NSW Syllabuses

# Mathematics K–10 - Stage 4 - Measurement and Geometry Volume

## Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-14MG

uses formulas to calculate the volumes of prisms and cylinders, and converts between units of volume

Related Life Skills outcomes: MALS-28MG, MALS-30MG, MALS-31MG

• Students:
• Draw different views of prisms and solids formed from combinations of prisms (ACMMG161)
• draw (in two dimensions) prisms, and solids formed from combinations of prisms, from different views, including top, side, front and back views
• identify and draw the cross-sections of different prisms
• recognise that the cross-sections of prisms are uniform (Reasoning)
• visualise, construct and draw various prisms from a given cross-sectional diagram
• determine if a particular solid has a uniform cross-section
• distinguish between solids with uniform and non-uniform cross-sections (Reasoning)
• Choose appropriate units of measurement for volume and convert from one unit to another (ACMMG195)
• recognise that 1000 litres is equal to one kilolitre and use the abbreviation for kilolitres (kL)
• recognise that 1000 kilolitres is equal to one megalitre and use the abbreviation for megalitres (ML)
• choose an appropriate unit to measure the volumes or capacities of different objects, eg swimming pools, household containers, dams
• use the capacities of familiar containers to assist with the estimation of larger capacities (Reasoning)
• convert between metric units of volume and capacity, using 1 cm= 1000 mm3, 1 L = 1000 mL = 1000 cm3, 1 m= 1000 L = 1 kL, 1000 kL = 1 ML
• Develop the formulas for the volumes of rectangular and triangular prisms and of prisms in general; use formulas to solve problems involving volume (ACMMG198)
• develop the formula for the volume of prisms by considering the number and volume of 'layers' of identical shape:
$$\text{Volume of prism}=\text{base area} \times \text{height}$$
leading to $$V=Ah$$
• recognise the area of the 'base' of a prism as being identical to the area of its uniform cross-section (Communicating, Reasoning)
• find the volumes of prisms, given their perpendicular heights and the areas of their uniform cross-sections
• find the volumes of prisms with uniform cross-sections that are rectangular or triangular
• solve a variety of practical problems involving the volumes and capacities of right prisms
• Calculate the volumes of cylinders and solve related problems (ACMMG217)
• develop and use the formula to find the volumes of cylinders:
$$\text{Volume of cylinder} = \pi r^{2}h$$ where $$r$$ is the length of the radius of the base and $$h$$ is the perpendicular height
• recognise and explain the similarities between the volume formulas for cylinders and prisms (Communicating)
• solve a variety of practical problems involving the volumes and capacities of right prisms and cylinders, eg find the capacity of a cylindrical drink can or a water tank

### Background Information

When developing the volume formula for a prism, students require an understanding of the idea of a uniform cross-section and should visualise, for example, stacking unit cubes, layer by layer, into a rectangular prism, or stacking planks into a pile. In the formula for the volume of a prism, $$V=Ah$$, $$A$$ refers to the 'area of the base', which can also be referred to as the 'area of the uniform cross-section'.

'Oblique' prisms, cylinders, pyramids and cones are those that are not 'right' prisms, cylinders, pyramids and cones, respectively. The focus here is on right prisms and cylinders, although the formulas for volume also apply to oblique prisms and cylinders provided that the perpendicular height is used. In a right prism, the base and top are perpendicular to the other faces. In a right pyramid or cone, the base has a centre of rotation, and the interval joining that centre to the apex is perpendicular to the base (and therefore is its axis of rotation).

The volumes of rectangular prisms and cubes are linked with multiplication, division, powers and factorisation. Expressing a number as the product of three of its factors is equivalent to forming a rectangular prism with those factors as the side lengths, and (where possible) expressing a number as the cube of one of its factors is equivalent to forming a cube with that factor as the side length.

The abbreviation for megalitres is ML. Students will need to be careful not to confuse this with the abbreviation mL used for millilitres.

#### Purpose/Relevance of Substrand

The ability to determine the volumes of three-dimensional objects and the capacities of containers, and to solve related problems, is of fundamental importance in many everyday activities, such as calculating the number of cubic metres of concrete, soil, sand, gravel, mulch or other materials needed for building or gardening projects; the amount of soil that needs to be removed for the installation of a swimming pool; and the appropriate size in litres of water tanks and swimming pools. Knowledge and understanding with regard to determining the volumes of simple three-dimensional objects (including containers) such as cubes, other rectangular prisms, triangular prisms, cylinders, pyramids, cones and spheres can be readily applied to determining the volumes and capacities of composite objects (including containers).

### Language

The word 'base' may cause confusion for some students. The 'base' in relation to two-dimensional shapes is linear, whereas in relation to three-dimensional objects, 'base' refers to a surface. In everyday language, the word 'base' is used to refer to that part of an object on, or closest to, the ground. In the mathematics of three-dimensional objects, the term 'base' is used to describe the face by which a prism or pyramid is named, even though it may not be the face on, or closest to, the ground. In Stage 3, students were introduced to the naming of a prism or pyramid according to the shape of its base. In Stage 4, students should be encouraged to make the connection that the name of a particular prism refers not only to the shape of its base, but also to the shape of its uniform cross-section.

Students should be aware that a cube is a special prism that has six congruent faces.

The abbreviation m3 is read as 'cubic metre(s)' and not 'metre(s) cubed' or 'metre(s) cube'. The abbreviation cm3 is read as 'cubic centimetre(s)' and not 'centimetre(s) cubed' or 'centimetre(s) cube'.

When units are not provided in a volume question, students should record the volume in 'cubic units'.

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA4‑14MG the sub-elements (and levels) of Number patterns and algebraic thinking (NPA9), Understanding units of measurement (UuM8-UuM9) and Understanding geometric properties (UGP3) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.