2. The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Mathematics and Further Mathematics to ensure that more students reach their potential in mathematics. To find out more please visit www.furthermaths.org.uk www.furthermaths.org.uk The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.

3. A Level Maths & Further Maths  Highly regarded and popular A levels  Facilitating subjects for universities  Many A level and university subjects require maths knowledge  Opens doors to a variety of careers, many of which are well paid  Enjoyable  Challenging  Useful

4. Careers that need maths qualifications  Engineer  Scientist  Teacher  Computer programmer  Games designer  Internet security  Logistics  Doctor  Dentist  Meteorologist  Finance  Accountancy  Business management  Nursing  Veterinary  Sports science  Physical Therapist  Pharmacist  Medical statistician And many more…

5. Fractals

6. What is a fractal?  A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop.

7. In nature…

8. Romanesco broccoli

9. Fjord in Norway

10. Fern Leaf

11. In Geometry…

14. The Sierpinski Triangle  The Sierpinski Triangle is made by repeatedly splitting a equilateral triangle into 4.  There is a lovely animation of an infinite Sierpinski Triangle here:  http://fractalfoundation.org/resources/what-are-fractals/ http://fractalfoundation.org/resources/what-are-fractals/  Or here  https://www.youtube.com/watch?v=TLxQOTJGt8c https://www.youtube.com/watch?v=TLxQOTJGt8c

15. Can you draw a Sierpinski Triangle?

17. Iteration 1

18. Iteration 2

19. Iteration 3

20. Iteration 4

21. Iteration 5

22. Questions. Look at the triangle after the first iteration. What fraction of the triangle did you NOT shade? What fraction of the triangle is NOT shaded after the second / third iteration? Do you see a pattern here? Use the pattern to predict the fraction of the triangle you would NOT shade in the fourth iteration. Can you confirm you prediction? What about the n th iteration?

23. Koch Snowflake –how is this made?

29. Koch snowflake zoom  https://www.youtube.com/watch?v=PKbwrzkupaU https://www.youtube.com/watch?v=PKbwrzkupaU

30. KOCH Questions  Assuming that the length of each side of the original triangle is 1 unit complete the following table: –Can you work out a formula for the perimeter at the nth stage? –Do you think the area of the snowflake curve is finite or infinite? STAGEPERIMETER 13 2 3 4 5 6

31. Do you think the area of the snowflake curve is finite or infinite?

32. Does this picture help you?

33. An infinite perimeter encloses a finite area... Now that's amazing!!

34. Let’s create a Dragon.  Draw a straight line, say 2cm long, but the larger this line, the more stages you can draw. This is the start of your dragon.  Now draw two lines so that the original line forms the hypotenuse of an isosceles right-angled triangle and erase the original line. This is the first stage. (See the diagram below.)

35.  For the second stage, replace each of the lines from the first stage with two new segments at right angles so that the lines from stage one form the hypotenuse of an isosceles right-angled triangle. The new segments are placed to the left then to the right along the segments of the first stage.  Continue this construction, always alternating the new segments between left and right along the segments of the previous stage. This generates the ‘dragon curve’.

36. Stage 1

37. Stage 2

38. Stage 3

39. Stage 4

40. If you keep going you get this.

41. You can see the Dragon Curve generated here: http://mathworld.wolfram.com/Drag onCurve.html

42.  Can you generate Pascal’s Triangle to line 15.  Colour in the odd numbers.  What do you notice? Extension Activity

44. Fractals get everywhere…..

45. What are Fractals useful for?  Nature has used fractal designs for at least hundreds of millions of years.  Recently engineers have been copying nature to use fractals.  Fractal geometry can also provide a way to understand complexity in “systems” as well as just in shapes. The timing and sizes of earthquakes and the variation in a person’s heartbeat and the prevalence of diseases are just three cases in which fractal geometry can describe the unpredictable.

46. There is an interesting article here:  http://www.bbc.co.uk/news/magazine-11564766 http://www.bbc.co.uk/news/magazine-11564766

47. Fractal Antenna  These are being developed for mobile phones.  Because of its folded self similar design, the antenna can be very small, while still about to receive signals across a range of frequencies.

48. A Level Maths & Further Maths  Highly regarded and popular A levels  Facilitating subjects for universities  Many A level and university subjects require maths knowledge  Opens doors to a variety of careers, many of which are well paid  Enjoyable  Challenging  Useful

49. Careers that need maths qualifications  Engineer  Scientist  Teacher  Computer programmer  Games designer  Internet security  Logistics  Doctor  Dentist  Meteorologist  Finance  Accountancy  Business management  Nursing  Veterinary  Sports science  Physical Therapist  Pharmacist  Medical statistician And many more…

50. The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Mathematics and Further Mathematics to ensure that more students reach their potential in mathematics. To find out more please visit www.furthermaths.org.uk www.furthermaths.org.uk The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.