Well these are things that some of the children have found out and they've been put together for ease of exploring.

Using Equilateral Triangles that are two units along each side - 3 dots for each side.

Dots | Lines | Triangles | |||

Around | Inside | Total | Unit length | Others | |

1/12 | 7 | 19 | 12 | 9 | 6 |

2/24 | 37 | 61 | 42 | 15 | 24 |

3/36 | 91 | 127 | 120 | 21 | 54 |

4/48 | 169 | 217 | 156 | 27 | 96 |

Well if you look at these sequences and explore them there's certainly a lot going on.

Then there are the patterns for equilateral triangles of 1 unit length sides.

One group of children were exploring the number of dots used in each successive size of hexagon and got: 1st 7

2nd 19

3rd 37

4th 61

5th 91

6th 127

They noticed that they are . . . . all PRIME !!!

Then one noticed that they were all related by taking one away and dividing by the order number. So 7 minus 1 divided by 1 is 6 And 19 minus 1 divided by 2 is 9 etc. and he got the sequence

6 9 12 15 18 21