Module 6 Geometry Flashcards
-Transformational geometry-
the study of the effects of various processes in transforming geometrical objects.
-Coordinate geometry
the field of mathematics linking geometry with algebra.
-Zimmerman and Cunningham (1991) have defined mathematical visualisation
process of forming images (mentally, or with pencil and paper, or with the aid of technology) and using such images effectively for mathematical discovery and understanding”.
-Mathematical visualisation-
is a means to understanding – not just in geometry but in all areas of mathematics.
Geometric reasoning
the invention and use of formal conceptual systems to investigate shape and space.
-Spatial reasoning-
‘the ability to “see” , inspect, and reflect on spatial objects, images, relationships, and transformations’ (Battista, 2007,pg 843).
-Spatial sense
two main spatial abilities: spatial orientation, and spatial visualisation and imagery.
-Additionally spatial sense involves
the ability to manipulate dynamic images, developing our store of images for shapes and other objects, connecting spatial knowledge to verbal/analytical knowledge.
Spatial sense is an important aspect of numeracy, being inherently mathematical and also essential for living and interacting in our world.
Spatial orientation-
involves knowing where you are and how to get around. For example, if you are looking at a map in a shopping centre to find a particular shop, you need to know where you are and then how to get to the new location.
-Geometry is mainly about ?
ideas. The nature of the ideas is that they are practical, and deal with relationships between real things.
Classification
enables us to isolate a concept. Initially, children have prototypes that they learn to recognise and label, for example, may regard an equilateral triangle as the prototypical triangle but a right angle triangle as half a triangle.
-Labels for shapes – 2D shapes
are those that lie on a plan. The two dimensions are length and width. 2D shapes with straight sides are known as polygons.
Solid or 3D shapes occupy space
The three dimensions are length, width and depth.
polyhedrons.
-Solids with flat faces
edges
lines where the faces of a polyhedron meet
Platonic solids
there are 5 polyhedrons that have faces made up of regular polygons that all meet at exactly the same angle. A regular polygon has congruent angles and sides. The 5 polyhedrons are tetrahedron (4 equilateral triangular faces), cube (6 square faces), octahedron (8 equilateral triangular faces), dodecahedron (12 pentagonal faces), and the icosahedron (20 equilateral triangular faces).
-Line symmetry-
occurs when every point on an object on one side of a line can be matched to another point the same distance from the line on the other side.
-Young children begin to spot symmetry from a very young age. The Australian curriculum: mathematics begins the study of transformational geometry at stage 2 although from the foundation year, children are expected to be describing position and movement. (Handy tip: when choosing shapes to demonstrate lines of symmetry, ensure that you include horizontal and oblique lines as well as vertical lines).
Rotational symmetry
often causes challenges for children. (Imagine an equilateral triangle orientated with the bottom side horizontal(the prototypical triangle). Now imagine that I am going to turn the triangle. If I turn the triangle a third of the way around a full turn (120degrees), the triangle will look the same as when I started. If I continue to turn another 120 degrees the same thing will happen.
-Developing visualisation
spatial visualisation is the ability to generate and manipulate images. Gutièrrez (1996) proposed a framework for visualisation that identified 6 main abilities; figure-ground perception, mental rotation, perception of spatial relationships, perceptual constancy, perception of spatial positions, visual discrimination
-Figure ground perception
the ability to identify and isolate a specific figure out of a complex background polygon.
Perceptual constancy
is the ability to recognise that some characteristics of an object are independent of ‘size, colour, texture,
or position. That is a a blue cup and a red cup are both cups and a cup upside down is still a cup.
Perceptual constancy
requires decisions about shapes that are on the border of a concept.
-Mental rotation
Defined by Gutièrrez as the ability to produce dynamic mental images and to visualise a configuration in movement (1996, pg10), provides considerable challenge even to adults. Note that the movement does not specifically have to be rotation. It can be any movement of mental imag
Perception of spatial position
is the ability of a person to relate objects (or pictures or mental images) to themselves.
Perception of spatial relationships
involves relating several objects to each other and to the person concerned.
Learning activities
should include problem solving situations that allow students to begin in an informal and investigative manner, with students exposed to a variety of regular and irregular 2D shapes and 3D objects. In the primary grades, those understandings should be extended in ways that encourage students to analyse shapes and objects by defining characteristics of different shapes and objects and considering these properties in relation to other 2D shapes and 3D objects. In time, students should be able to consider relationships and differences between classes of shapes and objects.
‘shape
refers to a two-dimensional figure
object
refers to a three-dimensional figure”
POLYGON-
A polygon is any two-dimensional shape bounded by line segments.
REGULAR SHAPE
A two-dimensional shape is said to be regular if all of its sides are the same length and all of its internal angles have the same measure
CLOSED SHAPE
Any two-dimensional shape that is formed when a line starts and ends at the same point is called a closed shape. The shapes shown below are all closed.
OPEN LINE
Any line curved, straight or a mixture of the two that does not join up with itself
Transformations
are processes that are used to manipulate shapes, examples include enlargements, reflections, rotations and translations. Students in Stage 1 are required to identify and name 2D shapes presented in different orientations, according to their number of sides. For this reason it is important for students to develop an understanding of how shapes can be manipulated. Students are introduced to the terms flip, slide, turn, clockwise, anticlockwise, half turn, full turn and quarter turn.
This is further developed in Stage 2 when the correct mathematical terminology for some transformations is introduced. Translate replaces slide, rotate replaces turn and reflect replaces flip. A square may be translated, reflected or rotated.
Symmetry:
- By the end of Stage 1 children should be able to recognise the connection between line symmetry and performing a flip. Paper folding exercises are extremely beneficial in assisting students to visualise this.
- Stage 2 children should be able to identify symmetry in the environment as well as be able to draw lines of symmetry on 2D shapes.
- In Stage 3 they are introduced to rotational symmetry as a means by which 2D shapes can be manipulated. Students may develop and apply understandings of transformations and symmetry across curriculum areas of art, craft, technology, science and geography.
Angles
Are introduced informally in Stage 2 as the ‘amount of turning’ between two arms.
Learning activities should provide students with an opportunity to create, identify and describe a range of angles. An example might include bending a pipe-cleaner to form a set angle and then using this ‘angle tester’ to compare the size of angles that occur in the classroom. From this, students can be introduced to different types of angles including acute, right, obtuse, straight, reflex and revolution.
-In Stage 3, children are introduced to the protractor as an implement used to measure the size of an angle in degrees. By the end of Stage 3 their understanding of angles should have progressed to the point where they are able to confidently recognise and identify special angles embedded in diagrams, including angles on a straight line, angles at a point and vertically opposite angles.
Position:
Is introduced in Early Stage 1 where students are taught how to describe the position of an object in relation to themselves, or another object. The language developed should include position, between, next to, behind, inside, outside, left, right and directions.
-Simple maps are introduced in Stage 1 and then expanded on in Stage 2 when students develop skills in using grid reference systems, using a legend to locate specific objects and using scales to calculate distances. Children should also be introduced to compass directions. In Stage 3, children should be able to describe routes between locations on maps using compass directions. They should also be able to find and/or describe locations on maps using their grid reference.
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.
