Taking a Die for a Walk
2 out of 3
Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.
Problem
We often roll a die to find a number from $1$ to $6$ for a game that we're playing. We wait until the die stops and then see what number is on top. What about if we slow it down and have the die slowing rolling along? What if we make it even slower and we move the die to a new face one at a time?
In this way we can move the die forward/backward or right/left. You might like to think about it as North, South, East or West.
You can then move the die around and take it on a journey. In this investigation, we're going to think of a journey as never re-visiting a place where the die has already been. So it can't cross back over its path or go back over where it has just been.
When I first thought about this I thought of the die leaving a kind of "footprint" of the bottom number wherever it went on its journey. You can do this if you want to.
I then found it easier to look at the number on the top as each new "step" was taken.
So with a die which might show:-
- or maybe you have one with numbers on it:-
we can go on a little journey:-
So, starting with $1$ on the top we move down. (I make sure that $2$ is going to come up by moving the die around before starting.) I then got $2$ followed by $6$. I then rolled it over to the right a few times. $3$ shows up, then $1$ again, followed by $4$.
Another journey could be:-
We start off as before. But, when I've rolled to the right a couple of times, I go up and it shows a $2$, and then left and it shows a $3$. It's a bit difficult to see where I've gone so I've added an arrow to show the direction.
What I CANNOT do now it to go further by rolling left from that last $3$ which would give a $5$ on top of the $2$ I had from my first step. Remember - no crossing back again!
So, explore a little and see what things happen as you roll around different paths and you write the numbers down.
Getting Started
It would be useful to have a real dice and some squared paper to record on.
You could begin by always starting with $1$ on top and making a certain number of rolls, for example. How many different paths can you make?
What do they have in common? What is different about them?
Student Solutions
There are lots of answers to this problem, depending on what questions you choose to ask. Have a go yourself, and if you discover anything interesting, email primary.nrich@maths.org
to tell us what you've done! Please don't worry that your solution is not "complete" - we'd like to hear about anything you have tried.
Teachers - you might like to send in a summary of your children's work.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem could be used when the four directions, north, south, east and west are being introduced or discussed. It could also be helpful if learners are apprehensive about number work as it gives an easy "open door" to number exploration and enquiry. Its openness means that it can be approached by a very wide attainment range. This problem presents an interesting context for encouraging children to visualise the die and predict the path of numbers.
Possible approach
Once they have had more time to work, bring the whole group together once more to show their various "dice walks" and to discuss what they have discovered. What do they notice about the patterns they have created? They could compare, for example, adjacent numbers in two different "walks". What is the same and what is different about them? Can they give explanations for their observations?
Key questions
Is the die going north, south, east or west now?
Possible extension
These sheets give many ideas about how the investigation could be taken further. You may want to give them out to children, or simply suggest something else for them to pursue. Also go Inky Cubes
Possible support
Some pupils might find it helpful to start with the die rolling in just one direction to get the idea of how to record and to become more familiar with the way a die is constructed. Larger dice and paper with $2$ cm squares might help those with difficulties in using fine motor skills.