Exploratory Problems of the Week
I was catching up on some blogs I enjoy when I came across f(t)’s recent post on the rectangular diagonal problem. I’d seen this problem before through the work of Dr Frederick Stevenson, a professor at the University or Arizona, and his work with Exploratory Problems in Mathematics (which is one of the best books I own). This, in turn, reminded me of a structure I’m starting in my classroom and might be able to turn into a consistent blogging effort: Explorator Problem’s of the Week.
Some background: I’m working at a school with an interesting bell schedule – every period meets every day, but for some days certain classes are longer than others. On Monday, my first and second period is 80 minutes each and the rest are 50ish. On Tuesday, third and fourth period are 80 minutes; on Thursday, fifth and sixth are 80 minutes. Wednesday and Friday, all classes are the same length. It’s almost a block schedule, but since every class still meets every day, I can’t plan equitable instruction for every period (all of which are geometry, by the way). So, the question is: what do I do with that extra half-hour that some classes get and some don’t?
The answer: The whole idea behind this strange schedule is that the extra time be used as mandatory intervention for struggling students (as modeled by current Professional Learning Community literature). However, I also intend to use it for enrichment and group projects – such as Exploratory Problems of the Week.
An exploratory problem, as defined by Dr Stevenson and to which I agree, is a problem which has two phases: an inductive phase and a deductive phase. The inductive phase involves gathering data and looking for patterns (which a good exploratory problem will have plenty of). The actual problem is usually part of a whole family of problems, so the inductive phase usually includes examining different cases of the entire family. For the diagonal intruder problem, this means seeing what happens with different sized rectangles and how that affects how many squares the line passes through.
Once data has been gathered, the student should be able to make a conjecture about the data – see a pattern and wonder if it holds true. This is the most exciting part of the problem – noticing a pattern, wondering if it continues, then wondering why this is happening. From there, we enter the deductive phase of the problem – can we explain the pattern? What’s going on mathematically that justifies our conclusions? Can we prove it for all cases? Because if I can prove it for a general case, then I can prove it for my original problem. Some of these answers for the diagonal intruder problem come from properties of slope and the gcd of the dimension of the rectangle, which is now something that the student is capable of applying.
This isn’t anything completely new or unheard of, by the way. It’s just usually found in Math Circles and enrichment activities for clubs and summer programs. In my opinion, every student should have the opportunity to experience these types of problems – to increase their confidence and independence on solving problems, as well as giving them a chance to be successful at one of the most natural aspects of human curiosity: finding patterns
My First Problem of the Week: Suppose I have a Square Lattice Grid. By connecting the dots, how many squares can I draw in the grid?
Handout for the classroom: http://bit.ly/plz2kV