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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.
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.
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.
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?
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.
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.
Find a great variety of ways of asking questions which make 8.
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
Noah saw 12 legs walk by into the Ark. How many creatures did he see?