• building and exploration of 3-D solids - some familiar, like pyramids and prisms, some newer like anti-prisms and bi-pyramids. And some even more exciting like Platonic and Archimedean solids.

• decomposition of 3-D solids into nets – to learn a number of age appropriate activities to investigate the connection between 2 and 3-dimensions. How many different nets are possible for a given 3-D solid? When are two nets different? Is it always possible to make a 3-D solid out of a given 2-D arrangement? Why? This part of the workshop presents rich opportunity to experience activities that train visual and spatial imagination, memory, spatial reasoning and coordination.

• how to demonstrate spatial concepts in concrete, touchable way? Learn, for example, how to make diagonals in a cube, observe and touch where the diagonals intersect; make and touch a height in a pyramid or a prism; show orientation of parallel, perpendicular and skew lines using edges of different solids.

• new ways to easily demonstrate and explore rotation and mirror symmetry in 3-D. Here we would work with drinking straws appropriately inserted into the solids to act as rotation axes. For exploring the mirror symmetry, we will work with 3-D fragments of solids and reflect them in mirrors. Exciting and mind challenging!

• review the ideas of how 3-D geometry connects with science, engineering, art and history. The structures of molecules, viruses, crystals, minerals and bridges are made up of the same 3-D solids that are built during the workshop. Paintings of Salvador Dali, etchings of M.C. Escher and works of modern sculptors will be linked with the ancient symbolism of Platonic Solids.

Workshops for university or collage students delve into more advanced 3-D concepts, such as relationships between various polyhedra and 3-D crystal structures. So, after building Platonic solids they would find duals and observe the relationships between their symmetry elements. Subsequently they would derive most of the Archimedean solids through vertex and edge truncation of the Platonic solids, investigate space filling with polyhedra and build models of crystal structures.

Classroom activites

Activity 1:

In this initial activity students typically work in groups to construct any structure they like and to write instructions to make it. Subsequently they present and describe their creations, usually also in groups, but sometimes individually. With older students (grades 6-8) the variation of the exercise is to afterwards swap the instructions among the groups and follow with construction of the structure from the instruction of another group. Subsequent comparison between the two structures is always interesting and an excellent occasion to learn the importance and challenge of technical communication.

Activity 2:

In the next activities the students learn one thing first - to describe 3-D solids by specifying the kind and number of polygons that meet at a vertex. For example - in cube there are 3 squares that meet at each vertex, in pentagonal prism there is one pentagon and two rectangles in each vertex (in special case these might be squares), and so on. Armed with this understanding the children, even quite young, are able to make remarkably complex solids all by themselves. This systematic and pattern oriented approach to building structures is very constructive and immensely satisfying for children. It also easily leads to important insights into the nature and relationships between 3-D solids. Although we only rarely had occasions to conduct enough workshops in one classroom to be able to fully explore and practice all the possibilities, the feedback from teacher workshops is very enthusiastic.

 

Activity 3:

Another super exciting activity for a whole classroom of grade 3 or 4 students is to find all different nets for a solid. Most of the time we worked with 6 squares per student (or two students) to make a cube and to discover all 11 nets for it. As children were drawing their proposed nets on the blackboard the whole class would be frantically checking if this was a correct net, and if none already on the blackboard was the same - as a byproduct they were practicing mental rotations and reflections of complex patterns and having lots of fun with it. The more nets on the blackboard the harder it was to come up with a new possibility and the excitement was running high. Once all 11 nets were found the students copied them to their journals. Finding a systematic pattern between the nets was the next task.

The first workshop for parents and children was presented in May 2011 at a conference of Ontario Association for Mathematics Education (OAME) in Windsor, Ontario. Here is a glimps of the workshop.