Tuesday, January 20, 2015

Section 1-3: Ordered Pairs as Points (Day 95)

No, that's not a typo. This week we begin coordinate geometry, and in order to do that, we need to review coordinates. This is given in Section 1-3 of the U of Chicago text. When we started the school year, we began with Section 1-4, so we haven't actually done Section 1-3 yet.

But first, there's a few more things I wanted to say about how I would change Common Core, testing, and tracking if I were in charge. First of all, my computer adaptive replacement to Smarter Balanced has the computer keep track of the students' answers not only as they are taking the test, but also from year to year. This would encourage both acceleration and remediation.

But one major criticism of Common Core testing is that it requires the computer to keep track of student information, which may be seen as a violation of privacy. Even traditionalists who favor tracking might be opposed to it if government computers are the ones doing the tracking. If asked, they'd probably say that they'd prefer the teacher whose been with the student the whole year to be the one to determine on which track to place the student.

My problem is that this is where the bias seeps in -- the bias that doomed the old form of tracking. It is only human nature to have even subconscious biases, and such biases may result in students placed on the tracks based on demographics rather than what's best for the students. And so, to satisfy the MLK Jr. dream ideal, I'd much prefer that the computer be the one to place students on tracks, with as few humans with their subconscious biases involved as possible.

Now, here's the other thing I want to address here. I've set the Dickens age, the age at which students can leave school, to be 16. But now, some people are wondering, how can I claim that my plan satisfies the MLK ideal, when MLK himself graduated school at 15?

Actually, age 15 was my first choice for the Dickens age. I changed it to 16 because this fits the Singapore standards better. But now, on second thought, I'm going to set the Dickens age back to my first choice of 15. A bright student like MLK may complete all the high school requirements by this age, and the normative student will have completed ninth grade by then.

For now, I'll still set the graduation standard to 800 -- that is, the completion of the seventh grade (Singapore Secondary One) standards. This means that a standardized test is given every year until the score 800 is reached, and then again at the Dickens age to show what has been achieved when the student is ready to leave school. The normative student will have ended up taking a test each year from grades 3-7 and then 9.

So now I'd have to reorganize my ninth grade standards. Once again, it's too bad that I don't have access to Singapore New Elementary 3. The standards link that I gave in my last post shows that the Secondary Three/Four standards are combined. All I know is that all this material is taught in three semesters -- corresponding to all of 9th grade and the first half of 10th grade -- and the second semester of 10th grade is for review.

I could change it so that the second half of 9th grade -- the last semester before the Dickens age -- is now the review half. So now all I have to do is choose which of the Singapore standards to fit into the first semester of 9th grade. Since there are three major topics -- Numbers and Algebra, Geometry and Measurement, and Statistics and Probability -- it would make sense just to include the first topic, Numbers and Algebra. This topic includes the sub-topic "Applications of mathematics in practical situations," including utilities bills, hire-purchase, and simple/compound interest. This is a great topic to include the final year before the Dickens age. But it also includes "Matrices" -- and this seems a bit advanced for this level.

Or perhaps one can keep up the Integrated Math spirit of my standards, and perhaps include a little of all three topics. "Applications" is great from the Numbers strand, "Congruence and similarity" is good from the Geometry strand (notice that the eighth grade course introduces congruence, but SSS, SAS, and ASA don't appear until the ninth grade course), and I'd include everything from the Statistics and Probability strand.

Notice that if we wanted to divide the grades into paths, now the division K-1, 2-3, 4-5, 6-7, 8-9 makes a little more sense. From 4-5 to 6-7 is the jump from elementary to middle school, 6-7 to 8-9 is the jump from Pre-algebra to Algebra, and 8-9 to 10-12 is the jump across the Dickens age.

OK, that's enough about MLK and my tracking plan. Let's get into the new chapter. We are now beginning coordinate geometry. Unlike similarity, where my unit is based completely on Dr. Wu, coordinate geometry will be based completely on Dr. Franklin Mason's geometry course.

