Sir Cumference and the First Round Table

Number spirals
Number patterns
Crop circle patterns
Using software

Start your project work by looking at all the tables you can find. What shapes are they? Are they all rectangular, or can you find some that are different shapes? How about if you look at home as well as at school - do you have different shape tables there?

Patterns in circles

In the Islamic culture the circle is the basis for the organization of space. It is a starting point in architecture, poetry, music and even calligraphy. From a circle it is possible to construct many regular polygons.

The decimal system we use did not appear as a standard until the eighth century CE. Before exact units of measurement were used, people measured exact distances using geometric patterns. For instance, Egyptian rope-stretchers and temple surveyors developed a method using pegs and cords to trace circles and straight lines on sand, and so established a way of generating precise and accurate constructions.

1. Using a pair of compasses and a pencil, draw a circle.


2. Keeping the same radius for the circle (so keeping the compasses set at the same width), draw another circle with its centre on the circumference of the first circle.


3. Draw a triangle with its vertices (corners) on the two centres and one of the points where the circles intersect (cross over).
   What kind of triangle is this?
   How do you know?
   Why does it have to be this kind of triangle?


4. Draw another two intersecting circles (the same as in 2 above), then, still keeping the compasses set to the same radius (width), draw another circle with its centre at one of the intersection points.


5. Repeat this over and over, to make a pattern of intersecting circles - using colour will make your pattern more interesting.
   What symmetry does your pattern have?
   If you think it has rotational symmetry, what angle do you have to rotate the pattern through for it to look the same?
   If you think it has reflective symmetry, draw on the lines of symmetry.



6. Add triangles, as in step 3 above. Does this pattern have the same symmetries?
   Are the triangles all the same?
   Why do you think this should be?
   What are the angles in the triangles?

   
   

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Number spirals

Pick a number from 2 to 9 and write this on the top corner of your worksheet. Put your pencil on the star and count from there the number of spaces you chose. For example, if you chose 4 count round to four and draw a straight line with your ruler from the star to the number four. Now start on number four and count (clockwise) four more spaces. You will draw a line to number eight. You will continue to do this all the way around until you end up back at the star. The star is zero. Once you have completed the patterns in pencil, trace over it with a coloured felt tip or marker. Now choose a larger number and draw it on the same circle. How many different designs can you make?

Then focusing on the different shapes created by different number patterns, discuss:

• What do you notice about patterns that have been created with smaller numbers?
• What do you notice when you look at patterns drawn with larger numbers?

You could also cut out your patterns and paste them on construction paper. This is particularly effective with related sets of numbers, such as 2, 4 and 6.

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Number patterns round a circle

Investigate sequences of numbers using a 20-point circle (large or small circles) or a 40-point circle (large or small circles). On the right, the sequences 0, 3, 6, 9, ... and 0, 5, 10, 15, ... are plotted on the 40-point circle. You could draw one sequence on a circle, concentrating on the shape produced and the numbers reached or not reached.

• Which numbers have a closed spiral, which go round the circle more than once before they close?
• What happens if you start a sequence in threes at 1 or 2 or 3 or 4 ... instead of 0?
• How many different paths are there?

You could also draw more than one sequence on a single circle.

• Which numbers do sequences meet at?
• Are there any pairs of numbers which do not meet at all going round the circle?
• What about if you have three sequences on a single circle?

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Crop circle patterns

Visit http://www.papercropcircles.com/ for some great suggestions on using crop circles in various shapes to look at different geometrical shapes.

Using software to create circular patterns

Visit http://www.adobe.com/education/digkids/lessons/pattern.html for suggestions on using graphics software to create circular patterns. Although the instructions given are for Photoshop, you could create patterns using Paint if you don't have any other suitable software. Paint won't do everything suggested here, but it can be used for some very effective patterns.