NSW Syllabuses

# Mathematics K–10 - Stage 2 - Measurement and Geometry Two-Dimensional Space

## Two-Dimensional Space 1

### Outcomes

#### A student:

• MA2-1WM

uses appropriate terminology to describe, and symbols to represent, mathematical ideas

• MA2-3WM

checks the accuracy of a statement and explains the reasoning used

• MA2-15MG

manipulates, identifies and sketches two-dimensional shapes, including special quadrilaterals, and describes their features

### Content

• Students:
• Compare and describe features of two-dimensional shapes, including the special quadrilaterals
• determine the number of pairs of parallel sides, if any, of each of the special quadrilaterals (Reasoning)
• use measurement to establish and describe side properties of the special quadrilaterals, eg the opposite sides of a parallelogram are the same length
• identify and name the special quadrilaterals presented in different orientations, eg
• explain why a particular quadrilateral has a given name, eg 'It is a parallelogram because it has four sides and the opposite sides are parallel' (Communicating, Reasoning)
• name a shape, given a written or verbal description of its features (Reasoning)
• recognise the vertices of two-dimensional shapes as the vertices of angles that have the sides of the shape as their arms
• identify right angles in squares and rectangles
• group parallelograms, rectangles, rhombuses, squares, trapeziums and kites using one or more attributes, eg quadrilaterals with parallel sides and right angles
• identify and describe two-dimensional shapes as either 'regular' or 'irregular', eg 'This shape is a regular pentagon because it has five equal sides and five equal angles'
• identify regular shapes in a group that includes irregular shapes, such as a regular pentagon in a group of pentagons, eg
(Reasoning)
• explain the difference between regular and irregular two-dimensional shapes (Communicating, Reasoning)
• recognise that the name of a shape does not change if its size or orientation in space is changed (Reasoning)
• draw representations of regular and irregular two-dimensional shapes in different orientations
• construct regular and irregular two-dimensional shapes from a variety of materials, eg cardboard, straws, pattern blocks
• determine that a triangle cannot be constructed from three straws if the sum of the lengths of the two shorter straws is less than the length of the longest straw (Reasoning)
• compare the rigidity of two-dimensional frames of three sides with the rigidity of those of four or more sides
• construct and manipulate a four-sided frame and explain how adding a brace can make a four-sided frame rigid (Communicating, Reasoning)
• Identify symmetry in the environment (ACMMG066)
• identify lines of symmetry in pictures, artefacts, designs and the environment, eg Aboriginal rock carvings or Asian lotus designs
• identify and draw lines of symmetry on given shapes, including the special quadrilaterals and other regular and irregular shapes
• determine and explain whether a given line through a shape is a line of symmetry (Communicating, Reasoning)
• recognise and explain why any line through the centre of (and across) a circle is a line of symmetry (Communicating, Reasoning)

### Background Information

The special quadrilaterals are the parallelogram, rectangle, rhombus, square, trapezium and kite.

Regular shapes have all sides equal and all angles equal. In Stage 2, students are expected to be able to distinguish between regular and irregular shapes and to describe a polygon as either regular or irregular, eg a regular pentagon has five equal sides and five equal angles.

It is important for students to have experiences with a variety of shapes in order to develop flexible mental images. Students need to be able to recognise shapes presented in different orientations.

When constructing polygons using materials such as straws of different lengths for sides, students should be guided to an understanding that:

• sometimes a triangle cannot be made from 3 straws
• a figure made from 3 lengths, ie a triangle, is always flat
• a figure made from 4 or more lengths need not be flat
• a unique triangle is formed, if a triangle can be formed, from 3 given lengths
• more than one two-dimensional shape can result if more than 3 lengths are used.

When using examples of Aboriginal rock carvings and other Aboriginal art, it is recommended that local examples be used wherever possible. Consult with local Aboriginal communities and education consultants for such examples.

### Language

Students should be able to communicate using the following language: shape, two-dimensional shape (2D shape), circle, triangle, quadrilateral, parallelogram, rectangle, rhombus, square, trapezium, kite, pentagon, hexagon, octagon, regular shape, irregular shape, orientation, features, properties, side, parallel, pair of parallel sides, opposite, length, vertex (vertices), angle, right angle, symmetry, line (axis) of symmetry, rigid.

