New Year, Problems: old, new, tricky
The problems are actually
- Real world, we’ve seen before
- Real world, math
- tricky math question from a kid
The first, how do we keep ourselves safe when the people who run the schools don’t? I don’t have the answer. I dutifully checked for masks and test kits, like a good UFT chapter leader. The principal was already handing out test kits. Clarified the mask situation (each teacher gets one KN95 each week).
But that’s not enough. There are too many people in school, and too much virus hanging around. We are going to get each other sick. The Department of Education is going to get people sick. It almost certainly already has. And the associated stress!
A new problem: I have often criticized the Regents for creating problems with lousy artificial context. Easy to criticize. Have I ever created a real world problem without that issue? Today I did:
If the positivity rate is among those who get tested… and those tested are in lines to get tested… and the positivity rate in a neighborhood is 44.4% What’s the chance the person in front of you AND behind you in the PCR line will be positive?
https://twitter.com/Jd2718x/status/1477975444511207425?s=20
Before you jump in to say what a great problem, but those numbers are not real world, let me point out that my testing center is at 42%, my school is at 45%, and my apartment is at 43% (adjacent zip codes) – so 44% is a fair stab at an average. On the other, I should point out that the “Math Teacher Blog O Sphere” #MTBOS already has a polite attack on the appropriateness of the question. Also, in case you didn’t notice, I used the same problem yesterday, but with lower numbers. So goes omicron.
Tricky problem. We were talking about (I was leading a discussion) of some basics in (x,y,z) coordinates. Nice ideas about adding dimensions. And then I set up a standard imaginary coordinate system in the room, with the head of the kid sitting under the projector the origin (0,0,0) and units 1 foot. We found a few points, and then I wrote y = 0 on the board, and with some prodding the students (70% of whom were in class) visualized and described the graph, with words and gestures. They got easier: z=0, z=-3, then harder, x+y=0. And we developed a 3D distance calculation. But before all that, the origin asked if a line could be perpendicular to a 3D object, and I was stumped. I don’t THINK so – but is anyone more certain?
JD2718 
I’m not certain I understand the student’s question, but here are two paths to answer:
1) in math, we get to try out definitions and see if they are useful. Particularly in this case, does it extend a definition from a simpler case that we already have explored? What might we mean be saying a line is perpendicular to a 3d object? How does that relate to similar notions in 2d? Does it look like a useful extension?
2) my instinctive notion would be: line L and object B, both in 3 space, are perpendicular if L is perpendicular to the plane tangent to the boundary of B at the point of intersection between L and the boundary of B.
Can the students find problems with this definition?
Problems I see:
What if the line has many points of intersection with the boundary of B (usually there will be at least two)?
What if there’s no tangent plane at the intersection (like a vertex of a polyhedron)?
I’m sure they can find more.
Maybe it makes sense to ask the question in lower dimensions, first? Can a line make an angle with a 2-D region in the plane? My personal inclination is “no, just with other one-dimensional objects in the plane” (like the boundary of the 2-D region), because at any point on the interior, the region can be found in all possible directions simultaneously — not very angle-like. Of course, if you go into higher dimensions, then more things can happen — two three-dimensional subspaces in 4-space can make an angle, etc.