Flags From A ShipTask 141 ... Years 2  10SummaryReasoning and counting come together in this problem. The scenario is coded messages displayed on a ship's mast using four flags in different orders. The mathematical challenge is to organise the search for the total number of messages. 
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IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
This 'clever counting' problem requires the data to be organised and recorded in a pattern to have the best chance of finding all the solutions. For example, a student might reason that if the first flag is blue and the second flag is yellow, then the possibilities are: Y Y R W W R But the second flag could have been red. Then the possibilities would be: R R Y W W Y Or the second flag could have been white and then the possibilities would be: W W Y R R Y So, if blue is the first flag, then 6 arrangements are possible. But each of the other 3 colours could have been first and for each of those another 6 arrangements are possible. So, there are four lots of 6 arrangements, which is a total of 24 ways to arrange the flags. Another student might reason that this is a problem requiring the strategy of breaking into simpler parts. They might argue that if there was only:
However, this may not be enough data to realise that each number in the 'Arrangements' column is the product of its 'No. of Flags' multiplied by each of the 'No. of Flags' above it. For example, in Row 3 the 6 Arrangements could be calculated using 3 x 2 x 1. If this was a correct way of reasoning, then the result for four flags would be 4 x 3 x 2 x 1 = 24, which confirms the first student's reasoning. Some of the students may see a connection between this problem and Cars In A Garage or Ice Cream Flavours, in which case they can be complimented for applying the mathematician's strategy Have I seen a similar problem?. Extensions

Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 
Most schools have sufficient coloured cubes (or squares) to be able to easily turn this task into a whole class lesson. Perhaps begin with drawing of a mast on the whiteboard and four coloured squares of paper to attach with stickytac. Set up the storyshell and ask students to come to the board to make a few different arrangements. Our first challenge today is to make and record all the possible flag arrangements.Set the students to work with their own cubes and look for opportunities to highlight aspects of the Working Mathematically process. Challenge the students to use what they learn about each other's reasoning to work out how many arrangements are possible for five different flag colours. Extend further using the suggestions above. (We look forward to seeing any outcomes of student flag designs.) At this stage, Flags From A Ship does not have a matching lesson on Maths300. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Flags From A Ship task is an integral part of:
