--- title: "Area problems" subtitle: "Area, volume, circles and Pythagoras · Area · Lesson 3" --- # Area problems **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **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 Write both lengths in metres first. :::keep-together **A room is 4 m long and 250 cm wide. What is its area?** ☐ 1000 m² ☐ 10 m² ☐ 1000 cm ☐ 6.5 m² My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 10 m². 250 cm = 2.5 m, and 4 × 2.5 = 10 m². 1000 mixes metres with centimetres; 6.5 adds. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Choose, convert, split :::keep-together ### Three moves Here are the three moves. 1. $\tfrac{1}{2} \times 10 \times 4 = 20 \text{ cm}^2$ — Choose: the base and the height at a right angle to it. Ignore the slanted 5 cm side. 2. $250 \text{ cm} = 2.5 \text{ m}, \quad 4 \times 2.5 = 10 \text{ m}^2$ — Convert: both lengths in metres, then multiply. 3. $24 + 9 = 33 \text{ m}^2$ — Split: a rectangle (6 × 4) plus a triangle (½ × 6 × 3). 4. $8 \times 5 - 3 \times 2 = 34 \text{ m}^2$ — Or subtract: a rectangle with a corner missing is the big rectangle minus the corner. Your working can be short notes. Then choose the reason for the key step. **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: {{field-15}} ::: :::teacher-note **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) 24 + 9 = 33 m². ::: :::keep-together ### 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 ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: 50 is in centimetres and 2 in metres. Change 50 cm to 0.5 m: 0.5 × 2 = 1 m². ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** The base 10 cm and the height 4 cm ½ × 10 × 4 = 20 cm². The 5 cm side is extra information. ::: :::typical ## One change at a time **A sail.** Each task changes one thing. **Take a part away.** **Square units.** **Two steps: paint.** **A sensible unit.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Area problems :::keep-together ### 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: {{field-15}} ::: :::teacher-note **Answer:** 46 40 + 6 = 46 cm². ::: :::keep-together ### Convert **A path is 300 cm long and 2 m wide. Its area, in m²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 300 cm = 3 m, and 3 × 2 = 6 m². ::: :::keep-together ### Extra information **A right-angled triangle has base 12 cm, height 5 cm and sloping side 13 cm. Area, in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 30 ½ × 12 × 5 = 30 cm². ::: :::keep-together ### Review: percentages **Review: find 25% of 48.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 A quarter of 48: 12. ::: ## Make: a tall sail Then work out the cost of the sailcloth at $15 per m². Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A shape is a 7 m by 4 m rectangle with a triangle on top (base 7 m, height 2 m). Find its area.** {{lines-3}} ::: :::teacher-note **Answer:** 28 + ½ × 7 × 2 = 28 + 7 = 35 m². :::