Triangular hexagons
Investigate these hexagons drawn from different sized equilateral
triangles.
Problem
Well it all started by drawing some equilateral triangles so that they made a regular hexagon:
Image
Then we made a bigger one:
Image
Well there was the thought about how many dots there were in various places.
On the circumference there were 6 and then 12 on the second one.
But what about the number inside? And, how do you count them?
Well we counted 1 in the first and 7 in the second one.
We'll leave you to try the third size and count the dots.
Then there are the lines.
We counted just the number of straight lines ignoring their lengths.
There were 12 in the first and 15 for the second. How many for the next we wondered?
If you remember this all started with a hexagon made out of six equilateral triangles.
So we wondered how many lines of unit length the same as the side of one of those triangles there are? Well we counted 12 on the first and 42 in the 2nd one. So what about the 3rd one?
There are lots of paths for you to investigate here. Let us know what patterns you discover and any explanations that you work out.
Then there was the idea of what if the equilateral triangles had longer sides?
The first would look like this:
Image
And the second:
Image
So again we thought about how many dots there were in various places.
On the circumference there were 12 and then 24 on the bigger one.
But what about the number inside? And, how do you count them?
Well we counted 7 dots inside the first.
The second one was a bit harder and we tried it different ways and finally agreed on 37.
So the total number of dots for the first one is 19. And the number of dots for the bigger one is 24+37=61.
We have not noticed those numbers before coming up in a number pattern, so we need the next bigger hexagon with the lengths of three triangles on each of the six edges. I'm not going to draw that one but you might like to ...
Then there are the lines.
We counted just the number of straight lines, ignoring their lengths.
There were 9 for the first hexagon and 15 for the bigger one.
How many for the third?
If you remember this all started with a hexagon made out of six equilateral triangles.
So we wondered how many lines of length the same as the side of one of those triangles there are?
We counted 12 in the first one and 42 in the second one.
It looked as if there was going to be a large number for the third largest hexagon.
Using triangles of course made it obvious to count how many we used. 6 for the first, 24 for the second, how many for the third one?
Again, you can investigate the dots, lines and triangles for this larger size too.
What do you find?
Can you explain the patterns?
Getting Started
Have you drawn the next hexagon?
How about making a table to show the numbers you've found?
Perhaps think about the number of dots first. Can you explain why they increase in the way they do?
How about making a table to show the numbers you've found?
Perhaps think about the number of dots first. Can you explain why they increase in the way they do?
Student Solutions
We haven't received any solutions for this investigation yet, but we'd love to hear from you. Please don't worry that your solution is not "complete" - we'd like to know about anything you have tried. Teachers - you might like to send a summary of your children's work.
Teachers' Resources
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