Treasure to ship

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Purpose

This problem solving activity has an algebra focus.

Achievement Objectives
NA2-8: Find rules for the next member in a sequential pattern.
Student Activity
Decorative image of a treasure chest.

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?

 

Specific Learning Outcomes
  • Use equipment to work out a problem.
  • Make and describe a repeating pattern to solve a problem.
  • Devise and use problem solving strategies to explore situations mathematically.
Description of Mathematics

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.

Required Resource Materials
Activity

The 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

  1. Introduce the problem with your chosen song, video, poem, or story.
  2. Pose the problem to the class.
  3. Ask the students to describe the problem in their own words to make sure that they understand what is required.
  4. 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).
  5. Let the students continue the problem in pairs.
  6. 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 trips?
    Do you think that you can use a smaller number of trips?
  7. Share the written solutions to the problem.

Extension

What if there are 8 treasure chests?

Solution

There are 9 trips from island to ship.

  1. 2 pirates go to the ship
  2. 1 pirate returns for treasure
  3. 1 pirate takes 1 treasure chest to the ship
  4. The pirate returns to island
  5. The pirate takes the 2nd treasure chest to the ship
  6. The pirate returns to island
  7. The pirate takes the 3rd treasure chest to the ship
  8. The pirate returns to island
  9. The pirate takes the last treasure chest to the 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 the 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. Note that students are not expected to write these kinds of expressions at Level 2.

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Level Two