Introduction: A pond has a line of lily pads from left to right. To start with, each brown frog is on its own lily pad on the left side of the pond, each green frog is on its own lily pad on the right side of the pond, and there is an empty lily pad between the two colours of frogs. Each problem below requires the brown frogs to end up on the right side of the pond and the green frogs to end up on the left side of the pond by following these rules:

• Only one frog can jump at one time (during one move).
• Frogs can either jump onto the next lily pad (if it is empty) or they can jump over a different-coloured frog (e.g., green frogs can't jump over green frogs). When they jump over a frog they must land on the empty lily pad right beside that frog.
• You can never have more than one frog on a lily pad.
• Frogs can only move forward (brown frogs move to the right and green frogs move to the left).

Problem:

1. Suppose there is one brown frog on the left, one green frog on the right, and one empty lily pad in the middle. It takes three moves to change the frogs' positions from B, __, G to G, __, B. Show the position of the frogs after each move.

   

2. Repeat the problem, this time with one brown frog and two green frogs. This time it takes five moves to change from B, __, G, G to G, G, __, B. Show these moves.
3. If you start with two frogs of each colour, how many moves will it take to change the frogs' positions from B, B, __, G, G to G, G, __, B, B. Show the moves.
4. Do you have a system or strategy for figuring out the moves? Show the moves to change the frogs from B, B, B, __, G, G, G to G, G, G, __, B, B, B. Explain your strategy.

Materials: two different-coloured counters (4 of each colour); pencil, paper, calculator

Extra Questions: You could repeat the problem with different numbers of frogs of each colour (e.g., 4 brown and 4 green frogs or 4 brown and 5 green frogs).

See if you can find a pattern for predicting the number of moves when there is the same number of each colour of frogs. Then see if you can predict the number of moves if you start with 9 frogs of each colour.

Try to find the pattern for predicting the number of moves when there is one less brown frog than green frog. How many moves are needed if you start with 7 brown frogs and 8 green frogs?

Numbers and operations, patterns, critical and creative thinking, communication

  Activity Type:

Group      Individual 

  Assessment:

Scale:    Levels 1-5   
Self     Peer     Teacher 
Include in your portfolio?
Yes     No     Optional 
Toward Your Marks?
Yes     No     Optional 

  Hints:

It may be helpful to use the two-colour counters to show the movement of the frogs.