Lesson 27 Accelerating the acceleration

Episode 2

Reasoning Resources: Worksheet 2, Worksheet 3 for Extension, enlarged graph from Worksheet 1, Calculators
Whole class and pair work
The intention is to extend the agenda of TM23: Rates of change from linear and quadratic relations to their first acquaintance with the weird properties of an exponential.

It is one further link to the mathematical modelling which is used, without explanation or background,
in science.
Give out Worksheet 2. Pupils work in groups, using calculators, on question 1 first, and only after they have completed the grains column go to question 2 and 3. Encourage them to see the pattern of doubling the number of grains in the middle column. As they are working, go around helping pupils to round answers to two significant figures. They should notice the advantage of using 2"° as a convenient unit in its own right. (10 steps, 2° = 1024, which can be rounded to 1000 to 2 s.f. with a small error of about 2%.) (Some pupils may know that this is what we call kg.)

Ask pairs and groups to summarise their conclusions, and deal with any computational difficulties. For question 2, write a summary of the conversion values of kg/tonne on the board:

200 grains weigh about 100 g, so 2000 grains weigh about 1000 g = 1 kg and 2000000 grains weigh about 1000 kg = 1 tonne. A large lorry carries about 50 tonnes.

For question 3 different pairs would explain the differences between the rocket function and the grain function in their own ways. Display the graph from Worksheet 1 with vertical axis extended to, say, 500 (for 29). Plot the points and sketch the graph up to (10, 2°). Where will the next point be? Now compare this new graph with the linear and quadratic patterns already there.
Here we want them to see that there are two different kinds of acceleration. nThe first, which in TM23 was applied to the motion of a rocket, is the kind they will get in physics (constant acceleration: e.g. using sensors to capture data for an accelerating trolley, and noting the increasing distance between the dots on the tape). But the second is more usually met in biology, e.g. the growth of yeast. It is also another kind of acceleration, which does more than increase. The rate of increase goes on increasing at the same rate of increase as the acceleration itself.

The questions allow for imagination and puzzlement.
Scientists are always trying to make other rockets, and some are thinking of a rocket that rises 1 metre in the first second, twice that in the second, and twice that again in the third and so on. They call it the DD rocket. (Some argue this is impossible. We will discuss that later!) Some pupils may be interested in issues related to the exponential concepts handled. You may wish to discuss the questions in a whole class mode with brief periods for pair consultation. Questions 1 and 2 on Worksheet 3 use data from the table and allow comparison between distance to the Moon and Sun.

Question 3a may be discussed starting with explosives. A small detonator is first triggered, which triggers 3 or 4 others, each of which triggers 3 or 4 more, etc. all within a fraction of a second. With atomic fission something like that happens at the basic unit of matter, but can the process be controlled? Question 3b would bring out ideas of whether there is friction in space to raise the temperature of the rocket, and whether cosmic dust particles could count as atmosphere. Question 3c leads to discussing the speed of light (about 300 000 km per second) and our current knowledge of what happens to matter at such speeds. The questions should allow flights of imagination rather than be based on closure and giving final answers.
End of Lesson Reflection
One analogy is of the physical constraints of the amount of grain with the other physical constrains of maximum speed. Regardless of how far the class have progressed in this activity, pupils should consider other situations where exponential growth is witnessed. For example:

  • A baby starts as a single cell: then this cell splits into two cells.
  • In bread mixture, each of the yeast particles splits into two, then each of those splits into two, every 20 seconds .....
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