Tuesday, February 20, 2018

Lesson 11-2: The Distance Formula (Day 112)

Lesson 11-2 of the U of Chicago text is called "The Distance Formula." In the modern Third Edition of the text, the Distance Formula appears in Lesson 11-5.

Today is the first day of the Big March -- the long stretch of school between President's Day and Easter when there are no days off from school. For many students -- and even teachers -- the Big March is the toughest time of the year. In fact, I failed to survive the Big March last year. Of course, teaching and classroom management during my first year was a struggle all year, but everything definitely fell apart at the Big March.

At some schools, the Big March doesn't start until next week, because students get the entire week of President's Day off from school. This is true in New York City, and last year I linked to a Northern California teacher whose school observes a week of February break. Actually, this year I found a Southern California district that has a mid-winter break as well. But that district clearly isn't one of the two districts where I'm employed as a sub.

In fact, sometimes I wonder whether things would have gone better for me last year if my charter school had a February week off. Many of the problems that I described earlier on the blog actually occurred on the first four days after President's Day.

After that first week, I wrote that I'd stop blogging for the rest of the Big March. If I were teaching full-time this year, I might have taken off the entire Big March again this year, just to make sure that I work hard to survive it rather than spend so much time posting on a blog. Since this year I'm just a sub, I'll continue to blog throughout the Big March.

Indeed, today I subbed at a high school. Since this was a math class, I'll do a full "Day in the Life" post for today (which is Day 104 in this district).

7:00 -- This teacher has a first period class -- and recall that in this district, first period classes are like "zero period" classes at other high schools.

This is the first of three Honors Algebra II classes. We know what distinguishes Honors Algebra II from regular Algebra II -- trigonometry. And it seems as if the tough trig lessons always begin right at the Big March -- and this class is no exception. (In previous years on the blog, I'd even post my trig lessons for Geometry right at the Big March.)

The lessons are posted completely online, and so there's not much for me to teach. Since this is the first trig lesson, students are introduced to the three main trig functions sine, cosine, and tangent. Of course, the famous mnemonic SOH-CAH-TOA is used.

Most of the classes are sophomores. This is somewhat expected in an honors class, since sophomores who take Algebra II are generally honors students. There are a few juniors, and even one or two advanced freshmen in each Algebra II class. (We'd expect there to be more juniors if this were a regular Algebra II class.)

7:55 -- First period leaves and second period arrives. This is the only math class of the day that isn't an Algebra II class.

The class has a strange name, "Math Studies," and the students are mostly juniors, with a few seniors as well. As it turns out, this is an International Baccalaureate (IB) course.

I've mentioned IB twice on the blog before. Two years ago, I had a traditionalist post in which I mentioned a proposal in Kansas to eliminate all out-of-state tests, including AP and IB. (I believe that the proposal failed, since I haven't heard about it since.)

My other reference to IB was early last school year. As part of Tina Cardone's "Day in the Life" project (that's right -- Cardone was the inspiration for the "Day in the Life" format that I still follow today, including in this very post), a New York teacher wrote about the IB program:


My first class of the day is a senior IB Studies Year 2 class. There are only 13 kids in the room, which is great, but also kind of surreal. I can check in with everyone and get a feel for who they are and what they’re thinking. We did one and a half three-act lessons just so the first day wasn’t only “Here’s the syllabus” nonsense. The questions weren’t too hard, so a lot of kids could participate. This was also the third year in a row I’ve taught some of these kids so I really know them pretty well by now I think.

I have more to say about this class and IB in general later in this post. Suffice it to say that today, the students have eight problems to solve, similar to questions they might see on the real IB exam. The problems are posted around the room, and the answers are given as QR codes -- that is, the students must work the problems and then scan the QR codes to check their answers.

8:50 -- Second period leaves and third period arrives. This is the second Honors Algebra II class.

9:45 -- Third period leaves and tutorial arrives. I mentioned "tutorial" two weeks ago -- the middle school I subbed at that day had tutorial built into the day, and so does this high school. Naturally, I help any students who are working on math questions.

10:25 -- The students leave for snack.

10:45 -- Fourth period arrives. This is the last Honors Algebra II class.

