This island focuses on adding money and involves strategy in placing coins to ensure the greatest amount is obtained. What is nice is that they have to continually think about their strategy. It might change a bit from example to example.

A money bag is the total of the two coins next to it. To start, the children can simply find what the amount would be in these money bags.

What about now? What would these money bags be? What would the total of both money bags be?

What if the two money bags couldn’t contain the same amount? What is the highest total that I could make using the coins above?

I deliberately avoided using £1 and £2 coins as I wanted the regrouping over £1. I’m making it more challenging and more relevant to them by developing their fluency of exchanging pence to pounds. It’s really important to emphasise that it isn’t the fact we can’t have the same coin twice but the same amount in a money bag twice. A noteworthy question here might be to ask why we can’t have 50p twice.

They should work out fairly easily that the 50p needs to go in the middle. It makes the most sense, we want to put the coins worth the most in the middle to have the greatest impact. We get £1.30 in total. As always, we start with a small part of the island and draw them in, then we go off and explore.

**What if we used four coins (and three money bags)?**

The obvious next step is to increase the number of boxes that we can put coins in. Whenever a task involves giving them a high score, I use dry erase pockets and give them two versions on a page so that they can have their best score and one they are experimenting with. Before they start, ensure they conjecture on strategies based on the first example with 3 coins as conjectures. These are really good to refer back to when reviewing it. Particularly if you name the conjectures after the children that make them.

Immediately, they will likely continue with their previous strategy of putting 50p in the middle (especially if you have emphasised it).

We can now get two fifty pence pieces in. We are still using a combination of 50p’s, 20p’s and 10p’s. This is an interesting discussion point as to whether they think this will continue to be the case.

Along the way there is the potential for some great mathematical thinking and reasoning. For example, proving with words why two 50 pence pieces couldn’t go as above. Proof doesn’t have to be this crazy, scary thing filled with algebra. We can explain this with words. No matter what we put in the middle, the two money bags adjacent to it will be the same amount. Therefore, we can definitively say that two coins the same can’t be two spaces apart. This is useful in future strategies.

There are two possible solutions for this one. Notice that the money bags are the same amounts but in different orders. It would be logical to imagine that three 50p’s would be worth more but the key is that in both cases, they only impact three money bags.

Whether you continue to six coins is up to you. The real fun begins by asking a ‘what if…?’ question to take things deeper.

**What if we used grids rather than rows?**

Obviously, you wouldn’t expose them to all these at once but things now start getting more challenging to think about. This is the whole value of the Phases of an Inquiry. These problems by themselves would have been too difficult but we are steadily building to them instead. Different children might choose to try different complexities of grids at this point and there is the potential for more than once lesson here.

In the one below, the obvious choice would be to put the 50p in the middle. However, this would then mean we couldn’t put another 50p anywhere else. Would it therefore be better to put a lower-value coin in the middle to try and get more 50p coins? It’s never about providing the answers to them. Whatever they come up with, even if it isn’t the absolute optimal solution, it is there solution. Don’t be tempted to show them at the end if they don’t reach it. Encourage different, creative ideas but don’t rob them of the opportunity to explore themselves. Dan Finkel explains this well in his 5 principles of extraordinary maths teaching.

**What if we played it as a game?**

You could play this as a game with a big grid and if the same total was made in two money bags, that player lost.

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

## Search

The use of the Search skills becomes more important as the grids get bigger. It would be easy for them to focus on a particular avenue or not consider the placement of the higher-valued coins. Emphasis on this, by steadily increasing the complexity of the problems is really essential to the success of the island.

## Organise

The Compare skill can be used to understand why the impact of the placement of certain coins is important in getting the highest totals.

## Discover

Conjectures can be formed on the number of 50p pieces possible in a grid. This is an interesting thing to focus on because it assumes that getting the most 50p pieces will get the greatest total – itself a conjecture.

## Investigate

Investigations focus on the placement of the 50p coins – trying out new ideas to maximise the total. An interesting discussion is how many examples you believe enough to know that you have the highest total. Realistically, there is no need to try the examples without any 50p pieces. What other ideas could they eliminate when trying to create a new maximum?

## Argue

Explanations of why they are putting coins in particular places lead to lots of potential discussion. This is particularly true if they are linking ideas between grids.

## Explore

Initially, it follows a fairly natural progression of just making the grid bigger. I normally introduce the grid instead of a row and often get a nice reaction from the children at the prospect. Further directions are possible, but the difficulty with this one is that having the grids ready for them saves them a lot of time.

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