'Get moving around to solve this problem and deduce the algebraic sequence behind it!'
Generalising about algebraic sequences is at the heart of good mathematical investigation and this problem gives you a great opportunity to try it out. The male and female frogs have got to exchange places with each other following some important rules about how they can move! It is not as easy as it sounds! First you have to do it, then you have to try and do it in the smallest number of moves and then you have to look at the underlying patterns to try and predict the number of moves required for any number of frogs to change places! So, assume the positions and try it out!
The video below gives you a quick idea of what is coming! Watch students doing this problem with up to 8 frogs on each side at 4 x speed!
You need a bit of open space to play the problem out in. If possible you need enough space to put a row of 11 chairs. Having played with the problem will need the Frogs worksheet to help you record your thoughts. You may also make use of this Frogs virtual manipulative to help you continue your investigations. Teachers can read more about this activity in the Frogs Teacher notes.
- You have 5 chairs
- The one in the middle is empty
- On one side you have two boys, the other two girls
- The aim is for the boys and girls to swap places in the least number of moves
- Boys can only move in one direction, girls only in the other
- You can slide 1 space in to an empty seat
- You can jump over a person of the opposite sex (only one at a time)
- Now vary the number of people on each side and see if you can predict the number of moves it will take!
The frogs manipulative, linked to above could help to check some of your results! The screencast below just demonstrates how it works.
Part 1 – Equal numbers
AIM – Find a way to predict the number of moves it will take to exchange any equal number of boys and girls from one side to the other. The table on the worksheet may help you to do so, but is not essential if you prefer another approach.
Write any general conclusions you have here!
Part 2 - Jumps and Slides
There are two types of moves that are allowed – ‘Jumps’ and ‘Slides’. Go back over some of the initial scenarios and try to count the jumps and slides separately. Can you find a way of telling how many jumps and how many slides are required to exchange any equal number of boys and girls from one side to the other? What should happen if you add these together? Does it?
Write any general conclusions you have!
Part 3 – Unequal numbers
Now imagine that you don’t have to have the same number of people on each side , just one space in the middle. So you could say that you have ‘b’ boys and ‘g’ girls. Investigate how many moves will be required to swap the boys and girls with the same rule! HINT a really complete answer will be an algebraic expression in terms of ‘b’ and ‘g’.