Lesson 12 Functions

# Functions

Overview Resources
The context of meshed cogs generates a continuous multiplicative relation of a different kind to earlier lessons, but continuity can remain implicit.

Proportionality is also implicit, but can easily be addressed if appropriate.
Worksheets 1 - 3
• Develop understanding of ratio and proportion
• Look at a function in multiple representations.
• Multiplicative relations
• Graphs
• Generalised number
• EPISODE 1
The two variables in a cog pair
Given a set of random pairs of data in a context, pupils construct a table to help them make sense of the mathematical relationship between the values. It helps to view one value as an input and the other as an output - some pupils may be familiar with the idea of a ‘function machine’ They can continue the table. EPISODE 2
Coordinating two relations Now pupils find examples and pairs of values in their table from Episode 1 where the input value doubles or trebles, and look at what happens to the output value. This coordination of two relations allows an extension of the proportional relationship that is more intuitive than the procedures normally taught. EPISODE 3
Graphing relationships
Looking first at the table of values, pupils discuss main features of the relationship as a whole, i.e. that there is only one output value for each input and the pattern of increase, and predict the shape of a graph. They plot the graph and discuss its features: constant rate of increase, and the steepness corresponding to the relationship. Pupils may discuss what they think their results mean in terms of the cogs and the number of teeth in each. Some pupils may find some way of expressing this function in terms of generalised number.
As an extension they can look at another pair of cogs with a different ratio, to compare.