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Teacher Guidance A simple (hand-shaking type) investigation based on a set of Mystic Roses. Students may or may not already be familiar with the sequence of triangular numbers and some groups may be aware of the general formula. The slight complication here of course is that the L = (p -1)p/2. Obviously it is a matter of tailoring the presentation to suit the particular group. You may wish to add in formulas etc as you see fit. Low ability groups can still get quite a bit out of it and if all else fails slide 8 will keep them occupied (and hopefully thinking). Printable worksheets are available at slides 6 and 7 should you wish to use them. There is more advanced work on triangular numbers in the presentation “Stairs and Steps”.

Lines and Dots Dots Lines 1 2 3 4 1 3 6 Draw the patterns shown plus the next two in the sequence and investigate the relationship between the number of lines and the number of dots. Dots Lines 1 2 3 4 1 3 6

Lines and Dots Dots Lines 1 2 3 4 6 5 10 6 15 7 8 21 28 2 3 4 6 5 10 6 15 7 8 21 28 Complete the table for the diagrams shown.

Black 5 18 Point Mystic Rose 1 + 2 + 3 + … +15 + 16 + 17 = 153 Can you use the rule that you found to calculate the number of lines in this 18 point Mystic Rose? or (17 x 18)/2 if you are familiar with the formula for triangular numbers.

18 Point Mystic Rose 1 + 2 + 3 + … +15 + 16 + 17 = 153