The term 'polygon' (derived from the Greek words meaning 'many angles') refers to closed shapes with three or more angles and sides. While the angles are the focus for the general naming system used for shapes, polygons are more usually understood in terms of their sides. Students are not expected to use the term 'polygon'. However, some students may explore other polygons and so benefit from being introduced to the collective term.

Students could explore the language origins of the names of polygons.

The term 'diamond' is often used in everyday contexts when describing quadrilaterals with four equal sides. However, 'diamond' is not the correct geometrical term to name such quadrilaterals; the correct term is 'rhombus'.

## Two-Dimensional Space 2

### Outcomes

#### A student:

• MA2-1WM

uses appropriate terminology to describe, and symbols to represent, mathematical ideas

• MA2-2WM

selects and uses appropriate mental or written strategies, or technology, to solve problems

• MA2-3WM

checks the accuracy of a statement and explains the reasoning used

• MA2-15MG

manipulates, identifies and sketches two-dimensional shapes, including special quadrilaterals, and describes their features

### Content

• Students:
• Compare and describe two-dimensional shapes that result from combining and splitting common shapes, with and without the use of digital technologies (ACMMG088)
• combine common two-dimensional shapes, including special quadrilaterals, to form other common shapes or designs, eg combine a rhombus and a triangle to form a trapezium
• describe and/or name the shape formed from a combination of common shapes (Communicating)
• follow written or verbal instructions to create a common shape using a specified set of two or more common shapes, eg create an octagon from five squares and four triangles (Communicating, Problem Solving)
• use digital technologies to construct a design or logo by combining common shapes (Communicating)
• split a given shape into two or more common shapes and describe the result, eg 'I split the parallelogram into a rectangle and two equal-sized triangles'

• compare the area of the given shape with the area of each of the shapes it is split into, eg if a pentagon is split into five equal triangles, describe the area of the pentagon as five times the area of one triangle, or the area of one triangle as $$\frac{1}{5}$$ of the area of the pentagon (Communicating, Reasoning)
• record the arrangements of common shapes used to create other shapes, and the arrangement of shapes formed after splitting a shape, in diagrammatic form, with and without the use of digital technologies
• record different combinations of common shapes that can be used to form a particular regular polygon, eg a hexagon can be created from, or split into, many different arrangements, such as

(Communicating, Problem Solving)
• Create symmetrical patterns, pictures and shapes, with and without the use of digital technologies (ACMMG091)
• use different types of graph paper to assist in creating symmetrical designs (Communicating)
• use digital technologies to create designs by copying, pasting, reflecting, translating and rotating common shapes (Communicating, Problem Solving)
• apply and describe amounts of rotation, in both 'clockwise' and 'anti-clockwise' directions, including half-turns, quarter-turns and three-quarter-turns, when creating designs (Communicating, Problem Solving)
• describe the creation of symmetrical designs using the terms 'reflect', 'translate' and 'rotate' (Communicating, Reasoning)
• create and record tessellating designs by reflecting, translating and rotating common shapes
• use digital technologies to create tessellating designs (Communicating)
• determine which of the special quadrilaterals can be used to create tessellating designs (Reasoning)
• explain why tessellating shapes are best for measuring area (Communicating, Reasoning)
• identify shapes that do and do not tessellate
• explain why a shape does or does not tessellate (Communicating, Reasoning)
• draw the reflection (mirror image) to complete symmetrical pictures and shapes, given a line of symmetry, with and without the use of digital technologies

### Background Information

Students should be given the opportunity to attempt to create tessellating designs with a selection of different shapes, including shapes that do not tessellate.

### Language

Students should be able to communicate using the following language: shape, two-dimensional shape (2D shape), triangle, quadrilateral, parallelogram, rectangle, rhombus, square, trapezium, kite, pentagon, hexagon, octagon, line (axis) of symmetry, reflect (flip), translate (slide), rotate (turn), tessellate, clockwise, anti-clockwise, half-turn, quarter-turn, three-quarter-turn.

In Stage 1, students referred to the transformations of shapes using the terms 'slide', 'flip' and 'turn'. In Stage 2, they are expected to progress to the use of the terms 'translate', 'reflect' and 'rotate', respectively.