Distance Time Graphs
Real Time Graphs
This is a rough outline of an activity that can be used to introduce the concept of Calculus. This is obviously a really key moment. I think it is very difficult to make progress with this topic if you don't have some grasp of what it is really all about. This activity is based on the creation of distance time graphs that are made in real time through experiments.
I have tried this activity in a number of ways but probably the most successful is by using some of my sons 'Hot Wheels' gear, a golf ball, some counters and a piece of string.
The video below shows students making a distance time graph for a golf ball traveling along the track. Each student has responsibility for placing a point on the graph after a given number of seconds. The time is being read out by another student. In the first instance, the point is placed on the vertical axis (distance) and then moved across to the corresponding time value on the x - axis. The points are then linked together with a piece of string to complete the distance time graph.
The creation of the graph in real time gives the graph and its context a level of meaning that really facilitates the conversation that follows. Obviously, the conversation can be run in a number of different ways - the following is just a suggestion.
Question 1 - Does the golf ball's speed vary and if so how?
Students should explore different sections of the graph trying to describe what is happening to the speed of the golf ball and why? It is nice if you can tease out the articulation of the idea that steepness is the measure of speed.
Question 2 - How fast is the golf ball traveling after 1 second?
This is the starting point from which the key principles can be drawn. Probably, students will converge on the idea that the speed can be deduced by dividing the distance by the the time and that this is a measure of average speed for that second.
Question 3 - How fast is the golf ball traveling after 2 seconds?
Now students might debate whether or not they should consider the average speed over the first 2 seconds or take a measure of what happened over the last second. You should work towards the conclusion that the latter is a more accurate measure of the speed, but still only an approximation, because the speed still varies in that time.
From here you can investigate the idea of smaller intervals by interpolation and more accurate approximations which, in turn can lead to the idea of small changes in time, limits and the instantaneous speed being the gradient of the curve at the point.
I have really enjoyed this activity and the idea that students make something that becomes the source of the sophisticated idea. Often this conversation is played out informally while students are sat around the edge of their graph and interacting with it by going in to it and moving things around.
Give it a try!