This island involves starting with shapes made with elastic bands on Geoboards and ‘pulling’ the sides out over the nearest peg – trying to avoid making right angles for as many turns as possible.

We start focused on a scalene triangle – specialising on this shape altogether as an entry point to the island. The Math Learning Centre has an online Geoboard manipulative that works really well with this island. It allows you to save and share examples. I have done so with the starting point here.

The video below shows. an example of 3 ‘pulls’ before a right angle is formed. It’s an island that’s easier to see in action to understand. When I’ve done this, I’ve had little strips of paper on the Geoboard to create a boundary for the 5×5 grid so that they can see it clearly.

This could also work as a game where you’d lose if you made a right angle.

Try and ensure that children are recording each step. Dry erase pockets are an amazing resource for this (and in so many other cases) because they enable trial and improvement approaches easily. A sheet such as the one below can be made to allow for them to record their highest scoring result and one they can experiment with to try and beat it.

There is plenty of room for investigation in the original premise. This is an island that benefits from going in a similar direction to others as it can be hard to make connections between different starting points as there are just so many things that you could change. You could have groups go in similar directions and then have everyone report back. Here are a few examples of possibilities. Focus on making connections back to the original example, does the new possibility allow for more ‘pulls’ or fewer? This creates an additional problem-solving element to new lines of inquiry.

**What if we started the triangle in a different position?**

Is there an optimal position for the triangle?

**What if we started with a different triangle?**

Which type of triangle (equilateral, scalene, isosceles) allows the greatest number of pulls?

**What if we started with a quadrilateral?**

**What if we used a 6×6 grid?**

**What if we used a triangular grid?**

Many Geoboards have triangular grids on one side which allows for an intriguing new direction.

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

## Search

Roaming will be important early on but as time goes on, by drawing attention to certain properties of the sides, they can be more systematic. For example, if they make a 45 degree angle between two pegs (with no pegs in between), it can’t be pulled out any more.

## Organise

The difficulty with organising in this case is that there are going to be so many different variations of moves. The children could focus on which initial move works best (making their Search more systematic as well) and you could group results based on this.

## Discover

This is the sort of island that conjectures are a bit harder to predict but with the culture established, they will definitely come. Making links between the different starting conditions could be one way that you explicitly draw conjectures out of them. For example, how will a 6×6 grid impact things? Like a scientific investigation, the important thing in this case then is not to try and change too much too quickly. Stick to small changes initially so that those comparisons can be made.

## Investigate

When Investigating, it could be really useful to display a sequence of moves from a child on the board and have the whole class investigate them in some way. For example, it might be an optimal solution that they believe to be true but you might get the children to focus on flipping the order of the moves to see if that has an impact on what is possible.

## Argue

A good opportunity for them to explain things might be why two moves are better done in a particular order. From the starting shape for example, they could focus on which side they want to pull out first and why.

## Explore

There is so much opportunity for them to go in a new direction but resist the temptation to go too soon. This is an island that can easily go beyond a single lesson. I generally focus the first lesson on small changes in a fairly guided manner and then give them a second lesson to go in more unique directions.

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