Outcomes
A student:

 MA41WM
communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

 MA42WM
applies appropriate mathematical techniques to solve problems

 MA414MG
uses formulas to calculate the volumes of prisms and cylinders, and converts between units of volume
Related Life Skills outcomes: MALS28MG, MALS30MG, MALS31MG
Content
 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 crosssections of different prisms
 recognise that the crosssections of prisms are uniform (Reasoning)
 visualise, construct and draw various prisms from a given crosssectional diagram
 determine if a particular solid has a uniform crosssection
 distinguish between solids with uniform and nonuniform crosssections (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^{3 }= 1000 mm^{3}, 1 L^{ }= 1000 mL = 1000 cm^{3}, 1 m^{3 }= 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 crosssection (Communicating, Reasoning)
 find the volumes of prisms, given their perpendicular heights and the areas of their uniform crosssections
 find the volumes of prisms with uniform crosssections 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 crosssection 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 crosssection'.
'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 threedimensional 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 threedimensional 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 twodimensional shapes is linear, whereas in relation to threedimensional 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 threedimensional 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 crosssection.
Students should be aware that a cube is a special prism that has six congruent faces.
The abbreviation m^{3} is read as 'cubic metre(s)' and not 'metre(s) cubed' or 'metre(s) cube'. The abbreviation cm^{3} 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 subelements (and levels) of Number patterns and algebraic thinking (NPA9), Understanding units of measurement (UuM8UuM9) and Understanding geometric properties (UGP3) describe observable behaviours that can aid teachers in making evidencebased decisions about student development and future learning.
The progression subelements and indicators can be viewed by accessing the National Numeracy Learning Progression.