11 3D Shapes and Capacity PDF

## 11 3D shapes ### Getting started 1. Match each of these shapes to its correct drawing on isometric paper. - A - B - C - D - i - ii - iii - iv ### Continued 2. Li draws these nets on squared paper. - i - ii - iii - iv - v - vi - a) Which of the nets will make an open cube? - b) Which of th...

## 11 3D shapes ### Getting started 1. Match each of these shapes to its correct drawing on isometric paper. - A - B - C - D - i - ii - iii - iv ### Continued 2. Li draws these nets on squared paper. - i - ii - iii - iv - v - vi - a) Which of the nets will make an open cube? - b) Which of the nets will make a closed cube? - c) Which of the nets will not make a cube? 3. Order the capacity of these bottles from least to greatest. - **A** 630 ml - **B** 1.01 litres - **C** 1.36 litres - **D** 950 ml - **E** 1.17 litres - **F** 1040 ml - **G** 190 ml - **H** 1980 ml In this unit you will learn to sketch nets for different shapes. This is a really important skill if you are designing gift boxes to hold things such as chocolates, toys or jewellery. ## 11.1 Shapes and nets We are going to ... - identify, describe and sketch compound 3D shapes - identify and sketch different nets for cubes, cuboids, prisms and pyramids - look at the relationship between area of 2D shapes and surface area of 3D shapes. When you see a person delivering a parcel to someone, what is the usual shape of the box? You will probably say a cuboid, but it is possible to have boxes that are cubes, pyramids or prisms as well. If you are working in a factory that makes the boxes, you need to know what to do to make the different shape boxes! ### Worked example 1 Describe this compound shape. This compound shape is either: - a cube and a cuboid - two cuboids Think how you can split the compound shape into simpler 3D shapes that you know. ### Exercise 11.1 1. Describe these compound shapes. - **a** - **b** - **c** 2. Classify these shapes into two groups. - **Group 1: simple shapes** - **Group 2: compound shapes** - **A** - **B** - **C** - **D** - **E** - **F** - **G** - **H** 3. Sketch a compound shape that is made from these simple shapes. - **a** two different cuboids - **b** a cuboid and a square-based pyramid - **c** two different cylinders ### This is part of Deema's homework. **Question:** Describe and sketch a net of this cuboid. **Solution:** A cuboid has six faces. In this cuboid four of the faces are rectangles and two are squares. Use the same method as Deema to describe and sketch a net of these shapes. - **a** cube - **b** square-based pyramid - **c** cylinder - **d** triangular-based pyramid **Tip:** There is more than one net for each of these shapes but you only need to sketch one of them. ### Think like a mathematician 1 **a** Marcus asks this question: "How do you work out the surface area of a cuboid?" What do you think Marcus means by the surface area of a cuboid? How do you think he could work it out? **b** How could you work out the surface area of: - i a cube - ii a square-based pyramid? **c** Copy and complete this general rule: The surface area of a 3D shape is the total area of all its _______. **d** Discuss your answers to parts a to c with other learners in your class. ### This diagram shows a triangular prism. **a** Copy and complete this description of the triangular prism. A triangular prism has a total of ______ faces. Two of the faces are ______ and ______ of the faces are ______. **b** Sketch a net for the triangular prism. ### Match each of these shapes to the correct net. - **A** pentagonal prism - **B** octagonal prism - **C** hexagonal prism - **i** - **ii** - **iii** ### Think like a mathematician 2 This shape is made of unit cubes. **a** What is the smallest number of unit cubes that must be added to the shape to make a cuboid? **b** Write down the method that you used to work out the answer to part a. **c** Discuss your method with other learners in your class. Critique each other's methods. Can you now improve on your method? ### Write down the smallest number of unit cubes that must be added to these shapes to make cuboids. - **a** - **b** - **c** ### Think like a mathematician 3 **a** Choose a simple 3D shape and draw a net for that shape on a piece of paper. Cut the net out, using a pair of scissors, and fold your net to make the shape. What do you think of your net? Did it fold together accurately to make the shape or did some corners not meet? Did you have any faces missing, or faces that were the wrong shape? **b** Give yourself a score out of 10 for your net, with 1 being not very good and 10 being perfect. How could you improve your score if you made the net again? **c** Discuss your answers to parts c and d with a partner. ### Look what I can do! - I can identify, describe and sketch compound 3D shapes. - I can identify and sketch different nets for cubes, cuboids, prisms and pyramids. - I can understand the relationship between area of 2D shapes and surface area of 3D shapes. ## 11.2 Capacity and volume We are going to ... - look at the difference between capacity and volume. When you are cooking or baking you need to measure out ingredients. Solid ingredients such as rice, pasta or vegetables can be weighed on kitchen scales. When you measure liquid ingredients such as milk, oil or water you will need to use a measuring jug. If you want the perfect pancakes, you need to measure the correct amount of milk! ### Worked example 2 The diagram shows some water in a jug. - **a** What is the capacity of the