Section 1-5: Drawing in Perspective (Day 2)

Section 1-5 of the U of Chicago text is the last section before the real geometry -- by which I mean the geometry taught in most other textbooks -- begins. But like the previous section, Section 1-5 contains some material that may be suitable as an Opening Activity.

Now recall that yesterday, I recommended -- following Fawn Nguyen -- that the lesson on Points in Networks should be the Opening Activity spanning the entire first week of school. So we don't really need a second Opening Activity. But still, I present Section 1-5 as an alternative to the activity in 1-4, so that one has a choice whether to open the course with 1-4 or 1-5. And as it turns out, Nguyen covered something similar to 1-5 in her class as well:

http://fawnnguyen.com/designing-buildings/

Now the U of Chicago text focuses on perspective drawings. Many introductory art classes teach the concept of one-point perspective -- and possibly two- and three-point perspective as well. But the text admits that most mathematicians don't draw, say, a cube in perspective.

Nguyen's lesson takes a different approach to drawing three-dimensional figures. For one, the focus on this lesson is on buildings. Her lesson begins by having some buildings already drawn and the students counting the "rooms" and "windows." (As it turns out, one "room" is one cubic unit of volume, and one "window" is one square unit of lateral area.)

I like the way that Nguyen's lesson begins. Unlike the bridge problem, where I wanted to avoid beginning the school year with a problem that's impossible to solve, here we begin with a very solvable problem. The only issue I have is with the second question, because it requires materials. I work from the assumption that most classrooms don't have the blocks and isometric dot paper that Nguyen's classroom has.

(As an aside, notice that cubes drawn on isometric dot paper are definitely not in perspective. This is because, while edges perpendicular on the cube intersect at 120 degrees on the iso dot paper, edges parallel on the cube remain parallel on the paper. Therefore there are no vanishing points.)

I started to design my own worksheet for this activity, but decided that it wouldn't be much of an improvement from Nguyen's. The one difference I'd make is that if blocks and iso dot paper aren't available, I could simplify the task by having the students imagine a fixed number of blocks (say eight) and consider all the different ways they could draw the blocks to make eight rooms, and compare the different numbers of windows that these eight-room buildings would have.

Or one can simply do yesterday's bridge problem instead.