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Secondary Magazine - Issue 68: Focus on


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Created on 01 September 2010 by ncetm_administrator
Updated on 13 September 2010 by ncetm_administrator

 

Secondary Magazine Issue 68Quadrigon
 

Focus on...Quadrigon

A mathematical game invented during the 1970s by a mathematics teacher, W. Ransome, and described in Mathematics Teaching, Number 75, is played on a hexagonal board showing part of a tessellation of 30˚, 60˚, 90˚ triangles.

Quadrigon blank board 

In the game, called Quadrigon, 2 or 3 players alternately fit coloured triangular counters, which are copies of the smallest right-angled triangles, onto triangles on the board.

W. Ransome devised a particular set of rules for the game, in which the object is to complete quadrilaterals on the grid – hence the name ‘Quadrigon’. But many different rules could be thought up and explored.

The great richness of this situation is in the opportunities that the board itself provides for students to explore their own examples of mathematical objects and relationships between them, ask their own questions, and consolidate what they know in creative and aesthetically satisfying ways.

I have observed students seeing in the Quadrigon board, and investigating, many different objects and relationships, such as those suggested by the following questions – which are just a few of the many questions that students might ask themselves.
 

orange circles

In what ways are different right-angled triangles on the board related to each other?

For example:

 different right-angled triangles on the Quadrigon board

  • It is easy to see, by noticing the number of the smallest right-angled triangles composing each larger triangle, that the areas of the similar right-angled triangles shown in this diagram are in the ratios 1:3:4:9:12. Can I show using Pythagoras’ theorem that their lengths are in the ratios 1:√3:2:3:2√3?
  • Is it possible to obtain each triangle shown in the diagram from every other one that is shown by combining mathematical transformations of rotation, reflection and enlargement?
green circles

What trapezia are on the board? Many pairs of trapezia are congruent, but are any pairs similar?

 What trapezia are on the Quadrigon board? Many pairs of trapezia are congruent, but are any pairs similar?

purple circles

Many shapes are reflections of other shapes.

For example, can I identify one of these five trapezia from which every one of the other four can be obtained by a reflection in a line shown on the board? If so, which line on the board is the mirror line for each reflection?

 Trapezia on the Quadrigon board

blue circles

What kinds of special quadrilateral can be found on the board? What kinds can’t be found? Why not?
 

orange circles

What symmetrical shapes are possible? What kinds of symmetry are possible/not possible? Why?
 

green circles

You and your students could use this sheet of Quadrigon boards to explore possibilities, which will prompt you to ask new questions of your own.

‘Quadrigon’ should not be confused with ‘Quadriagon’, which is the name that Wolfgang Von Wersin, a German glassware designer, gave in 1956 in The Book of Rectangles, Spatial Law and Gestures of The Orthogons Described, to one of the 12 special rectangles known as orthogons – of which the Golden Rectangle is the most familiar.

To produce a Quadriagon first draw a square. Then rotate a half-diagonal of the square to create an arc. Extend two sides of the square to form a rectangle – the Quadriagon – that includes the arc, and one side of which is tangential to the arc.

Quadriagon diagram

The longest side of the Quadriagon created from a unit square is (1 + √2)/2, whereas the longest side of the Golden Rectangle created from a unit square is (1 + √5)/2, which is the Golden Ratio.

I can find no information about W. Ransome other than that in 1976 this person was teaching mathematics at Highgate School, and might possibly be Wilson Ransome who is the author of several books about mathematics.

 

 
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