11:35 -- Fourth period leaves. As it turns out, the teacher has a fifth period conference, which combines with lunch to provide me with a very long break.

2:10 -- It is now seventh period. Ordinarily, a teacher who has a first period wouldn't have a seventh period as well. The problem is that this teacher is a coach, and all athletics are seventh period. She is the basketball coach, for both the boys and the girls. It's rare to see a woman coaching a boys' team, yet that's exactly the case here.

But actually, basketball season has just ended (and I mean just barely, as in a week or two ago). Only teams that have qualified for the playoffs are still playing basketball -- and apparently, these teams are not in the playoffs.

In the end, this class becomes another tutorial, except that it's being held in the gym. I assist any students who are working on math, including Algebra I and Geometry.

3:00 -- Seventh period leaves, and I finally complete one of my longest days of subbing ever.

Let me say more about the IB class. I'd argue that this Math Studies class is nearly equivalent to a Pre-Calculus class. Indeed, many of the students take the Honors Algebra II class as a sophomore, then IB Math Studies as a junior, and finally AP Calculus AB as a senior. (Students in that New York IB program mentioned above apparently take two years of IB Math Studies, not AP Calc.)

During my 2 1/2-hour break, I find an IB study guide on the shelf. I learn that the IB exam consists of seven different topics:

1. Numbers and Algebra
2. Descriptive Statistics
3. Logic, Sets, and Probability
4. Statistical Applications
5. Geometry and Trigonometry
6. Mathematical Models
7. Introduction to Differential Calculus

Let's look at Topic 5 in more detail -- not only because this is a Geometry blog, but also because this is what the second period students are currently reviewing:

5.1 Equation of lines
5.2 Trigonometric ratios
5.3 Sine and cosine rule; area of a triangle
5.4 Three-dimensional solids
5.5 Volume and surface area of three-dimensional solids

So we see that just as with the SAT, the IB gives Geometry short shrift. Subtopic 5.1 is really just Algebra in disguise. Subtopics 5.2 and 5.3 go beyond the trig we learn in Chapter 14 of the U of Chicago text. Subtopics 5.4 and 5.5 are equivalent to Chapters 8-10 -- I include the area chapters since the volume and surface area formulas are derived from area formulas. The rest of the text doesn't appear on the IB exam at all.

The last topic of the year introduces a little Calculus. It goes perhaps a little beyond the introduction we might see in the last chapter of a Pre-Calc text.

Let me post some of the questions the second period students are working on today:

1. The straight line L_1 has equation 2y - 3x = 11. The point A has coordinates (3, 1).
a. Does L_1 pass through A?
b. L_2 is a line perpendicular to L_1. L_2 passes through A, find the value of c.

2. The straight line L passes through the points A(-4, 1) and B(8, -5).
a. Calculate the gradient of L.
b. Find the equation of L. Write your answer in slope-intercept form.

5. The straight line, L_1, has the equation y = (-1/3)x - 2.
a. Write down the gradient & y-int of L_1.
b. Find the equation of L_1. Give your answer in the form ax + by + d = 0.
c. The line L_2 is perpendicular to L_1 and passes through the point (3, 7). Find the equation L_2. Give your answer in slope-intercept form.

6. Triangle ABC is such that AC is 7 cm, angle ABC is 65 degrees, and angle ACB is 30 degrees.
a. Calculate the length of AB.
b. Find the area of triangle ABC.

We notice that since this is the International Baccalaureate exam, many of the terms used on the exam are Commonwealth rather than American. The word "gradient," for slope, is interesting. In Multivariable Calculus (a college-level class, the next step beyond Calc BC), we learn about a generalization of the derivative called the "gradient." We now see that "gradient" is the international word for "slope" even in Algebra problems.

The equation of a line is y = mx + c, not y = mx + b. (And I still don't know what the letter m has to do with slope!) Notice that y = mx + c is still called "slope-intercept form," not "gradient-intercept." I guess since c appears in the slope-intercept form, we have to use d in the standard form of a line instead, so it's ax + by + d = 0.

Other international terms include "sine rule" and "cosine rule" -- of course, American texts would call these "Law of Sines" and "Law of Cosines."

