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'More Magic Potting Sheds' printed from https://nrich.maths.org/
This problem requires students to work systematically and
challenges them to arrive at further generalisations. It is an
excellent exemplar of a context where a full understanding of one
problem is really useful when applied to the next problem. It is
useful as an object lesson in mathematical process skills.
The problem can be tackled on paper but access to the interactivity
will speed up the process of exploring different possibilities. It
is important to have paper available for students to record the
results of the different trials.
Teacher's Notes
Start with Magic
Potting Sheds
Once students have found several answers to this problem, bring the
class together.
Suggest that if they can understand why the solutions involve
multiples of $7$ and $8$ they will find it a lot easier to solve
similar problems.
Ask students if they can explain why the solutions are multiples of
$7$ and $8$.
When I have used this with students, I've found that a particularly
effective intervention at this point is to offer the following way
of picturing the situation:
Imagine Mr McGregor places his
$7$ plants on three shelves:
$4$ on the bottom shelf, ready
for planting the next day
$2$ on the middle shelf, ready
for planting the day after
$1$ on the top shelf, ready for
planting on the last day.
On the first night they become
$8, 4$ and $2$ and the $8$ are planted in the first garden.
That leaves $4$ on the middle
shelf and $2$ on the top shelf.
The following night they become
$8$ and $4$, and the $8$ are planted in the second garden.
That leaves $4$ on the top
shelf.
On the last night the $4$ become
$8$, and they are planted in the third garden.
So $7$ is significant because it
is the sum of $1, 2$ and $4$ (the smallest triple of numbers that
have the required doubling relationships).
Equally, Mr McGregor could have
placed $5, 10$ and $20$ plants on his three shelves (multiples of
$\{ 1, 2, 4 \}$):
The $20$ will become $40$ after
one night.
The $10$ will become $40$ after
two nights (double and double again).
The $5$ will become $40$ after
three nights (double and double and double again).
Students can now be set to work on More Magic Potting Sheds, using
this or other insights to help them along the way. Encouraging
students to reflect on the solutions to the original problem may
help them tackle the follow-up questions in a more informed way.
The intention is to avoid the situation in which students just sit
at the computers typing any value in until they eventually hit upon
the answer. It may be a good idea to set students working on the
follow-up questions with paper and pencil, and only allow them to
move to the computers once they have clear ideas about what they
should try.