--- title: "Irregular shapes" subtitle: "Area, volume, circles and Pythagoras · Area · Lesson 13" --- # Irregular shapes **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Look at the squares the edge passes through. :::keep-together **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 My prediction, and why: {{lines-3}} ::: :::teacher-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. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Why half? :::keep-together ### 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 ::: :::teacher-note **Answer:** Some are mostly in and some mostly out, so on average about half is inside The edge cuts each part square somewhere: on average, half of it is inside. ::: :::keep-together ### 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. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: a square the edge passes through is only partly inside. Count each as half: 20 + 7 = 27 km². ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** A little shorter: a straight piece is the shortest way between its two points Each straight piece cuts the corner of the curve, so the total is a little short; more pieces get closer. ::: :::typical ## Closer estimates **More pieces.** **The finest grid.** **A leaf.** **Precision.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Estimating :::keep-together ### Count **A shape covers 30 full squares and part of 12 more. Each square is 1 m². Estimate its area in m².** Answer: {{field-15}} ::: :::teacher-note **Answer:** 36 30 + 12 ÷ 2 = 36 m². ::: :::keep-together ### 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: {{field-15}} ::: :::teacher-note **Answer:** 12.5 Each square is 0.5 × 0.5 = 0.25 m². (40 + 10) × 0.25 = 12.5 m². ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** 24 short straight pieces round its edge More, shorter pieces follow the curve more closely. ::: :::keep-together ### 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: {{field-15}} ::: :::teacher-note **Answer:** 36 8 × 5 − 2 × 2 = 40 − 4 = 36 m². ::: ## Make: an island Then write your estimate with its precision, and estimate the coastline with 32 pieces. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A shape covers 18 full squares and part of 10 more on a 1 cm grid. Estimate its area.** {{lines-3}} ::: :::teacher-note **Answer:** 18 + 10 ÷ 2 = 23 cm². :::