Mathematics

Autumn 1: Reasoning in Angles (3 weeks) and Probability (3 weeks)

Skills

• Review of angles including angle-chasing accompanied with chains-of-reasoning (finding quantities by deductive reasoning about properties) and is a useful introduction to proof.

Derive and utilise circle theorems for increased complexity angle-chasing accompanied with chains-of-reasoning is a useful introduction to proof: deductive reasoning about properties

Use experimentation, observation, frequency tables, relative frequency as well as theoretical probability to make predictions about expectations of future events.

• Use sample spaces, two-way tables, tree diagrams and Venn diagrams to visualize combinations of independent or dependent events and to calculate conditional probability, probabilities of repeated events and repeated events without replacement.

Knowledge

• The central idea of mathematics in general and geometry in particular, is the study of invariance. Most theorems in geometry can be seen as resulting from the study of what change is permitted that leaves some relationships or properties invariant.  For example, knowing that the angle in a semicircle, is an invariant 90 degrees provided that the point being moved remains on the circle; if it moves inside the circle, the angle is greater than 90 degrees, if it is outside it is less, which could mean that a valid definition of a circle would be the locus of points that subtend an angle of 90 degrees to the ends of a line segment.
• Geometry consists of three kinds of cognitive process: visualisation processes (for example, the visual representation of a geometrical statement), construction processes (when using tools, be these ruler and compass, or computer-based), and reasoning processes. Knowing that circle theorems require spatial thinking and visualisation and as well as deductive reasoning involving transformation or congruency arguments. These aspects are not separate; each gives rise to the other and each only exists in relation to the other.

Rationale

Geometry is one of the longest established branches of mathematics and remains one of the most important. Its development can be traced back through a wide range of cultures and civilisations with its origins in the surveying of land and in the design of religious and cultural artefacts.  If anything, geometry is becoming more important across many fields. This is not only because of the wide-ranging applications of geometry in everything from robotics to CGI (computer generated imagery) movies, from crystallography to architecture, from neuroscience to the very nature of our universe, but also because new geometrical ideas are being generated within such diverse fields as these.

The challenge of making sense of situations that are unpredictable pervades everyday living.  Situations that lack information create conditions for uncertainty. Probability is one approach to reason about such uncertain situations. Uncertainty in some games and sports contests can be compelling and fun.  A player who can reason about uncertainty will be able to adopt strategic approaches, which will improve the player’s success rate in the long term.  Uncertainty is also apparent in everyday decision making.