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NSW Syllabuses

Mathematics K–10 - Stage 3 - Measurement and Geometry Three-Dimensional Space

Three-Dimensional Space 1

Outcomes

A student:

  • MA3-1WM

    describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions

  • MA3-3WM

    gives a valid reason for supporting one possible solution over another

  • MA3-14MG

    identifies three-dimensional objects, including prisms and pyramids, on the basis of their properties, and visualises, sketches and constructs them given drawings of different views

Content

  • identify and determine the number of pairs of parallel faces of three-dimensional objects, eg 'A rectangular prism has three pairs of parallel faces'
  • identify the 'base' of prisms and pyramids L
  • recognise that the base of a prism is not always the face where the prism touches the ground (Reasoning)
  • name prisms and pyramids according to the shape of their base, eg rectangular prism, square pyramid  l
  • visualise and draw the resulting cut face (plane section) when a three-dimensional object receives a straight cut CCT
  • recognise that prisms have a 'uniform cross-section' when the section is parallel to the base
  • recognise that the base of a prism is identical to the uniform cross-section of the prism (Reasoning)
  • recognise a cube as a special type of prism (Communicating)
  • recognise that pyramids do not have a uniform cross-section when the section is parallel to the base
  • identify, describe and compare the properties of prisms and pyramids, including: L
  • number of faces
  • shape of faces
  • number and type of identical faces
  • number of vertices
  • number of edges
  • describe similarities and differences between prisms and pyramids, eg between a triangular prism and a hexagonal prism, between a rectangular prism and a rectangular(-based) pyramid (Communicating, Reasoning) CCT
  • determine that the faces of prisms are always rectangles except the base faces, which may not be rectangles (Reasoning) CCT
  • determine that the faces of pyramids are always triangles except the base face, which may not be a triangle (Reasoning) CCT
  • use the term 'apex' to describe the highest point above the base of a pyramid or cone L
  • Connect three-dimensional objects with their nets and other two-dimensional representations (ACMMG111)
  • visualise and sketch three-dimensional objects from different views, including top, front and side views CCT
  • reflect on their own drawing of a three-dimensional object and consider how it can be improved (Reasoning) CCTPSC
  • examine a diagram to determine whether it is or is not the net of a closed three-dimensional object CCT
  • explain why a given net will not form a closed three-dimensional object (Communicating, Reasoning)
  • visualise and sketch nets for given three-dimensional objects CCT
  • recognise whether a diagram is a net of a particular three-dimensional object (Reasoning) CCT
  • visualise and name prisms and pyramids, given diagrams of their nets CCT
  • select the correct diagram of a net for a given prism or pyramid from a group of similar diagrams where the others are not valid nets of the object (Reasoning) CCT
  • show simple perspective in drawings by showing depth CCT

Background Information

In Stage 3, the formal names for particular prisms and pyramids are introduced while students are engaged in their construction and representation. (Only 'family' names, such as prism, were introduced in Stage 2.) This syllabus names pyramids in the following format: square pyramid, pentagonal pyramid, etc. However, it is also acceptable to name pyramids using the word 'based', eg square-based pyramid, pentagonal-based pyramid.

Prisms have two bases that are the same shape and size. The bases of a prism may be squares, rectangles, triangles or other polygons. The other faces are rectangular if the faces are perpendicular to the bases. The base of a prism is the shape of the uniform cross-section, not necessarily the face on which it is resting.

Pyramids differ from prisms as they have only one base and all the other faces are triangular. The triangular faces meet at a common vertex (the apex). Pyramids do not have a uniform cross-section.

Spheres, cones and cylinders do not fit into the classification of prisms or pyramids as they have curved surfaces, not faces, eg a cylinder has two flat surfaces and one curved surface.

A section is a representation of an object as it would appear if cut by a plane, eg if the corner were cut off a cube, the resulting cut face would be a triangle. An important understanding in Stage 3 is that the cross-sections parallel to the base of a prism are uniform and the cross-sections parallel to the base of a pyramid are not.

Students could explore these ideas by stacking uniform objects to model prisms, and by stacking sets of seriated shapes to model pyramids, eg

The image shows 2 stacks of objects; one models a rectangular prism and the other a pyramid.

Note: such stacks are not strictly pyramids, but they do assist understanding.

In geometry, a three-dimensional object is called a solid. The three-dimensional object may in fact be hollow, but it is still defined as a geometrical solid.

Language

Students should be able to communicate using the following language: object, shape, three-dimensional object (3D object), prism, cube, pyramid, base, uniform cross-section, face, edge, vertex (vertices), apex, top view, front view, side view, depth, net.

In Stage 1, students were introduced to the terms 'flat surface' and 'curved surface' for use in describing cones, cylinders and spheres, and the terms 'faces', 'edges' and 'vertices' for use in describing prisms and pyramids.

National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA3‑14MG the sub-elements (and levels) of Understanding geometric properties (UGP2-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.

Three-Dimensional Space 2

Outcomes

A student:

  • MA3-1WM

    describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions

  • MA3-14MG

    identifies three-dimensional objects, including prisms and pyramids, on the basis of their properties, and visualises, sketches and constructs them given drawings of different views

Content

  • create prisms and pyramids using a variety of materials, eg plasticine, paper or cardboard nets, connecting cubes
  • construct as many rectangular prisms as possible using a given number of connecting cubes (Problem Solving) CCT
  • create skeletal models of prisms and pyramids, eg using toothpicks and modelling clay or straws and tape CCT
  • connect the edges of prisms and pyramids with the construction of their skeletal models (Problem Solving)
  • construct three-dimensional models of prisms and pyramids and sketch the front, side and top views
  • describe to another student how to construct or draw a three-dimensional object (Communicating) L
  • construct three-dimensional models of prisms and pyramids, given drawings of different views

Background Information

In Stage 3, students are continuing to develop their skills of visual imagery, including the ability to perceive and hold an appropriate mental image of an object or arrangement, and to predict the orientation or shape of an object that has been moved or altered.

Refer also to background information in Three-Dimensional Space 1.

Language

Students should be able to communicate using the following language: object, shape, three-dimensional object (3D object), prism, cube, pyramid, base, uniform cross-section, face, edge, vertex (vertices), top view, front view, side view, net.

National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA3‑14MG the sub-elements (and levels) of Understanding geometric properties (UGP2-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.