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2. Teacher Guidance The presentations in the Investigation Category are intended to be relatively short in nature, perhaps taking a single lesson or possibly two to complete. This could be the case with this one (hence the brief class introduction with some suggestions at slide 5 for order 3/4/5 spirals -- the order 5 spiral is inserted for added interest --). However, you could easily tailor this investigation to all ability groups and spend several lessons on it if you wished. It is generally open in nature so that students can investigate their own ideas. Some suggestions are supplied at slide 5 but you should edit the text on this slide to suit your particular group. The main objective of the slides is simply to provide you with a resource for showing the class how the spirals are generated. Hopefully the attractiveness of the completed spirals will motivate them in getting started. In addition you have a variety of (non- animated) spirals of various orders should you wish to use them. For students that need a prompt there is a list of spirals that they might want to draw (next slide). These correspond to the order 2/3/4 spirals shown in the table below with a few extra order 2 ones. The spirals are drawn throughout on a square dotty background (available to print: see last slide) but you may prefer the students to use square grid paper. There are some notes to look at on slides 7/8/9 that you may like to look at/delete. Students trying to explain the closed nature of the order 3 spirals may do so in terms of directions moved before each arm is completed E/N/W/S etc. Similarly order 4 arms E/E/E/E so cannot get back. Similar explanations for extended work on higher order odd spiral and even spirals of the form 4n may be offered.. Other things that you might want to consider are: Looking at different permutations of digits, rotational symmetry and centres of rotation. Less able students might wish to use the NAME SPIRAL (See slide 6). The investigation could be extended by looking at higher order spirals/using logo/using a different turn angle (perhaps 60 o on triangular spotty). Order 2Order 3Order 4Order 5Order 6Order 7Order 8Order 9 1,21,2,31,2,3,41,2,3,4,51,2,3,4,5,61,2,3,4,5,6,71,2,3,4,5,6,7,81,2,3,4,5,6,7,8,9 1,2,41,1,2,22,3,4,5,62,3,4,5,6,61,2,2,3,3,4,41,2,2,3,3,5,5,5 1,2,51,3,3,12,2,3,3,42,2,3,3,4,51,2,3,4,4,4,5 1,1,21,3,4,52,3,2,3,5 2,2,32,3,2,3 2,2,41,1,1,2 2,2,5 2,3,5 2,3,6 1,2,7 2,3,2 3,4,10 All these spirals are included in this presentation

3. Starting Points Order 2Order 3Order 4 1,11,2,31,2,3,4 1,21,2,41,1,2,2 3,21,2,51,3,3,1 4,41,1,21,3,4,5 1,22,2,32,3,2,3 2,2,41,1,1,2 2,2,5 2,3,5 2,3,6 1,2,7 2,3,2 3,4,10 Starting Points Order 2Order 3Order 4 1,11,2,31,2,3,4 1,21,2,41,1,2,2 3,21,2,51,3,3,1 4,41,1,21,3,4,5 1,22,2,32,3,2,3 2,2,41,1,1,2 2,2,5 2,3,5 2,3,6 1,2,7 2,3,2 3,4,10 Order 2Order 3Order 4 1,11,2,31,2,3,4 1,21,2,41,1,2,2 3,21,2,51,3,3,1 4,41,1,21,3,4,5 1,22,2,32,3,2,3 2,2,41,1,1,2 2,2,5 2,3,5 2,3,6 1,2,7 2,3,2 3,4,10 Order 2Order 3Order 4 1,11,2,31,2,3,4 1,21,2,41,1,2,2 3,21,2,51,3,3,1 4,41,1,21,3,4,5 1,22,2,32,3,2,3 2,2,41,1,1,2 2,2,5 2,3,5 2,3,6 1,2,7 2,3,2 3,4,10 Starting Points Order 2Order 3Order 4 1,11,2,31,2,3,4 1,21,2,41,1,2,2 3,21,2,51,3,3,1 4,41,1,21,3,4,5 1,22,2,32,3,2,3 2,2,41,1,1,2 2,2,5 2,3,5 2,3,6 1,2,7 2,3,2 3,4,10 Order 2Order 3Order 4 1,11,2,31,2,3,4 1,21,2,41,1,2,2 3,21,2,51,3,3,1 4,41,1,21,3,4,5 1,22,2,32,3,2,3 2,2,41,1,1,2 2,2,5 2,3,5 2,3,6 1,2,7 2,3,2 3,4,10

4. Demo 1,2,4 2,2,5 1,2,3,4 1,3,4,5,6 Watch carefully how each of the spirals are drawn below. All turns are 90 o to the right. 1 2 4 3 Order 3 Order 4 Order 5

5. Investigation Investigate 2, 3 and 4 digit spiral patterns using your own numbers. Some are open and some are closed. Some look like windmills and some look like boxes. This is an open investigation and so you should come up with some ideas and questions of your own but you might want to consider some points listed below: When are the spirals closed? When are spirals open? What happens when you change (or move) a digit in the sequence? Areas contained within order 3 spirals (windmill arms and holes). Length of a spiral. Predicting what a spiral will look like before you draw it. Extend your investigation if you wish by considering 5,6,7 or any other higher order spirals and asking some questions about them.

6. Name Spiral 123456789 ABCDEFGHI JKLMNOPQR STUVWXYZ Alice  1,3,9,3,5 123456789 ABCDEFGHI JKLMNOPQR STUVWXYZ Ben  2,5,5

7. Order 2 and 3 1,2 1,2,3 1,1,2 1,2,4 1,2,5 2,2,5 2,3,5 2,2,3 2,2,4 2,3,6 1. Three digits always gives a closed spiral 2. For windmill spirals, sail size = A x B 3. Hole size for windmill = C – (A+B) 4. Two repeated digits give a square boundary : for example if C – (A+B)  0 and A = B

8. Order 3 1,2,7 3,4,10 2,3,2 1. Three digits always gives a closed spiral 2. For windmill spirals, sail size = A x B 3. Hole size for windmill = C – (A+B) 4. Two repeated digits give a square boundary : for example if C – (A+B)  0 and A = B

9. Order 4 1,2,3,4 1,1,2,2 1,3,4,5 1,1,1,2 2,3,2,3 1,3,3,1 Four digits give open spiral unless they are of the form: A,A,A,A or A,B,A,B in which case we get a square/rectangle.

10. Order 5 1,2,3,4,5 2,2,3,3,4 2,3,2,3,5 2,3,4,5,6

11. Order 6 1,2,3,4,5,6 2,3,4,5,6,6 2,2,3,3,4,5

12. Order 7 1,2,3,4,5,6,7 1,2,2,3,3,4,4 1,2,3,4,4,4,5

13. Order 8 1,2,3,4,5,6,7,8 1,2,2,3,3,5,5,5

14. Order 9 1,2,3,4,5,6,7,8,9

15. Worksheet