Geometry, Common Core Style: Lesson 11-1: Proofs with Coordinates (Day 111)

Today I subbed in a high school history class. It is in my new district. Since it is a high school class and not math, there is no need for "A Day in the Life" today -- especially considering that it's another one of those classes where the regular teacher teaches from home, and so I'm only there to supervise.

It's an even period day. Second period is senior Government while fourth period is junior U.S. History, and sixth period is conference. The seniors are learning about our voting structure, and the juniors are studying World War II.

As is typical for upper high school, the classes are small as most students have opted out. I see only four seniors and two juniors in-person today.

Today is Sunday, the third day of the week on the Eleven Calendar:

Resolution #3: We remember math like riding a bicycle.

It's another one that doesn't work well in a history class -- except through the song I sing. I perform only for the juniors today, during the twenty-minute tutorial at the end of fourth period.

Today I choose the "Runnin'" song from Hidden Figures, for several reasons. First of all, the song does take place during a moment in our nation's history, so it does in fact fit a U.S History class. Notice that while Katherine Johnson was clearly alive during World War II (the period the juniors are now studying), she didn't join NACA/NASA until after the war. She married her first husband, James Goble, just prior to the conflict, and she was at home raising the couple's three daughters. I wouldn't be surprised if Goble fought in the war, as did nearly all males who were of prime fighting age at the time. (I say all of this to the two juniors who are in the room at the time.)

The second reason I sing "Runnin'" is all about the future, not the past. One line from the song is "They want the moon, I'm on Mars." NASA lands the Mars rover Perseverance today -- and this fuels optimism that there will soon be a manned mission to the Red Planet.

The final reason I choose "Runnin'" is because running -- as in Cross Country -- is on my mind. During sixth period conference, I make my way down to the track, where several XC runners are doing an interval workout. They inform me that their first race is coming up this weekend. The rules here in California is that counties in the purple tier (which is now nearly all counties except for a few rural ones) can only have fall sports that have enough social distancing to be placed in the purple tier. Only one sport meets the criteria -- Cross Country.

In fact, this week my alma mater held its first XC race of the season. Our home course is usually at a park, but it wasn't approved for XC this year, and so the race was held at the high school itself. In recent years, our league has had "cluster meets" for every school in the league, but just as I suspected, the dual meet, with only two teams competing, is suddenly back in vogue during the pandemic. (Recall the dual meet between McFarland and Clovis in the movie.)

There's only one problem this week -- the race was noncompetitive, because our opponents had fewer than seven runners. I know this because I drove right past the school and saw only our own school's jerseys -- I saw no one from the other school. (Even back in the days when I was running XC, that school had very few runners.) Thus as far as I'm concerned, what I saw was a time trial, not a race. I hope that we'll have a competitive race either next week or the week after.

Lesson 11-1 of the U of Chicago text is called "Proofs with Coordinates." In the modern Third Edition of the text, proofs with coordinates appear in Lesson 11-4.

Coordinate proofs are mentioned in the Common Core Standards:

CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

In Lesson 11-1, we are given the coordinates of the vertices of a polygon, and we are asked to prove that the polygon is a parallelogram, right triangle, or rectangle. The key to these coordinate proofs is to find and compare the slopes of the sides.

But here's another Common Core Standard:

CCSS.MATH.CONTENT.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

So students are supposed to use similarity to prove the properties of slope. David Joyce, whom we mentioned throughout Chapter 9, also endorses the use of similarity to prove slope -- and indeed, he has harsh words to say about the treatment of coordinate geometry in most Geometry texts:

In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10. At this point it is suggested that one can conclude that parallel lines have equal slope, and that the product the slopes of perpendicular lines is -1. The only justification given is by experiment. (A proof would require the theory of parallels.)

And in our text, similar triangles don't appear until Chapter 12. Thus to follow the Common Core and David Joyce, we should wait to teach Chapter 11 until after Chapter 12. Students need to have mastered similar triangles before they can begin learning about slope.

This has been the source of many headaches in my blog posts over the years. First of all, I'd start with Chapter 12, then go back to teach slope and some other Chapter 11 topics -- but I'd never actually reach Lesson 11-1.

The problem, of course, is that slope is an Algebra I topic. High school students are thus going to see slope well before they ever see similarity, because they take Algebra I before Geometry.

I also wrote extensively about Integrated Math courses. But even Integrated Math usually covers slope before similarity -- indeed, slope is a Math I topic, while similarity is a Math II topic. I once tried to devise my own Integrated Math courses that teach similarity before slope, but I failed. It's difficult to justify teaching similarity (from the second half of Geometry) before slope (which is from the first half of Algebra I).

In fact, we notice that the Common Core Standard requiring students to use similarity to prove the slope properties is an eighth grade standard, not a high school Geometry standard. This now makes sense -- students are introduced to slope in eighth grade in order to prepare to study it in more detail in Algebra I.

I think back to the eighth grade class from three years ago. Of course, student behavior and classroom management were issues. But another problem was that I began teaching translations, reflections, and rotations -- and rotations, understandably, confused some students. The extra time spent on isometries meant less time on dilations -- and dilations are the bridge to similarity and slope.

I now sometimes wonder whether it's better to teach only one of the isometries -- perhaps reflections, since they generate all isometries (i.e., all isometries are the composite of one or more reflections) -- and then skip directly into dilations. But this contradicts the Common Core Standards that explicitly mention translations and rotations -- and these might appear in PARCC or SBAC questions.

At any rate, if the connection between similarity and slope is covered in eighth grade, then it doesn't need to be introduced in high school Geometry. And so we can write about slope in Chapter 3 without having to prove anything about similarity first. As I mentioned before, Chapter 3 is a great time to teach slope, since it's a review topic from Algebra I, and Chapter 3 is often taught right around the time of the PSAT (where slope questions will appear).

When David Joyce wrote about slope and similarity, he forgot that there's a class called "Algebra I" where students learn many things about slope and coordinates without proving everything. In the end, I did say that this year I'd adhere to, not Joyce's suggestions, but the order of the U of Chicago text.

And by the order of the text, I mean the order of the old Second Edition. Earlier, I wrote that Lesson 11-1 appears as Lesson 11-4 of the new Third Edition. So what exactly appears in the first three lessons of the modern version?

Well, Lessons 11-1 through 11-3 of the new text correspond to Lessons 13-1 through 13-4 of the old version of the text. Indeed, the new Chapter 11 is called "Indirect and Coordinate Proofs." You might recall that Chapter 13 of the old text has been destroyed, and its lessons are now included as parts of different chapters. And so the first half of the old Chapter 13 now forms the first part of 11. (There are now only three lessons instead of four because the old Lesson 13-2, "Negations," has now been incorporated into the other three lessons.)

Otherwise Chapter 11 remains intact in moving from Second to Third Edition. Chapter 11 of the old edition has six lessons, and these correspond roughly to Lessons 11-4 through 11-9 of the new text.

Let's finally take a look at the new Lesson 11-1 worksheet. We begin with the two examples from the text -- the first problem lists four ordered pairs and asks us to prove that they are the vertices of a parallelogram, while the second lists three pairs that may be the coordinates of a right triangle. In each case, students are to calculate the slopes of the sides formed by adjoining vertices, and show that these slopes are either equal or opposite reciprocals.

CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

The second part of that standard, on circles, will have to wait until later in the chapter. To complete the rectangle question, students must calculate the four slopes, and show that slopes of opposite sides are equal, while slopes of adjacent sides are opposite reciprocals.