Volume of prisms: a week of ready-to-teach maths

Five lessons for Year 7 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus prism nets to fold, prism problem cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets: prism nets (Days 1 and 3), one per student, and prism problem cards (Day 4), one per pair.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A right prism has the same cross-section all along its length. Fill it in layers, each the shape of the base: one layer holds as many cubic units as the base has square units, and the height says how many layers there are. So the volume of any right prism is base area × height, which is the same as cross-section area × length. For a box that is length × width × height; for a triangular prism, the triangle’s area, ½ × b × h, times the length of the prism. Volume is in cubic units, and 1000 cm³ fill exactly 1 L.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write volume

Write the base area first, then multiply: V = A × h = 8 × 6 = 48 cm³. Name the unit every time: cm² for the base, cm³ for the volume. Diagrams are not drawn to scale unless they are on a grid, and a dashed line is a height. Calculators are welcome on Day 5 only.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 counts cubes in layers; Day 2 turns the layers into base area × height; Day 3 uses a triangle for the base; Day 4 takes the same rule to any cross-section; and Day 5 links cubic units to capacity. It builds on the area of triangles and parallelograms (AC9M7M01) and on capacity in litres from Year 6, and it leads into the surface area and volume of prisms in Year 8.

Materials for the week (one trip)

• From the classroom: pencils, rulers, scissors, sticky tape, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per student: a box net and a triangular prism net (Days 1 and 3).
• Cut-out sheet 2, one per pair: 8 prism problem cards (Day 4).
• Unit cubes or blocks if the school has them (Days 1 and 2, not essential); calculators for Day 5.

Words for the word wall

volume, capacity, cubic unit, cubic centimetre (cm³), cubic metre (m³), millilitre, litre, prism, right prism, cross-section, base, base area, height, length, layer, net, face, edge, rectangular prism, triangular prism

Dear families

This week in maths, Year 7 is finding the volume of prisms, solid shapes with the same cross-section all the way along, such as boxes, wedges and tents, and linking cubic centimetres to litres.

Try this at home

• Find a box at home. Measure its length, width and height, and work out how many 1 cm cubes would fill it.
• Read the volume on a milk carton or juice box. Measure the carton: is its volume close to what it holds?
• Look for prisms: a tent, a chocolate bar with triangular ends, a slice of cheese, a loaf of bread.
• Ask: if every edge of a box doubled, how many of the old boxes would fit inside? (Eight.)

Volumes this week

Fill one row a day with a prism from class, its base area and height, and its volume with the unit.

Printed from the free seegongsik Volume of prisms teaching pack · seegongsik.com/au/y7/measurement/AC9M7M02/pack

Boxes of cubes

Volume counts the unit cubes that fill a solid. Students count cubes in layers, see why length × width × height counts them all at once, fold a box, and find the volume of boxes from their lengths.

We are learning to

• count the cubes in a box by layers, not one by one,
• find the volume of a box with V = l × w × h,
• write the answer in cubic units.

Success criteria

• I can explain why l × w × h counts every cube.
• I can find the volume of a box and work back to a missing length.

You need

The box net from cut-out sheet 1 (one per student), scissors, tape, unit cubes if you have them, the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after folding; start the second with Part A.

Watch for these ideas

• Counting only the cubes that can be seen. The hidden ones are there too: count a layer, then the layers.
• Adding the three lengths instead of multiplying them.
• Writing cm or cm² for a volume. Volume is in cubic units, cm³.

Answers

• Box net: 5 × 2 × 2 = 20 cubes.
• Part A: 5 × 2 × 3 = 30; 4 × 4 × 2 = 32; 5 × 3 × 3 = 45 cubes.
• Part B: 7 × 5 × 3 = 105 cm³; 9 × 4 × 2.5 = 90 m³; 4 × 4 × 4 = 64 cm³.
• Part C: height = 96 ÷ 24 = 4 cm; width = 150 ÷ 30 = 5 m.
• Part D: 1 × 1 × 24, 1 × 2 × 12, 1 × 3 × 8, 1 × 4 × 6, 2 × 2 × 6, 2 × 3 × 4: 6 boxes.

Boxes of cubes

A. Count by layers

Each box is filled with unit cubes, hidden ones too. How many cubes in a layer? How many layers? How many cubes?

B. Boxes from their lengths

Not drawn to scale.

C. A missing length

1. A box has a volume of 96 cm³ and a base 4 cm by 6 cm. Find its height.   ____________
2. A box has a volume of 150 m³ and a length of 10 m and a height of 3 m. Find its width.   ____________

D. Stretch

A box is filled with exactly 24 unit cubes, and every edge is a whole number of units. List every box it could be. How do you know you have them all?

Base area × height

One layer of a prism holds as many cubes as its base has squares, and the height counts the layers. Students use V = A × h, stand one box on each face, and work back from a volume to a base area or a height.

We are learning to

• explain V = A × h as layers of the base,
• find a volume from a base area and a height,
• work back to a base area or a height.

