From MathWiki


One activity explored the reasoning that arises when asked to explore a quadrilateral (not drawn) with the information: AB=CD and BC=DA. This activity was designed to highlight the role of symmetries in understanding our reasoning, as well as the possibility of moving from 2-D to 3-D to expand the reasoning.

An associated GSP Sketch, activity sheet, and handout can be found at:

Here is the blurb, at the bottom of that page:

Exploring the Parallelogram via Symmetries Presenter: Walter Whiteley

Presentation Description: In practice, almost all proofs of properties of parallelograms rely on a congruence -- the half-turn symmetry. We explore this definition of a parallelogram as a quadrilateral with a half-turn symmetry. We see how this symmetry gives easy access to all of the properties and how most alternative proofs/definitions use this symmetry as the core congruence. We also provide a number of extensions to other settings (e.g. the sphere, 3-space) and to other classes of quadrilaterals. A key point is role of symmetry in providing a mathematical basis for extensive reasoning with Sketchpad, with physical objects (provided) and with mental processes.

The Popcorn Box

Since the workshop, there has been a substantial expansion of both the manipulatives we used, and of the class of problems which can be investigated by these methods.

The activity has also become the center of some investigations of the use of manipulatives, and what reasoning, and connections, the use of the models affords. I have used the activities in multiple places: with a room full of folk writing support material for the Ontario Grade 12 curriculum; with a group of students between Grade 8 and Grade 9; with a class of Grade 12 students; with a class of in-service masters students. A former Masters student has used them as part of a professional development session with a group of high school teachers.

One of the extensions has been to move first to other regular polygons (as the original sheet of paper) and finally, the visual analysis was refined to a local analysis which applies to any polygonal paper which has a single incircle tangent to all the edges. In all these cases there is a single geometric result: the base which forms the maximum volume is a 2/3 dilation of the original shape, towards the center of the incircle. Translated to how the cut works along a side, it turns out that the residual side is also 2/3 of the original side.

GSP sketch for exploration of basic box

GSP sketch for more general box shapes

In the zipped package below, I enclose a discussion paper by a student, as well as the two GSP sketches above which explore (a) the square based example; (b) a set of regular polygons, ending with the general analysis at each vertex which actually applies to the general case above. Note that this extension is primarily visual/geometric, given that we have 'learned' that the optimum must occur when the area of the sides equals the area of the base of the box. The analysis is a geometric comparison of these areas, using geometric techniques of folding and overlap. I am interested in feedback on these ideas.

Materials and GSP sketches for exploration

Finally, being unable to resist 'what if', one of my students evoked an exchange through which we realized that the analogous problem in 4-space (with a hypercube) results in an optimum where the 'base' is 3/4 of the original. So we 'see' a pattern: in dimension d, with a d-1 cube as the 'paper' the optimum is found by taking a base which is a (d-1)/d dilation of the original material!