Starry-eyed - Geometry level 4

These activities involve investigating and creating patterns made by joining points around the circumferences of circles. Copymasters are provided to assist the students.

Activity 1

The students are unlikely to have any problems with questions 1a-b, but they may have trouble explaining (for question 1c) why the 8/5 and 8/3 star polygons are the same. The reason can be seen from the diagram. Counting around 3 from one direction is the same as counting around 5 from the opposite direction. This is true because 3 + 5 = 8. The only difference, therefore, between the two star polygons is the order in which the chords (lines) are drawn.

Before they begin to draw the star polygons in question 2, the students could predict which ones will be the same shape (10/3 and 10/7; 10/6 and 10/4) and why they expect that this will be the case (because 3 + 7 and 6 + 4 both equal 10).

When doing question 3, your students could again predict which 9-point stars will be the same, and why, before they begin the drawings. If they are able to predict that 9/8 and 9/1, 9/7 and 9/2, 9/6 and 9/3, and 9/5 and 9/4 will each be the same, they can predict that there will be 4 different 9-point stars. They can confirm this by drawing them all.

Question 4 tests the students' understanding of the principles explored in the earlier questions as they predict the outcomes and then draw the stars to check their predictions.

Activity 2

It will take the students some time to complete this intricate design. The task is not difficult, but it demands a sharp pencil and a good degree of accuracy. If your students colour it as suggested, they may find that coloured pencils give them a finer result than felt-tip pens.

To answer question 2, the students will need to experiment with various possibilities using another 12-point circle. If they do this, they should discover that the pattern is based on a 12/5 or 12/7 star polygon with every second point coloured to give a 6-pointed star. The students could make the pattern, starting with the same cluster of circles as for question 1.

As a further activity, the students could draw a number of identical star polygons and colour the different repeated parts of the shape (perhaps using felt-tip pens this time) to reveal a variety of patterns. For example, they will discover a regular dodecagon in the centre of a 12/5 star polygon, surrounded by various triangles and quadrilaterals, all with their own reflective symmetry.

A website featuring star polygons is Star Polygon - Wolfram Mathworld.

Activity 1

1.

a. Practical activity

b. Practical activity

c. Although the lines are drawn in a different order, the two stars are identical. This is because 3 + 5 = 8.

2.

3.

There are 4 if you include the regular polygon, 9/1 (9/8). The others are 9/3 (9/6), 9/4 (9/5), and 9/2 (9/7).

4.

a.

i. A regular 16-sided polygon.

ii. An asterisk with 8 lines crossing at a central point. (If you don't lift your pencil, you will get a single straight line through the centre.)

iii. 16/11

iv. 16/2 (16/14), 16/4 (16/12), and 16/8.

v. 8 if you include 16/1 and 16/8; 6 if you don't.

vi. 16/7 (16/9). 16/8 is not a polygon (because it does not enclose a space).

b. Practical activity

Activity 2

1.

Practical activity

2.

This diagram shows how the 6-pointed stars in the design were made. Begin with a cluster of 12/5 star polygons. Colour in every second point on each star polygon. Remove the construction lines.