uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures
generalises mathematical ideas and techniques to analyse and solve problems efficiently
uses deductive reasoning in presenting arguments and formal proofs
draws, interprets and analyses graphs of physical phenomena
- Solve problems involving direct proportion; explore the relationship between graphs and equations corresponding to simple rate problems (ACMNA208)
- interpret distance/time graphs when the speed is variable
- match distance/time graphs to situations, and explore whether they are accurate, appropriate and possible (Problem Solving, Reasoning)
- match distance/time graphs to appropriate descriptions and give reasons for choices (Communicating, Reasoning)
- record the distance of a moving object from a fixed point at equal time intervals and draw a graph to represent the situation, eg move along a measuring tape for 30 seconds through different activities that include variable speeds, such as running fast, walking slowly, and walking slowly then speeding up (Communicating, Problem Solving)
- analyse the relationship between variables as they change over time, eg draw graphs to represent the relationship between the depth of water in containers of different shapes when they are filled at a constant rate
- interpret graphs, making sensible statements about the rate of increase or decrease, the initial and final points, constant relationships as represented by straight lines, variable relationships as represented by curved lines, etc
- decide whether a particular graph is a suitable representation of a given physical phenomenon (Reasoning)
- describe qualitatively the rate of change of a graph using terms such as 'increasing at a decreasing rate'
- sketch a graph from a simple description, given a variable rate of change
Rate of change is considered as it occurs in practical situations, including population growth and travel. Simple linear models have a constant rate of change. In other situations, the rate of change is variable.
This work is intended to provide students with experiences that will give them an intuitive understanding of rates of change and will assist the development of appropriate vocabulary. No quantitative analysis is needed in Stage 5.3.
When describing graphs of rates of change, teachers should model the various words and language structures before independent work is required, eg 'The population is increasing at a decreasing rate'.