Bring life to this classic sequences problem by getting students out of their chairs and jumping around to solve the problem. This is a terrific problem for generating and investigating a quadratic sequence. It can be looked at from a number of angles and demonstrating the way they link together gives a very satisfying result. This problem has been around for a while and this activity is really about getting the most out of it. This can be done by getting students to generate the sequence using themselves as the frogs and then by looking at all the different angles. This is a good example of an activity that is better done in a practical context than on computers. The physical reality of the problem helps students to engage with it and definitely adds a lot of fun, helping to make it a memorable experience!
Another benefit of avoiding computers for this activity is that it means the class has to come up with its own ideas for analyzing the results. For example, there are a number of manipulatives that count the ‘Jumps’ and ‘Slides’ for you automatically and it is actually really nice to try and get students to come up with this idea for themselves by thinking about how to break the problem down. I am often surprised during this type of activity by students making observations that are new to me and I think this is more likely to happen the more freedom students are given to play with the problem.
This problem really takes off well if you start with students on chairs, sliding and jumping over each other. This is the hook. It’s a lot of fun and has so many other benefits.
Once students have got the idea and have succeeded I think it is a good idea to have a good discussion about what has happened. These discussions can yield some great results with students making observations and conjectures that help the class to think about how to go on from there. The longer this conversation goes on the more students will want to try things out and write stuff down and it is great if they reach that conclusion for themselves.
At this point they can start working more individually using the worksheet (paper or digital) or not if they want to structure things for themselves. At this stage they need to check their results and conjectures and try to break the problem down.
It is easy to get carried away with playing this game and it is important to manage the transition from trial to prediction. With this in mind, a lot of progress can be made in an hour and there after it depends on the records you want to keep and whether or not you want to use the opportunity for students to practise writing about mathematics.
As suggested above, there is an opportunity for students to practise writing about mathematics here and produce a report on this problem, but this is by no means essential. A good record might be some photographs and a video (like the one on the activity page) of students trying to solve the problem. This helps to make it a memorable experience.
What to expect
To start with, expect a bit of fun.
Students usually struggle in the beginning to exchange 3 frogs and at this point the teacher needs to be prepared to chip in with hints if it looks like students are struggling too much. From my experience they do get there by themselves.
Once trying to predict the problem, students come up with a variety of different approaches to predicting the problem and this is where much of the richness in this problem comes. The following are some of the more common approaches to generalising
Term to term
- Students notice that the differences between the terms are consecutive odd numbers and become able to predict from term to term.
Position to term
- Students notice that the number of moves are always one less than consecutive square numbers, leading to (n+1)²-1.
- Students look at constructing the number of moves by noticing that each term is constructed with the following pattern, 1 x 3, 2 x 4, 3 x 5 and so on, leading to n(n+2)
- Breaking the problem into slides and jumps, students can notice that one is modeled by n² and the other by 2n.
At this point there is a golden opportunity to ask students to explain why all of those things must be the same by looking at their algebraic equivalence!