Problem
John, Jo and Chris have got seats for the pictures. In fact their seats are F5, F6, F7. In how many ways can they sit in those seats?

What is this problem about?
This problem is first of all about being systematic. This can be done using a number of strategies. You might make an organized list, draw a picture or use equipment. The difficult thing for many children is to keep track of all the possibilities as they go along. This is especially difficult if they use equipment.

The maths behind the problem involves counting all the possible combinations.

Achievement Objectives
Number (Level 3)
- write and solve problems which involve whole numbers and which require a choice of one or more of the four arithmetic operations

Mathematical Processes
- devise and use problem solving strategies (act it out, draw a picture, organised list)
- use equipment appropriately

Resources
classroom chairs
paper
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific Learning Outcomes
The children will be able to:
- work systematically to find all the possible combinations

Teaching Sequence

1. Use 3 children and 3 chairs to pose the problem.
2. Ask the children to work in 3’s to solve the problem.
3. Circulate to see that the children are keeping track of their solution.
   How are you recording your work?
   How do you know that you have found all the possible ways?
   How could you convince someone else that you have found all the ways?
4. Sharing of solutions
5. Focus on the methods that children have used to be systematic.

Extension to the problem
What would the answer be if another two had joined the three friends?
What if your whole class went?
What if the seats were in a circle?

Answer to the problem
There are 6 ways for the children to sit on the seats.

Here we give every child a chance to sit in F5.  There are 3 choices for this seat.  For each particular child in F5 we then have two choices for F6.  Then we have only one choice left for F7.  Note that 3 x 2 x 1 = 6 is the final answer.

The extension is possibly ambiguous. We meant to mean that five friends had five seats, though some children might take it to mean that the five children had just three seats. Either way is worth considering.

Here five friends can be seated in five seats in 5 x 4 x 3 x 2 x 1 = 120 ways. If there are 25 in your class, then there are 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. This is a very big number. You might like to let the class calculate it.