Year 9 Common Task: Painted Cube
A first lesson:
Discuss with the class first what it means
to do a piece of maths coursework. What skills/techniques are required,
etc. Make a poster of the class' ideas.
'You now need to put in practice all the skills
we've talked about. The problem you are going to work on starts
with a visualisation … Put all your pens down … You
may want to put your head in your arms so you can close your eyes
to do this … [Then said slowly and with feeling! …]
Picture in your mind a cube …Try to spin it around a bit so
you can see it from all angles … The cube must be solid, made
of some kind of white material … Now colour the outside of
the cube red … Spin it around again … I now want you
to make a series of cuts in your cube so that you make it into cubelets
… so you are going to do a set of vertical cuts, then another
set of vertical cuts and then a set of horizontal cuts … so
that your original cube is split up into a whole load of cubelets
… Can anyone say anything about these cubelets, and in particular
about how many of their sides are red?'
- You don't know how many cuts to make
- There will be eight cubelets with 3 sides painted
[Responses will vary with the type of student you teach …
if they are able to make general comments, work with them for as
long as possible on this. Otherwise you will need to look at particular
examples, eg start with 2 cuts and a 3by3by3 cube. Try to get students
visualising this, again for as long as possible and set up the challenge
of finding how many cubelets have 0,1,2,3 sides red for any number
of cuts or any size starting cube.
To help students who are struggling, isometric
paper and or multi-link can be invaluable.]
Where this can go
This problem can run for a standard length piece of coursework,
ie up to 2 weeks of lessons. Bring the class back together regularly
to hear from individuals about what they have found out, or what
approach they are using to begin the problem, or what justification
they are using. With all students push them to justify their findings
and rules by appealing to the structure of the cube (this is the
key to accessing mark 5 at GCSE, or level 7 at KS3). See exemplar
piece, which we use with year 11 students, for what a project can
look like.
High attaining students should get on to looking
at cuboids and also expanding out their algebraic rules to check
they come to eg n3 for nbynbyn cubes.
At the end of the project you could swap over
scripts between students and get them to write positive and constructive
comments about someone else's work before you mark it.
Guidelines for notes at the end of the topic
Get students to reflect on the process of doing
this piece of coursework and come up with ideas/resolutions/tips
for next time. This may be best done after you have handed back
the projects and marked them.
Vocabulary
Counter-example
Valid (validity)
Proof
Justification
Typical mathematical content
Approaching tasks in an organised way
Drawing on isometric paper
Using algebra to express rules
Justifying rules by appealing to the mathematical structure from
which they have arisen
Extending problems
Providing algebraic justifications for rules
A homework to be done by everyone
All homework during this project will be
to continue the classwork.
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