Lesson 1-3: Ordered Pairs as Points (Day 13)

This is what Theoni Pappas writes on page 244 of her Magic of Mathematics:

"Mechanics is the paradise of mathematical science because here we come to the fruits of mathematics." -- Leonardo da Vinci

Yes, Pappas just wrote about Leonardo and the human body last week, and now she quotes him again in this, the introduction to her architecture chapter. Of course, the idea is the same -- science and technology are the fruits of mathematics, which is why students should study it.

According to Pappas, math has been a tool in construction for millennia. Architects eliminate trial and error in building by using math -- indeed, this is true in many fields. Those who don't learn math are forced to use trial and error techniques -- and in construction, this would be quite expensive.

Pappas writes:

"As comprehensive as it may seem, this is but a partial list of some of the mathematical concepts which have been used in architecture over the centuries."

The list actually extends to page 245, but I'll give some of the items on this list anyway:

-- pyramids
-- optical illusions
-- polyhedra
-- Pythagorean Theorem
-- spheres, hemispheres
-- angles
-- symmetry
-- parabolic curves

The chapter introduction actually extends several pages, so it will be well into next week before we actually reach the first proper section of this chapter. Part of this is because there are so many photos in the intro. On page 244 is the first item on the list, pyramids -- it's the Temple of Kululkan, Chichen Itza, Yucatan. And as for the second item, optical illusions, we wrote about these last week as we were reading Serra's text. Yes, op art appears in architecture as well!

Lesson 1-3 of the U of Chicago text is called "Ordered Pairs as Points." (It appears as Lesson 1-2 in the modern edition of the text.) The main focus of the lesson is graphing points on the plane. Indeed, we have another description of a point:

Third description of a point:
A point is an ordered pair of numbers.

The idea of graphing points on a coordinate plane is a familiar one. But sometimes I wonder whether we should make students graph points and lines so soon in their Geometry course.

Once again, here's how I think about it -- the students coming to us just finished Algebra I. Some of them struggled just to earn a grade of C- or D- (whatever the lowest allowable Algebra I grade is in your district is so that the students can advance to Geometry). The students who just barely passed Algebra I are tired of seeing algebra. They may look forward to Geometry where they won't have to see so much algebra -- and then one of the first things we show them is more algebra.

Then there's also the issue, first brought up by David Joyce, that students should use similarity to show why the graph of a linear equation is a line. This idea appears in the Common Core standards for eighth grade, but it's awkward in high school. Graphing linear equations is a first semester Algebra I topic while similarity is a second semester Geometry topic -- and it's difficult to justify delaying graphing linear equations by three semesters just to conform to Joyce's wishes.

In the past, I've tried -- and failed -- to teach linear graphs after similarity. (This includes last year, when I tried to follow the eighth grade standards, but I left the class before graphing equations.) This year, my plan is simple -- I will conform to the order of the U of Chicago text. The U of Chicago text introduces linear graphs in Lesson 1-3, and so that's when I'm teaching it.

The bonus question asks about longitude and latitude. I've already located my own coordinates as being near 34N, 118W.

This blog isn't following the LAUSD calendar, so today isn't the Admission Day holiday. On the other hand, Monday is definitely Labor Day, so my next post will be on Tuesday.

Oh, and I can't let the day pass without mentioning that today, September 1st, 2017, is the Harry Potter equivalent of "Back to the Future Day," since the epilogue is set today. Fans gathered at King's Cross Station in London -- at Platform 9 3/4, of course -- and met Warwick Davis, the actor who played Professor Flitwick in the films. Recall that Flitwick is the example I gave of Lee Canter's "nonassertive" classroom manager. Sorry, Warwick Davis, but I need to be less like your character in order to become a successful, assertive manager.