--- title: "Composite shapes" subtitle: "Area, volume, circles and Pythagoras · Area · Lesson 12" --- # Composite shapes **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** An L-shaped bedroom, a house front with a pointed roof, a photo frame: none is a single rectangle, but each is easy once you cut it up. :::teacher-note **In one sentence:** Students cut composite shapes into rectangles and triangles, or take pieces away from a rectangle, and find the area and the perimeter (AC9M8M01, node 1). **Students should be able to say:** I can cut a composite shape into pieces I know and add their areas. I can find an area by taking pieces away from a big rectangle. I can work out a missing side and the perimeter. **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 Sketch it, then decide: cut it up, or take a piece away? :::keep-together **An L-shaped room is 8 m wide and 6 m tall, with a 5 m by 3 m corner missing. What is its area?** ☐ 18 m² ☐ 28 m² ☐ 33 m² ☐ 48 m² My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 33 m². The whole 8 by 6 rectangle is 48 m², and the missing corner is 15 m²: 48 − 15 = 33. (28 is the perimeter, in metres.) ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Lengths and pieces :::keep-together ### Missing lengths Here is the routine. 1. $\text{an L } 10 \text{ m by } 7 \text{ m, a } 6 \times 4 \text{ corner missing}$ — A different L, as an example. 2. $\text{pieces: } 4 \times 7 \text{ and } 6 \times 3$ — Cut into shapes you know, with each piece's own lengths. 3. $28 + 18 = 46\ \text{m}^2$ — Add the pieces. 4. $10 \times 7 - 6 \times 4 = 70 - 24 = 46\ \text{m}^2$ — Or: the big rectangle, take away the missing corner. 5. $10 + 3 + 6 + 4 + 4 + 7 = 34\ \text{m}$ — The perimeter adds every side, in metres. Now find the step. **The L-shape is 8 m along the bottom and 3 m along the top. How long is the step across in the middle?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5 The bottom (8 m) = the top (3 m) + the step: 8 − 3 = 5 m. ::: :::keep-together ### Find the mistake **Ella finds the area of the L-shape. Click the line where her working goes wrong.** ``` 1 The L-shape is 8 m wide at the bottom and 6 m tall on the left. 2 Ella cuts it up and down into a piece 3 m wide and a piece 5 m wide. 3 Left piece: 3 × 6 = 18 m². 4 Right piece: 5 × 6 = 30 m². 5 Total: 18 + 30 = 48 m². ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: the right piece is only 3 m tall, not 6 m. 5 × 3 = 15, so the total is 18 + 15 = 33 m². ::: :::keep-together ### Three ways **Ali cuts the L across, Bea cuts it up and down, and Cal takes a corner away from a rectangle. Who gets the right area?** ☐ Only Cal ☐ All three, if each uses the right lengths ☐ None: you must use a formula for L-shapes ☐ Only Ali and Bea ::: :::teacher-note **Answer:** All three, if each uses the right lengths The area does not depend on how you cut it, only on using each piece's own lengths. ::: :::typical ## Other shapes **Take away.** **A house.** **Missing sides.** **Units.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Composite shapes :::keep-together ### Rectangle and triangle **A shape is a rectangle 10 cm by 4 cm with a triangle on top (base 10 cm, height 3 cm). What is its area in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 55 10 × 4 + ½ × 10 × 3 = 40 + 15 = 55 cm². ::: :::keep-together ### A frame **A photo frame is 30 cm by 20 cm on the outside, with a 24 cm by 14 cm hole for the photo. What is the frame's area in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 264 30 × 20 − 24 × 14 = 600 − 336 = 264 cm². ::: :::keep-together ### Perimeter shortcut **An L-shape fits exactly inside an 8 m by 6 m rectangle (one corner cut out). What is its perimeter in metres?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 28 The steps in the L add up to the missing sides of the rectangle, so the perimeter is 8 + 6 + 8 + 6 = 28 m. ::: :::keep-together ### Review: triangles **Review: what is the area of a triangle with base 6 cm and height 5 cm, in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 15 ½ × 6 × 5 = 15 cm². ::: :::keep-together ### Another L **An L-shape is 10 m wide along the bottom and 7 m tall on the left. Its left part is 4 m wide; its right part is 6 m wide and only 3 m tall. What is its area in m²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 46 4 × 7 + 6 × 3 = 28 + 18 = 46 m². ::: ## Make: design a garden Then find your garden's perimeter, and say how many metres of edging it needs. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A rectangle 12 m by 5 m has a 4 m by 2 m corner cut out. Find its area.** {{lines-3}} ::: :::teacher-note **Answer:** 12 × 5 − 4 × 2 = 60 − 8 = 52 m². :::