**Summary:** Teaching geometry to develop visual thinking, benefits from usual proofs for understanding basic concepts, from challenging hands-on materials and from the ingenuity of well designed everyday life objects. We will show why and how.

## INTRODUCTION

Teaching mathematics implies to combine in a clever way numbers, symbols, words, diagrams, real objects and stories. Among diagrams we find illustrations, graphs, pictures, charts, figures, etc. Among real objects we find natural resources, designed tools and hands-on materials. Our aim in this presentation is to show the pedagogical value of three concret resources for teaching geometry: images that facilitate visual proofs, hands-on materials that facilitate visual thinking and objects from our everyday life that show how geometry is applied to solve real challenges. These strategies correspond to the NCTM standards on problem solving, reasoning, proofs, communication, connections and representation.

## IMAGES WHICH ARE PROOFS WITHOUT WORDS

Following our work with Roger Nelsen (Alsina and Nelsen, 2006), we consider that good images in geometry may open the possibility of proving. We can restrict our attention to proofs which facilitate a better understanding of properties or concepts. But by means of appropriate images we open at the same time ideas on proving and the possibility of developing visual thinking.

Fig 1. If a/b<c/d with a,b,c,d>0 then a/b<(a+c)/(b+d)<c/d; Fig. 2. 1+2+…+n=n(n+1)/2. Fig. 3. If r<1, a+ar+ar^{2}.+…=a/(1-r). Fig. 4, Viviani’s theorem (Kawasaki’s proof): the perpendicular to the sides from a point on the boundary or within an equilateral triangle add up to the height of the triangle.

## CHALLENGING HANDS-ON MATERIALS

In the case of space geometry we can use appropriate hands-on material to make representations, to experiment with real bodies and to solve problems. Just with paper, pencils, scissors, boxes, cups, soap, water, plastic models, etc., we can produce very challenging questions. The following are four examples of geometrical problems that require this practical approach:

- If you can measure a box from outside, which is the most efficient way to calculate its main interior diagonal?
- In box of interior measures 39 inches x 39 inches x 10 inches, how many cubes of edge 10 inches can we locate?
- Find the shape of the hole to be made in a sheet of paper in such a way that forming a cylinder with it another cylinder of identical measures can be inserted in the first one.
- On the two faces of a sheet of paper mark two point. Form a cylinder with the sheet. Which is the minimal path on the cylinder which may connect the two points?

## THE GEOMETRICAL CHARMING OF EVERYDAY LIFE OBJECTS

This is a new idea on resources for teaching that I have been developing in the last years. Industrial design provides us with a great collection of geometrical objects that we can use to study shapes, measures and transformations. The following table gives many examples of this kind of objects.

Polyhedra |
Everyday life objects |

Cubes | Standard dice, bouillon cubes, Rubik’s cube, boxes, etc. |

Tetrahedra | 3D puzzles, tripods, tetrahedral dice |

Octahedra | Mineral crystals, octahedral dice |

Dodecahedra | Paperweight, desk calendar, dodecahedral dice |

Icosahedra | MAA logo, icosahedral dice, domes, sculptures |

Prisms | Toblerone™ package, candy boxes, pencils |

Pyramids | Egyptian pyramids, top of an obelisk, plumb bob |

Bipyramids | Toy tops, jewels |

Other polyhedra | Jewels and jewelry, soccer balls, puzzles |

Polygons |
Everyday life objects |

Triangles | Traffic yield sign, danger sign, musical instrument |

Quadrilaterals | Sheets of paper or cardboard, tiles |

Pentagons | Chrysler logo, tables, Department of Defense building, paper strip tied into an overhand trot |

Hexagons | Tiles, plates, cross-section of a pencil, hex bolts and nuts |

Octagons | Stop sign, trays, tables |

n-gons | Some clock, faces, foreign coins |

Star gons | Star fish, Star of David |

Curves |
Everyday life objects |

Circles | Rimof plate or glass, coin, wheel, ring |

Ellipses | Liquid in a inclined glass, circle viewed at an angle |

Parabolas | Cable on a suspension bridge |

Hyperbolas | Profile of some bells, six arcs at the point of a sharpened pencil |

Sine curves | Snake’s path, sea waves |

Cycloids | Trajectory of a point on a wheel |

Catenaries | Power lines, hanging chain |

Spirals | Grooves in vinyl record or CD, tape in a cassette, coiled rope |

Quadric surfaces |
Everyday life objects |

Circular cylinders | Radio Ekco AD65 of Wells Coats (1934) for E.K. Cole Cylindra-Line Collection of Arne Jacobsen (1967) for Skelton. Pencils, water glasses, containers, bottles, paper rolls; tubes, hats,… |

Circular cones | Il conico (tea pot) Aldo Rossi (1988) for Alessi Ice-cream cones; glasswares; road markers; hats;… |

Ellipsoids and spheres | Il Ballitore of Richard Sapper (1983) for Alessi; Balls (rugby, soccer, ping-pong,…); ornamental balls; boxes; meat boilers; hats;… |

Paraboloids of revolution | Collator of Max de Chinois (1990) for Alessi; TV antennas; headlamps for cars;… |

Hyperboloids of 1 sheet | Bells; baskets; drums; metallic musical instruments,… |

Hyperbolic paraboloids | Seats on horses; park structures; fried potatoes,… |

Transformations |
Everyday life objects |

Translations | Telescopic stairs; Repetition of seats; friezes, screens;… |

Rotations | Can’s opener; nut-cracker; clocks; folding table; lamps… |

Symmetries | Left-hand designs; shoes; scissors; folding forms; |

Helicoidal transformations | Bottle’s opener; screws; folding forms; steps;… |

Similitudes | Beach umbrellas; lamps, decorative fans; models;… |

With all these designs that we have at home we can appreciate interesting applications of geometry but at the same time we can induce “problem posing” activities as well as “problem solving”. The following are good examples of rich tasks:

- Take glasses of wine and cups. For any one conjecture the relation between the altitude and the perimeter of the superior cercle. Then make appropriate measures (John Mason).
- How can you distribute holes in a sphere in a uniform way?
- Close one eye and mark on a mirror the profile of your face. Which measures will have this representation? (Anton Aubanell).
- How to divide a rectangular cake (with chocolate icing on top and all the sides) between five people in a fair way?

## A FINAL REMARK

If we want students to enjoy geometry we need to give them the pleasure of discovering its many applications, how fruitful is experimentation and how exciting is to develop visual strategies. Less chalk, less writing,… more images, more objects. Thanks for making all this possible!

* We do maths with our brain, we must teach maths with our heart! *

### REFERENCES

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C. Alsina, “Less Chalk, less Words, less symbols… more objects, more context, more actions” in *ICMI Study 14: Applications and Modelling in Mathematics Education*. H.W. Henn and W. Blum eds., Springer, Berlin, 2006.

C. Alsina, “Mathematical Proofs in the Classroom: the Role of Images and Hands-on Materials”, *Mathematikunterricht im Spannugnsfeld von Evolution und Evaluation- Festschrift für Werner Blum*, W. Henn and G. Kaiser, G. eds., Frazbecker, Hildesheim (2005) pp 129-138.

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