MONDAY: Mathematically, this task involves a nice context, though one that is perhaps not so familiar to today's school students - you might therefore have to tease out the crucial fact that audio tape has to travel at a constant speed (as it passes over the tape-head which picks up the sound). Most students will realise that as the left hand spool unwinds, the width (diameter) of the reel of tape decreases. This means the circumference also decreases and so each successive turn of the spool will feed out less and less tape. So if the tape passes the tape-head at a constant speed, the spool will rotate more and more quickly, and the width of the reel will decrease more and more rapidly.
This task should generate rich classroom discussion about these geometric aspects of the context and about the shape of the graph: the graph clearly slopes downwards, but is it a straight line?
Note: The curve through A and B is a parabola. It turns out that it passes roughly through the point (76, 0) and that the relationship between M minutes and W mm, is given by M ≂ 76 – (W²/30) [or, more precisely, W = 78.83 – (12W²/345)*, if we assume the given measurements are accurate].
*You might like to verify this formula (please correct me if I've got it wrong).
Note 2: Here is an alternative wording for the task. Would this be better?
TUESDAY: In this task we assume that the pile of sand keeps its exact shape as it gets bigger. Students might initially think that if the time for which the sand flows is doubled (from 1 minute to 2 minutes), then the height of the pile will be doubled too. (It turns out that it takes 8 minutes rather than just 2, for the height to reach 2 metres - something we look at in Wednesday's task.)
However, it shouldn't take much discussion for students to realise that as the pile gets higher it gets wider too, so ever more sand is needed to raise the height by a given amount. This means that the graph will consist of a curve with an ever flatter positive slope. In turn this means that at the 2 minute mark, the height of the graph will be substantially less than 2 (m). However, at this juncture we wouldn't expect students to have a more precise sense of what that height will be. This is something we consider in a more analytic way in Wednesday's task.
WEDNESDAY: We consider what the height of the pile of sand will be after 2 minutes, and use this information to draw a more accurate graph, which we then use to interpolate the height at 2 minutes.
A good approximation for the value of the height at 2 minutes is 5/4. This is close to the cube-root of 2, since (5/4)-cubed is 125/64 = 1.953125.
THURSDAY: We step onto a travelator and think about some (nice, simple, straight line) distance-time graphs.
FRIDAY: We relate our distance-time graphs to speed-time graphs. This may not be easy - they look so very different!
Marco's and Mario's speeds are constant for most of the time (except for the brief moments when their speeds change). This means their speed-time graphs will consist primarily of horizontal straight lines. As such, the graphs look very different from the distance-time graphs, where the increase in distance (from Lisa) over time is shown by the upward slope of the graphs. In contrast to this, on the speed-time graphs, the increasing distance is shown by the increase in area under the graphs as the time increases. This looks far less dramatic.