Success criteria

• I can say what the base area and the height each count.
• I can show that a box has one volume whichever face it stands on.

You need

The box folded on Day 1, the worksheet, and the unit open on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after turning the box; start the second with Part A.

Watch for these ideas

• Multiplying the base area by another side of the base. The base area already holds two of the lengths.
• Thinking the base must be the bottom face. Any face can be the base, with the height at right angles to it.
• Dividing the wrong way when working back: height = volume ÷ base area.

Answers

• Part A: 25 × 6 = 150 cm³; 18 × 2.5 = 45 m³; 40 × 0.5 = 20 cm³.
• Part B: 15 × 2, 10 × 3, 6 × 5: 30 every way.
• Part C: height = 252 ÷ 36 = 7 cm; base area = 200 ÷ 8 = 25 m².
• Part D: 20 × 9 = 180 and 24 × 7 = 168, so the first holds more; the volume stays the same, because the doubling and the halving cancel.

Base area × height

A. Layers of the base

1. A prism has a base area of 25 cm² and a height of 6 cm. Find its volume.   ____________
2. A prism has a base area of 18 m² and a height of 2.5 m. Find its volume.   ____________
3. A prism has a base area of 40 cm² and a height of 0.5 cm. Find its volume.   ____________

B. One box, three ways

C. Work back

1. A prism has a volume of 252 cm³ and a base area of 36 cm². Find its height.   ____________
2. A prism has a volume of 200 m³ and a height of 8 m. Find its base area.   ____________

D. Stretch

1. One box has a base area of 20 cm² and a height of 9 cm. Another has a base area of 24 cm² and a height of 7 cm. Which holds more?
2. The base area of a prism doubles and its height halves. What happens to its volume? Explain.

Triangular prisms

A triangular prism follows the same rule as a box: area of the triangle, then times the length. Students fold a triangular prism, find triangle areas first, and catch the two usual slips: forgetting to halve, and using a slanted side.

We are learning to

• find the area of the triangular end with ½ × b × h,
• multiply by the length of the prism,
• tell the triangle’s height from the prism’s length.

Success criteria

• I can find the volume of a triangular prism in two steps.
• I can say which given lengths are not needed.

You need

The triangular prism net from cut-out sheet 1, scissors, tape, the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after folding; start the second with Part A.

Watch for these ideas

• Forgetting to halve: b × h × length is the volume of a box twice the size.
• Using a slanted side as the triangle’s height.
• Mixing the triangle’s height with the prism’s length. Call the second one the length.

Answers

• Net: ½ × 4 × 3 = 6 squares, × 7 = 42 unit cubes.
• Part A: ½ × 6 × 5 = 15, × 9 = 135 cm³; ½ × 10 × 4 = 20, × 12 = 240 m³; ½ × 8 × 3 = 12, × 4 = 48 cm³.
• Part B: Lee forgot to halve; ½ × 6 × 4 × 11 = 132 cm³.
• Part C: length = 90 ÷ 15 = 6 cm; triangle area = 200 ÷ 10 = 20 cm², so its height = 2 × 20 ÷ 8 = 5 cm.
• Part D: ½ × 3.5 × 3 × 20 = 105 cm³; each half is 54 cm³, half of 4 × 3 × 9.

Triangular prisms

A. Triangle first, then the length

Not drawn to scale. Some lengths are not needed.

B. Find the slip

Lee found the volume of a triangular prism whose triangle has base 6 cm and height 4 cm, and which is 11 cm long, as 6 × 4 × 11 = 264 cm³. What went wrong? What is the volume?

C. Work back

1. A triangular prism has a volume of 90 cm³, and its triangle has an area of 15 cm². How long is it?   ____________
2. A triangular prism is 10 cm long with a volume of 200 cm³. Its triangle has a base of 8 cm. What is the triangle’s height?   ____________

D. Stretch

1. A chocolate box has triangular ends with base 3.5 cm and height 3 cm, and it is 20 cm long. Find its volume.
2. A box 4 cm by 3 cm by 9 cm is cut corner to corner along its length into two triangular prisms. What is the volume of each?

Any prism

Every right prism is one cross-section carried along a length, so its volume is cross-section area × length. Students find the area of trapezium, house and pool cross-sections first, then work through the prism cards.

We are learning to

• find the area of a prism’s cross-section, splitting it if needed,
• use V = cross-section area × length for any right prism,
• work back to a missing length.

Success criteria

• I can find the volume of a prism whose end is not a rectangle or a triangle.
• I can say which face of a prism is its cross-section.

You need

Prism problem cards from cut-out sheet 2 (one set per pair) and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the carousel; start the second with Part A.

Watch for these ideas

• Multiplying all the lengths on the picture. Find the cross-section area first.
• Taking the wrong face as the cross-section. It is the face that stays the same all the way along.
• Forgetting a piece of a split cross-section, such as the roof of the shed.

