The people who laid out the city of Philadelphia between the Delaware and the Schuylkill Rivers had visions of creating a great, orderly city. Chief in creating this order was a large grid of streets crossing each other at right angles at regular intervals. Before the Parkway was paved, cutting diagonally through these streets, and before hundreds of alleys and side streets disrupted the grid's evenness, this grid, it was dreamed, would provide regularity and structure to people's daily activities. It was the hope, manifested, for a prosperous city. In this setting, a city block becomes the natural measure by which to gauge distances. A pedestrian walking from Rittenhouse Square at 18th and Walnut to Franklin Square at 7th and Race travels eleven blocks east and four blocks north for a total distance of fifteen blocks. If, however, we could travel as the crow flies, Pythagoras tells us that the distance is shorter, only $\sqrt{11^2+4^2}=\sqrt{137}$, which is less than twelve city blocks. To bring this geometry to high school freshmen, I mixed a little of the old with a little of the new. I took an old map of Philadelphia's grid system and overlaid it with the locations of presentday landmarks, as if the city planners could imagine what would one day be built in their city. I asked them to imagine a helicopter flight over 17thcentury Philadelphia and a rendezvous that could only happen years later. Why shouldn't a math lesson have a little history mixed in? Enjoy!
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Today a student asked me, "Can you write the number zero in scientific notation?" This got me thinking. As a refresher, when we write a number $x$ in scientific notation, we write $x$ as $$x = a\cdot 10^n$$ where $1\le a < 10$ is a real number and $n$ is an integer. We can do this uniquely for any number we like. Except zero. The student had recognized that if we want to write $0$ in scientific notation as $0=a\cdot 10^n$, then $a$ has to equal $0$, violating the inequality above, and also $n$ could be any number (shifting the decimal place on $0.00$ doesn't change the number). She saw that we're stuck with expending the concept of scientific notation in a messy way or with leaving $0$ as the oddball out. How do you think we should resolve this? In the opening days of the school year, my geometry class was discussing the basic building blocks of geometry: points, lines, and planes. We got to talking about points that are coplanar and points that aren't, and our discussion went like this. Me: I have drawn four points on the board. Are the coplanar? Are they all on the same plane? Students: Of course! Me: Can someone draw another point that is not coplanar with these points? Student 1: I'll do it! (Walks to the white board) Student 1: (to me) Is it right? Me: (to the class) What do we think? Is that red point in the same plane as the others? Students: Yes, they're still coplanar. They're on the same white board. Me: Does someone else have an idea about how to make noncoplanar points? Student 2: Put a dot on the other white board. Me: Great idea. What do we think? Is that black dot on the other white board coplanar with the other five points? Me: Even though they're on two separate boards, a plane keeps going forever, so these points are still coplanar. Can anyone think of where to put a point somewhere in this room that isn't coplanar with the others? Student 3: Put a point on the other wall. Me: I love it! This black dot way over here can't be on the same plane as the other points. If the wall the original points are on kept going, it wouldn't hit this black dot! I really enjoyed how students built on each other's ideas and that they were willing to put a guess out there and then let their classmates discuss it. I hope the discussion was as much fun for them as it was for me.

About Me
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome. Archives
June 2017
