Lesson 23 Rates of change

# Rates of change

Overview Resources
Worksheets 1, 2 and 3
An activity on comparing linear and quadratic functions in terms of rate of change as well as their general patterns, something pupils have accessed earlier in TM10, TM13 and TM15. The starting point is algebraic formulas, which are evaluated in tables and then drawn as graphs. Pupils then use this algebraic process in a context they can easily visualise, but still need to interpret mathematically.
Concept of rate of change. Graphs; gradient of functions.

(The National Curriculum places ‘familiarity with graphs of the major functions’ and ‘tangent to curve measuring slope’ at its highest levels. Here we approach these concepts in largely intuitive terms.)
EPISODE 1
Step and rise in a linear graph
Starting from a one-step linear function b = 4a, pupils explore different values of variables, then complete a table focusing on the change (rise) in the function b for unit step of change in the variable a. They then plot the graph and exchange between them the most appropriate language to describe it. The ‘step and rise’ pairs at different points on the graph would hopefully arise through pupils’ offerings, in whatever language form they prefer. Pupils should be able to connect elements of the table to the graph, including the steepness or the constant rise for the same step across all values of a.

Some groups may wish to explore other linear functions, e.g. b = 2a, b = 8a etc.
EPISODE 2
Step and rise in a quadratic graph
Working in a similar way to Episode 1 on the function b = a 2, pupils explore the patterns of change in the table, and how that is shown on the graph by ‘step and rise’ pairs. They put in their own words how different starting points for the step produce different values for the rise. They also recognise a pattern in the changes.
EPISODE 3
The balloon and the rocket
Pupils’ insights and language in handling variables and patterns of change are now employed in the real-life contexts of a balloon and a rocket rising vertically. The focus is on comparison of the two mathematical patterns, either through visually seeing them on the graphs, or by looking at number values in tables, or in both.

Some pupils will benefit from airing a common error in reading the graphs as pictures of the events, i.e. seeing the straight line as the trajectory of the rocket, and the curve as a trajectory of a balloon rising with the wind blowing. They contrast that with understanding graphs of functions as relations of height versus time in vertical movements for both rocket and balloon.

Some more able pupils would realise they are dealing with velocity and acceleration. Pupils may talk in their own ways about the balloon showing a linear function (having reached its terminal rising velocity). A few pupils may also discuss the rocket climbing vertically with constant thrust as a use of constant acceleration, shown in the pattern of change.