I do off-topic problem solving, as I can fit, with my students. I claim that not knowing what technique or skill to use, and having to figure out what to try, has value, helps develop students at mathematical thinkers.

Now I have students take the problems I gave them, and generate their own problems. Instead of confronting a problem with a pretty solution, they may introduce interesting complexity. They may understand the original problem better. And they really don’t know where they are going. I claim that tackling this sort of problem that you make up, instead of one the teacher gives, pushes mathematical creativity.

I make certain that they can solve the problems they propose, or come close. I give them some time in class to work on the problems (they are grouped in 3s and 4s).  And then they write up their work, using this structure:

• Original Problem – Understand the Problem
• Devise a Plan
• Carry out the Plan
• Look Back – include the proposed extension
• Devise a Plan for the extension
• Carry out the Plan
• Look Back – include, at a minimum, recommendations for future work, or if the problem is still unsolved, tips for the next people to attempt the problem

There can be more than one plan, as the first often does not work. And this needs to be reflected in the write up. And, importantly, they do not need to complete their problems to do well.

One of the two best problems we solved this year was Ghost the Bunny.

Laura’s pet bunny, Ghost, hops up a flight of 12 stairs. Ghost hops up one step or two steps at a time, and never hops down. How many ways can Ghost reach the top step?

And in each class when we discussed extensions, someone mentioned altering the size of the hops, as an example, so it’s no surprise that several proposals did just that. Here they are:

Ghost the Bunny wants to go up 12 steps. He can jump 1 or 2 steps at a time and has the option to jump up 4 steps only one time on his journey. How many ways of going up are there?

How many ways can Phantom the Ferret get to the 15th step taking 1, 2, or 4 steps? He cannot hop backwards, and every different order of number of steps counts as a different way. Ghost must land on the 15th step, and cannot hop any further.

Laura’s pet bunny, Ghost, hops up a flight of 12 steps. Ghost hops up one or three steps at a time, and never hops down. In how many ways can Ghost reach the top step?

Laura’s pet bunny, Ghost, hops up a flight of twelve steps. Ghost hops up one, two, or three steps at a time, and never hops down. In how many ways can Ghost reach the top step?

Laura’s pet bunny, Nemo, hops up a flight of ten steps. Nemo can hop up any amount of steps at a time, but can’t hop down steps. In how many ways can Nemo reach the top step?

Ghost the bunny needs to hop up a flight of 12 stairs. Ghost can jump 1 and 2 steps at a time, 1, 2, and 3 steps at a time, 1, 2, 3, and 4 steps at a time, 1, 2, 3, 4, and 5 etc. How many ways can Ghost hop up for each set of numbers? What about for any set of numbers?

Thoughts on these problems?  On this activity?

I will share more of the extensions in the coming days. Some students, you will see, moved much further from the original problem…