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This problem offers students an opportunity to relate numerical
ideas to spatial representation, and vice versa.

Thinking about different ways of counting the number of lines
in a mystic rose can lead to a better understanding of the general
formula for triangle numbers.

This problem works very well in conjunction with
Picturing Triangle Numbers and
Handshakes. The whole class could work on all three problems
together, or small groups could be allocated one of the three
problems to work on, and then report back to the rest of the
class.

Start by showing the animation of the seven point mystic rose.
Then reset it and ask the students to describe to their partners
what they saw. Choose a different mystic rose and show the
animation, pausing it as it plays. Ask the class to predict what
will happen at each stage. Can they predict how many lines will be
drawn in total?

Set the class the challenge of working out how many lines are
needed to draw 8, 9, and 10 point mystic roses. Allow
them some time to work on this. Bring the class together to discuss
their answers and methods, and more importantly, how their methods
relate to the construction of the mystic rose.

Set the class another challenge, this time to work out how
many lines are needed to draw a much larger mystic rose
(e.g. a 161 point mystic rose). Allow them
some time to work on this. When they report back, discuss the need
for efficient ways of working this out. Draw attention to
"Charlie's method" in the problem, if no-one has suggested it, and
encourage students to think about how this method relates to the
image of the completed mystic rose.

For a class that has been introduced to algebra, students
could express "Alison's method" and "Charlie's method"
algebraically.

Finally, ask them to work out which of the following numbers
of lines could be used to draw mystic roses:

- 4851
- 6214
- 3655
- 7626
- 8656

What is special about the numbers of lines needed for
different sizes of mystic rose?

How do the different ways of working out the number of lines relate to the construction and final image of the mystic rose?

How do the different ways of working out the number of lines relate to the construction and final image of the mystic rose?

Can you draw a mystic rose using 9, 19, 29, 39, ... lines? Are
these impossible? How do you know?

Will there ever be a mystic rose constructed from a multiple
of 1000 lines?

Students could construct their own mystic roses using
different colours for the lines from each point, to build up an
understanding of their structure. Circle templates with dots evenly
spaced on the circumference can be found
here.