Totals Turned to Ten

Starting with four dice, we rotate them so that the number facing us is now on top. However, we must avoid any number of dice in a row, whether two, three or four, totalling ten. You could change the total or the number of dice to adjust it to work with different children. There are then many opportunities to explore further.

Set out four dice as above. Play a game. Each turn, the player rotates a die so that the number facing us (see the video) is now on top. If any pair, three or four dice in a row adds up to ten, that player loses.

Here, 6 + 3 + 1 = 10 so it results in a loss.

2 + 3 + 3 + 2 = 10 so again, a loss

This is fine as no total makes ten.

If the dice were put in the positions above after a move, the player that did so would lose as the first three, all four and the middle two total ten respectively.

What if the players worked together?

What would be the longest chain that could be created without totaling and ten and without any of the dice returning to their original values?

In investigating possible chains, if we record the total of each combination and the number of the dice that was rotated, the children can make links between the dice number and the totals.

By not allowing them to return to the original values of the dice, we are limiting their possibilities and I think it makes them think more about the choices they make. It sets the goal of trying to rotate each dice three times. How many ways are there to do it?

What if we changed the starting orientations of the dice?

In doing this, I like the idea of trying to find a starting orientation that makes it impossible for children to rotate each dice three times. Obviously, any starting position cannot have a total of ten from any of the dice.

What if we used five or six dice?

This is the natural progression. They can think about why this becomes more challenging.

There are lots of creative ideas for the children to explore new islands here. I’ve got a few suggestions but as always, they can come up with ideas themselves.

What if we didn’t allow the sides facing us to total ten either?

Is it still possible with four dice?

What if we investigated the number of possible combinations using the numbers one to six once in a line?

If we have the numbers one to six, not through turning the dice, we will try and line them up so that no total of ten is made. By forcing them to use each number once, we are reducing the number of possibilities so that they can focus on harder-to-find combinations.

One example that works is: 1, 4, 3, 5, 6, 2

What if we tried to have six dice in a circle rather than a line?

In this, like the last one, we are using the numbers one to six once. We are looking for any combination of dice where no totals of ten are made. I’m pretty certain this is impossible but I think, in many ways, that is why it is worthy of exploration. The children can consider why it is impossible. Above, my combination doesn’t work as 6 + 4 = 10, 5 + 1 + 4 = 10 and 5 + 3 + 2 = 10. You could then think about which combination in a circle has the most number of totals of ten instead.

What if we laid out the dice in a grid? How many dice would it be possible to have?

We are going quite far away from the original island with this but sticking to the original concept of avoiding a total of ten. With this one, I like the idea of the grid being like sudoku where you can’t have the same number twice in a row or column. How many dice can they place whist not making a total of ten in any combination of dice in each row or column?

When doing this, you might want to think about having a maximum grid size. In the example above, I’ve made a grid seven-dice wide. By building a staircase shape, it would be very easy to create a very large grid. You could wait until they notice the approach before introducing further constraints to make it more challenging for them.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

In a sense, the whole island is based on the use of Seeking to avoid ten and Combing to make small changes. However, they can take this further by considering the impact of having particular dice. For example, there are more combinations that make ten with a five than there is a one. By looking at the same starting position multiple times, we can ensure that they are not just randomly choosing dice to rotate. The recordings they make will also encourage them to go back and make small changes.

Organise

The children’s recording and Organising plays a crucial role with this island. As the dice will quickly change, they won’t remember the sequence they have used. It is important to have recording in some sort of list so that the children can repeat and adjust what they have produced previously. It will also help spot times when they fail to spot totals of ten.

Discover

The deeper the children go into the island and the more they focus on particular cases rather than just going randomly, the more likely that they will Notice patterns and make Conjectures. This is why their use of the Organisation skills are so important here. The Archipelago phase encourages specific focus on more confined cases that will promote conjectures.

Investigate
Investigate inquiry reasoning skill in maths

This island isn’t necessarily going to involve the use of a lot of Investigation skills initially. However, who knows what kinds of conjectures are made so it is always hard to predict exactly when they will be needed.

Argue

With some aspects of the Islands such as the idea of the six numbers in a circle are impossible, Arguing skills can be employed as to why. With the circle, they might choose to investigate specific pairings of numbers next to each other and why there are no possible combinations with the remaining six numbers as a away of Proving why it is impossible.

You can also make comparisons between starting configurations and why some are perhaps more challenging than others.

Explore

There are so many possibilities to extend this further. I’ve included some that I think are interesting but, given the time and culture, the children can come up with many of their own. For example, they could think about how to manipulate the dice differently. Perhaps the order of the dice can change somehow. What would it look like with different dice?

Leave a Reply

Your email address will not be published. Required fields are marked *