This island focuses on using multiplying and dividing by ten. There are so many interesting ways to deepen the inquiry and go in different directions with this one. It essentially involves creating a cycle of numbers, multiplying and dividing by powers of ten between each one. It builds a deeper understanding of inverse operations in the cycle and the impact of multiplying and dividing by each power of ten.

To make a Power of Ten Cycle, we multiply or divide by 10, 100 or 1000 so that we make a complete cycle of operations starting and ending at our original number.

Initially, we just want a completed. Often, children will realise that if they just reverse the first two steps for the final two, they can cycle back to the beginning.

**What if every number in the cycle had to be different?**

By adding in this rule, it breaks that possibility and they have to try again. So much of the Isles of What If…? is about creating managed chaos once regulation has been reached. This is a perfect example. We had a strategy that worked and all seemed simple until we threw a spanner in the works. Of course, as the example above shows, they can reverse the order of the final two operations to avoid duplicating a number. But that is good mathematical thinking.

There are many ways of changing the constraints to explore deeper into the island.

**What if we had a cycle of 5 numbers?**

The value of odd-numbered cycles is that they can’t simply use the direct inverse of the operations in the first half. These kind of peculiarities are what we want to test conjectures against with the Expand skill. A conjecture that worked for four-numbered cycles, might not work for ones with five numbers. Sometimes, with this one, children want to multiply by ten four times and then divide by 10,000. That’s great, it shows their depth of understanding, but once it has been realised, I think it is better to stick to 10, 100 and 100. They have found a strategy that works and it would work for any cycle, but once discovered it does not need to be pursued further. This is where the teacher has to exhibit their role as an arbitrator of genuine mathematical investigation.

We can extend further by looking at 6 numbers, 7, etc. We are looking to generalise about possible strategies across different cases and connect them.

**What if we couldn’t use the direct inverse of a calculation?**

Essentially, if we multiplied by ten between two numbers, we then wouldn’t be able to divide by ten at any point. This is attempting to draw out the understanding of the associative law – that x10x100 = x1000. It is more challenging than before but that is why you build things up and gradually increase the breadth and depth of the inquiry.

**What if we had two cycles?**

There is lots of room for some creative ideas with this one. One child I taught came up with the idea of using two cycles and that apart from the starting number in the blue circle, no numbers in the same-coloured circle could be the same. I thought it was intriguing! Here is a solution for two four-numbered rings. We then explored how many rings we could have whilst making this true.

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

## Search

The main skill they will be using is Seeking. However, there is an element of trial and improvement through Roaming. This isn’t a task where being systematic (and using Combing) is as important.

## Organise

Compare can be a really powerful skill to use with this island at the beginning as you can look at the differences between the divisions and multiplications and as a result, deepen their understanding of inverses.

## Discover

The Conjecturing that they do will relate to the strategies they are using to complete the cycle. If they move on to cycles of more numbers, they may think about strategies that go beyond just a specific amount of numbers.

## Investigate

The Investigating here will mainly be Digging. They can come up with a strategy and make up new cycles using that strategy – possible with a different amount of numbers.

## Argue

The Persuading that can be done here is to convince people of why a strategy works – looking at the power of inverses and the associative law.

## Explore

This is definitely an island that the children can come up with their own ‘What if…?’ questions. It’s important though that they don’t try and cover too much. Focus on perhaps a specific amount of numbers after they have looked at the original one.

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