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

Five lessons for Year 8 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 every solid drawn, twelve solid cards to sort, two nets to fold and fill, 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. Every solid is drawn on the worksheets.
3. Print the two cut-out sheets, one copy per pair: the solid cards are for Day 1, the nets for Day 4 (print the nets at full size).
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 the way along its length, and its sides stand square to its two ends. So one rule gives the volume of every right prism: the area of the cross-section times the length, V = A × L. For a box the cross-section is a rectangle and the rule becomes length × width × height; for a triangular prism the cross-section area is half base times height; for an L-shaped bar it is the composite area from the unit before. Capacity is what a container holds, and it follows straight from volume: 1 cm³ holds 1 mL, 1000 cm³ holds 1 L, and 1 m³ holds 1000 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.

Units in this pack

Volumes are in cubic units (mm³, cm³, m³) and capacities in mL, L, kL and ML. Before multiplying, all three lengths go in the same unit: a garden bed 2 m long and 40 cm deep is 2 × 0.4, not 2 × 40. Cylinders, which need π, come in a later unit; here they appear only as a solid that is not a prism.

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 learns to see a prism by its cross-section and counts cubes layer by layer; Day 2 turns the layers into length × width × height; Day 3 shows the same rule works whatever the cross-section; Day 4 links volume to what a container holds; and Day 5 chooses units and solves longer problems. It builds on the volume of rectangular and triangular prisms in Year 7 and the areas of composite shapes from the unit before, and prepares cylinders and surface area in Year 9.

Materials for the week (one trip)

• From the classroom: rulers, sharp pencils, calculators, scissors, sticky tape or glue sticks, this pack printed.
• Cut-out sheet 1, one per pair: 12 solid cards (Day 1).
• Cut-out sheet 2, one per pair, printed at full size: two nets, an open box and an open triangular prism (Day 4).
• For Day 4: about a cup of dry rice or lentils per pair, and a measuring jug if you have one.

Words for the word wall

right prism, cross-section, uniform, end face, length, base, rectangular prism, triangular prism, cube, layer, volume, cubic centimetre, cubic metre, capacity, millilitre, litre, kilolitre, megalitre, net, fold

Dear families

This week in maths, Year 8 is finding the volume of prisms: solids like boxes, tents and building blocks that have the same shape all the way along. The rule is the area of the end shape times the length. They also link volume to capacity, what a container holds, using 1 cm³ = 1 mL and 1000 cm³ = 1 L.

Try this at home

• Find a box in the pantry. Measure its length, width and height in centimetres and multiply: that is its volume in cm³.
• Ask: a cereal box is 20 cm by 7 cm by 30 cm. What is its volume? (4200 cm³.)
• Ask: a carton holds 1 L. How many cubic centimetres is that? (1000.)
• Look for prisms: a tissue box, a chocolate bar shaped like a triangle, a pencil before it is sharpened.

My prisms this week

Fill one row a day. Name a prism you find, sketch its cross-section, and give its volume or capacity with units.

Printed from the free seegongsik Volume of prisms teaching pack · seegongsik.com/au/y8/measurement/AC9M8M02/pack

Prisms and cross-sections

A right prism keeps one shape, its cross-section, all the way along. Students sort solids, draw cross-sections, and count cubes in boxes layer by layer, ready for V = A × L.

We are learning to

• decide whether a solid is a right prism, and give the reason,
• draw and name the cross-section of a prism, whichever way it lies,
• find the volume of a box of cubes as one layer times the number of layers.

Success criteria

• I can say why a pyramid is not a prism but a pencil is.
• I can find the volume of a box of cubes without counting every cube.

You need

The worksheet and a pencil each; the solid cards from cut-out sheet 1, one set per pair, cut out.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Thinking the cross-section must be the bottom face: a prism can lie on its side.
• Counting only the cubes you can see, and missing the hidden ones inside.
• Calling a cylinder a prism: it keeps one shape, but its face is curved.

