Monday, May 15, 2017

Lesson 15-4: Locating the Center of a Circle (Day 154)

This is what Theoni Pappas writes on page 135 of her Magic of Mathematics:

"How well the entire garden was shaping up and exploding with new growth! Admiring the new green leaves..."

This page is right in the middle of a new section, "The Mathematically Annotated Garden." Pappas writes this section as a story told from the perspective of a gardener admiring all of the various plants in her backyard. On this page, she notices the symmetry of the maple leaf, the phyllotaxis nature uses to arrange the leaves on branches in stems, and the tessellation of carrots in the patch.

Symmetry and tessellations are both Common Core topics -- they refer to figures that are invariant with respect to reflections and translations respectively. Phyllotaxis is a new topic, though. It refers to the observation that the number of leaves on many plants is a member of the Fibonacci sequence -- 1, 1, 2, 3, 5, 8, and so on.

Before I get to today's lesson, I want to mention the news that First Son Barron Trump's school for the fall has been announced. Of course, I try to avoid politics during school year posts, but I've written so much about Presidential Consistency that I want to write about this now. Recall that Presidential Consistency stems from the frequent accusation that Common Core is knowingly terrible, yet politicians promote the Core because they know that they'll send their children to private schools that don't use the Core.

Therefore, Presidential Consistency is the notion that whenever a new president with school aged children is elected, the curriculum at the children's new school is automatically defined to be the new Common Core.

Barron's new school is St. Andrew's Episcopal. Here's a link to its curriculum:

Many people have noted that unlike the children of previous presidents, Barron waited until a brand new school year before moving to the DC area. Since he's eleven years old, this means that he'll be a sixth grader -- so waiting for summer makes it a clean transition from elementary to middle school.

As we look over the curriculum at St. Andrew's, there's much for traditionalists to like:

-- Unlike Sidwell Friends, there is no integrated pathway. All students must take the traditional classes Algebra I, Geometry, and Algebra II to graduate from St. Andrew's.
-- The highest class offered is Multivariable Calculus. Therefore students can take Algebra I as early as seventh grade in order to make it to Multivariable Calc.

I notice that there's a class called Functions and Trig -- I assume that this corresponds to Algebra III at Sidwell and other schools. There's also a non-AP Calculus class in addition to both AB and BC. I find it interesting that even though a B+ in Algebra II is required for Honors Pre-Calc, where a B is required to advance to AP Calc, any Calc class (including non-AP Calc) may be used to advance to Multivariable Calc!

Now let's look at what traditionalists aren't going to like -- the middle school curriculum:

-- The sixth grade class is Connected Math. As described on the St. Andrew's website, "CMP is a problem-centered curriculum where concepts are sequenced and developed to allow students to explore them in depth."

The traditionalist Barry Garelick doesn't have much to say about Connected Math, but Katharine Beals was outspoken in her opposition to this curriculum. Here's a link to her old website, where at this time four years ago, she featured Connected Math in her "Math problems of the week" series:

In many ways, CMP 6 and CMP 7 correspond to Common Core 6 and 7, respectively, but the eighth grade course is called Pre-Algebra. So there's more algebraic content ("operations with polynomials") and a little less geometric content than in Common Core 8. Unfortunately for traditionalists, the word "project" appears three times in the description of the Pre-Algebra course.

All students take Pre-Algebra before Algebra I. Therefore students on the Multivariable Calc track take Pre-Algebra in sixth grade and those on the AP Calc track take Pre-Algebra in seventh -- and can take the CMP 7 class in sixth grade.

We can't be sure which class Barron will take next year. We note that his father is an outspoken opponent of Common Core, but even if Barron takes Pre-Algebra in sixth grade he'll still have to complete those three darned "projects" that Core opponents dislike.

Suppose we wanted to implement Presidential Consistency and adjust the Common Core so that it matches the St. Andrew's curriculum. The best way to do so would be to abolish the Common Core 8 standards and replace it with St. Andrew's Pre-Algebra. The resulting PARCC and SBAC tests for Pre-Algebra can be administered to students in any middle school grade. Students who score sufficiently high on the Pre-Algebra PARCC can take the Algebra I End of Course PARCC the following year rather than any middle school test. Those in SBAC states who score high enough in Pre-Algebra can be exempt from taking any more SBAC math tests until high school.

Notice that transformational geometry isn't mentioned in normal Geometry -- but it is mentioned under Honors Geometry. The final unit of Honors Geometry is on spherical geometry. Perhaps in the name of Presidential Consistency, transformations should be dropped from all Geometry courses unless they are Honors courses.

St. Andrews has a Labor Day Start Calendar, so Barron will start there in September. The school uses trimesters for all grades -- including high school, where trimesters are a rarity. Usually, trimester high schools are on a 5 x 3 block schedule. but this doesn't seem to be the case at St. Andrews. According to the website, high school students do indeed take five classes -- which means five academic classes, each of which is a full year. The bell schedule has eight periods, but one of these is lunch. The remaining seven periods are for the five academic classes, and then two periods for non-academic courses like art, P.E., and religion, which are the only trimester-length classes.

By the way, note that my own middle school doesn't offer any sports. St. Andrew's is the exact opposite -- the only P.E. offered to middle school students is competitive sports! So Barron will have to choose sports for at least the first and third trimesters -- the second trimester, which is in the winter, is optional. Some people speculate that Barron will choose golf and soccer -- but these are both fall sports, so he'll need to choose another sport for the third trimester.

Lesson 15-4 of the U of Chicago text is "Locating the Center of a Circle." According to the text, if we are given a circle, there are two ways to locate its center. The first is the perpendicular bisector method, which first appears in Lesson 3-6. (Recall that the perpendicular bisectors of a triangle are a concurrency required by Common Core.) This section gives the right angle method:

1. Draw a right angle at P (a point on the circle). AB (where angle sides touch circle) is a diameter.
2. Draw a right angle at Q (another point on the circle). CD is a diameter.
3. The diameters AB and CD intersect at the center of the circle.

This method is based on the fact that a chord subtending a right angle is a diameter -- a fact learned in the previous lesson. Indeed, "an angle inscribed in a semicircle is a right angle" is a corollary of the Inscribed Angle Theorem.

Notice that unlike the perpendicular bisector method, this is not a classical construction. That's because the easiest way to construct a right angle is to construct -- a perpendicular bisector, which means that if we have a straightedge and compass, we might as well use the first method. The text writes that drafters might use a T-square or ell to produce the right angles, while students can just use the corner of a sheet of paper.

There are a few things I want to say about today's worksheet. One of the questions from the text mentions a fictional school, Emmy Noether High School. This isn't just a random name -- it's in fact the name of a famous mathematician. I wrote about her on the blog two years ago:

"At Göttingen, after 1919, Noether moved away from invariant theory to work on ideal theory, producing an abstract theory which helped develop ring theory into a major mathematical topic. Idealtheorie in Ringbereichen (1921) was of fundamental importance in the development of modern algebra."

The other thing I want to mention is the bonus question. This is an Exploration question in the text:

"Each of the three circles below intersects the other two. The three chords common to each pair of circles are drawn. They seem to have a point in common. Experiment to decide whether this is always true."

As it turns out, these three chords are indeed concurrent, except for a few degenerate cases such as if the circles have the same center or if the centers are collinear. (The concurrency of perpendicular bisectors has the same exceptions.) The students are asked to experiment rather than attempt to prove the theorem that these three lines (called radical lines) intersect at a common point (radical center, or power center). The name "power center" refers to "power of a point" -- a dead giveaway that we must wait until Lesson 15-7 before we can attempt to prove the theorem.

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