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# Volume in layers

Area, volume, circles and Pythagoras · Volume

<!-- Speaker note: 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 (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** 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.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> 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.

<!-- Speaker note: **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 $4 \times 3 = 12$ cubes. Every layer above is the same, so the number of layers is the height: $$V = \text{area of the base} \times \text{height} = A \times h$$ -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A box is 4 cubes long, 3 cubes wide and 5 cubes high. How many cubes fill it?**

- 12
- 47
- 60
- 94

<!-- Speaker note: What happens: 60: 5 layers of 12. 12 is one layer; 94 counts the squares on all six outside faces, which is not the space inside. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 94 counted the outside squares; 12 is one layer. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fmeasurement%2Fvolume-layers%2Fstep-2&code=widget%3A+prism%0Al%3A+4%0Aw%3A+3%0Aslider+h%3A+0%2C+6%2C+0%0Aunits%3A+cm%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/measurement/volume-layers/?full=step-2#step-2)

<!-- Speaker note: The top layer is orange. Count the cubes in it, then the layers. -->

<!-- Speaker note: Teacher: ask how many cubes are hidden in the middle. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 60: 5 layers of 12. 12 is one layer; 94 counts the squares on all six outside faces, which is not the space inside.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Layers

Here is the idea, then the routine.

Now try one: a 6 by 2 by 3 box.

$$4 \times 3 = 12$$

1) $12 \times 5 = 60 \text{ cm}^3$ — 5 layers of 12.
2) $V = A \times h = l \times w \times h$ — For a box, the base area is length × width.
3) $V = 6 \times 2 \times 3 = 36 \text{ cm}^3$ — Volume is in cubic units: cm³.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 4 × 3 = 12 and say "One layer covers the base: 12 cubes, as many as the base has squares." Line 1: write 12 × 5 = 60 cm^3 and say "5 layers of 12." Line 2: write V = A × h = l × w × h and say "For a box, the base area is length × width." Line 3: write V = 6 × 2 × 3 = 36 cm^3 and say "Volume is in cubic units: cm³." -->

<!-- Speaker note: Teacher: link V = A × h to area: the base is an area, the height counts layers. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Layers

**A box is 6 cm long, 2 cm wide and 3 cm high. What is its volume, in cm³?**

* Answer: 36

<!-- Speaker note: Base 6 × 2 = 12 cm², 3 layers: 12 × 3 = 36 cm³. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Finn counts the cubes that fill a box. Click the line where his method goes wrong.**

```
 1  A box is 4 cubes long, 3 cubes wide and 2 cubes high. How many cubes fill it?
 2  I count the squares I can see: top 12, front 8, side 6.
 3  12 + 8 + 6 = 26
 4  So 26 cubes fill the box.
```

* Answer: line 2

<!-- Speaker note: Line 2: the squares you can see are only the outside. The box is full: 4 × 3 = 12 cubes in a layer, 2 layers: 24. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Why it works

**Why is the volume of a box the area of its base times its height?**

- Because the base is the biggest face
- Because volume is always three numbers multiplied
- Each layer has one cube for each square of the base, and there are as many layers as the height

* Answer: Each layer has one cube for each square of the base, and there are as many layers as the height

<!-- Speaker note: Each layer is the base (A cubes), and there are h layers: V = A × h. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Volume 24:** Each task changes one thing.
- **A base area:** 
- **Backwards: the height:** 
- **A decimal:** 
- **Same volume:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/volume-layers/?full=step-4#step-4)

<!-- Speaker note: What this step is for: 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. Hints, in order: 2 layers, so each layer holds 12. / Which base is 12 squares? For students who need more support: Build with real cubes; use the board's layers for every question. For early finishers: Find every box made of 24 unit cubes (whole-number sides). Which has the least outside surface? 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. -->

<!-- Speaker note: Teacher: for the missing height, ask how many layers of 20 make 60. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A cube

**A cube has edges of 5 cm. What is its volume, in cm³?**

* Answer: 125

<!-- Speaker note: 5 × 5 × 5 = 125 cm³. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Units

**A fish tank is 50 cm long, 20 cm wide and 30 cm high. What is its volume?**

- 100 cm³
- 1000 cm³
- 30 000 cm³
- 30 000 cm²

* Answer: 30 000 cm³

<!-- Speaker note: 50 × 20 = 1000 cm² base, × 30 = 30 000 cm³. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Backwards

**A box holds 48 cm³. Its base is 4 cm by 3 cm. How high is it, in cm?**

* Answer: 4

<!-- Speaker note: 48 ÷ 12 = 4 cm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: triangles

**Review: a triangle has base 8 cm and height 3 cm. Its area, in cm²?**

* Answer: 12

<!-- Speaker note: ½ × 8 × 3 = 12 cm². -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: 36 cubes, stacked

Then find another box with the same volume and a different height.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/volume-layers/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building a box with a set volume and more than one layer means finding a base and height that multiply to it. Hints, in order: Which numbers multiply to 36? / Choose a height first, then a base that fits. -->

<!-- Speaker note: Teacher: collect boxes with volume 36 and sort them by height. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A box is 5 cm by 4 cm by 3 cm. Find its volume. A box with base 6 cm² has volume 42 cm³: how high is it?**

* 5 × 4 × 3 = 60 cm³. 42 ÷ 6 = 7 cm.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this 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. -->