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# Irregular shapes

Area, volume, circles and Pythagoras · Area

<!-- Speaker note: Students estimate the area of irregular shapes by counting full and part squares on finer and finer grids, and the perimeter with straight pieces round the edge (AC9M8M01, node 2). About 35 minutes (Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min). -->

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## Today

**We are learning to** a lake or a leaf has no formula. Lay a grid over it: count the full squares, add half the part squares, and multiply by the area of one square. A finer grid gives a better estimate; straight pieces round the edge estimate the perimeter.

- I can estimate an irregular area by counting full squares and half the part squares.
- I can explain why a finer grid gives a better estimate.
- I can estimate a curved perimeter and say how precise my answers are.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need composite areas (previous lesson). The one idea: count full squares, add half the part squares, multiply by one square's area; finer grids and more straight pieces get closer. The common mistake is counting every square the shape touches as whole (Sam's mistake). The board's measured area (shown on the finest grid) comes from the exact outline; the estimates are within about 1 km² there. Straight pieces always fall a little short of a curve. Wider: estimate a real area from a map or an aerial photo (a school oval, a local lake) with a printed grid. -->

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## Why it matters

> How big is a lake? Councils, farmers and park rangers estimate areas like this from maps all the time, with a grid and some careful counting.

<!-- Speaker note: An **irregular shape** has curved or uneven edges, so no formula fits. Estimate instead: - **Area:** put a grid over the shape. Count the **full squares**, and count each **part square** as half (some are mostly in, some mostly out, so on average half). Multiply by the area of one square. A **finer grid** gives a closer estimate. - **Perimeter:** join points on the edge with **straight pieces** and add their lengths. More, shorter pieces follow the curve more closely. Always say how precise your estimate is ("about 25 km², to the nearest whole km²"). -->

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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

**You count only the squares that are completely inside a lake. Is that the area, too little or too much?**

- Too little
- Too much
- Exactly the area

<!-- Speaker note: What happens: Too little. The squares on the edge are partly inside the lake, and their parts are left out. Counting each as a half fixes most of that. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what the orange squares are worth. -->

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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%2Firregular-shapes%2Fstep-2&code=widget%3A+composite%0Amode%3A+estimate%0Ablob%3A+lake%0Asize%3A+8%0Aslider+n%3A+4%2C+32%2C+8%2C+4%0Aunits%3A+km%0A&fresh=1}}

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

<!-- Speaker note: **Count the squares** colours full squares green and part squares orange. -->

<!-- Speaker note: Teacher: ask how much the estimate changed with the finer 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?

* Too little. The squares on the edge are partly inside the lake, and their parts are left out. Counting each as a half fixes most of that.

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

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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>

## Half a square

**Why is each part square counted as half a square?**

- So the answer is smaller
- Because the grid is half a kilometre
- Some are mostly in and some mostly out, so on average about half is inside
- Because half of them are inside

* Answer: Some are mostly in and some mostly out, so on average about half is inside

<!-- Speaker note: The edge cuts each part square somewhere: on average, half of it is inside. -->

<!-- Speaker note: Teacher: ask Sam to look at one orange square. -->

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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

**Sam estimates his lake's area. Click the line where his method goes wrong.**

```
 1  Sam's lake covers 20 squares completely and part of 14 more.
 2  Each square is 1 km by 1 km, so 1 km².
 3  He counts all 34 squares as 1 km² each.
 4  The lake's area is 34 km².
```

* Answer: line 3

<!-- Speaker note: Line 3: a square the edge passes through is only partly inside. Count each as half: 20 + 7 = 27 km². -->

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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>

## Straight pieces

**You estimate a curved edge with straight pieces between points on it. How does the estimate compare with the real perimeter?**

- A little longer
- A little shorter: a straight piece is the shortest way between its two points
- Exactly the same
- Twice as long

* Answer: A little shorter: a straight piece is the shortest way between its two points

<!-- Speaker note: Each straight piece cuts the corner of the curve, so the total is a little short; more pieces get closer. -->

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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>

## Closer estimates



- **More pieces:** 
- **The finest grid:** 
- **A leaf:** 
- **Precision:** 

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

<!-- Speaker note: What this step is for: More pieces and finer grids close in on the true values; a different shape tests the method; the answer states its precision. Hints, in order: Watch the perimeter estimate as n grows. / Press Finer grid twice. / Then Count the squares. For students who need more support: Give a printed grid and two coloured pencils: one for full squares, one for part squares. For early finishers: Trace a leaf from outside onto 1 cm grid paper. Estimate its area on 1 cm squares and then 0.5 cm squares. How much did the estimate change? The hard part, on purpose: Deciding what a part square is worth. Ask students to look at several orange squares and say how much of each is inside. When students get stuck: A student counts every square the shape touches as whole. Ask: Is this orange square all inside? Fix: Part squares count as half. / A student forgets the size of a square on a finer grid. Ask: What is the area of one 0.5 by 0.5 square? Fix: Multiply the count by one square's area: 0.25. / A student reports many decimal places. Ask: How precise can a 1 km grid be? Fix: Round to what the grid can tell you, and say so. -->

<!-- Speaker note: Teacher: ask when the extra effort of a finer grid is worth it. -->

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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>

## Count

**A shape covers 30 full squares and part of 12 more. Each square is 1 m². Estimate its area in m².**

* Answer: 36

<!-- Speaker note: 30 + 12 ÷ 2 = 36 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>

## Half-metre squares

**On a grid of 0.5 m squares, a shape covers 40 full squares and part of 20 more. Estimate its area in m².**

* Answer: 12.5

<!-- Speaker note: Each square is 0.5 × 0.5 = 0.25 m². (40 + 10) × 0.25 = 12.5 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>

## Perimeter

**Which gives the best estimate of a pond's perimeter?**

- Counting the squares inside it
- 24 short straight pieces round its edge
- 4 straight pieces round its edge
- Multiplying its length by its width

* Answer: 24 short straight pieces round its edge

<!-- Speaker note: More, shorter pieces follow the curve more closely. -->

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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: composite shapes

**Review: a rectangle 8 m by 5 m has a 2 m by 2 m square cut from one corner. What is its area in m²?**

* Answer: 36

<!-- Speaker note: 8 × 5 − 2 × 2 = 40 − 4 = 36 m². -->

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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: an island

Then write your estimate with its precision, and estimate the coastline with 32 pieces.

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

<!-- Speaker note: What this step is for: Estimating to a stated precision brings the counting method and the finer grid together. Hints, in order: Count on each grid, coarsest first. / The measured area appears on the finest grid. -->

<!-- Speaker note: Teacher: ask how they will say how precise their estimate is. -->

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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 covers 18 full squares and part of 10 more on a 1 cm grid. Estimate its area.**

* 18 + 10 ÷ 2 = 23 cm².

<!-- 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: This completes AC9M8M01. Where it leads: estimating areas under graphs (Years 11 and 12), and scale drawings and maps. -->