# Stars and Polygons

Purpose: promote interest in geometry, demonstrate the connection between geometry and group theory (clock groups), discover a diagram to promote the concept of infinite progressions.

Background: this lesson was used successfully in a 1st-2nd grade multi-age classroom setting. Students enjoyed drawing the different patterns and discovered the relationship between the pentagon and pentagram on their own.

Written 2001:

Reformatted: January 2010

### Introduction:

The following steps may be Introduced:

1. Arrange points around a circle.
2. Create a simple rule for going from one point to another, such as skip a point.
3. Draw in lines that follow your rule until you reach your starting point
4. Describe your results. Compare your results to other rules.

### Examples:

[A] 5 Points around a circle & Rule: skip 1   start step 2 final product

[B] 5 Points around a circle & Rule: skip 0  second line final product

[C] 6 points around - Rule: skip 1  This will be the actual result. What change in the rule will give this preferred result?

[D] 6 points around - Rule: skip 2  This is both the first and the actual result, and, of course every step in between. What rule change will give you this result?

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Questions:

• How are all these results similar?
• What do you think makes each one different? How can you test your conjecture?
• What rules will give you a polygon, a star, a line?
• When will two different rules give you the same picture?

### Extensions:

This lesson may be expanded to visually present the idea of an infinite progression. Notice how when we put the skip 0 rule and the skip 1 rule together we get the pentagon inside the pentagram inside the pentagon. This sequence may be extended ad infinitum - in the class room drawn ad nauseum..

 This lesson may be used to demonstrate addition in clock groups. For example: If it is now 10:00 what time will it be in 5 hours? Questions: How do you think these diagrams can be used to show the last digit (one's place of an addition? How do you think these diagrams can be used to help you predict the results of repeated additions? Eg: 5+5+5+5+5+5 =? 