Answers

• Part A: right prisms 1, 2, 4, 5, 8, 10, 11; not right prisms 3, 6, 7, 9, 12. Reasons: square pyramid, it narrows to a point; cylinder, a curved face; it has its own name, though V = A × L still works for it; cone, it narrows to a point; leaning square prism, the sides lean, so they do not stand square to the ends; ball, no flat face at all.
• Part B: 1. a T-shape; 2. a right-angled triangle; 3. a cross (plus sign); 4. a parallelogram.
• Part C: 1. 4 × 3 = 12 in a layer, 2 layers, 24 cubes; 2. 5 × 2 = 10 in a layer, 3 layers, 30 cubes; 3. 3 × 3 = 9 in a layer, 4 layers, 36 cubes.
• Part D: a sharpened pencil is no longer a prism: near the point the cross-section gets smaller.

Prisms and cross-sections

A. Sort the solid cards

Write each card number in a pile. For every card that is not a right prism, give the reason in a few words.

B. Draw the cross-section

Draw the shape that runs all the way along each prism, then name it.

C. Boxes of cubes

Each box is built from 1 cm cubes, with no gaps inside. Find the cubes in one layer, the layers, and the volume.

D. Stretch: the sharpened pencil

A new hexagonal pencil is a right prism. Is it still one after you sharpen it? Explain using the word cross-section.

The volume of a box

A box is a prism whose cross-section is a rectangle, so its volume is the base area times the height. Students find volumes, work back from a volume to a missing edge, and order soil for a garden bed in cubic metres.

We are learning to

• find the volume of a rectangular prism as its base area times its height,
• find a missing edge by dividing the volume by the area of the face we know,
• put all three lengths in the same unit before multiplying.

Success criteria

• I can find the volume of a box and explain it as a base area times a height.
• I can find a missing edge when I know the volume and the other two edges.

You need

The worksheet and a calculator each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Adding the three edges instead of multiplying them.
• Multiplying lengths in different units, such as 2 × 1.5 × 40.
• Writing cm² for a volume: volume is in cubic units.

Answers

• Part A: 6 × 4 × 5 = 120 cm³; 10 × 8 × 3 = 240 cm³; 2.5 × 2 × 1.2 = 6 m³; 15 × 15 × 15 = 3375 cm³.
• Part B: 96 ÷ (8 × 4) = 3 cm; 200 ÷ (10 × 4) = 5 cm; 1.8 ÷ (3 × 1.5) = 0.4 m.
• Part C: 2 × 1.5 × 0.4 = 1.2 m³; 1.2 × $64.00 = $76.80.
• Part D: 1 × 1 × 24, 1 × 2 × 12, 1 × 3 × 8, 1 × 4 × 6, 2 × 2 × 6, 2 × 3 × 4; the 2 × 3 × 4 is the closest to a cube.

The volume of a box

A. Volumes

Find the volume of each box. Show the base area, then the volume, with units.

B. The missing edge

C. Soil for a garden bed

A raised garden bed is 2 m long and 1.5 m wide, and is filled with soil 40 cm deep. Soil costs $64.00 a cubic metre. How much soil is needed, and what does it cost?

D. Stretch: boxes of 24 cubes

List every box with whole-centimetre edges and a volume of 24 cm³ (a 2 by 3 by 4 box is the same box as a 4 by 3 by 2 one). Which is the closest to a cube?

Any right prism: V = A × L

One rule works for every right prism: the cross-section area times the length. Students find the volumes of triangular prisms, of prisms whose cross-section is a step or a T, and of a pool with a sloping floor.

We are learning to

• find the area of the cross-section first, then multiply by the length,
• use the triangle formula and the composite-area methods for a cross-section,
• pick out the prism’s length when the prism lies on its side.

Success criteria

• I can find the volume of a triangular prism as ½ × base × height × length.
• I can find the volume of a prism whose cross-section is made of rectangles.

You need

The worksheet and a calculator each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Forgetting the half in a triangle’s area.
• Using a sloping edge as the triangle’s height.
• Multiplying by the wrong length, such as the triangle’s base, instead of the prism’s length.

