Geoboard
On a mostly square boards nails are so taken that a square lattice is formed. The number of nails is at least 9 (3 × 3 grid ), but as a rule 16 (4 × 4 grid ) or 25 (5 × 5 grid) and is bounded above only by a workable size of the board. On these boards geometric figures can be clamped and tested for their properties with different colored rubber bands.
The geoboard was invented in the early 1950s by the Egyptian mathematician Caleb Gattegno and educators.
Applications
In mathematics education the geoboard is mainly used for the investigation of plane geometric figures in their area calculation as well as the geometric transformations of the plane ( translation, rotation, mirroring, stretching and their sequential execution ).
Investigation of lines
The number of lines in an (n x n ) lattice obtained by means of a binomial coefficient to
An explicit formula for calculating the number of lines in an (n x n ) lattice is not known; there are recursive formulas.
The following formula is used for calculating the greatest common divisor of the values
Another recursion formula calculates the values by means of Euler's φ - function
Examination of figures
The focus here is primarily
- The investigation of symmetry properties of geometric figures,
- The position and number of inner and outer, as well as grid points on the sides of polygons, as well as
- The determination of the possible type or number of simple geometric figures.
Results for the number of different triangles and squares on an (n × n) - grid ( for 2 ≤ n ≤ 5):
The number of triangles on an (n × n) - grid is calculated according to the formula
The triangles can be the one on the angular sizes in the disjoint classes of acute, right or obtuse-angled triangles divided, on the other hand by the side lengths in the disjoint classes of irregular ( non-equilateral ) triangles and isosceles triangles - the latter include the equilateral triangles, but which can not occur on the geoboard.
Obviously, therefore,
The squares can be divided into disjoint classes of concave and convex quadrilaterals because of their shape. Both of which can be further divided on the basis of symmetry properties, wherein the sub-classes overlap in the case of convex quadrilaterals.
Obviously, therefore,
Furthermore, the following relations between the convex and concave quadrilaterals are:
- Squares rectangles = ∩ diamonds
- ⊂ ⊂ squares rectangles parallelograms ⊂ ⊂ Trapeze convex quadrilaterals ⊂ squares
- ⊂ ⊂ squares rhombuses parallelograms ⊂ ⊂ Trapeze convex quadrilaterals ⊂ squares
- Squares ⊂ ⊂ diamonds dragon ⊂ ⊂ convex quadrilaterals squares
- Arrow squares ⊂ ⊂ concave quadrilaterals squares
This allows the numbers of quadrilateral types which only the subordinate relation that (about rectangles, squares are not simultaneously) easily determined by difference.
Area calculation
To calculate the surface area of lattice polygons Pick's theorem (1899 ) is used: A Gittervieleck with grid points on the boundary and inner grid points has a surface area of grid squares.
Literature and collections of problems
- Caleb Gategno: Geoboard geometry. New York: Educational Solutions Worldwide Inc., 1971, ISBN 978-0-87825-020-2. .
- Karl- Heinz Keller: On discovering geo- board geometry. A basic course in geometry. Offenburg: Mildenberger, 2002 ISBN 978-3-619-02520-6. .
- Judith and Ulrich Lüttringhaus: The great geoboard. Vol 1: Geometric constructions. Augsburg: Brigg, 2009 ISBN 3-87101-427-3. .
- Hans -Günter Senft Life: Questions for the large geoboard. Hamburg: Rittel, 2001 ISBN 3-93644-301-7.
- Horst Steibl: geoboard in the classroom. Hildesheim; Berlin: Franz Becker, 2006 ISBN 3-88120-417-2. .