Frogs: Teachers' Notes
Introduction
This is a great problem as it can be used in a variety of ways. As well as one or two students exploring the problem on a computer screen it can be done as a class with students acting out the problem. I have found that students of all ages as well as pre- and in-service teachers get a great deal of fun out of this problem. As a bonus it contains a great deal of very rich mathematics.
These notes are not intended to be a sequential description of how to use this activity with students but they should provide you with further information that may help you to better understand the problem and how it might be used to provide a rich mathematical experience for your students. You should also read the Students’ notes for this problem, as they provide specific hints on completing the Bright Sparks activity online.
About the problem
It’s always a good idea to start these problems by just playing with the situation. Try to interchange the frogs and see what happens.
In this phase students should realise that
- Frogs can’t go back
- There are hops – where a frog moves onto an empty lily pad
- There are jumps – where a frog jumps over another frog onto an empty lily pad on the other side of the jumped frog
- That, in the middle stages of the problem, you should avoid have two frogs of the same colour together
Some possible solutions
For some students it may be enough to just succeed in getting the frogs past each other for 2, 3 and four frogs of each colour. However the really interesting maths starts to happen when you look at how many moves it takes to get the frogs to switch sides.
After a while your students will see that 3 frogs a side need 15 moves and 4 frogs a side need 24 moves. After experimenting with 5, 6 and even 2 frogs a side the obvious conjecture is that n frogs a side require n2 + 2n moves. Proving this is not easy. A counting argument should convince them that that many moves is needed but it is much harder to prove that it can actually be done in that many moves.
A justification of the number of moves required for any number of frogs of each colour is available on the Frogs proof page. You might take your students through the proof for a couple of specific values of n before you show them the general proof.
Extension
The question now is can we take the problem any further. What frog-like problems can we build around this one? It’s very important to have this discussion with students at all levels. It’s a fundamental way of working for mathematicians and student should know that. In the process they should know that mathematicians can’t always solve the problems that they create. Let me illustrate this with two things that I often do with classes.
- Different numbers of green and brown frogs: Here the whole process of the first three phases can be repeated with, say, 2 green frogs and 4 brown frogs. Can they be interchanged? (Try it.) How many moves are required? (Collect the data).
- Different jumping ability: Suppose that brown frogs can jump over two green frogs (they have more powerful legs) but not over one. Can the interchange take place? How many moves will it take?
- Two empty lily pads: Instead of one empty lily pad in the middle at the start put 2. What happens then?
- Mix up the frogs before you start. Can you get all the brown frogs ot one end and all the green frogs to the other?
