Fill a box with unit cubes layer by layer. Each layer has as many cubes as the base has squares, so the volume is the area of the base times the height: V = A × h. Volume is in cubic units, cm³.
How many sugar cubes fit in a box, how much water fills a fish tank, how much concrete for a path? All are volumes: how much space a solid takes up.
Volume is how much space a solid takes up, counted in unit cubes. A cube 1 cm along each edge is one cubic centimetre, 1 cm³.
A box (a rectangular prism; a prism is any solid with the same shape all the way through) fills up in layers. The bottom layer covers the base: a 4 cm by 3 cm base holds 4×3=12 cubes. Every layer above is the same, so the number of layers is the height:
V=area of the base×height=A×h
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Fill the box
How many cubes fit on the bottom? How many layers?
TryTry it on the board and compare the result with your prediction.
Build it up
The top layer is orange. Count the cubes in it, then the layers.
Try it yourselfYour edits are saved in this browser.
Build the box up to 5 layers. How many cubes in each layer, and in all? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Base times height
Here is the idea, then the routine.
Worked exampleBefore each step, say what you think comes next.
4×3=12One layer covers the base: 12 cubes, as many as the base has squares.
12×5=60 cm35 layers of 12.
V=A×h=l×w×hFor a box, the base area is length × width.
V=6×2×3=36 cm3Volume is in cubic units: cm³.
Now try one: a 6 by 2 by 3 box.
Read Finn's method. Is the box hollow?
A box is 6 cm long, 2 cm wide and 3 cm high. What is its volume, in cm³?
Finn counts the cubes that fill a box. Click the line where his method goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
2 layers, so each layer holds 12.
Another hint
Which base is 12 squares?
Try it yourselfYour edits are saved in this browser.
Build a box with volume 24 cm³ that is 2 layers high.
Use the board. This ticks when you get there.
A prism has a base of area 20 cm² and height 7 cm. What is its volume, in cm³?
A box has volume 60 cm³ and a base 4 cm by 5 cm. How high is it, in cm?
A box is 2.5 cm long, 4 cm wide and 2 cm high. What is its volume, in cm³?
PractiseA few on your own, to check you have it.
Volume
A cube has edges of 5 cm. What is its volume, in cm³?
A box holds 48 cm³. Its base is 4 cm by 3 cm. How high is it, in cm?
Review: a triangle has base 8 cm and height 3 cm. Its area, in cm²?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
36 cubes, stacked
Then find another box with the same volume and a different height.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Which numbers multiply to 36?
Another hint
Choose a height first, then a base that fits.
Build a box with volume 36 cm³ that is more than one layer high.
Use the board. This ticks when you get there.
For teachers
Students build rectangular prisms from layers of unit cubes, see that each layer is the base area, and use V = A × h with cubic units, including working backwards to a missing length (AC9M7M02, node 1). About 40 minutes.
I can explain why the volume of a box is the area of its base times its height.
I can find the volume of a box in cm³ or m³.
I can find a missing length from a volume.
Before class
Do the Run board yourself first. Students need the area of a rectangle (Area unit) and multiplying decimals.
The one idea: a prism fills up in layers, each a copy of the base, so V = A × h. The common mistake is counting only the outside (Finn's mistake), or adding the faces (a surface area). Cubic units (cm³) are new.
Running the lesson
Predict: students who chose 94 counted the outside squares; 12 is one layer.
Run: ask how many cubes are hidden in the middle.
Investigate: link V = A × h to area: the base is an area, the height counts layers.
Modify: for the missing height, ask how many layers of 20 make 60.
Make: collect boxes with volume 36 and sort them by height.
The hard part (on purpose)
Seeing the box as full, not a shell of faces. Ask students to describe the hidden middle layer before they calculate, instead of counting for them.
When students get stuck
A student counts only the cubes or squares they can see (Finn's mistake). Ask: What is under the top layer? Fix: Every layer is full: count one layer, then the layers.
A student adds the three lengths. Ask: How many cubes in one layer? Fix: Multiply length × width for a layer, then × height.
A student writes cm² for a volume. Ask: Are we counting squares or cubes? Fix: Cubes: cm³.
Stretch and support
Stretch: Find every box made of 24 unit cubes (whole-number sides). Which has the least outside surface?
Support: Build with real cubes; use the board's layers for every question.
Where it leads
Next: Prisms (node 2): triangular prisms and capacity.
Where it leads: volume of cylinders (Year 9: the circle's area times the height), and surface area.
What each step is for
fill the box: Committing to a count for a 4 by 3 by 5 box shows whether they picture it full of layers or count only the outside.
build it up: Building the box layer by layer shows each layer is the base area, so the volume is base area × number of layers.
base times height: V = A × h, because every layer is a copy of the base; Finn counts the outside squares he can see, not the cubes that fill the box.
one change at a time: One change at a time: building a given volume, a base area given directly, a missing height (backwards), a decimal edge, then two boxes with the same volume and different shapes.
volume: A cube, a box with units and a missing length, mixed; counting the outside is the planted slip; the review keeps triangle area fresh.
36 cubes, stacked: Building a box with a set volume and more than one layer means finding a base and height that multiply to it.
Now you can
Explain why the volume of a box is the area of its base times its height.