PAGE 1: Overview OVERVIEW ======== Volume is the amount of space an object takes up. In this sub-strand you will pile blocks into cubes and cuboids, then find the volume by counting the blocks in the pile, including the blocks you cannot see. ## Strand 2.0: Measurement Sub-strand 2.3: Volume Suggested lessons: 8 ## What this sub-strand covers • Volume of cubes and cuboids ## Source • KICD Grade 4 Mathematics Curriculum Design (Revised 2024) — Strand 2.0 Measurement, Sub-strand 2.3 Volume • KICD Grade 4 Mathematics radio lesson: Volume of cubes and cuboids ## Learning outcomes By the end of the sub-strand, the learner should be able to: a) pile objects into stacks of cubes and cuboids in real-life situations b) work out volume of cuboids by piling blocks to form a cuboid c) calculate the volume of cubes by piling blocks to form a cube d) appreciate use of piling method in working out volume in real life ## Key inquiry questions 1. Where can we apply the volume of cubes and cuboids in real- life situations? PAGE 2: How we will learn ## Learning experiences • discuss and carefully arrange blocks or objects on top of each other into cuboid and cube shapes • practise stacking objects or blocks to form cubes and cuboids of different sizes • share tasks while counting the number of objects in the pile that makes a cuboid to determine the volume • team up with peers to count the number of objects in the pile that makes a cube to determine the volume • use digital devices or other resources to safely play games involving stacking blocks to form cubes and cuboids ## Core competencies • Learning to learn: learner practises stacking objects or blocks to form cubes and cuboids of different sizes. • Critical thinking and problem solving: learner explores new ways of counting the number of objects in the pile that makes a cube to determine the volume. ## Values • Self-awareness: learner discusses and carefully arranges blocks or objects on top of each other into cuboid and cube shape. ## Pertinent and contemporary issues • Safety: learner observes safety precautions while stacking blocks to form cubes and cuboids. ## Links to other learning areas • The learner is able to relate arranging blocks or objects on top of each other into cuboid and cube shapes to the construction of food preservation equipment in Agriculture. PAGE 3: Lesson 1: Cubes, cuboids and piling — The cube LESSON 1: CUBES, CUBOIDS AND PILING =================================== ## What you will learn • Describe a cube and a cuboid. • Stack blocks safely to make a cube and a cuboid. Curriculum outcomes: a) pile objects into stacks of cubes and cuboids in real-life situations ## Introduction In a shop, loaves of bread, packets of milk and trays of eggs are often stacked. Putting objects neatly on top of one another is called piling. Some of those stacks form cubes and some form cuboids. ## Notes A cube and a cuboid are both solid shapes with flat faces. They are not flat drawings. You can hold them and stack them. ### The cube A cube has 6 faces, and every face is a square of the same size. A dice and some gift boxes are cubes. All the edges of a cube are equal. If you build a cube from small blocks, the number of blocks along the length, the width and the height is the same. ### The cuboid A cuboid also has 6 faces, but the faces are rectangles and they are not all the same. A chalk box, a matchbox and many book piles are cuboids. The length, the width and the height need not be equal. PAGE 4: Lesson 1: Cubes, cuboids and piling — Example 1 ### Piling Piling means arranging objects in layers, one layer on top of another. The bottom layer must sit on a flat surface. Each block should rest fully on the blocks below it, not on an edge. Build on the floor or on a low table. Do not stand on a chair to add a high layer. Keep your fingers clear when you lower a block. A stack that leans can fall. If it starts to lean, take it down and begin again. The same blocks can be piled into a cube or into a cuboid, depending on how many you put along each direction. ## Key words • cube — a solid with 6 equal square faces and equal edges • cuboid — a box-shaped solid with 6 rectangular faces • piling — stacking objects in layers, one on top of another ## Worked examples ### Example 1 A stack is 4 blocks long, 4 blocks wide and 4 blocks high. Is it a cube or a cuboid? Working: 1. The length, width and height are all 4 blocks. 