The initial island that focuses on drawing diagonals with one intersection within a hexagon will likely not take that long to explore. However, it leads to many possible new islands and the opportunity for children to go in their own directions and be creative. When using curved lines or dots in different places, the possibilities become endless and they have to consider their actions carefully.

We have a hexagon of dots. Draw a line between any two such as the one to the left.

**What if every line that followed had to connect two dots together and, at the time of drawing, intersect exactly one other line?**

How many lines would it be possible to draw?

As you can see in the image on the left, after the third line, the first line has two intersections. As long as it only intersects one line when it is drawn, it is within the rules.

Although the ordering and orientation may be different, the children will all likely be able to create six lines like this:

I like using different colours to show each line for effect and also to enable replication of the order that the children went in. You can add numbers but it becomes messy when done by hand for some children. Instead, everyone draws the first line blue, the second line green, the third red, etc.

There are continuations of this that follow the same pattern with 7 dots, 8 dots etc. However, one of the values of this island is the opportunity to be more creative in the directions they go. Instead of changing the number of dots, we can change the nature of the dots or lines.

**What if we allowed curved lines?**

With this, we need to add the restriction of not allowing two lines from the same pair of dots and that the lines stay within the bounds of the hexagon. However, it then adds more complexities in terms of trying to arrange the lines to allow a greater number of connections.

**What if we changed the position of the dots?**

What is the optimum placement of the dots?

**Children’s possible use of reasoning skills:**

## Search

There is lots of Roaming in this lesson. Particularly when they move onto curved lines. However, as they add lines in an attempt, they gradually move to Seeking in trying to find new possible lines as they have less space to work with and need to use more problem solving in generating lines with fewer available options.

## Organise

Initially, they will come up with the same solution but it will be at different orientations. Sharing these examples gives them the opportunity to Compare them. As things become more complex, getting them to also Compare examples, whilst they won’t be exactly the same, they might notice trends.

## Discover

Children are likely to conjecture about aspects of the pattern that emerges from the 6 lines from the initial problem. For instance, two sets of parallel lines.

When they move onto curved lines, you might focus on the number of lines coming from each dot. This might be something you explicitly draw out from the children. In making their optimal solution, are they using similar amount of lines from each dot? Is it possible to get a line from every dot going to every other dot?

There will also be conjectures about the maximum number of lines possible.

## Investigate

In investigating, Expanding can be a really useful skill here as they can be creative. Particularly when moving onto curved lines or their own arrangement of dots. There might be a maximum number of lines that are presumed and the only way to beat a score is by altering what they have drawn carefully or considering new approaches.

## Argue

Looking at the initial 6 lines, the children can make attempts to explain why it is that no further lines can be drawn looking at the three possible diagonals that aren’t possible.

This extends to being able to reposition the dots. They can attempt to argue why it is better to arrange the dots in a certain way or why a certain way won’t work as well.

## Explore

I have shown a couple of possibilities but this is an island where children can come up with plenty of their own directions and I’m sure they will with encouragement. It’s just important to focus on genuine mathematical investigation. There is no value in allowing multiple curved lines between the same two dots as they could then do lots and lots of adjacent lines between them. We want to keep that struggle.

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