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Twenty cent geometry*
* Reproduced in "Reflections" with permission from: Curriculum Ideas for Secondary School - Geometry, Part 1, SM13, 1986
1. Start with a 20 cent coin. How many other 20 cent coins can you fit around the outside of it? Try with other coins. Does the same thing happen?
2. Construct an accurate diagram of what you’ve been doing, carefully marking the centre of each circle.
A template comes in handy here.
3. Join the centres of the outside circles. What figure have you made?
4. Joint the centre of the middle circle to the centres of the surrounding circles. What type of triangles have you made?
Challenge
See if you can now explain why you can only fit six coins around the outside of your middle coin.
5. Carefully draw in the lines joining, but only just touching, each pair of adjacent circles. Extend these lines so that they intersect. Rub out your construction lines (the dotted lines in the diagram).
What two (2) types of figures do you get?
6. Extension work
(a) There are two regular hexagons in your diagram. See if you can construct another two.(b) How many sets of congruent equilateral triangles can you find? How many triangles are there in each set?
(c) There are many other sets of congruent figures ‘hidden’ in your diagram. Your job is now to find them, name them, and say how many of each type there are. Construct an exact copy of each type of figure you find.
Twenty cent geometry - notes for teachers
Syllabus Ref: G2.1, 2.2, 2.4, 3.5, 3.6, 4.1
Classroom use
1. The activity can be teacher-directed, using OHTs with different coloured overlays, but it lends itself much better to groupwork, with students having their own diagrams and using these to discuss the extension work. Perhaps one large diagram can be constructed by and for a group of 4—6 students, with their desks arranged so that they can all see it. Great for language work!2. In 6(c) above, look for equilateral triangles, isosceles triangles, isosceles trapeziums, kites, rhombuses, parallelograms, circles. Any more?
Results can be set out in table form, at right:
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3. Large versions of the pattern, on butcher’s paper or cardboard, would look great on display. ‘Creative colouring-in’ will make it look even better.
4. The idea of beginning with a basic geometric pattern and extending it as a basis for ‘polygon searches’ can also be used with pentagons (perhaps as a follow-up to ‘Donald Duck in Mathmagicland’), octagons, etc. Templates like Mathaid or Math-o-Mat will give you the initial shape.