2. All three measurements are equal, so the stack is a cube. Answer: A cube ## Worked examples ### Example 2 A stack is 6 blocks long, 2 blocks wide and 3 blocks high. Is it a cube or a cuboid? Working: 1. The three measurements are 6, 2 and 3. 2. They are not all equal, so the stack is a cuboid. Answer: A cuboid PAGE 5: Lesson 1: Cubes, cuboids and piling — Answers to practice ## Activity: Build a cube and a cuboid You will need: Equal small boxes, matchboxes or building blocks, A flat part of the floor 1. In a group, talk about which objects at home are piled, such as books, bricks or packets. 2. On a flat surface, stack blocks into a cuboid. Say how many blocks are along the length, the width and the height. 3. Use other blocks to stack a cube, with the same number of blocks in each direction. 4. Look at each stack from the side. If it leans, take it down safely and rebuild it. 5. Leave the floor clear when you finish so that nobody trips over the blocks. ## Practice 1. How many faces does a cube have, and what shape is each face? 2. A stack is 5 blocks long, 5 blocks wide and 2 blocks high. Is it a cube or a cuboid? 3. Name one safety rule to follow when you pile blocks. ## Answers to practice 1. 6 faces, and each face is a square. 2. A cuboid, because the height is not equal to the length and the width. 3. Build on a flat surface and stop if the stack starts to lean, so that it does not fall. ## Summary • A cube has equal edges and six equal square faces. • A cuboid is a box shape whose length, width and height need not be equal. • Pile blocks on a flat surface and keep the stack straight. PAGE 6: Lesson 2: Stacks of different sizes — Notes 1 LESSON 2: STACKS OF DIFFERENT SIZES =================================== ## What you will learn • Arrange the same number of blocks into different cuboids. • Explain that rearranging the same blocks does not change how many blocks there are. Curriculum outcomes: a) pile objects into stacks of cubes and cuboids in real-life situations ## Introduction Twelve bricks can make a long low wall or a shorter, taller stack. The shape changes. The number of bricks does not. ## Notes Start with a known number of equal blocks. Arrange them in one layer first. A single layer is a flat cuboid of height 1. Twelve blocks can be arranged in several ways: - 12 blocks in one row: 12 by 1 by 1 - 6 by 2 by 1 - 4 by 3 by 1 - 3 by 2 by 2 Each arrangement uses 12 blocks. The piles look different, but you have not added or lost a block. ### Rows, columns and layers Call the lines going back the rows, the lines going across the columns, and the levels from the bottom the layers. Number of blocks = rows × columns × layers. For 3 rows, 2 columns and 2 layers: 3 × 2 × 2 = 12 blocks. PAGE 7: Lesson 2: Stacks of different sizes — Example 1 ### Practise different sizes Try 24 blocks. One possible cuboid is 4 rows, 3 columns and 2 layers, because 4 × 3 × 2 = 24. Another is 6 rows, 2 columns and 2 layers, because 6 × 2 × 2 = 24. A cube needs the same number in each direction. Eight blocks make a cube of 2 by 2 by 2. Twenty-seven blocks make a cube of 3 by 3 by 3. You cannot make a cube with 12 equal blocks, because no whole number multiplies by itself three times to give 12. Build on a flat surface and keep each layer complete before you start the next one. ## Key words • layer — one flat level of blocks in a stack • rearrange — to build a different shape using the same blocks ## Worked examples ### Example 1 Show that 4 rows, 3 columns and 2 layers use 24 blocks. Working: 1. One layer: 4 × 3 = 12 blocks. 2. Two layers: 12 × 2 = 24 blocks. Answer: 24 blocks ## Worked examples ### Example 2 A group has 18 equal blocks. Can they make a stack of 3 rows, 3 columns and 2 layers? Working: 1. 3 × 3 × 2 = 18. 2. Yes, that stack uses exactly 18 blocks. Answer: Yes. The stack uses 18 blocks. PAGE 8: Lesson 2: Stacks of different sizes — Practice ## Activity: Same blocks, different stacks You will need: 24 equal blocks or small boxes, Flat floor space 1. Count out 12 blocks together so that everyone agrees on the number. 