jug? - **b** What is the volume of water in the jug? - **a** 600 ml - **b** 500 ml 600 ml is the maximum the jug can hold. The scale shows the water is at the 500 ml mark. ### Exercise 11.2 1. For each of these jugs write down: - i the capacity of the jug - ii the volume of water in the jug. - **a** - **b** - **c** 2. Read what Sofia says. You can write a volume of 2500ml as 2 litres 500 ml or 2.5 litres. - **a** Give a convincing reason to justify why Sofia is correct. - **b** Use Sofia's example to help you copy and complete this table. | millilitres | litres and millilitres | litres | |---|---|---| | 2500 ml | 2 litres 500 ml | 2.51 | | 3200 ml | 4 litres 300 ml | 3.71 | | 12 100 ml | 0 litres 800 ml | | **Tip:** Use the fact that 1 litre = 1000 ml to help you explain. ### Think like a mathematician 1 Marcus and Arun are looking at this question: What is the volume of water in this jug? Read their conjectures. I think the volume of water is 6.1 litres. I think the volume of water is 6.2 litres. - **a** Who is correct? Give a convincing reason to justify your answer. - **b** Explain the mistake that the other person has made. - **c** What volume of water must be added to the jug to fill it to capacity? - **d** Discuss your answers to parts a to c with other learners in your class. Discuss methods you can use to make sure that you read a scale correctly. Discuss the different methods that you can use to answer part c. ### For each of these jugs write down: - i the capacity of the jug - ii the volume of water in the jug. - **a** - **b** ### What volume of water must be added to these jugs to fill them to capacity? - **a** - **b** - **c** ### Chipo needs to measure out 2.3 litres of milk. She only has the measuring jug shown. Explain how she can use this measuring jug to measure out 2.3 litres of milk. ### Vishan buys a fish tank with a capacity of 120 litres. He pours water into the tank until it is 3/4 full. What is the volume of the water in the tank? ### Think like a mathematician 2 Work with a partner to answer this question. Mair has these four measuring cups, A, B, C and D. The capacity of each cup, in millilitres, is shown. - **A** 240 ml - **B** 160 ml - **C** 120 ml - **D** 60 ml ### Continued **a** Give a convincing explanation to show how Mair can use the cups to accurately measure out these volumes: - **i** 400 ml - **ii** 360 ml - **iii** 420 ml - **iv** 320 ml - **v** 180 ml - **vi** 600 ml **b** Discuss your answers to part a with other learners in your class. **Tip:** I think that if Mair wanted to accurately measure out 300 ml of water, she could fill cup A and cup D. ### Each of these containers (A-F) is marked with its capacity. Estimate the volume of liquid in each container. - **A** 1 litre - **B** 800 ml - **C** 3 litres - **D** 1800 ml - **E** 1.5 litres - **F** 2 litres **Tip:** Estimate what fraction of the container is full. Work out that fraction in millilitres or litres. Order the volumes of liquid from the least to the greatest. ### Copy and complete the Carroll diagram to sort the containers by their capacity and the volume of liquid they contain. | Capacity of 1 litre or less | Volume of 500 ml or less | Volume of more than 500 ml | |---|---|---| | Capacity of more than 1 litre | | | - **A** - **B** - **C** - **D** - **E** - **F** ### Think like a mathematician 3 **a** Jug A has a capacity of 3 litres. Jug B has a capacity of 4 litres. Investigate what amounts you can make with jug A and jug B when neither jug has a measurement scale. The combined capacity of the two jugs is 7 litres. You can make a volume of any whole number of litres, from 1 litre to 7 litres. You can fill a whole jug, pour water from one jug to the other or empty a jug. Investigate and record how each volume can be made. **b** Investigate which amounts of whole litres can be made with a 3 litre jug (A) and a 5 litre jug (B). **c** Predict what amounts you will be able to make with 3 litre (A) and 6 litre (B) jugs. Check your prediction and record which numbers of whole litres can be made and how they are made. **d** Suggest an explanation for what you have found out. Choose two jugs of your own to investigate. Make sure each jug is a different size. Predict what volumes can be made. Investigate your problem, record your results and check your prediction. Write what you have found out. Look back at the questions in this exercise. Which ones have you found the easiest and which ones have you found the hardest? Do you feel confident in answering all the different types of question? What can you do to increase your level of confidence? ### Look what I can do! - I can understand the difference between capacity and volume. ## Check your progress 1. Describe these compound shapes. - **a** - **b** - **c** 2. Sketch a compound shape that is made from a cuboid and a triangular prism. 3. This diagram shows a square-based pyramid. **a** Copy and complete this description of the square-based pyramid. A square-based pyramid has a total of ______ faces. Four of the faces are ______ and ______ of the faces is a square. The surface area of a square-based pyramid is the total area of all its ______. **b** Sketch a net for the square-based pyramid. 4. Write down the smallest number of unit cubes that must be added to this shape to make a cuboid. ### Continued 5. The diagram shows a jug containing water. **a** Write down: - **i** the capacity of the jug - **ii** the volume of water in the jug. **b** What volume of water must be added to the jug to fill it to capacity?