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# Area problems

Area, volume, circles and Pythagoras · Area

<!-- Speaker note: Students solve area problems by choosing the base and perpendicular height, converting to one unit, and splitting composite shapes into rectangles and triangles (adding or subtracting parts), with reasoning they own (AC9M7M01, node 2). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** pick the base and the height at a right angle to it, put both lengths in the same unit, and split a shape into rectangles and triangles. Area is in square units: m², cm².

- I can pick the right base and height when a shape has extra lengths.
- I can change lengths to the same unit before finding an area.
- I can find the area of a shape made of rectangles and triangles.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need the parallelogram and triangle formulas (previous lessons) and converting cm and m. The one idea: choose, convert, split. The common mistake is multiplying lengths in different units (Ivy's mistake); extra information (slanted sides) is planted in several tasks. 1 m² = 10 000 cm² surprises most students. The house-wall task is a reasoning check: the working is never marked, only the final area and the reason for the roof step. -->

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## Why it matters

> How much paint for the end wall of a house, with its triangle of roof? Real shapes mix rectangles and triangles, and real measurements mix metres and centimetres.

<!-- Speaker note: Area problems come with extra information. Three moves sort them out: - **Choose the base and its height.** The height is at a right angle to the base; a slanted side is not needed. - **Use one unit.** Change every length to the same unit (100 cm = 1 m) before you multiply. Area comes out in square units: m² or cm². - **Split the shape.** A shape made of rectangles and triangles has area equal to the sum of the parts (or a big shape minus a missing part). -->

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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 room is 4 m long and 250 cm wide. What is its area?**

- 1000 m²
- 10 m²
- 1000 cm
- 6.5 m²

<!-- Speaker note: What happens: 10 m². 250 cm = 2.5 m, and 4 × 2.5 = 10 m². 1000 mixes metres with centimetres; 6.5 adds. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 1000 m² did not convert. -->

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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%2Farea-problems%2Fstep-2&code=widget%3A+area%0Ashape%3A+rectangle%0Aslider+b%3A+1%2C+6%2C+1%0Aslider+h%3A+0.5%2C+4%2C+0.5%2C+0.5%0Aunits%3A+m%0A&fresh=1}}

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

<!-- Speaker note: Each grid square is 1 m by 1 m: one square metre. -->

<!-- Speaker note: Teacher: ask how the half-metre row shows on the grid. -->

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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?

* 10 m². 250 cm = 2.5 m, and 4 × 2.5 = 10 m². 1000 mixes metres with centimetres; 6.5 adds.

<!-- 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>

## Three moves

Here are the three moves.

Your working can be short notes. Then choose the reason for the key step.

$$\tfrac{1}{2} \times 10 \times 4 = 20 \text{ cm}^2$$

1) $250 \text{ cm} = 2.5 \text{ m}, \quad 4 \times 2.5 = 10 \text{ m}^2$ — Convert: both lengths in metres, then multiply.
2) $24 + 9 = 33 \text{ m}^2$ — Split: a rectangle (6 × 4) plus a triangle (½ × 6 × 3).
3) $8 \times 5 - 3 \times 2 = 34 \text{ m}^2$ — Or subtract: a rectangle with a corner missing is the big rectangle minus the corner.

<!-- 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 1 ÷ 2 × 10 × 4 = 20 cm^2 and say "Choose: the base and the height at a right angle to it. Ignore the slanted 5 cm side." Line 1: write 250 cm = 2.5 m, 4 × 2.5 = 10 m^2 and say "Convert: both lengths in metres, then multiply." Line 2: write 24 + 9 = 33 m^2 and say "Split: a rectangle (6 × 4) plus a triangle (½ × 6 × 3)." Line 3: write 8 × 5 - 3 × 2 = 34 m^2 and say "Or subtract: a rectangle with a corner missing is the big rectangle minus the corner." -->

<!-- Speaker note: Teacher: ask two students to share different splits of the house wall. -->

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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>

## Three moves

**The end wall of a house is a rectangle 6 m wide and 4 m high, with a triangle of roof on top: base 6 m, height 3 m. Find the area of the whole wall. Write your steps.**