2. Build a cuboid of 6 by 2 by 1. Then rebuild the same blocks as 3 by 2 by 2. 3. Count the blocks after each build to show that you still have 12. 4. With 8 blocks, build a cube. Check that the length, width and height are equal. 5. Take the stacks down carefully and return every block. ## Practice 1. How many blocks are in a stack of 6 rows, 2 columns and 2 layers? 2. A single layer has 4 rows and 3 columns. How many blocks are in that layer? 3. How many blocks make a cube that is 2 blocks along each edge? ## Answers to practice 1. 24 blocks 2. 12 blocks 3. 8 blocks ## Summary • The same blocks can be piled into different cuboids. • Rows × columns × layers gives the number of blocks. • A cube has the same number of blocks along every edge. PAGE 9: Lesson 3: Volume of a cuboid by piling LESSON 3: VOLUME OF A CUBOID BY PILING ====================================== ## What you will learn • Explain volume as the number of unit cubes that fill a solid. • Find the volume of a cuboid by counting the blocks in its pile. Curriculum outcomes: b) work out volume of cuboids by piling blocks to form a cuboid ## Introduction A chalk box takes up space on the table. The amount of space a solid takes up is its volume. When the solid is built from equal blocks, the volume is the number of those blocks. ## Notes Volume is the space occupied by an object. For a stack of equal blocks, you do not need a new measuring tool. Each small block is one unit cube, and the volume is how many unit cubes are in the stack. ### Count a cuboid in layers A cuboid pile has rows, columns and layers. 1. Count the blocks in one row. 2. Count how many such rows are in the bottom layer. Multiply to get the blocks in one layer. 3. Count the layers. Multiply the blocks in one layer by the number of layers. A cuboid with 5 rows, 3 columns and 2 layers has volume 5 × 3 × 2 = 30 unit cubes. ### Hidden blocks still count In a solid stack, blocks in the middle and at the back are hidden. They still fill space, so they are part of the volume. Do not count only the blocks you can see on the front. PAGE 10: Lesson 3: Volume of a cuboid by piling — The unit ### The unit Say what the unit cube is. If you used matchboxes, the volume is that many matchboxes. If you used centimetre cubes, the volume is that many cubic centimetres. Two people who use different blocks will get different numbers for the same big box. An egg tray with 6 rows and 5 columns holds 6 × 5 = 30 eggs in one layer. Four such trays piled up hold 6 × 5 × 4 = 120 eggs. The pile is a cuboid, and 120 is the number of egg spaces in it. [DIAGRAM] TYPE: svg ALT: A cuboid of 4 columns, 1 row and 2 layers has a volume of 8 unit cubes. PAYLOAD: CAPTION: A cuboid of 4 columns, 1 row and 2 layers has a volume of 8 unit cubes. [/DIAGRAM] PAGE 11: Lesson 3: Volume of a cuboid by piling — Summary ## Key words • volume — the amount of space an object takes up • unit cube — one equal block used to fill a cube or a cuboid ## Worked examples ### Example 1 A cuboid is piled with 5 rows, 3 columns and 2 layers of blocks. What is its volume? Working: 1. One layer: 5 × 3 = 15 blocks. 2. Two layers: 15 × 2 = 30 blocks. Answer: 30 unit cubes ## Worked examples ### Example 2 One egg tray has 6 rows and 5 columns. Four trays are piled. How many egg spaces are in the pile? Working: 1. One tray: 6 × 5 = 30. 2. Four layers: 30 × 4 = 120. Answer: 120 egg spaces ## Activity: Count the cuboid You will need: Equal blocks or small boxes, Exercise book 1. In a group, pile blocks into a cuboid with at least 2 layers. 2. Count the rows, the columns and the layers. Multiply to find the volume. 3. Take the stack apart and count every block. The two numbers should match. 4. Build a different cuboid and repeat. Record both volumes. 5. Point to a block that was hidden inside the first stack and explain why it was still counted. ## Practice 1. Find the volume of a cuboid with 4 rows, 4 columns and 3 layers. 2. A cuboid has 8 rows, 2 columns and 5 layers. What is its volume? 3. Why do hidden blocks inside a stack still count in the volume? ## Answers to practice 1. 48 unit cubes 2. 