Answers

• Part A: tent: ½ × 2 × 1.5 × 3 = 4.5 m³; doorstop wedge: ½ × 8 × 3 × 4 = 48 cm³; the space under a roof: ½ × 10 × 2.4 × 12 = 144 m³.
• Part B: steps 60 × 20 + 30 × 20 = 1800 cm², × 120 = 216 000 cm³ (0.216 m³); beam 20 × 4 + 4 × 16 = 144 cm², × 300 = 43 200 cm³.
• Part C: A = 10 × 1 + ½ × 10 × 1 = 15 m², so V = 15 × 4 = 60 m³.
• Part D: double the length, the volume doubles; double the triangle’s base and height, the area and the volume are 4 times as big; double all three, 8 times.

Any right prism: V = A × L

A. Triangular prisms

Find the area of the triangle, then the volume.

B. Steps and a beam

Each cross-section is in centimetres. Find its area, then the volume.

C. A pool with a sloping floor

The drawing shows the water in a pool from the side: 1 m deep at the shallow end and 2 m at the deep end. The pool is 4 m wide. How much water does it hold, in m³?

D. Stretch: doubling

What happens to the volume of the tent in Part A if its length doubles? If the triangle’s base and height both double? If all three double? Explain without working out every volume.

From volume to capacity

Capacity is how much a container holds, and it follows straight from volume. Students change between cm³, mL and L, find what tanks and cartons hold, and fold two prisms of the same height to see that one holds half as much as the other.

We are learning to

• change between cubic centimetres, millilitres and litres, and between cubic metres and litres,
• find the capacity of a box-shaped container from its inside measurements,
• explain why a prism with half the floor, and the same height, holds half as much.

Success criteria

• I can say how many litres a container 50 cm by 20 cm by 30 cm holds.
• I can explain why the triangular prism holds half as much as the box.

You need

The worksheet and a calculator each; per pair, cut-out sheet 2 printed at full size, scissors, tape and a cup of dry rice.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dividing by 100 instead of 1000 to change mL to L.
• Thinking 1 m³ is 100 L: it is 100 × 100 × 100 = 1 000 000 cm³, so 1000 L.
• Measuring the outside of a container when the question asks what it holds.

Answers

• Part A: 250 cm³ = 250 mL; 3500 cm³ = 3.5 L; 2.5 L = 2500 cm³; 0.4 m³ = 400 L.
• Part B: fish tank: 60 × 30 × 40 = 72 000 cm³ = 72 000 mL; juice carton: 7 × 4 × 9 = 252 cm³ = 252 mL; rainwater tank: 2 × 1 × 1.5 = 3 m³ = 3000 L.
• Part C: box 5 × 4 × 3 = 60 cm³ = 60 mL; triangular prism ½ × 5 × 4 × 3 = 30 cm³ = 30 mL; the box fills it 2 times.
• Part D: 2 L = 2000 cm³, and 2000 ÷ (20 × 25) = 4 cm.

From volume to capacity

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

A. Change the units

B. What does it hold?

The measurements are inside measurements. Find each volume, then the capacity in mL or L.

C. Fold and fill

Fold both nets from cut-out sheet 2. Fill the box with rice, level with the top, and pour it into the triangular prism.

The box filled the triangular prism ______ times. Why?

D. Stretch: how high does it rise?

2 L of water is poured into an empty tank whose floor is 20 cm by 25 cm. How deep is the water?

Units and problems

The same rule gives a medicine cup and a dam, so the unit matters. Students choose units from mm³ to ML, solve problems in several steps, find a units mistake, and play the self-check quiz as a class game.

We are learning to

• choose a sensible unit for a volume or a capacity,
• plan a problem in steps and change units before we multiply,
• check an answer by asking whether its size makes sense.

Success criteria

• I can choose between mL, L, kL and ML for a container and say why.
• I can find how many 250 mL glasses a 3 L jug fills.

You need

The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Changing units halfway through and losing track: change everything first.
• Choosing cm³ for a pool or ML for a cup: check the size.
• Leaving 200 minutes as it is when hours and minutes are easier to read.

