Geoboard

The geo board (or nail board ) is a common tool in geometry lessons in primary and secondary schools.

Self-made (5 × 5) geo board with three different colored rubber bands stretched as squares

On a mostly square board, nails are hammered in so that a square grid is created. The number of nails is at least 9 (3 × 3 grid), but usually 16 (4 × 4 grid) or 25 (5 × 5 grid) and the upper limit is only limited by the practical size of the board. Different colored rubber bands can be used to stretch geometric figures on these boards and examine their properties.

The geoboard was invented at the beginning of the 1950s by the mathematician and teacher Caleb Gattegno (1911–1988) of Egyptian descent .

Applications

In mathematics lessons, the geoboard is mainly used for the investigation of flat geometric figures , for their area calculation and for the geometric transformations of the plane ( displacement , rotation , mirroring , stretching and their execution one after the other).

Investigation of lines

number	(2 × 2) grid	(3 × 3) grid	(4 × 4) grid	(5 × 5) grid
Routes ( OEIS , A083374 )	6th	36	120	300
Straight lines (OEIS, A018808 )	6 (100%)	20 (55.6%)	62 (51.7%)	140 (46.7%)

The number of tracks on a (n × n) lattice obtained by binomial coefficient to

${\displaystyle \#s(n)={\binom {n^{2}}{2}}={\frac {n^{2}(n^{2}-1)}{2}}}$.

An explicit formula for calculating the number of lines on an (n × n) grid is not known; but there are recursive formulas.

The following formula uses the greatest common divisor to calculate the values

${\displaystyle \#g(n)=2\left(f(n,1)-f(n,2)\right)}$ With

${\displaystyle f(n,k)=\sum _{i=0}^{n-1}\sum _{j=0}^{n-1}(n-i)(n-j)}$if . ${\displaystyle {\mbox{ggT}}(i,j)=k}$

Another recurrence formula, the values calculated using the Euler's φ function to

${\displaystyle \#g(n)=2g_{1}(n)-g(n-1)+r_{1}(n)}$,

in which ${\displaystyle g_{1}(n)=2g(n-1)-g_{1}(n-1)+r_{2}(n)}$

with ; ${\displaystyle g(0)=g_{1}(0)=0}$

${\displaystyle r_{1}(n)=r_{1}(n-1)+4\left(\varphi (n-1)-e(n)\right)}$and , ${\displaystyle r_{1}(0)=0}$

${\displaystyle e(n)={\begin{cases}\varphi ({\frac {n-1}{2}}),&{\mbox{falls }}n{\mbox{ gerade}}\\0,&{\mbox{sonst}}\end{cases}}}$

and . ${\displaystyle r_{2}(n)={\begin{cases}(n-1)\cdot \varphi (n-1),&{\mbox{falls }}n{\mbox{ gerade}}\\{\frac {n-1}{2}}\cdot \varphi (n-1),&{\mbox{falls }}n\equiv 1\mod 4\\0,&{\mbox{sonst}}\end{cases}}}$

Investigation of Figures

The main focus here is on

• the investigation of symmetry properties of geometric figures,
• the position and number of inner, outer and lattice points on the sides of polygons , as well
• the determination of the possible type or number of simple geometric figures.

Results for the number of different triangles or squares on an (n × n) grid (for 2 ≤ n ≤ 5):

number	(2 × 2) grid	(3 × 3) grid	(4 × 4) grid	(5 × 5) grid
Triangles ( OEIS , A045996 )	4th	76	516	2148
Acute triangles (OEIS, A190019 )	0 (0%)	8 (10.5%)	80 (15.5%)	404 (18.8%)
Right triangles (OEIS, A077435 )	4 (100%)	44 (57.9%)	200 (38.8%)	596 (27.7%)
Obtuse triangles (OEIS, A190020 )	0 (0%)	24 (31.6%)	236 (45.7%)	1148 (53.4%)
Irregular triangles (OEIS, A190312 )	0 (0%)	40 (52.6%)	368 (71.3%)	1704 (79.3%)
Isosceles triangles (OEIS, A186434 )	4 (100%)	36 (47.4%)	148 (28.7%)	444 (20.7%)
Acute-angled isosceles triangles (OEIS, A190317 )	0 (0%)	8 (10.5%)	48 (9.3%)	164 (7.6%)
Right-angled isosceles triangles (OEIS, A187452 )	4 (100%)	28 (36.8%)	96 (18.6%)	244 (11.4%)
Obtuse isosceles triangles (OEIS, A190318 )	0 (0%)	0 (0%)	4 (0.8%)	36 (1.7%)
Different (non- congruent ) triangles (OEIS, A028419 )	1	8th	29	79
Various acute triangles (OEIS, A190021 )	0 (0%)	2 (25%)	8 (27.6%)	23 (29.1%)
Various right triangles (OEIS, A189979 )	1 (100%)	4 (50%)	9 (31.0%)	17 (21.5%)
Various obtuse triangles (OEIS, A190022 )	0 (0%)	2 (25%)	12 (41.4%)	39 (49.4%)
Various irregular triangles (OEIS, A190313 )	0 (0%)	3 (37.5%)	18 (62.1%)	57 (72.2%)
Various isosceles triangles (OEIS, A189978 )	1 (100%)	5 (62.5%)	11 (37.9%)	22 (27.8%)
Various acute angled isosceles triangles (OEIS, A190309 )	0 (0%)	2 (25%)	5 (17.2%)	11 (13.9%)
Various right isosceles triangles (OEIS, A108279 )	1 (100%)	3 (37.5%)	5 (17.2%)	8 (10.1%)
Various obtuse isosceles triangles (OEIS, A190310 )	0 (0%)	0 (0%)	1 (3.4%)	3 (3.8%)

The number of triangles on an (n × n) grid is calculated using the formula

${\displaystyle {\begin{array}{lcl}\#\triangle (n)&=&{\binom {n^{2}}{3}}+n{\binom {n+1}{3}}-2\sum _{i=2}^{n}\sum _{k=2}^{n}(n-i+1)(n-k+1)\operatorname {ggT} (i-1,k-1)\\&=&{\frac {1}{6}}(n-1)^{2}n^{2}(n+1)^{2}-2\sum _{i=1}^{n-1}\sum _{k=1}^{n-1}(n-i)(n-k)\operatorname {ggT} (i,k)\end{array}}}$

The triangles can be divided into the disjoint classes of acute, right or obtuse angled triangles according to their angular sizes, and according to side lengths into the disjoint classes of irregular (unequal-sided) triangles and isosceles triangles - the latter also include equilateral triangles, which cannot appear on the geoboard.

Obvious is therefore

${\displaystyle \#\triangle =\#\triangle _{\mathrm {spitzwinklig} }+\#\triangle _{\mathrm {rechtwinklig} }+\#\triangle _{\mathrm {stumpfwinklig} }}$

${\displaystyle \#\triangle =\#\triangle _{\mathrm {ungleichseitig} }+\#\triangle _{\mathrm {gleichschenklig} }}$

${\displaystyle \#\triangle _{\mathrm {gleichschenklig} }=\#\triangle _{\mathrm {spitzwinkliggleichschenklig} }+\#\triangle _{\mathrm {rechtwinkliggleichschenklig} }+\#\triangle _{\mathrm {stumpfwinkliggleichschenklig} }}$

number	(2 × 2) grid	(3 × 3) grid	(4 × 4) grid	(5 × 5) grid
Complete squares ( OEIS , A175383 )	1	78	1278	9498
Quadrilaterals (OEIS, A189414 )	1	94	1758	13698
Concave squares (OEIS, A189412 )	0 (0%)	24 (25.5%)	720 (41.0%)	6300 (46.0%)
Arrows (OEIS, A173502 )	0 (0%)	8 (8.5%)	64 (3.6%)	292 (2.1%)
Convex squares (OEIS, A189413 )	1 (100%)	70 (74.5%)	1038 (59.0%)	7398 (54.0%)
Trapeze (OEIS, A189415 )	1 (100%)	50 (53.2%)	490 (27.9%)	2618 (19.1%)
Parallelograms (OEIS, A189416 )	1 (100%)	22 (23.4%)	158 (9.0%)	674 (4.9%)
Kite (OEIS, A189417 )	1 (100%)	10 (10.6%)	58 (3.3%)	222 (1.6%)
Lozenges (OEIS, A189418 )	1 (100%)	6 (6.4%)	22 (1.3%)	66 (0.5%)
Rectangles (OEIS, A085582 )	1 (100%)	10 (10.6%)	44 (2.5%)	130 (0.9%)
Squares (OEIS, A002415 )	1 (100%)	6 (6.4%)	20 (1.1%)	50 (0.4%)
Various convex squares (OEIS, A181944 )	1	12	89	407
Various trapezoids (OEIS, A181945 )	1	9	43	141
Different parallelograms	1	6th	21st	55
Various kites (OEIS, A181946 )	1	4th	11	25th
Various diamonds (OEIS, A181947 )	1	3	6th	11
Different rectangles	1	4th	9	16
Different squares (OEIS, A108279 )	1	3	5	8th

