id: 06448936
dt: j
an: 2015d.00619
au: Foster, Colin
ti: Fitting shapes inside shapes: closed but provocative questions.
so: Math. Sch. (Leicester) 44, No. 2, 12-14 (2015).
py: 2015
pu: Mathematical Association (MA), Leicester
la: EN
cc: G40 D50
ut: plane geometry; rectangles; squares; solid geometry; cubes; cuboids;
problem posing; open-ended problems; open and closed questions
ci:
li:
ab: From the text: Mathematics teachers are often encouraged to try to turn
closed questions (such as ‘What is $8 \times 3$?’) into open
questions (such as ‘What numbers multiply to make 24?’) because
open questions are widely perceived to be richer and more productive.
However, sometimes a question that is technically closed ‒ even a
dichotomous yes/no question ‒ can lead to lots of interesting
discussion and thought. Also at the level of school mathematics, the
power of closed but provocative questions should not be underestimated.
Recently in this journal Chris Pritchard has given us a fascinating
series of articles on fitting shapes inside shapes. (November 2010
through to September 2013, with problem sets following to January
2015.) Continuing this theme, I offer here three questions ‒ all of
them closed ‒ which I hope that you may find provocative: 1. Will a
$1 \times 6$ rectangle fit completely inside a $5 \times 5$ square? 2.
Will a $2 \times 13$ rectangle fit completely inside a $9 \times 12$
rectangle? 3. Will a $1 \times 4 \times 8$ cuboid fit completely inside
a $6 \times 6 \times 6$ cube? Naturally, you are not allowed to break
up the shapes in any way! Of course, there is an implicit invitation in
these closed questions to justify your answers as well as to generalize
and invent ‘easy’ and ‘hard’ problems along these lines.
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