80 unit cubes 3. They fill space inside the cuboid, so they are part of the pile. ## Summary • The volume of a block cuboid is the number of unit cubes in the pile. • Multiply rows × columns × layers. • Count hidden blocks as well as the ones you can see. PAGE 12: Lesson 4: Volume of a cube by piling — Small cubes you can build LESSON 4: VOLUME OF A CUBE BY PILING ==================================== ## What you will learn • Build a cube from equal blocks. • Calculate the volume of a cube by multiplying three equal numbers. Curriculum outcomes: c) calculate the volume of cubes by piling blocks to form a cube ## Introduction A dice is a cube. If you could fill a large cube with tiny equal cubes, every edge would hold the same number. That equal number is the key to the volume. ## Notes A cube made of unit blocks has the same number of rows, columns and layers. Call that number the side of the cube, measured in blocks. ### Small cubes you can build - Side 1: 1 × 1 × 1 = 1 unit cube. This is just one block. - Side 2: 2 × 2 × 2 = 8 unit cubes. - Side 3: 3 × 3 × 3 = 27 unit cubes. - Side 4: 4 × 4 × 4 = 64 unit cubes. - Side 5: 5 × 5 × 5 = 125 unit cubes. PAGE 13: Lesson 4: Volume of a cube by piling — Example 1 ### How to multiply three numbers Multiply two of them first. For a cube of side 3: 3 × 3 = 9 blocks in one layer. Then 9 × 3 = 27 blocks in three layers. ### Do not count only the outside A cube of side 3 looks as if it has blocks only on its faces. There is also a block in the centre that you cannot see when the cube is solid. The volume is 27, not just the blocks on the shell. Taking the cube apart and counting all the blocks checks the multiplication. ### Cube or cuboid? If the three numbers are equal, the pile is a cube and you may multiply the side by itself and by itself again. If one number is different, the pile is a cuboid and you still multiply rows × columns × layers, but you do not call it a cube. ## Key words • side of a cube — the number of unit cubes along one edge • cubic unit — the volume of one unit cube ## Worked examples ### Example 1 Find the volume of a cube of side 3 unit cubes. Working: 1. One layer: 3 × 3 = 9. 2. Three layers: 9 × 3 = 27. Answer: 27 unit cubes PAGE 14: Lesson 4: Volume of a cube by piling — Build cubes and count ## Worked examples ### Example 2 A cube is 4 blocks along each edge. What is its volume? Working: 1. 4 × 4 = 16 blocks in one layer. 2. 16 × 4 = 64. Answer: 64 unit cubes ## Worked examples ### Example 3 Is a stack of 5 by 5 by 4 a cube? What is its volume? Working: 1. 5, 5 and 4 are not all equal, so it is a cuboid. 2. 5 × 5 × 4 = 100. Answer: It is a cuboid of volume 100 unit cubes. ## Activity: Build cubes and count You will need: At least 27 equal blocks, Flat surface, Exercise book 1. Team up and build a cube of side 2. Count the blocks and check that 2 × 2 × 2 gives the same number. 2. If you have enough blocks, build a cube of side 3. Before you take it apart, predict the volume. 3. Take it apart and count every block, including any that were hidden. 4. Try to build a cube with 12 blocks. Discuss why a cube does not form. 5. Stack and unstack slowly so that blocks do not fall on feet or fingers. PAGE 15: Lesson 4: Volume of a cube by piling — Practice ## Practice 1. What is the volume of a cube of side 2? 2. What is the volume of a cube of side 5? 3. A learner counts only the 9 blocks on the front face of a solid 3 by 3 by 3 cube. What is the correct volume? ## Answers to practice 1. 8 unit cubes 2. 125 unit cubes 3. 27 unit cubes ## Summary • In a cube, rows, columns and layers are equal. • Volume of a cube = side × side × side. • Blocks hidden inside the cube are part of the volume. LESSON 5: COUNTING LAYERS TO FIND VOLUME ======================================== ## What you will learn • Find the number of blocks in one layer, then multiply by the number of layers. • Use the same method for cubes and for cuboids. Curriculum outcomes: b) work out volume of cuboids by piling blocks to form a cuboid; c) calculate the volume of cubes by piling blocks to form a cube ## Introduction Once you know how many blocks make the bottom layer, every layer above it is a copy of that layer. You can find the volume without touching every block. PAGE 16: Lesson 5: Counting layers to find volume — A cuboid ## Notes The bottom layer of a pile is