Three ball line up
Problem
Two children are playing with three balls, one blue, one red and one green.
They toss up the balls, which run down a slope so that they land in a row of three.
In how many different ways could the balls land?
How do you know you have found them all?
You might like to use the interactivity below to explore the problem.
If you would like to try another task which involves finding all possible solutions, you could look at Inside Triangles.
You may be interested in the other problems in our Working Systematically Feature.
Getting Started
If you are not using the interactivity, perhaps you could find three differently coloured balls, counters or cubes to try out some ideas. Or you could just use coloured pencils and draw three spots.
If red landed in the middle, how could the blue and green fall?
Where else could red land if it wasn't in the middle? Can you use this idea to find all the ways?
Student Solutions
We received some very well explained solutions to this problem. Many of you described how you used ordered your solutions so you knew you had all the solutions. We can call this 'working systematically'.
Jack from Tetherdown Primary sent his solution and his teacher said:
Jack worked out that there were six ways that three different coloured balls could land. At first, he stopped at four ways but then realised that if there were two ways of a red ball landing first, then there would be two ways
of a blue ball landing first and two ways of a green ball landing first.
Here is a picture of Jack's recording:
AT from ASB in the Netherlands used the same method:
So, what I did was I first started with one colour, red and then I added blue and green. I did the same on the next one (still starting with red) but then green-blue.
So basically, I did the same for each of them and found six different outcomes...
It is easier if you just draw it on a piece of paper because then you can keep track of them easily!
Here is AT's drawing:
Pranaiv from Saint Bartholomews Primary School in Wolverhampton used a similar approach:
Thank you also to Roheena and Esther from Edgware Primary, who agreed with the six solutions.
Maryam from the International Community School explained how she worked on the task, which again, was similar:
The balls can land in 6 different ways; each coloured ball (green, blue, red) can be in three different positions - front, middle and end. So if (for example) the green ball is at the front, the red and blue need to follow it, creating one solution. Then swapping the red and blue ball (with the green ball still at the front), we created the second solution. If we do the same with a turn for each coloured ball at the front, we get six possibilities in total.
Her recording is very clear:
Mehetabel from Corpus Christi also organised her recording very clearly:
A pupil at St John the Baptist School in Wales shared this fantastic video explanation with us:
Thank you also to Charles from Clear Water Bay School in Hong Kong, Alex from Longsands School and Maria from Wimbledon High Junior School who all took the trouble to explain how they worked it out.
Aiste from VDU Rasos Gymnasium, Lithuania, wrote:
There are three balls and also three spaces for them.
So: _ _ _
We should put three different balls.
In the first place you can put all three balls. In the second place - 2 and at last in the third - 1.
So when we multiply these numbers and then will find the answer. 3x2x1 is 6.
I wonder why we multiply the numbers?
Teachers' Resources
Why do this problem?
This problem is a good context for encouraging learners to work systematically. Then they will be able to convince someone else that they have found all the possibilities.
Possible approach
You could introduce the task by showing the class the interactivity and, without saying anything at all, roll the balls three or four times. Ask children what they notice and what they wonder, and invite them to talk to a partner.
Take some noticings and questions from the whole group, but try not to respond yourself. Instead, rely on other members of the group to comment or answer questions.
You can then clarify what is happening in the simulation, and ask how many different ways the balls could land. Look back at the results you have recorded so far in the interactivity, and invite learners to explore whether there are any other combinations. As they work in pairs on this (perhaps using coloured counters to represent the balls), listen out for those who are developing a system to find all possibilities. You may like to draw attention to those who have created a good recording system too.
In the plenary, you could ask particular pairs to share what they have been doing and you could facilitate a whole class discussion so that a conclusion is reached. You may wish to ask learners to record individual solutions on individual strips of paper, then you can pin some of these up on the board. Invite the group to reorder the solutions to reflect a pattern and use this to create any solutions that are not yet displayed. Emphasise that although they have worked together using one pattern, or one particular way of working systematically, there is no 'right' way to work systematically. There will be many different systems in the room and that should be celebrated.
Key questions
If red landed in the middle, how could the blue and green fall?
Where else could red land if it wasn't in the middle?
Can you use this idea to find all the ways?
How will you remember the ways you have found so far?
Possible support
Pupils would benefit from having three differently-coloured counters to use while tackling this problem. If they are finding it difficult to work systematically, you could offer them 6 Beads first, which might be a more familiar context.
Possible extension
Encourage children to use four balls/counters. If learners would like another context in which to practise working systematically, Inside Triangles would be an appropriate task.