Construct and interpret plans and elevations of 3D shapes (1)
This is the first of two sets of tasks involving plans and elevations. In this set our 'shape' consists of various arrangements of three vertical rods. The tasks make substantial demands on students' visualisation skills. The first three tasks bring out the fact that an elevation, in isolation, is ambiguous. The fourth and fifth tasks explore a kind of elevation-view symmetry.
MONDAY: Here we start with a 3D view (technically, a parallel projection) of three vertical rods and remind students of what is meant by a plan view and elevation from a given direction (viewpoint). The task is quite demanding as we focus on a viewpoint (from C) that is substantially different from the one shown in the 3D-view. Also, students have to imagine the arrangement of the rods after one of them has been moved.
How do students interpret the various diagrams? For example, do they imagine rotating some of them, or do they imagine moving themselves to a different viewpoint (such as that of C)?
The elevation view of the blue rod from C is unchanged, so the rod must have moved in the direction C-to-A, ie moved one square away from C towards A:
TUESDAY: Here we are given elevations from two viewpoints which allows us to 'triangulate' (sic!) the position of the rod. However, as some of the rods are moved, the given drawing of the rods (the parallel projection) is of limited use: there is a lot of information to keep track of and to coordinate!
WEDNESDAY: Here we again have two elevations, but this time the result is still ambiguous as some rods are hidden by other rods.
The blue rod has again not moved, but the other two have.
The elevation from B tells us that the green rod is hiding behind the yellow or the green rod.
The elevation from C tells us that the yellow rod is hiding behind the blue or the green rod.
This results in three different possibilities as show by the three plan views, below.
THURSDAY: Here we look at arrangements of rods, this time all of the same colour, that produce identical elevation views. It turns out that this can be achieved even if a plan view of the rods is not symmetrical.
This is a challenging task, since we have to view a putative plan view from two directions to check whether the resulting elevations from B and C are the same. The diagram below shows two possible solutions. Are there any more?
FRIDAY: Here we start with an arrangement whose plan view is the second of the three 'symmetrical' views shown above. So the two elevations from B and C are the same but this time we are asked to move two rods rather than one, to produce two more identical elevations.
We can maintain identical elevations if the two rods are always moved in a 'symmetrical' way, as here (below), even when the arrangements of the rods themselves (as seen in the first three plan views) are not symmetrical.
When students take this further, by exploring other arrangements, they might discover that the simplest way to arrange the rods to produce identical elevations from B and C is to place them symmetrically about the diagonal running from bottom-left to top-right of the square, as in the 4th of the plan views above, and in this example, below:
