Abstract:

This paper describes an activity to develop geometry skills by using Origami. There are a lot of shared educational origami experiences to teach geometry in primary school and the one I will propose here is a co-operation between math and representation in the metacognitive process of concrete understanding of geometry.

The scale of representation is a subject of the geography discipline, in the third or fourth class of primary school: it is normally presented only as an aspect linked to the representation of maps but is also in direct relation with fractions. The fact of linking them and using the proposed experience to introduce reasoning related to linear and surface relationships makes - possible to realize a series of transversal applications that educate the student to make use of more skills and exit from the scheme that sees the skills related to the single discipline and not part of a wider culture.

The scaled representation is a task that makes use of both disciplinary approaches and the proposed activity with an origami sheet is born with the intention of visualizing its geometric transformations in a dynamic way. The model, designed with the intention of explaining the scale ratio 1: 2, ranging from linear to spatial relationships, was later revealed as a useful tool for displaying - 2^(-2n) scales and the comparison between equivalent areas.

In this sense, in this contribution it is possible to see a parallel between the geometrical transformations inherent in the bending path of the proposed model and the description of ‘the square’ by M. Montessori in her Psicogeometria, almost as if the model could represent the material figures to which the author refers explicitly with images at the end of the text, without however describing the physical construction procedure. Said that in the text the author normally makes use of cardboard figures, cut out according to the line of the perimeters, and therefore the Origami model described here is clearly far from this operating mode, its dynamism allows you to directly observe many geometric transformations and therefore accompanies the discovery in a privileged learning path.

Furthermore, the finished model lends itself to being an object to be decorated with a funny management of the layers of paper that create invisible decomposed figures.

Finally, the folding sequence can be cyclically repeated (respecting the limits of the thickness of the paper layers and their size) giving life to a fractal model. Starting from a square sheet of Area Q, in the preliminary step (n=0) the folding process leads to two squares: A0= Q/4 Area and B0=A0/4 Area. Repeating the folding process in square B0 this again turns into two squares: A1= B0/4= A0/16 Area and B1= A1/4 Area. At step n folding square Bn-1 two squares are observed: An=Bn-1/4=A n-1/16 and Bn = An/4.
Thus the geometric sequence An= [(1/16) ^(n)] A0 is produced.

It follows an algorithmic fractal process provided that at the final step n the square Bn is excluded: folding, you have to hide the last square Bn in the backside.

Although fractals are not included in the current ministerial guidelines for the primary school curricula, many mathematical topics would allow the description and use of an interdisciplinary approach - combining mathematics and representation and involving spatial intelligence as well as emotional intelligence - allows deepen them indirectly by offering useful bases for higher levels of schooling.

Keywords:

Scale ratio, fractions, origami, fractal, geometric sequence.