* Answer: Rectangle: $6 \times 4 = 24$ m² (a rectangle's area is length × width); Triangle: $\tfrac{1}{2} \times 6 \times 3 = 9$ m² (a triangle is half of base × height); Wall: $24 + 9 = 33$ m² (the whole is the sum of its parts)

<!-- Speaker note: 24 + 9 = 33 m². -->

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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

**Ivy finds the area of a rug. Click the line where her working goes wrong.**

```
 1  A rug is 50 cm wide and 2 m long. Find its area.
 2  Area = length × width
 3  = 50 × 2
 4  = 100 m²
```

* Answer: line 3

<!-- Speaker note: Line 3: 50 is in centimetres and 2 in metres. Change 50 cm to 0.5 m: 0.5 × 2 = 1 m². -->

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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>

## Which lengths?

**Which two lengths give this triangle's area?**

- All three: 10 + 4 + 5
- The height 4 cm and the slanted side 5 cm
- The base 10 cm and the slanted side 5 cm
- The base 10 cm and the height 4 cm

* Answer: The base 10 cm and the height 4 cm

<!-- Speaker note: ½ × 10 × 4 = 20 cm². The 5 cm side is extra information. -->

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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



- **A sail:** Each task changes one thing.
- **Take a part away:** 
- **Square units:** 
- **Two steps: paint:** 
- **A sensible unit:** 

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

<!-- Speaker note: What this step is for: One change at a time: a sail on the board, a shape with a missing corner (subtract), square metres to square centimetres, a two-step paint problem, then choosing a sensible unit. Hints, in order: The sail's base is along the boom (3 m) and its height up the mast (5 m). For students who need more support: Use squared paper; convert all lengths to the smaller unit and write them on the sketch. For early finishers: Find the area of the letter L, T or H drawn on squared paper in two different ways (adding parts, and subtracting from a rectangle). The hard part, on purpose: Planning a composite area before calculating. Ask students to sketch the parts and label each with its own lengths first; resist giving the split. When students get stuck: A student multiplies cm by m (Ivy's mistake). Ask: What unit is each length in? Fix: Change both to metres (or both to cm) first. / A student uses every number given. Ask: Which lengths meet at a right angle? Fix: Use the base and its height; ignore slanted sides. / A student cannot start a composite shape. Ask: Can you see a rectangle and a triangle in it? Fix: Sketch the parts, find each area, then add (or subtract a missing part). -->

<!-- Speaker note: Teacher: for 1 m², draw a 1 m square and ask how many 1 cm squares fit along one side. -->

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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>

## Split

**A flag is a 10 cm by 4 cm rectangle with a triangle on one end: base 4 cm, height 3 cm. Total area, in cm²?**

* Answer: 46

<!-- Speaker note: 40 + 6 = 46 cm². -->

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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>

## Convert

**A path is 300 cm long and 2 m wide. Its area, in m²?**

* Answer: 6

<!-- Speaker note: 300 cm = 3 m, and 3 × 2 = 6 m². -->

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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>

## Extra information

**A right-angled triangle has base 12 cm, height 5 cm and sloping side 13 cm. Area, in cm²?**

* Answer: 30

<!-- Speaker note: ½ × 12 × 5 = 30 cm². -->

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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: percentages

**Review: find 25% of 48.**

* Answer: 12

<!-- Speaker note: A quarter of 48: 12. -->

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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: a tall sail

Then work out the cost of the sailcloth at $15 per m².

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

<!-- Speaker note: What this step is for: Designing a sail with a set area and a shape condition means searching base and height pairs, not just using a formula forwards. Hints, in order: ½ × b × h = 12, so b × h = 24. / Pick a pair where the height is bigger than the base. -->

<!-- Speaker note: Teacher: compare designs, then cost them. -->

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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 shape is a 7 m by 4 m rectangle with a triangle on top (base 7 m, height 2 m). Find its area.**

* 28 + ½ × 7 × 2 = 28 + 7 = 35 m².

<!-- 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: Volume (AC9M7M02): the area of the base times the height. Where it leads: area of trapeziums, circles and composite shapes with circles (Year 8), and surface area (Year 9). -->