Circular Number Constructions

This island on place value is all about building the greatest number possible using a number circle. Thought needs to be placed into both the length of the number that they can make as well as the value of the digits that they use.

In the number circle above, we can select digits by going around it. If I started on one and went round four clockwise, I’d be on five.

Each time we go around the number circle, we will ‘step’ by a different amount. Starting and ending on zero, what is the largest number you could make in one cycle?

This is a nice starting puzzle in itself and emphasises that the number of digits is more significant than the values of the digits. The largest number possible is four digits:

We can make 4,790 by going 4 steps, 3 steps, 2 steps and then 1 step.

What if we tried to make the greatest number in two cycles?

This is trickier to visualise but in essence, we now have a total of 20 steps possible. However, we have the constraint of needing to return to zero.

1 + 2 + 3 + 4 + 5 + 6 = 21

Therefore, a six-digit number isn’t possible to make. As such, we want to explore the largest five-digit number that is possible.

If we go nine steps initially to maximise the most significant digit, that leaves us with eleven steps in our cycle and four digits needed. It is possible but thought needs to be put into the order of those eleven steps to maximise the number.

At this point, we could continue and explore three or four cycles or we could change the rules a bit.

What if we were allowed as many cycles as we liked but could not land on the same number more than once?

With this, I have only allowed nine as a maximum number of steps in each move to make it more challenging. I haven’t found a way to make a nine-digit number but maybe you can!

What if we went counter-clockwise?

This should not be a long exploration but it is a nice idea of a simple change to the rules that can warrant exploration and produce mathematical thinking.

What if we changed how many numbers there are on the circle?

It might be interesting to explore smaller circles. How would more than ten numbers on a circle work?

What if every time we passed zero, we had to remove our greatest digit?

A more radical example of asking ‘what if…?’ but it just shows that all kinds of things are possible once the children are in this mindset. This is a counter to going straight for that nine as it will be immediately removed. A limit can be set on the number of cycles as before and lots of thought needs to be placed into how they build up the number as much as possible.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

A lot of thought can be placed into each move and the order of the moves to ensure the highest numbers. As such, children should not be spending much time Roaming and instead looking to Comb and Seek. This is particularly true when working with more than one cycle.

Organise

The main Organising involved here is in the thought of how many steps left are possible and how that number can be split up into unique values.

Discover

Conjectures looking at the maximum number of digits possible are likely to be the biggest use of discover skills that you will see. These can be Investigated and explained as well.

Investigate
Investigate inquiry reasoning skill in maths

Whilst they cannot try every possible combination, they can discount a lot of possibilities reasonably quickly which should aid in their investigations into what is possible or not possible.

Argue

Reasoned arguments are definitely attainable with this island. Children should be encouraged to explain why they are making particular moves as well as why they are discounting others. With just one cycle, the steps are made in decreasing order with consecutive numbers: 4, 3, 2, 1. Why does that strategy not work for two cycles?

Explore

There is a lot of scope for children to think of further ideas to explore. The nice thing about the initial problem not being too big, time is allowed to do this! I’ve given a few examples of this but children can come up with so much more.

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