Thursday, August 27, 2015

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

We think back to what I posted for the second day of school last year. I stated that we could have an alternate activity for the first week of school. Let's look back at what I posted that day:

Lesson 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, Lesson 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 Lesson 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:

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.)

Now here's the difference between last year and this year -- in 2014, I didn't post a worksheet for this opening activity. Instead, I suggested that we can just use Fawn Nguyen's worksheet. But this year, I give a worksheet that we can use.

Then again, my worksheet is very similar to Nguyen's. On the front side, I gave the same example as she did and the three buildings for the students also come from the Ventura County teacher. I used two of her easier buildings -- A and B -- and the more challenging Building F.

The back side of my worksheet differs slightly from Nguyen's, though. Her worksheet specified the number of rooms and windows and asked the students to draw the buildings. Mine, on the other hand, simply has the students draw four different buildings with eight rooms and then asks them to count the number of windows in each one. When designing my worksheet, I took in consideration that my students might not necessarily have access to the manipulative interlocking cubes that are apparently available in Nguyen's classroom. Hopefully, this will be easier for the students.

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

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