The quadrilaterals can be divided into the disjoint classes of the concave and convex quadrilaterals based on their shape. Both can be further subdivided on the basis of symmetry properties, with the subclasses overlapping in the case of the convex quadrilaterals.

Obvious is therefore

${\displaystyle \#\square =\#\square _{\mathrm {konkav} }+\#\square _{\mathrm {konvex} }}$

In addition, the following relationships apply between the convex and concave quadrangles:

• Squares = rectangles ∩ diamonds
• Squares ⊂ rectangles ⊂ parallelograms trapezoids ⊂ convex quadrilaterals ⊂ quadrilaterals
• Squares ⊂ diamonds ⊂ parallelograms ⊂ ​​trapezoids ⊂ convex quadrilaterals ⊂ quadrilaterals
• Squares ⊂ diamonds ⊂ dragons ⊂ convex quadrilaterals ⊂ quadrilaterals
• Arrow squares ⊂ concave squares ⊂ squares

This makes it easy to determine the numbers of types of quadrilaterals that only fulfill the subordinate relationship (e.g. rectangles that are not squares at the same time) by calculating the difference.

Area calculation

Pick's theorem (1899) is used to calculate the area of ​​grid polygons : A grid polygon with grid points on the edge and inner grid points has an area of grid squares. ${\displaystyle r}$ ${\displaystyle i}$ ${\displaystyle A={\tfrac {1}{2}}r+i-1}$

Literature and collections of exercises

• Caleb Gategno: Geoboard geometry. New York: Educational Solutions Worldwide Inc., 1971. ISBN 978-0-87825-020-2 .
• Karl-Heinz Keller: Discover geometry on the Geo-Board. A basic course in geometry. Offenburg: Mildenberger, 2002. ISBN 978-3-619-02520-6 .
• Judith and Ulrich Lüttringhaus: The large geoboard. Vol. 1: Geometric constructions. Augsburg: Brigg, 2009. ISBN 3-87101-427-3 .
• Hans-Günter Senftleben: Collection of exercises for the large geoboard. Hamburg: Rittel, 2001. ISBN 3-93644-301-7
• Horst Steibl: Geoboard in class. Hildesheim; Berlin: Franzbecker, 2006. ISBN 3-88120-417-2 .

Web links

Commons : Geobrett  - collection of images, videos and audio files

• Natalie Bär, Nicole Bröll and Birgit Kühn: Geobrett. A WebQuest for children from 1st grade.
• Education server Hessen. Teaching material.
• The geoboard.
• The geoboard. (Montessori shop.)
• Margherita Barile: Geoboard. (MathWorld - A Wolfram Web Resource.)
• Alexander Bogomolny: Geoboard. (Virtual geoboard.)
• Tom Scavo: Geoboards in the classroom. (Lesson unit.)
• Utah State University. National Library of Virtual Manipulatives. (Virtual geoboard.)

Individual evidence

1. ↑ Caleb Gattegno: The Gattegno Geoboards. In: Bulletin of the Association for Teaching Aids in Mathematics 3 (1954).
2. ↑ Seppo Mustonen: On lines and their intersection points in a rectangular grid of points (PDF; 681 kB), pp. 11–15 (Appendix 1: Recursive formulas).
3. ↑ See OEIS , A018808 .
4. ↑ See OEIS , A045996 , A000938 , also A178208 , A008911 .
5. ↑ Georg Alexander Pick: Geometric to the theory of numbers. (Processing of a lecture given in the German mathematical society in Prague.) In: Reports of the meetings of the German natural science and medicine association for Bohemia “Lotos” in Prague 19 (1899), pp. 311–319.