--- title: "Volume in layers" subtitle: "Area, volume, circles and Pythagoras · Volume · Lesson 4" --- # Volume in layers **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** How many sugar cubes fit in a box, how much water fills a fish tank, how much concrete for a path? All are volumes: how much space a solid takes up. :::teacher-note **In one sentence:** Students build rectangular prisms from layers of unit cubes, see that each layer is the base area, and use V = A × h with cubic units, including working backwards to a missing length (AC9M7M02, node 1). **Students should be able to say:** I can explain why the volume of a box is the area of its base times its height. I can find the volume of a box in cm³ or m³. I can find a missing length from a volume. **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 How many cubes fit on the bottom? How many layers? :::keep-together **A box is 4 cubes long, 3 cubes wide and 5 cubes high. How many cubes fill it?** ☐ 12 ☐ 47 ☐ 60 ☐ 94 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 60: 5 layers of 12. 12 is one layer; 94 counts the squares on all six outside faces, which is not the space inside. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Base times height :::keep-together ### Layers Here is the idea, then the routine. 1. $4 \times 3 = 12$ — One layer covers the base: 12 cubes, as many as the base has squares. 2. $12 \times 5 = 60 \text{ cm}^3$ — 5 layers of 12. 3. $V = A \times h = l \times w \times h$ — For a box, the base area is length × width. 4. $V = 6 \times 2 \times 3 = 36 \text{ cm}^3$ — Volume is in cubic units: cm³. Now try one: a 6 by 2 by 3 box. **A box is 6 cm long, 2 cm wide and 3 cm high. What is its volume, in cm³?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 36 Base 6 × 2 = 12 cm², 3 layers: 12 × 3 = 36 cm³. ::: :::keep-together ### Find the mistake **Finn counts the cubes that fill a box. Click the line where his method goes wrong.** ``` 1 A box is 4 cubes long, 3 cubes wide and 2 cubes high. How many cubes fill it? 2 I count the squares I can see: top 12, front 8, side 6. 3 12 + 8 + 6 = 26 4 So 26 cubes fill the box. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: the squares you can see are only the outside. The box is full: 4 × 3 = 12 cubes in a layer, 2 layers: 24. ::: :::keep-together ### Why it works **Why is the volume of a box the area of its base times its height?** ☐ Because the base is the biggest face ☐ Because volume is always three numbers multiplied ☐ Each layer has one cube for each square of the base, and there are as many layers as the height ::: :::teacher-note **Answer:** Each layer has one cube for each square of the base, and there are as many layers as the height Each layer is the base (A cubes), and there are h layers: V = A × h. ::: :::typical ## One change at a time **Volume 24.** Each task changes one thing. **A base area.** **Backwards: the height.** **A decimal.** **Same volume.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Volume :::keep-together ### A cube **A cube has edges of 5 cm. What is its volume, in cm³?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 125 5 × 5 × 5 = 125 cm³. ::: :::keep-together ### Units **A fish tank is 50 cm long, 20 cm wide and 30 cm high. What is its volume?** ☐ 100 cm³ ☐ 1000 cm³ ☐ 30 000 cm³ ☐ 30 000 cm² ::: :::teacher-note **Answer:** 30 000 cm³ 50 × 20 = 1000 cm² base, × 30 = 30 000 cm³. ::: :::keep-together ### Backwards **A box holds 48 cm³. Its base is 4 cm by 3 cm. How high is it, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4 48 ÷ 12 = 4 cm. ::: :::keep-together ### Review: triangles **Review: a triangle has base 8 cm and height 3 cm. Its area, in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 ½ × 8 × 3 = 12 cm². ::: ## Make: 36 cubes, stacked Then find another box with the same volume and a different height. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A box is 5 cm by 4 cm by 3 cm. Find its volume. A box with base 6 cm² has volume 42 cm³: how high is it?** {{lines-3}} ::: :::teacher-note **Answer:** 5 × 4 × 3 = 60 cm³. 42 ÷ 6 = 7 cm. :::