These isles focus on finding multiples of numbers utilising the divisibility rules and lots of problem solving. There is both the scope to spend an extended period of time focusing just on the divisibility rules for one to five and the possibility of exploring large numbers that are multiples of six to nine. That gives it lots of flexibility in terms of the children that can access it.

Using the digits 1, 2, 3, 4, 5 and 6 to create numbers, can we make multiples of…

- 2
- 3
- 2 and 3
- 2, 3 and 4

**What if we used 1, 2, 3, 4, 5 and 6 as digits exactly once across the numbers?** **Can we create different multiples of four consecutive numbers?**

1 | 2 | 3 | 4 |

1 | 6 | 45 | 32 |

2 | 3 | 4 | 5 |

34 | 6 | 12 | 5 |

You could go on with this in its own right but I think it is more interesting to explore an additional possibility:

**What if we could use common multiples? **

These combinations would then be a possibility:

1 | 2 | 3 | 4 |

6543 | 12 | 12 | 12 |

351 | 264 | 351 | 264 |

654312 | 654312 | 654312 | 654312 |

This serves a number of purposes. It allows for a greater range of combinations and encourages the use of the divisibility rules as much greater numbers are now possible.

**What if we tried to find multiples of 2, 3, 4 and 5 using 1 to 6 as digits?**

For 2, 3, 4 and 5 we could have examples such as these:

2 | 3 | 4 | 5 |

624 | 315 | 624 | 315 |

312 | 312 | 312 | 645 |

24 | 24 | 24 | 6315 |

Then for 3, 4, 5 and 6, it is an interesting observation that the solutions above work as well since anything that is a multiple of 2 is also a multiple of 6. Making it explicit that they could try previous solutions with new combinations will likely result in a conjecture that all solutions for 2, 3, 4 and 5 will also work for 3, 4, 5 and 6.

3 | 4 | 5 | 6 |

624 | 624 | 315 | 624 |

312 | 312 | 645 | 312 |

24 | 24 | 6345 | 24 |

The children might expect that to carry on with 4, 5, 6 and 7. One of my sets does, but equally, new solutions are required. An interesting discussion could be had around why there are fewer solutions than with 3, 4, 5 and 6.

4 | 5 | 6 | 7 |

624 | 315 | 624 | 315 |

6124 | 35 | 6124 | 35 |

24 | 15 | 24 | 63 |

With this, once you reach 7, 8, 9 and 10, no solutions are possible. Without a zero, we cannot make a multiple of ten. However, it offers up an accessible opportunity for the children to explain why so it does not need to be avoided.

**What if we had more than four consecutive numbers?**

How many consecutive numbers could we make multiples of using 1 to 6 as digits?

Instead of just finding many different solutions for each set of four numbers, there is a lot of scope for trying to focus on particular goals with each set. Each goal has a different focus.

**What if we tried to find the greatest number possible?**

With this, since we are aiming to find large numbers the focus is on using the divisibility rules efficiently. As such, it is likely better to focus on using a set of four consecutive numbers below seven. Of course, there is nothing wrong with them attempting it though.

**What if we tried to find the lowest common multiple possible?**

You could explore this in terms of the solution with the lowest common multiple for two numbers, three numbers or possibly all four consecutive numbers.

**What if we tried to find the greatest mean possible?**

I like this as it brings in the mean and brings in discussions of whether or not we should value combinations where all four multiples are consistently high rather than just one. It demonstrates the impact of those two situations. In this case, it might be worthwhile adding the condition that all four numbers can’t be the same value.

**What if we tried to find the lowest difference possible between the multiples?**

With this, we are focusing on common multiples across multiple numbers as a difference of zero would be possible if we found a common multiple of all four consecutive numbers.

In exploring these ideas, you might do a lot of hopping between them. As always, the children can come up with their own ideas to explore. Some of the best ‘what if…?’ questions come from them.

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

## Search

These isles are all about Seeking. However, there is a lot of Combing as well. Children can be encouraged to split the digits between the multiples one way and then make small changes, thinking about divisibility rules, and try again. In this case, whilst some children will do that of their own accord, explicitly highlighting how it is done will benefit everyone. There will need to be elements of Roaming but this can be quickly followed by the other two Search skills. Having digit cards for the numbers one to six will massively aid them in making these subtle changes as they can quickly move the digits around.

## Organise

Particularly when looking at different ‘scoring’ systems (whether we are finding the greatest number etc.), comparisons can be made between them. Having them together in a table like the ones I have used above will help to encourage this.

## Discover

This isn’t necessarily an inquiry that focuses on conjecturing a lot but with a culture of them, some will definitely come. One way that you could explicitly draw them out is by looking at the size of the numbers in the Landing Spot when each number has to be different. They might conjecture that a number above a certain size is impossible. This also then leads into why it is relevant to allow common multiples for further exploration.

As explained above though, you can also look at sets of consecutive numbers that share solutions.

The treasure amongst these islands is more being able to find new solutions to each set of consecutive numbers.

## Investigate

With the sharing of solutions from 2, 3, 4 and 5 and 3, 4, 5 and 6 (explained above), the conjecture might come that the same is possible with 4, 5, 6 and 7. However, it is likely that a counter example for this has already been found. As there is a lot of Seeking required with this island, there is potential for Investigating as conjectures will be made on limited data.

## Argue

As mentioned in the Discover section, they might think about why numbers above a certain size are impossible and the explanation of needing to spread digits across four numbers is an accessible explanation to them. They can also explain why common multiples makes this more possible.

With the set of 1, 2, 3 and 4, they can Persuade why a multiple of one should be left until last since every integer is a multiple of one.

As well as this, they can Prove why sets that include 10 (such as 7, 8, 9 and 10) are impossible to make. Lots of potential for Arguing here.

## Explore

This is definitely isles that you want to encourage deeper Exploration and the connections to be discussed between them. I have given several ideas for problem solving above but they could equally investigate their own idea. It might be you only introduce one of my ideas above and see what they come up with next. You always have the possibility to introduce a second idea from above if they are struggling. Equally though, they might come up with something better!

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