Coinality

This is based on the fantastic game, Totality, by NRICH. Go here for the original (and of course all the other fantastic tasks and articles they have). It has always been a favourite of mine not just for the tactical nature of it but because I have found it so easy to adapt and investigate in different contexts. In this case, I have used the context of money but it would work equally well with numbers or other units of measures.

Instead of using the original game board, we are going to use one with coins on and adapt the rules slightly.

Ideally, you will have coins to place on each point. To play the game, the first player will choose the starting point. That coin will be taken off and put in a ‘Total’ to the side. A counter will replace it:

The next player can then move the counter to one of the connected coins. Let’s go to the 50p. It goes to the bank and the total is now 70p.

If a player causes the total to go over £1, they will lose.

Straight a way, they might think about where the best starting position is – if there is such a thing. Are they better starting on a higher or lower total? What is the longest game that it is possible to play? Can we make a conjecture on what the different amounts are that can be made when we cross the boundary to £1?

I often like the idea of starting with a game and then slowing it down by investigating the possibilities that lie within it by bringing in aspects of problem solving.

What if each time we landed on a coin, instead of totaling it, we formed a chain? What if, nowhere along that chain could there be consecutive coins that total exactly £1?

To understand this let’s look at this example:

I have picked up the coins along the path in the order shown. The last five coins that I have added to my chain (10p, 20p, 50p, 5p and 10p) total 95p. Therefore, I couldn’t pick up the 5p next as that would make a total of £1 with a set of consecutive coins on my chain. How long can the chain be? Rather than focusing on the whole board, different children could work on different starting positions to remove any competitive elements to it. Does it matter at all where we start anyway?

What if no set of consecutive coins could make ANY multiple of £1?

This adds another layer of challenge and adds a lot of mental calculations in checking various totals.

What if we had a web rather than a chain?

Taking this idea further, rather than a one-dimensional chain, we can have a web. In this instance, we aren’t taking the coins off the board but putting them down instead. The next coin has to connect to the previously-placed one. They have to use the same coins as the board above: 4x50p, 5x20p, 5x10p and 5x5p.

In the example above, I started with the 50p in the middle, then the 5p, the 20p, the 10p and then the 50p. Now instead of just joining in a chain with the previously selected coin, they form a web. No connected group that is a subset of the web can form one that makes a total of £1. Tricky!

There are so many directions that this can go in and that’s why I love Totality as a game. Here is a blank board if you would like to print one:

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

Children have to continuously think about the placement of the coins carefully. There is going to be lots of trying things, seeing if it works and trying again. Sometimes, their paths will be limited and sometimes they will have lots of choices.

Organise

It will be useful for the children to think about the combinations of coins that can form one pound.

Discover

With any problem involving scores, there are the discoveries involving which scores are possible. They can conjecture about which paths are possible from different starting points.

Investigate
Investigate inquiry reasoning skill in maths

The children can investigate different starting positions and first moves from those positions. Having this focus helps ensure that they are thinking about each step rather than moving randomly.

Argue

Particularly following on from playing it as a game, this is perfect for working in pairs which lends itself to discussion about the paths forwards.

Explore

They can switch the coins around and change the totals – all sorts!

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