This is Chapter 1 – Relationships Between Variables for the Year 11 Mathematical Methods course.

This opening chapter establishes the modelling mindset that underpins Mathematical Methods. Rather than solving equations only as isolated algebraic tasks, students begin analysing how one quantity changes in response to another and how those relationships can be represented using equations, tables and graphs.

We begin with linear relationships, revisiting gradient and intercepts while placing greater emphasis on interpretation. Students learn to view gradient as a rate of change and to connect the parameters of a linear equation with the behaviour of the corresponding graph.

Inverse proportion then introduces a different type of relationship in which one quantity decreases as another increases. Students develop equations from contextual information, identify the constant of proportionality and use the model to solve practical problems. This creates an early contrast with linear models and helps students recognise that different contexts can produce very different patterns of change even when the same two variables are involved.

Reciprocal functions extend this reasoning into function behaviour. Students examine domain restrictions, intercepts and asymptotes and learn that some graphs approach particular values without reaching them. Rational functions then broaden these ideas by combining algebraic expressions and restrictions within more varied graph shapes.

These relationships provide important preparation for later chapters on functions, polynomials, trigonometry and calculus. Mathematical modelling of variables is also widely used in science, economics and engineering, where equations are used to describe how real quantities respond to one another. Students are encouraged to compare models rather than memorise graph shapes in isolation, building a stronger sense of why a particular relationship suits a given situation.

Subchapters:

• 1A: Linear Relationships
  
• 1B: Inverse Proportion
• 1C: Reciprocal Functions
• 1D: Rational Functions

By the end of this chapter, students will be able to construct, graph and interpret linear, inverse, reciprocal and rational relationships while explaining how changes in one variable affect another.