³; 60 litres. |
| Faded example | 280 cm³. |
| 1. Volume and its unit | A. B has the wrong unit, C adds, D doubles for the two ends. If many choose B, ask which of the three numbers could be a length, which an area, and which a volume. |
| 2. Volume of a triangular prism | 180 cm³. 15 × 12 = 180. |
| 3. Volume and litres | 80 litres. 80000 ÷ 1000 = 80. |
| 4. Missing length | 15 cm. 450 ÷ 30 = 15. |
| 5. A regular polygon end | 936 cm³. 93.6 × 10 = 936. |
| 6. An L-shaped end | 1320 cm³. 88 × 15 = 1320. |
| 7. A pool seen from the side | 100 m³. 20 × 5 = 100. |
| 8. Scaling a volume | 5000 cm³. 40 × 125 = 5000. |
| 9. Half used twice | A. B halves again, C accepts the double half, D adds. The point is the difference between “the end has area 15” and “the end has base 6 and height 5”. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1 | Gives the right number in the wrong unit. | Ask which of the three quantities is measured in cm³. |
| 360 or 15 in question 2 | Forgets the half, or stops at the area of the end. | Draw the end alone and ask for its area, then for the volume. |
| 80000 or 800 in question 3 | Forgets to convert, or divides by 100. | Write “1 litre = 1000 cm³” and divide by that. |
| 7.5 or 13500 in question 4 | Forgets the half, or multiplies. | Ask which number times 30 is 450. |
| 1872 or 93.6 in question 5 | Forgets the half, or stops at the area. | Ask for the area of the hexagon first, and for the unit. |
| 1620 or 300 in question 6 | Ignores the corner, or gives only the corner. | Shade the end and find its area on its own. |
| 200, 150 or 20 in question 7 | Forgets the half, uses one depth, or stops at the side view. | Draw the side view and find its area. Then ask what the width is for. |
| 200 or 1000 in question 8 | Multiplies once, or twice. | Ask how many lengths are in a volume. Three. |
| B, C or D in question 9 | Halves again, takes the double half, or adds. | Ask whether 15 is a base or an area. |
Questions worth asking
- Why does the shape of the end not matter? Every slice has the same area and the same thickness. Listen for “same layer, repeated”.
- How many cubic centimetres fit in one litre? 1000, a cube 10 cm by 10 cm by 10 cm. Listen for the cube.
- Why does the volume grow by 8 when the lengths double? Three lengths are doubled. Listen for “2 times 2 times 2”.
- Which is the cross-section of the pool? The side view: it is the same all the way across the width. Listen for “the side”.
Differentiation
A way in for anyone stuck
- Give stacks of cut-out paper shapes, or cubes arranged in an L, so students can count layers.
- Write the three units cm, cm² and cm³ on the board and ask which one each number needs.
- Draw the end on its own and find its area on its own line before multiplying.
Stretch
- A cube 4 cm on each side and a box 16 cm by 4 cm by 1 cm have the same volume. Compare their surface areas. (Both 64 cm³; the cube 96 cm², the box 2 × (64 + 16 + 4) = 168 cm².)
- A cube has a volume of 8 cm³. Another cube has a volume of 64 cm³. By what factor is the edge multiplied, and the surface area? (Edge 2 times, surface area 4 times.)
Students learning in English as an additional language
“Cubic” and “cube” share a root: a cubic centimetre is a cube 1 cm on each side. “Cross-section” is the shape you see when you cut straight through. “Capacity” is how much a container holds, usually in litres.
| Word | Plain meaning | Example |
|---|---|---|
| volume | the amount of space inside a solid | How much water a tank holds |
| cross-section | the shape you see when you cut straight through a solid | The end of a prism |
| cubic centimetre | the volume of a cube 1 cm on each side | A small sugar cube |
| litre | 1000 cubic centimetres | A bottle of water |
| capacity | how much a container can hold | The capacity of a tank is 60 litres |
| layer | one slice of a prism, all the same shape | One level of a stack of books |
Sentence starters for explaining aloud or in writing:
- The cross-section is a … with area …
- The volume is the area of the end times …
- The unit is cubic … because …
Two things not to say
“Volume is base times height times width.” True for a box, but a triangular prism has no single ‘width’ that works. Say “area of the end times length”.
“Just multiply everything you are given.” Extra lengths, such as sloping sides, are not used. Ask which numbers describe the end and which the length.
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 prism has an end of area 18 cm² and a length of 7 cm. Find its volume.
126 cm³. - FluencyA triangular prism has an end with a base of 10 cm and a height of 6 cm, and is 8 cm long. Find its volume.
240 cm³.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tank in Lagos is a box 60 cm by 50 cm by 40 cm. How many litres does it hold?
120 litres: 60 × 50 × 40 = 120000 cm³. - SpacingFrom lesson 4: a circle has a radius of 5 cm. What is its area? Use 3.14.
78.5 cm².
3Challenge
Reasoning, generalizing and working backwards.
- MissingA prism has a trapezium end with parallel sides of 4 cm and 8 cm and a height of 5 cm. Its volume is 150 cm³. How long is it?
5 cm: the end is 30 cm² and 150 ÷ 30 = 5. - ReasoningA prism has a volume of 60 cm³. Every length is halved. What is the new volume?
7.5 cm³: one eighth, because three lengths are halved.
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.