Answers

• Part A: a grain of sand, mm³; a dose of cough medicine, mL; a house brick, cm³; a skip bin of rubbish, m³; a bath full of water, L; a backyard pool, kL; an Olympic pool, ML.
• Part B: 1. 2000 ÷ 250 = 8 cups; 2. 6 × 4 × 3 = 72 cubes; 3. 8 × 1.2 × 0.1 = 0.96 m³; 4. 3000 ÷ 15 = 200 minutes = 3 h 20 min.
• Part C: Ari multiplied 2 m × 50 cm × 30 cm as if all were metres. In metres: 2 × 0.5 × 0.3 = 0.3 m³, which holds 300 L.
• Part D: 1 m³ holds 1000 L, so 1000 one-litre cartons; at about 1 kg a litre, the water has a mass of about 1000 kg, a tonne.

Units and problems

A. Which unit?

Choose mm³, cm³, m³, mL, L, kL or ML for each. Use each one once.

B. Problems in steps

Estimate first, then work it out. Write the units at every step.

1. How many 250 mL cups can be filled from a full 2 L jug?
2. How many 10 cm cubes fit in a crate 60 cm by 40 cm by 30 cm?
3. A path 8 m long and 1.2 m wide is made of concrete 10 cm thick. How much concrete is that?
4. A 3000 L tank fills at 15 L a minute. How long does it take to fill, in hours and minutes?

C. Find the mistake

Ari found the volume of a trench 2 m by 50 cm by 30 cm and wrote 2 × 50 × 30 = 3000 m³. What went wrong? Give the right volume in m³ and in litres.

D. Stretch: a cubic metre of water

How many 1 L cartons would it take to fill a cube 1 m on each side? A litre of water has a mass of about 1 kg. About how heavy is a cubic metre of water?

Solid cards

For Day 1, one set per pair: twelve solids

Sort the cards into right prisms and solids that are not right prisms. Be ready to give a reason for every card.

Teacher note: right prism: 1 (box), 2 (hexagonal pencil, unsharpened), 4 (L-shaped bar), 5 (tent), 8 (trapezium ramp), 10 (house-shaped prism), 11 (triangular prism standing on its end). not a right prism: 3 (square pyramid), 6 (cylinder), 7 (cone), 9 (leaning square prism), 12 (ball).

Two nets to fold and fill

For Day 4, one sheet per pair, printed at full size

Cut along the solid lines, fold along the dashed lines with the printing outside, and tape each grey tab inside the wall next to it. Both prisms are open at the top and 3 cm tall. Check with a ruler: the box floor should be 5 cm by 4 cm. If it is smaller, the sheet printed shrunk, but the triangular prism still holds half as much as the box.

What we know: Volume of prisms

Work on your own. Use a calculator only for Questions 4 to 7. Show your working and give units.

1. Which solid is a right prism? Why is the other one not a right prism? ______________________
   A

   B
2. Find the volume of a box 8 cm by 5 cm by 4 cm. ______________________
3. A box has a volume of 90 cm³ and a base 6 cm by 5 cm. How tall is it? ______________________
4. Find the volume of this triangular prism. ______________________
5. A bar 10 cm long has this L-shaped cross-section, measured in cm. Find its volume. ______________________
6. A tank is 50 cm by 40 cm by 30 cm inside. How many litres does it hold? ______________________
7. Change 1.5 L to mL and to cm³. Change 2.4 m³ to litres. ______________________
8. Mia says, “If you double every edge of a box, you double its volume.” Is Mia right? Show an example. ______________________

Answers and marking guide

Answers with working

1. A is a right prism (a pentagon all the way along); B narrows to a point, so it is a pyramid, not a prism.
2. 8 × 5 × 4 = 160 cm³.
3. 90 ÷ (6 × 5) = 3 cm.
4. ½ × 6 × 5 = 15 cm², × 9 = 135 cm³.
5. 6 × 2 + 2 × 3 = 18 cm², × 10 = 180 cm³.
6. 50 × 40 × 30 = 60 000 cm³ = 60 L.
7. 1.5 L = 1500 mL = 1500 cm³; 2.4 m³ = 2400 L.
8. No: a 1 by 1 by 1 box has volume 1, a 2 by 2 by 2 box has volume 8. Doubling every edge multiplies the volume by 2 × 2 × 2 = 8.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and the right cubic or capacity unit.

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: prisms and cross-sections, boxes, V = A × L, capacity, and units and problems.