Shape and Space in Geometry

Geometry Geometry is the mathematics of shape and space. It is about the properties of objects (their angles and surfaces, for instance) and the consequences of how these objects are positioned (where their shadows fall, how people must move around them). (NCTM, 1989)

Importance of Geometry World is build of shapes and space – geometry is the mathematics of this Application of geometry go beyond its field Informal geometry is good preparation for abstraction and proof later on For people who think visually more easily, geometry can be their way in to mathematics (NCTM, 1989)

Spatial Understanding Spatial understandings are necessary for interpreting, understanding, and appreciating our geometric world. Insights about two- and three-dimensional shapes and their characteristics, the interrelationships of shapes, and the effects of changes to shapes are important aspects of spatial sense. (NCTM, 1989, p.48)

About Shape… We first meet geometry through shapes and their properties. Geometry and spatial sense require developing deep understanding that takes years and encompasses many subfields Symmetry, coordinates, and proportion are the core ideas of the following three activities, which approach shape and properties of shape from different directions. (NCTM, 1989, p.48)

About Space… Visualization is an important part of geometrical thinking. It is the skill you use when you pretend to be somewhere else and imagine how that place looks, or when you fancy how a situation would look if things were just a little bit different. (NCTM, 1989, p.48)

About Space… But…visualization is especially problematic in three dimensions—perhaps because mathematics curricula do not emphasize three- dimensional geometry so… Some people have a hard time, for example, rotating an object in their minds to see how it would look from a different angle. When looking at a map, some find it hard both to imagine where they are on the map and to grasp the relationships of the map objects around them. (NCTM, 1989, p.48)

Visualising What is communicated through diagrams? Mason and Johnston-Wilder, 2005

Visualising 3D objects in 2 dimensions Mason and Johnston-Wilder, 2005

Isometric projection

Times, 3 rd Feb, 2015.

Plot Plans…construct one for this Front, side and top view

Casting Shadows… Can you judge an object by its shadow? For example, suppose you had a shadow that was a square. What shape could cast a square? What shape could not cast a square?

What about curved shapes? Mw

Gaudi – master of the curve?

Things to consider when beginning to map a lesson sequence… Properties of shapes – staying the same Language use and viewpoints Shadows Reasoning and explaining Visualising and representing

Mapping the bigger picture… perimeter, area, volume formula/units of measurement/conversion geometric objects in different dimensions angles/symmetry/ patterns and tessellations/shadows transformations- rotations/translations/reflections constructing/drawing/visualising

An open investigation – linking space and shape Use this object to develop a lesson sequence State your intended purpose and audience Don’t limit your investigation Include manipulation/handling of something Demonstrate some differentiation

Pedagogic Aspects Also consider how students will be systematic in the recordings Can they discover rules? Can they make predictions? Can they pose further lines of inquiry Work in groups of 5 – you have 30 minutes before a 5 minute presentation.

Group leaders… Work in groups of 5 – you have 30 minutes before a 5 minute presentation.

SE2 Think about learning theory Think about what is driving student engagement Think about being the teacher you want to be…try one new thing everyday and reflect on whether it is a keeper or not ….best of luck!

“Shape is a vital, growing and fascinating theme in mathematics with deep ties to classical geometry…the study of shape can from a central component of mathematics education, a component that draws on and contributes to not only mathematics but also science and the arts” ( Marjaorie Senechal, 1990)