Why do this problem?
This
problem presents a series of three-dimensional challenges
which encourage the learner to visualise a solid and then use
two-dimensional representations to help them to reach a solution.
On the way to a solution, there are opportunities to practise
using trigonometry and Pythagoras, as well as formulas for
volume.
Possible approach
Each part of this problem could be tackled by small groups who
could then present their solution to other groups. For each
problem, it is best to take some time to visualise what is
being asked, and then draw some diagrams to see what
calculations will be necessary.
The first problem is the simplest of the three, requiring only
visualisation of a right-angled triangle from a section of the
cube, and an application of Pythagoras's theorem.
For the second problem, learners will first need to imagine how
an equilateral triangle could be constructed by cutting through
the cube, and from this work out the dimensions of the largest
such equilateral triangle. The information given is the area of
the triangle so learners will need to come up with a
relationship between the area and the side length.
The third problem is about volume and surface area. In order to
work these out, learners will have to calculate the dimensions
of a tetrahedron cut from the corner of a cube.
Key questions
What two-dimensional diagrams can be drawn to help to solve the
problems?
If I know the side of an equilateral triangle how can I find
its area?
Possible extension
Possible support
Learners who have not met or are not
confident with trigonometry and Pythagoras could solve parts of
the problem using scale drawing.