a rectangle of unit cubes. Its area, in unit squares, is also the number of cubes in that layer. Blocks in one layer = rows × columns. Volume = blocks in one layer × number of layers. ### A cuboid A bottom layer has 8 cubes, and there are 5 layers. The volume is 8 × 5 = 40 unit cubes. You should still be able to say the rows and columns. If the layer is 4 by 2, then 4 × 2 × 5 = 40 as well. ### A cube Each layer of a cube of side 4 has 4 × 4 = 16 cubes. There are 4 layers, so the volume is 16 × 4 = 64 unit cubes. ### Two ways, one answer Count all the blocks if the pile is small. Multiply if the pile is large or if some blocks are hidden. The two ways agree when every layer is complete. A missing block in the middle means the pile is not a full cuboid, so you cannot multiply as if every layer were full. Fill the gap or subtract the missing block: a full 3 × 3 × 2 cuboid has 18 blocks, and if 1 block is missing the solid part has 17. This layer method is still the piling method. You are counting the pile by groups instead of one block at a time. PAGE 17: Lesson 5: Counting layers to find volume — Example 3 ## Key words • bottom layer — the first complete level of blocks, resting on the surface • complete layer — a layer with every space filled, and no missing block ## Worked examples ### Example 1 One layer of a cuboid has 8 blocks. There are 5 layers. What is the volume? Working: 1. Each layer matches the bottom layer. 2. 8 × 5 = 40. Answer: 40 unit cubes ## Worked examples ### Example 2 A cube of side 4 is built in layers. Find the volume from the layers. Working: 1. One layer: 4 × 4 = 16. 2. Four layers: 16 × 4 = 64. Answer: 64 unit cubes ## Worked examples ### Example 3 A full cuboid of 3 rows, 3 columns and 2 layers has one block missing. How many blocks are actually in the pile? Working: 1. Full volume: 3 × 3 × 2 = 18. 2. 18 − 1 = 17. Answer: 17 blocks PAGE 18: Lesson 5: Counting layers to find volume — Practice ## Activity: Layer by layer You will need: Equal blocks, Exercise book 1. Build only the bottom layer of a cuboid. Count the blocks and write the number. 2. Add two more layers that match the bottom layer. Predict the volume, then count to check. 3. Build a cube layer by layer. Record the side, the blocks in one layer and the volume. 4. Remove one block from a cuboid and explain why you must subtract instead of using the full multiplication. 5. Compare your method with another group. A different order of multiplying should give the same volume. ## Practice 1. A layer has 6 rows and 4 columns. Six such layers are piled. What is the volume? 2. How many blocks are in one layer of a cube of side 5, and what is the volume of the cube? 3. A cuboid of volume 30 unit cubes is made of 3 equal layers. How many blocks are in each layer? ## Answers to practice 1. 144 unit cubes 2. 25 blocks in one layer, and the volume is 125 unit cubes. 3. 10 blocks ## Summary • Find the blocks in one layer, then multiply by the number of layers. • The same method works for a cube, where every side is equal. • Multiply only when every layer is complete. PAGE 19: Lesson 6: Using volume in everyday piles — Notes 1 LESSON 6: USING VOLUME IN EVERYDAY PILES ======================================== ## What you will learn • Describe situations where people stack objects into cubes or cuboids. • Work out how many items are in a real pile of equal layers. Curriculum outcomes: d) appreciate use of piling method in working out volume in real life ## Introduction A grocer stacks packets so that customers can see them and so that they do not fall. A builder stacks bricks. Knowing the number in the pile tells them how much they have, without unpacking every layer. ## Notes The piling method is useful whenever equal objects are stacked in a cuboid. - Packets of flour or sugar stacked in a store - Bricks stacked at a building site - Books packed in a carton in equal rows - Crates of bottles arranged in layers - Boxes used to keep produce off a damp floor In each case, count one layer and multiply by the number of layers. A store has 4 packets along the row, 3 packets across and 5 layers. The stack holds 4 × 3 × 5 = 60 packets. ### Same items, different stack Sixty packets can also be arranged as 6 × 5 × 2. The volume in packets stays 60, but a lower stack may be safer. A tall stack of bottles can fall and break, so a wide, low cuboid is often the better pile. ### What volume tells you Volume tells you how much space the stack fills and how many unit items are inside it. It does not tell you the mass of the items or the price. A large pile of empty boxes can have a big volume and a small mass. People who pack goods, build stores or make containers rely on careful stacking. Counting honestly, and stacking safely, makes the piling method worth using. PAGE 20: Lesson 6: Using volume in everyday piles — Example 1 ## Key words • stack — a pile of objects arranged in layers • packing — arranging items so that they fit in a space without wasting room ## Worked examples ### Example 1 Packets are stacked 4 along a row, 3 across and 5 layers high. How many packets are in the stack? Working: 1. One layer: 4 × 3 = 12 packets. 2. Five layers: 12 × 5 = 60 packets. Answer: 60 packets ## Worked examples ### Example 2 The same 60 packets are restacked as 6 rows, 5 columns and 2 layers. How many packets is that? Working: 1. 6 × 5 × 2 = 60. 2. The number of packets has not changed. Answer: 60 packets ## Activity: Pack a model stack You will need: Equal boxes, books or blocks, Exercise book 1. Discuss places where you have seen cubes or cuboids stacked, such as a shop, a store or a building site. 2. Pile 24 equal objects into a cuboid that is only 2 layers high. Record rows, columns and layers. 3. Work out the volume by multiplying, then count to check. 4. Talk about why a lower stack can be safer than a tall one. 5. If a digital device is allowed, play a stacking game and write the volume of one cuboid you built. Stop if the stack in the classroom becomes unsteady. ## Practice 1. Bricks are piled in 8 rows, 5 columns and 4 layers. How many bricks is that? 2. A carton holds books in 3 rows, 4 columns and 3 layers. How many books is that? 3. Give one reason for finding the volume of a stack instead of unpacking it. 4. A cube stack of packets is 3 packets along each edge. How many packets is that? ## Answers to practice 1. 160 bricks 2. 36 books 3. You can find how many items are in the stack by counting one layer and multiplying, without taking the stack apart. 4. 27 packets ## Summary • Shops, stores and building sites pile equal objects into cuboids. • Rows × columns × layers gives the number of items. • A lower stack of the same items can be safer than a tall one. PAGE 21: Revision quiz REVISION QUIZ ============= Answer all the questions, then mark yourself on the Answers tab. 1. Which stack is a cube? A. 3 by 3 by 2 B. 4 by 4 by 4 C. 5 by 2 by 5 D. 6 by 1 by 4 2. How many blocks are used in a stack of 3 rows, 2 columns and 2 layers? 3. What is the volume of a cuboid with 5 rows, 3 columns and 2 layers? A. 10 unit cubes B. 30 unit cubes C. 25 unit cubes D. 15 unit cubes 4. One layer has 7 rows and 4 columns. There are 3 layers. What is the volume? 5. Four egg trays, each with 6 rows and 5 columns, are piled. How many egg spaces are in the pile? 6. What is the volume of a cube of side 3? A. 9 unit cubes B. 12 unit cubes C. 27 unit cubes D. 18 unit cubes 7. What is the volume of a cube of side 5? 8. A cube of side 4 is counted layer by layer. How many unit cubes are in one layer? 9. Packets are stacked 6 by 4 by 3, then restacked as 8 by 3 by 3. What happens to the number of packets? A. It becomes 72 B. It stays 72 C. It becomes 36 D. It becomes 24 10. Why is a low, wide stack often safer than a tall stack of the same blocks? PAGE 22: Quiz answers ANSWERS ======= 1. B. 4 by 4 by 4 A cube has the same number of blocks along the length, the width and the height. Only 4 by 4 by 4 is equal in all three. 2. 12 blocks 3 × 2 × 2 = 12. 3. B. 30 unit cubes 5 × 3 × 2 = 30 unit cubes. 4. 84 unit cubes 7 × 4 × 3 = 84. 5. 120 egg spaces 6 × 5 × 4 = 120. 6. C. 27 unit cubes 3 × 3 × 3 = 27. 7. 125 unit cubes 5 × 5 × 5 = 125. 8. 16 unit cubes One layer is 4 × 4 = 16. The full cube is 64. 9. B. It stays 72 6 × 4 × 3 = 72 and 8 × 3 × 3 = 72. Rearranging does not change the number. 10. A tall stack can lean and fall, while a lower stack is steadier. The number of blocks can stay the same while the shape of the pile changes.