Answers

• Part A: (a) (8 + 4) ÷ 2 × 5 = 30, × 7 = 210 cm³; (b) 8 × 4 + ½ × 8 × 3 = 44, × 10 = 440 cm³; (c) (1 + 2) ÷ 2 × 25 = 37.5 m², × 10 = 375 m³.
• Cards: 1 72 m³, 2 100 cm³, 3 60 cm³, 4 84 m³, 5 180 cm³, 6 144 m³, 7 216 cm³, 8 120 cm³.
• Part C: 96 ÷ 12 = 8 m.
• Part D: 30 ÷ 1.5 = 20 slices; 90 × 30 = 2700 cm³, which is also 20 slices of 135 cm³.

Any prism

A. Cross-section first

Not drawn to scale. The dashed lines are heights.

B. Card carousel

C. A missing length

A prism has a cross-section of area 12 m² and a volume of 96 m³. How long is it?   ____________

D. Stretch

A loaf of bread is a prism 30 cm long. It is cut into slices 1.5 cm thick, each with an area of 90 cm². How many slices are there, and what is the volume of the loaf?

Units and capacity

Volume is in cubic units, and capacity in millilitres and litres: 1 cm³ holds 1 mL, 1000 cm³ hold 1 L, and 1 m³ holds 1000 L. Students solve tank, box and path problems, change units, and then play the quiz.

We are learning to

• choose cm³ or m³ for a volume, and mL or L for a capacity,
• change cm³ to mL and L, and m³ to L,
• put every length in one unit before multiplying.

Success criteria

• I can find how many litres a box-shaped tank holds.
• I can explain why 1 m³ is 1000 L.

You need

The worksheet, calculators, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Thinking 1 m³ is 100 L or 1000 cm³. It is 1000 L, and 1 000 000 cm³.
• Multiplying metres by centimetres. Change the 10 cm depth to 0.1 m first.
• Writing a capacity in m². Volume is cubic; capacity is mL or L.

Answers with working

1. 60 × 30 × 40 = 72 000 cm³ = 72 L.
2. 2 × 1.5 × 1 = 3 m³ = 3000 L.
3. 6 × 4 × 10 = 240 cm³, which is 240 mL.
4. 2 × 2 × 0.3 = 1.2 m³.
5. 8 × 1.2 × 0.1 = 0.96 m³; 10 cm is 0.1 m.
6. a dose of medicine in a syringe: mL; the water in a swimming pool: m³; a bottle of water: mL or L.
7. 1 m is 10 lengths of 10 cm, so 1 m³ holds 10 × 10 × 10 = 1000 cubes of 1 L: 1000 L.
8. 8000 L = 8 m³, and 2 × 2 × 2 = 8, so each edge is 2 m.

Units and capacity

1 cm³ holds 1 mL · 1000 cm³ hold 1 L · 1 m³ holds 1000 L

Problems (7 and 8 are the stretch)

1. A fish tank is 60 cm long, 30 cm wide and 40 cm high. Find its volume in cm³, then how many litres it holds.
2. A rainwater tank is a box 2 m by 1.5 m by 1 m. Find its volume in m³ and in litres.
3. A juice box is 6 cm by 4 cm by 10 cm. How many millilitres does it hold when full?
4. A sandpit is 2 m by 2 m, filled with sand 0.3 m deep. How much sand is needed, in m³?
5. A concrete path is 8 m long, 1.2 m wide and 10 cm deep. How much concrete is needed, in m³?
6. Which unit would you use for a dose of medicine in a syringe, the water in a swimming pool, and a bottle of water: mL, L or m³?
7. Stretch. How many litres does 1 m³ hold? Explain with 10 cm cubes.
8. Stretch. A cube-shaped tank holds 8000 L when full. How long is each edge?

Prism nets to fold

Cut out each net along its outside edge, fold along every line, and tape the edges. Each square is 1 unit. Day 1: the box. Day 3: the triangular prism.

Prism problem cards

Cut out the 8 cards. Day 4: find each prism’s volume, with its unit. Not drawn to scale; the dashed lines are heights.

What we know: Volume of prisms

Work on your own. Show your working and write each answer with its unit. Diagrams are not to scale.

Answers and marking guide

Answers with working

1. 7 × 2 = 14 in a layer, 2 layers: 28 cubes.
2. 9 × 4 × 3 = 108 cm³.
3. 16 × 7 = 112 cm³.
4. ½ × 7 × 4 = 14 cm², × 5 = 70 cm³; the 6.4 cm side is not used.
5. 126 ÷ (6 × 7) = 126 ÷ 42 = 3 cm.
6. (7 + 3) ÷ 2 × 4 = 20 m², × 11 = 220 m³.
7. 50 × 40 × 30 = 60 000 cm³, and every 1000 cm³ is 1 L: 60 L.
8. 8 times: 2 × 2 × 2 = 8. Twice as many cubes in each direction, so 2 × 2 in a layer and twice as many layers.

A quick three-level guide

Eight questions, four ideas. A student at standard finds the cross-section area first, multiplies by the length, and writes every volume in cubic units, changing to litres where asked.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: boxes of cubes, base area times height, triangular prisms, any prism, and units.