Dr. M named his final Chapter 13 "Analytic Geometry." His lessons are organized as follows:

Section 13.1 -- The Coordinate Plane
Section 13.2 -- Distance
Section 13.3 -- The Midpoint Formula
Section 13.4 Pt. 1 -- Slope
Section 13.4 Pt. 2 -- Parallels and Perpendiculars
Section 13.5 -- Equations of Lines

Let's convert these into U of Chicago sections, and this will give my plan for the next two weeks:

Today, January 20th -- Ordered Pairs as Points (Section 1-3)
Tomorrow, January 21st -- The Distance Formula (Section 11-2)
Thursday, January 22nd -- The Midpoint Formula (Section 11-4)
Friday, January 23rd -- Activity
Monday, January 26th -- Slope (Section 3-4)
Tuesday, January 27th -- Parallel Lines and Perpendicular Lines (Sections 3-4 cont., 3-5)
Wednesday, January 28th -- Equations of Lines (no exact correspondence in geometry)
Thursday, January 29th -- Review for Chapter 11 Test
Friday, January 30th -- Chapter 11 Test

Notice that officially, I stated that I would be covering "Chapter 11" after Chapter 12. But as we see above, we jump around among Chapters 1, 3, and 11 of the U of Chicago text. Indeed, only two sections -- the ones on the Distance and Midpoint Formulas -- actually come from Chapter 11.

Today's lesson is straightforward. I simply remind students about the coordinate plane that they surely learned in a previous course -- Algebra I, if not earlier.

The main concern is about tomorrow's lesson. Tomorrow I am scheduled to teach the Distance Formula -- but this is supposed to be proved using the Pythagorean Theorem, which I haven't taught on this blog yet. If I teach the Distance Formula now, I'm no better than the Prentice-Hall and Glencoe texts that try to teach the Distance Formula in Chapter 1.

The problem is that with all of my jumping around back and forth, the Pythagorean Theorem has been orphaned without a chapter. The U of Chicago gives the Pythagorean Theorem in Chapter 8, on Measurement Formulas. It appears after area, since the text uses area in its proof of the theorem. But not only are we delaying Chapter 8 until March, but an area proof isn't even how the Common Core wants us to prove the theorem anyway:

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

(We've already proved the first two in Section 12-10, as the Side-Splitting Theorem.) So we see that the Pythagorean Theorem is to be proved using triangle similarity, not area.

Dr. M proves the Pythagorean Theorem using similarity in his Chapter 8. Now then, his Chapter 8 corresponds roughly to the U of Chicago's Chapter 14. In Section 14-2 of the U of Chicago, Question 11 asks the students to give a proof of the Pythagorean Theorem using Similarity. As it turns out, this proof, buried in an exercise in the U of Chicago, is the primary proof that Dr. M and Common Core want the students to learn.

I could give that proof right now, since we have already covered similarity. The problem is that the proof uses geometric means, and that's covered earlier in Section 14-2. Dr. M devotes a whole section, 8.1, to the geometric means and then 8.2 is the Pythagorean Theorem. I could simply let tomorrow's lesson be Section 14-2 and teach both geometric means and Pythagoras. But then the primary topic of tomorrow -- the Distance Formula -- would be buried.

The only other choice would be to reshuffle the chapters once again -- just teach U of Chicago Chapter 14 right now and let Chapter 11 be last, just as Dr. M's Chapter 13 is last. But I've already said that I would do Chapter 11 now, and I feel that I've already delayed the Distance and Midpoint Formulas long enough.

So the plan for tomorrow is to start with a proof of the Pythagorean Theorem. Since we haven't given geometric means yet, it will just set up the necessary proportions directly without mentioning geometric means at all. Then it ends with the Distance Formula -- the emphasis on the fact that the Distance Formula is simply the Pythagorean Theorem applied to the coordinate plane. My hope is that students will understand both Pythagoras and the Distance Formula better if they are taught in the very same lesson.

As for today's lesson, it's always tough when I need to create a worksheet on graph paper. I chose to get the grid from another site:


I hope that this will show up on the page when printing.

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