The Pentagonal Ring is made up of 10 regular pentagons, forming a ring with a decagon (polygon with 10 sides) in the interior. Constructing a pentagon is very simple, if you wre using The Geometer Sketchpad, just as any regular polygon is. The compass and straightedge construction is more complicated. If compass and straightedge are used, it would be best to ask the students to each construct one pentagon, and then to trace the remaining 9 pentagons. Repeating the pentagon construction 2 or perhaps 3 times is good practice, and would certainly help the student to remember the construction. However, nine repetitions of the same construction might be tedious and counter-productive.
Although you may save time using the pattern with the theorems already on it, this project is much more valuable to the students if they write the theorems themselves. In doing so, the process of writing the theorems helps the students to remember them, and gives them a sense of ownership of both the project and the information they write on it. Xeroxing the pattern onto different colors of paper adds a nice touch as well.
To make the "PentaRing", construct 10 congruent pentagons, each one attached to the next one, as shown below. In this example, three of the circle theorems have been typed on the pattern. The double arrows are the "double implies" symbol, meaning that congruent arcs implies congruent chords, and congruent chords imply congruent arcs, for example. This means, of course, that if two arcs of the same or congruent circles are congruent, the their corresponding chords are also congruent, and that the converse of this is also true. It is a very clever way to write three related theorems and their converses.The theorems themselves are written in full at the bottom of this page.
The front of this "reviewee" is shown below. Students should use scissors to cut the pattern out of plain paper, including cutting out and removing the center ten-sided polygon ("decagon").
If time permits, students can draw pictures on the back, in each pentagon, write other theorems for review, or glue photos on the back to "personalize" their project as shown below:
The PentaRing should be constructed on plain paper, and then the students can fold in alternating "valley" and "mountain" folds all the way around; scoring the folds first will result in a crisper fold. This project would make a fine review sheet for any group of theorems; 10 would be preferable as there are ten pentagons. But the 3 circle theorems and their converses as shown with the "double implies" arrow works very well. The PentaRing folds into a tidy pocket-sized review package, or "Reviewee", as shown below.
