Loopy Polygons
Octagons can be fitted together to make loops. Try it for yourself with this applet!
How many octagons do you need to make a loop? Are there certain numbers of octagons that will not make a loop? What is the inside perimeter of the shape? What is the outside perimeter? The perimeters form and interesting pattern. Can you find a formula that links the perimeter and the number of octagons?
Can you make loops with other polygons? What are the properties of the polygon that mean that it can form loops?
All these questions and many more can be explored in this investigation about loopy polygons.
This page addresses the following items from the syllabus:
Sequences
- Continue a given number sequence or pattern
- Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences.
- Find and use the nth term of sequences.
This investigation is based on an old GCSE coursework, 'Octagon Loops'.
The following animation should give an idea of what the investigation entails:
The Rules
Here are the only important rules for making loops from polygons
1. Edges must match up exactly
2. Only one space can be created in the middle of the loop
3. Each polygon must touch exactly two others
Instructions
This investigation can be attempted with just the animation and the rules above.
A more structured investigation can be completed using the following resources:
- Here is a
structured investigation where students are encouraged to complete tables and draw graphs to find the formula. - Tiles that can be cut out and manipulated by printing the following: octagons hexagons squares triangles
- Students can be given help with writing up their investigation.
Click on the following photos to see some students working on the investigation in class:
Manipulatives
Here are some applets that can be used to explore the loops in this investigation. Paper versions of these can be printed out from above.
Hexagon Loops
You may wish to use this manipulative to make loops with equilateral hexagons.
The bottom right vertex rotates the hexagon.
Square Loops
You may wish to use this manipulative to make loops with squares.
Triangle Loops
You may wish to use this manipulative to make loops with equilateral triangles.
The bottom right vertex rotates the triangle.
Tags: conjecturing; investigation;manipulatives; formulae; nth term; sequences; graphing