# Flags From A Ship

### Task 141 ... Years 2 - 10

#### Summary

Reasoning 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.

#### Materials

• 4 cubes, each a different colour

#### Content

• counting
• algebraic patterns
• mathematical language of position and order
• visual patterns, including patterns of colour
• multiplication, multiplicative thinking, multiplication principle of counting
• permutations & combinations
• logical reasoning
• problem posing and solving
• recording mathematics

#### Iceberg

A 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:

B        B
Y        Y
R        W
W        R
thereby realising the key to all combination problems, which is that two items can only be ordered 'this way or that way'.

But the second flag could have been red. Then the possibilities would be:

B        B
R        R
Y        W
W        Y

Or the second flag could have been white and then the possibilities would be:

B        B
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:

• one flag in one colour there would be 1 arrangement
• two flags in two colours there would be 2 arrangements
• three flags in three colours there would be 6 arrangements
and then use a table to try predicting for four flags in four colours.

 No. of Flags Arrangements 1 1 2 2 3 6 4 ...

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

1. If each arrangement is a coded message, what could the 24 different messages mean? For example the arrangement in the picture on the card might mean 'turn to starboard'.
2. Find out more about International Maritime Signal Flags and how they are coded.
3. International Signal Flags include one flag for each letter of the English alphabet. Each flag has a geometric design but there appears to be no sense in which flag stands for which letter. Students could be asked to design their own alphabet flag system using, for example, only squares and rectangles and a limited set of fractions such as whole, half, third, quarter and fifth. What is the simplest (easiest to remember) system they could design?

#### Whole Class Investigation

Tasks 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 sticky-tac. 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 hands-on learning kits available from Years 3-10 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:

• MWA Pattern & Algebra Years 3 & 4