There are 2 pirates and 4 treasure chests on an island.

The pirates have 1 small boat to take the treasure to their ship.

The boat can take 2 pirates or 1 pirate and 1 chest of treasure.

How many trips do the pirates have to take to get all the treasure and both pirates onto the ship?

This problem develops a strategy that works a number of times, as students use equipment or a diagram, and logic to solve the problem.

This problem is an introduction to algebra where the notation gives a way of expressing the repetition of a strategy.

Copymaster of the problem (Māori)

Copymaster of the problem (English)

Paper and pencil

2 orange rods (pirates) and 4 yellow rods (treasure chests)

### Problem

There are 2 pirates and 4 treasure chests on an island. The pirates have 1 small boat to take the treasure to their ship. The boat can take 2 pirates or 1 pirate and 1 chest of treasure.

How many trips do the pirates have to take to get all the treasure and both pirates onto the ship?

### Teaching Sequence

- Introduce the problem by reading a poem about pirates.
- Pose the problem to the class.
- Ask the students to describe the problem in their own words to make sure that they understand what is required.
- Using 2 students act out a trip to the ship with the treasure. Discuss the ways that you could keep track of the trips taken (draw, list).
- Let the students continue the problem in pairs.
- As the students work ask questions that focus their thinking on the steps they are taking.
*How many trips have you taken?*

How did you work out who to put in the boat?

Can you see any patterns in what you are doing? Describe them.

How are you keeping a record of the strips?

Do you think that you can use a smaller number of trips? - Share the written solutions to the problem.

#### Extension to the problem

What if there are 8 treasure chests?

#### Other Contexts for the Problem

Fairytales: 3 Bears and 5 bowls of porridge (Baby Bear taking bowls of porridge to the dining room table);

3 Pigs and 10 bricks (Porky Pig moving bricks).

These problems also extend the pirate problem in that they require the same strategy but use larger numbers. The problem can be further developed using any number of people (or animals) and any number of objects to be transported.

### Solution

There are 9 trips from island to ship.

- 2 pirates go to ship
- 1 pirate returns for treasure
- Pirate takes 1 treasure chest to ship
- Returns to island
- Takes 2nd treasure chest to ship
- Returns to island
- Takes 3rd treasure chest to ship
- Returns to island
- Takes last treasure chest to ship

Note that steps 1 and 2 can occur at any time that the small boat is on the land or steps 3 to 9 can be followed by ‘returns to land, 2 pirates go to ship’.

#### Solution to the extension

Each treasure chest requires two trips, one to the ship and one back to the land. So with 8 chests the pirates will need 8 x 2 trips with the chests and 1 trip to take the extra pirate. This means 17 trips.

(With c chests and two pirates there will need to be 2c + 1 trips.)