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Space Crossing

Achievement Objectives:

Purpose: 

This activity has a logic and reasoning focus.

Specific Learning Outcomes: 

follow a chain of reasoning

recognise logical arguments

Description of mathematics: 

This is a pure logic problem that doesn’t seem to relate to any other part of the mathematics curriculum. Nevertheless it is an important part of the curriculum. We have included two more problems along the same line. They can be found under Lake Crossing I and Lake Crossing II.

Your students will likely rebel at the suggestion but they should write down all the steps they make in this problem. There is no other way to be sure that they have got things right otherwise. So an answer can’t be accepted unless it is written down. Writing clear solutions is a vital piece of the mathematical puzzle and is necessary for every problem on the web site. Until a solution is there in black and white it is possible that there is a mistake lurking around. Complete written solutions should be encouraged at all stages.

The reason that we mention it here is that students seem to like to do logic problems like this ‘in their heads’ or with equipment. Along the way they tend not to write anything down. So at the end, there is no way to check what they have done. Many of them will claim to have solved a problem and it will only be when you go through with them step by step that you will see an error.

Now we don’t expect that the answer will be written down in everyday language in all it’s gorey detail. Some sort of diagrammatic answer will do. (See the Solution below.)

We suggest also that acting it out might be a good strategy, at least for one group of students to explain their answer to the rest of the class.

Required Resource Materials: 
Copymaster of the problem (English)
Copymaster of the problem (Māori)
Objects to represent Galpons and three Exetrarts.
Activity: 

The Problem

As you all probably know, even in the newly Federated Universe, the Galpons and the Exetrarts do not get on. Indeed the Exetrarts will dematerialise any Galpon they are in close contact with, provided they are in the majority. On the other hand, the Galpons are fairly peaceful and can be trusted not to harm any Exetrarts they find themselves with.

Now unfortunately, three Galpons and three Exetrarts have found themselves on a crippled space transport next to the planet Jeeboh. They have to evacuate the transport and head to the planet on a space shuttle. BUT, only two creatures can travel in the space shuttle at a time. What’s more, although all of the Galpons can drive the shuttle only one Exetrart can.

Can they all get to the surface of Jeeboh safely?

Teaching sequence

  1. Pose the problem and ask for initial thoughts on the problem.
  2. Get the students to make suggestions for the first couple of "moves".
    How could we keep track of the moves taken? (diagrams, symbols and words)
  3. Let the students continue working on the problem. When the students have solved the problem ask them if this is the quickest way to get everyone across.
  4. Can you get everyone across in fewer moves?
  5. Remind the students that they need to justify their answer. This means that they will need to record the steps taken.
  6. Share answers. You may like to get one of the groups to demonstrate with objects while the rest of the class counts the trips taken.
    Did anyone else use the same number of trips? Did you use the same method?
    Did anyone get a different number of trips?
    Are you convinced that you have found the fewest number of trips? How do you know?
  7. Look at the various written records. Get students to see if they can follow the reasoning used by other pairs.

Extension

Can you generalise this problem? Can you solve them?

Solution

In the shorthand solution below, g1, g2, g3 are the Galpons and e1, E2, E3 are the Exetrarts and e1 has done the space shuttle driver’s course.

On the space transport

In the shuttle

On Jeeboh

(g1, g2, g3, e1, E2, E3 temporarily)

   

g1, g3, e1, E3

g2, E2 (to Jeeboh)

(g2, E2 temporarily)

(g1, g2, g3, e1, E3 temporarily)

g2 ( to transport)

E2

g1, g2, g3

e1, E3(to Jeeboh)

(e1, E2, E3 temporarily)

(g1, g2, g3, e1 temporarily)

e1 (to transport)

E2, E3

g1, e1

g2, g3 (to Jeeboh)

(g2, g3, E2, E3 temporarily)

(g1, g2, e1, E2 temporarily)

g2, E2 (to transport)

g3, E3

g2, E2

g1, e1 (to Jeeboh)

(g1, g3, e1, E3 temporarily)

(g1, g2, E2, E3 temporarily)

g1, E3 (to transport)

g3, e1

E2, E3

g1, g2 (to Jeeboh)

(g1, g2, g3, e1 temporarily)

(e1, E2, E3 temporarily)

e1 (to transport)

g1, g2, g3

E3

e1, E2 (to Jeeboh)

(g1, g2, g3, e1, E2 temporarily)

(e1, E3 temporarily)

e1 (to transport)

g1, g2, g3, E2

 

e1, E3 (to Jeeboh)

 
   

g1, g2, g3, e1, E2, E3

Can you do this in fewer than 13 trips?
Extension: We’d like to hear your generalisations.

AttachmentSize
SpaceCrossing.pdf59.92 KB
SpaceCrossingMaori.pdf73.53 KB