Geometry:
key ideas and important concept knowledge
Key ideas
The overarching key ideas of estimation, benchmarks, visualisation, equality and equivalence, language and strategies need to be considered when developing units of work in geometry. The specific key ideas in geometry are: features, properties, classification, orientation, transformation, point of view and symmetry.
Features
Features, sometimes called attributes, are visual characteristics of a shape or object that can be seen.
Properties
Shapes and objects have distinguishable
characteristics and are named because of their properties.
The properties of two-dimensional (2D) shapes typically include relationships among the number, length and relative position of sides, as well as the number and angle of corners, lines of symmetry, convexity and concavity.
The properties of three-dimensional (3D) objects typically include the shape and relative position of faces, surfaces, edges and vertices (corners).
Classes (i.e. categories) of shapes are defined by their properties, which are relationships among features of similar shapes.
Properties are relationships connecting characteristics of shapes or objects.
Some properties are defining; that is, they describe what a shape must have to belong to a class.
Classification
Classification involves establishing criteria to group shapes by their common properties. Classification is about working with relationships among properties, where individual shapes are examples of the classes and are hierarchical.
Orientation
Orientation is the position of a shape on a plane or an object in space, in particular the direction that the features of the shape or object are facing.
The orientation of a shape remains unchanged as it is translated (i.e. shifted) or enlarged, but changes when a shape is reflected or rotated through an angle not equal to 360°.
Transformation
Transformation is the change in the size, shape or position of a shape or object.
Isometric transformations
Isometric transformations include translations, rotations and reflections. Note: these do not change the size or proportions of a shape or object.
1. Translation (slide): the movement of a shape to a new position. The length of sides and angles remain unchanged.
2. Rotation (turn): a change to the position of a shape by rotating it about a fixed point through a given angle. The point may be inside or outside the shape.
3. Reflection (flip): a change to the position of a shape by reflecting it along a mirror line. The line may run through the shape or be external to it.
Non-isometric transformations
Non-isometric transformations include enlargements and reductions (dilations).
A dilation is a change to the size of a shape about a point. The lengths of the sides and/or the angles are changed.
Point of view
Objects appear differently depending upon the position from which they are viewed.
Symmetry
A shape has symmetry if it maps onto itself by a transformation, particularly through reflection and/or rotation.
Reflective symmetry
In reflective symmetry, the locations of the mirror are called lines of symmetry.
Rotational symmetry
In rotational symmetry, the point is known as the centre of rotation and the angle of rotation is the measure of turn that maps the shape onto itself.
Important concept knowledge
Geometric reasoning
Geometric reasoning is thinking with the properties of shapes.
Point
A point is a single location in space. A point is usually represented as a dot, although, theoretically, a point has no area.
Line
A line is a set of points that extend infinitely in two directions. A line is usually represented by a straight segment. Theoretically, a line has only one dimension (length) and no area.
Types of lines include:
• horizontal – lines that are parallel to the horizon
• vertical – lines that are at a right angle to a horizontal line
• oblique – lines that are neither vertical nor horizontal
• perpendicular – lines that meet at a right angle.
Ray
A ray is a set of points that extend infinitely in one direction. A ray is usually represented by an arrow or vector beginning at a fixed point.
Plane
A plane is a flat surface that extends infinitely in two dimensions; width and length.
Two-dimensional shape
A two-dimensional shape exists on a flat surface, so it possesses length and width. Two-dimensional shapes include polygons, such as triangles and quadrilaterals, and simple closed curves, such as circles.
Polygon
A polygon is a two-dimensional, planar shape that is bounded (i.e. enclosed) by line segments. The line segments form the sides of the polygon. A regular polygon has sides and angles of equal measure.
Three-dimensional object
A three-dimensional object exists in real life, so it possesses three dimensions: length, width and depth. Polyhedra, such as prisms and pyramids, and closed surfaces, such as spheres and cylinders, are solid objects that have three dimensions.
Polyhedron
A polyhedron is a three-dimensional object bounded (i.e. enclosed) by polygons and called a solid.
Solids or objects enclosed by polygons are called polyhedra and those polygons comprise the faces.
A regular polyhedron has the same regular polygons for all of its faces.
Curved solids, such as cones, spheres, cylinders and others are not classified as polyhedra.
Net
A net is a flat shape created by unfolding a three-dimensional solid.
Congruence
Congruent shapes are exactly the same shape and size.
Similar shapes
Similar shapes are exactly the same shape (angles and side ratios), but are a different size.
Regular shapes
A polygon is a regular shape if all of its sides are the same length and all of its angles have the same measure.
Tessellation
A tessellation is the tiling of a plane in a repeated pattern with no gaps or overlaps of the shapes.
Apex
An apex is the highest point above the base of a cone or a pyramid.
Cross-section
A cross-section is the flat surface created when a cut is made through an object, parallel to the base. The cross-section of cylinders and prisms are uniform, whereas the cross-section of pyramids and cones are not uniform.
Truncated object
A truncated 3D object has a vertex removed by cutting along a plane.
Right object
A 3D object is ‘right’ when the top face or the apex is centered above the base and is perpendicular to the centre point of the base.
Oblique object
An oblique object is a 3D object that does not fulfill the criteria for 'right'.
Angle
An angle is a figure formed by two rays joining at a common endpoint (P), which is used to represent a turn of one ray from another about P. Angles assist in defining the properties of classes of shapes.
Adjacent angles
Adjacent angles are two angles next to each other that share a common ray.
Complementary angles
Complementary angles are adjacent and add up to 90°.
Supplementary angles
Supplementary angles are two adjacent angles that add up to 180°.
Vertically-opposite angles
Vertically-opposite angles refer to two lines intersecting so that the angles opposite to each other are the same measure.
Angles at a point
Angles at a point refers to angles surrounding a point (with no gap or overlap) that add up to 360°.
This property is particularly important for establishing which polygons, or sets of polygons, will tessellate.
Parallel lines
Parallel lines extend infinitely in two directions but never meet. A real-life analogy is railway tracks.
Transversals
A transversal is a line that intersects parallel lines, forming different sets of angles.
Corresponding angle
A corresponding angle is an angle that is on the same side of the transversal and in a like position. Corresponding angles are equal in measure.
Alternate angle
An alternate angle is an angle that is on the opposite side of the transversal and inside the two intersected lines. Alternate angles are equal in measure.
Co-interior angle
A co-interior angle is an angle that is on the same side of the transversal and inside the two intersected lines. Co-interior angles add up to 180°.
