Lesson 27 Accelerating the acceleration
|Pupils contrast simple acceleration, as in a quadratic function, with exponential growth, using tables of values at unit time intervals. By looking at ‘change in the change’ values they can see the pattern of increase in unit time (constant for the linear relation, and steadily increasing for the quadratic). But with the exponential even the rate of increase of the increase goes on increasing. Here the emphasis is on the real-life implications of these different mathematical models.||Worksheets 1, 2 and 3 Calculators with 10 digit display|
|Multiplicative relations; metacognition and bridging.||
|Familiar speed and acceleration functions|
|Pupils compare the pattern of speed of a balloon and a rocket rising vertically, based on the graph produced in TM23: Rates of change. They talk in everyday language about the fact that speed does not change in the linear function, while it changes in the quadratic function. They look at the pattern of change in the quadratic function, and compare the two graphs.
They also see how the relationship in each case is not simply a sequence, but can be calculated for any value of time.
|The doubling function|
|Pupils consider the doubling function in the context of the wheat grains on successive chessboard squares. They compare its pattern of change with those of the balloon and rocket. Because of the huge numbers involved, the pupils round numbers to two significant figures.
Extension problems are suggested, including scientific concepts that make the doubling function unrealistic, and making an analogy with the problem of doubling the acceleration of a rocket. This allaws some discussions of the physical constraints of speed.