Sometimes you hear the perfect word to describe something you were already doing, but didn’t quite realize you were doing. My most favorite recent case of this came from openmiddle.com.
We’ve all heard of open-ended problems, the genuine, intriguing problems that can take us on journeys to unexpected answers. A lesson based around an open-ended problem can be a beautiful thing, and done right, there’s hardly a more exciting educational experience you can have.
The trouble is, it doesn’t always go so well. Guiding a class through a discussion and exploration of an open-ended problem requires tremendous insight and understanding on the part of the teacher, and even the most experienced teachers sometimes have them go awry. Will this student’s comments lead in a productive direction, or just muddle the issue, and leave us all frustrated? Is this problem so hard that we can’t hope for more than a partial understanding of it, or is there some key idea that we’re all missing? When you have to make assessments on the fly, things don’t always go well.
One trick to coax these explorations in productive directions is to game them just a little bit. The teacher can go in with a plan, or an arc of where they expect the lesson to go. So they can “discover” something like the fact that the midpoints of quadrilaterals seem to form parallelograms and ask, “that doesn’t always happen, does it?”, all the while knowing that it is, in fact, true, and with a couple of good ideas about how to prove it. This little bit of lesson planning can greatly improve the chances of an individual lesson feeling like a success, even if it gives up a little bit of the total free of open ended lessons.
Enter the Open Middle. The idea of Open Middle problems is that the beginning (the problem) and the end (the answer) are both clearly well-defined, but the middle of the problem—the part where you look for the solution—is wide open. The middle is where you cast around for ideas, structures, tools, or just blind guesses about what’s going on. Because Open Middle problems have such a well-defined structure, there’s much less of a chance of them going off the rails. This makes them a fantastic tool for teachers who might be taking their first steps into less regimented math teaching, or who don’t feel like taking a chance on an exploration yielding a broad discussion that doesn’t go anywhere. (Read more about Open Middle problems here.)
I produced a number of these Pyramid Puzzles recently, and I think they’re great examples of Open Middle problems. it’s relatively easy to adjust their difficulty as well.
Pyramid Puzzle 1.
Each number must be the sum of the two directly below it in the pyramid. Fill in the blanks.
This is a puzzle that seems easy, but the solution is just out of reach. We can put a 7 above the 2 and 5, but what then? There’s no obvious next step, and our best guess is to take a stab at it, and see what happens. Some people like to work from top to bottom, and try taking a guess of what two numbers might go below the 30 (say, 12 and 18?), while some prefer to work up from the bottom, putting a number in the blank and going from there.
Suppose I put a 10 in the bottom blank. Then I’d have this:
A 10 on the bottom leads to a 51 at the top, which is a problem. But it’s also a useful mistake, since now I know the number in the bottom must be smaller than 10. One wild guess gives me traction, and it’s only a matter of time before I solve this problem.
Pyramid Puzzle 2.
Each number must be the sum of the two directly below it in the pyramid AND no number can appear more than once. Fill in the blanks with positive integers so that the top of the Pyramid is 20.
Bonus: Could the 20 at the top of the pyramid be replaced with a smaller number and the pyramid still be solved? Show how if it’s possible, or show why it’s impossible.
This is a much subtler puzzle, and solving the bonus problem in particular requires you to delve into the workings of these puzzles. Is there some kind of theory that can tell us what the maximum number at the top of the pyramid can be, given the rules that we’re filling them only with positive integers, and never the same one twice? What about differently-sized pyramids?
This is clearly the smallest number that can be atop a pyramid 2 stories high.
And this seems to be the smallest number that can be on top of a pyramid three stories high.
So what’s the smallest for four stories high, or more?
We’re in the territory of open-ended problems now, which is where I always seem to end up. But that’s the exciting thing about mathematical exploration: there are always questions pointing in directions you’ve never gone before.
Check out openmiddle.com for more examples of Open Middle problems.