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Lake Crossing 1

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 link to any of the "content" strands of the mathematics curriculum. Nevertheless logic is an important part of mathematics. We have included two more problems along the same line. They can be found under Lake Crossing II and Space Crossing.

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 or be convinced that their method is correct.

Now we don’t expect that the answer will be written down in everyday language in all it’s gory 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)
Activity: 

The Problem

Tui Martin has a problem. She needs to row her pet ferret, pet rabbit and a bale of hay across to the island where she lives. The problem is that she can’t handle more than one thing at a time.

Now if she leaves the ferret with the rabbit, the ferret will kill the rabbit. Then if she leaves the rabbit with the hay, the rabbit will eat the hay. This will cause a problem on either side of the water.

Can she successfully get ferret, rabbit and hay across to her island without losing anything?

Teaching sequence

  1. Ask for four volunteers to play act each of the characters. (A name tag to hang around their neck helps the students keep track of who is who.)
  2. Pose the problem and ask for initial thoughts on the problem. (The graphic is not intended as a hint – Rabbits can not scuba dive!)
  3. 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)
  4. Let the students continue working on the problem in pairs. When the students have solved the problem ask them if this is the quickest way to get everyone across.
  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 and discuss the various written records.
    Are you convinced that you have found the fewest number of trips? How do you know?

Extension

The next time that Tui goes off her island, she needs to bring back her dog, another ferret, another rabbit and some hay. The dog would the kill the ferret if Tui is not around and we know all of the other difficulties.

This time Tui can take two things across the water with her. Can she get them all across successfully?

Can you make up some more problems like this? Can you solve them?

Solution

In the shorthand solution below, T = Tui, F = ferret, R = rabbit and H = hay.

On the land

On the water

On the island

T, F, R, H

   

F, H

T, R (to island)

 
 

T ( to land)

R

H

T, F (to island)

 
 

T, R (to land)

F

R

T, H (to island)

 
 

T ( to land)

F, H

 

T, R (to island)

 
   

T, F, R, H

Can you do this in fewer than 7 crossings?

Extension:

On the land

On the water

On the island

T, D, F, R, H

   

D, R

T, F, H (to island)

 
 

T (to land)

F, H

 

T, D, R (to island)

 
   

T, D, F, R, H

That was a lot easier than the original problem. Can it be done if Tui can only take one thing across with her?

The answer to that is no. If she takes the dog or the hay, there will be trouble with the ferret and the rabbit. If she takes the ferret the rabbit will eat the hay. If she takes the rabbit the ferret and the dog will be left together.

AttachmentSize
LakeCrossing.pdf58.66 KB
LakeCrossingMaori.pdf65.06 KB