Unfortunately, two of the QR-coded answers contain errors. A simple sign error occurs in the solution to Question #5c above. Let's try to solve it ourselves:

5c. The slope (excuse me, gradient) of L_1 is -1/3, and so the slo -- gradient -- of L_2 is 3. We now calculate the y-intercept:

y = mx + b (Did I say b? Of course I meant c!)
7 = (3)(3) + c
7 = 9 + c
c = -2

So the correct equation is y = 3x - 2. Unfortunately, the QR-coded solution is y = 3x + 2. This is understandable, as we all make sign errors all the time.

But a more noticeable error occurs in Question #1b above, a similar type of problem. We begin by rewriting the equation for L_1 in slope-intercept form. (Oops, I mean gradi -- oh, I was right the first time, slope-intercept form.)

2y - 3x = 11
2y = 3x + 11
y = (3/2)x + 11/2

But here's the kicker -- the QR-coded answer to Question #1b is c = 11/2. This is indeed a y-intercept, but it's the y-int of L_1 when the question asks for that of L_2. In fact, I help a few second period students by showing them all the correct steps to find the y-int of L_2 -- then they check the QR code, and wonder why I've shown them so many extra steps after finding 11/2 a long time ago! Of course, I inform the teacher of these errors.

I don't have time to check the trig solutions for errors during second period. At tutorial time I do one of the trig problems with a girl who is struggling with some of the questions. She makes one mistake and tries to plug the wrong angle into the Law of Cosines, but after I correct her, she completes the problem perfectly. She and I obtain the same answer, but we never check our answer against QR.

Oh, and this leads to the next New Year's Resolution I want to discuss, namely the fifth:

5. Engage the students in the learning process instead of lecturing excessively.

The students are learning on Google Classroom, but still, I catch myself needing to look at the resolution more closely. This girl I just mentioned tried to solve the problem above by "subtracting cosine" from both sides of the equation, and I jump in to correct her. It turns out that she knows what she's doing -- "minus cosine" just reminds her to write "cos^-1" (inverse cosine) on the next line. In order to keep the fifth resolution, I should have given her the opportunity to explain what she's doing with "minus cosine" instead of jumping in.

Meanwhile, I do have some opportunities to work on classroom management today (first resolution), which includes making sure the students organize the Chromebooks (by number, of course) and keeping the basketball players in their correct spots in the gym.

OK, that's enough about the IB Math Studies class. Meanwhile, the Geometry problems on the Pappas calendar don't stop. Today on her Mathematics Calendar 2018, Theoni Pappas writes:

The shortest side in this figure is opposite which degree?

(There are two triangles with a side in common. The angles of one triangle are 50-60-70, and the angles of the other are 20-33-127. The 70- and 127-degree angles are adjacent, as well as the 60- and 20-degree angles.)

The key to this problem is the Unequal Angles Theorem of Lesson 13-7:

Unequal Angles Theorem:
If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

In some texts, this is called the "Triangle Angle Side Inequality," and I've used the abbreviation TASI in past years.

It's too easy to say that since the smallest angle on this diagram is 20 degrees, the side opposite this angle must be the shortest side. But this is flawed -- the Unequal Angles Theorem only works within a single triangle. Otherwise -- let's see, since last week was the new moon, there were a time over the weekend (waxing crescent) when the angle formed by the sun, earth, and moon was 20 degrees. So according to this reasoning, the side of the triangle opposite the 20-degree angle (the distance from the sun to the moon) must be shorter than the side opposite the 50-degree angle in the drawing on the Pappas calendar, since 20 is less than 50.

Of course this is is ludicrous -- the theorem only works within one triangle. Typically, there is a way to determine which side is the shortest in a Pappas question even with multiple triangles. To see how, let's label the vertices A, B, C, and D. The two triangles have side BC in common:

Angle A = 50
Angle ABC = 70
Angle ACB = 60
Angle DBC = 127
Angle DCB = 20
Angle D = 33

In triangle ABC, side BC is the shortest because it's opposite 50 degrees. But in triangle BCD, side BC is not the shortest side, because it's opposite 33 degrees in that triangle. Instead, the shortest side is side BD because it's opposite 20 degrees. Side BD is shorter than BC (and CD), and BC is itself shorter than AB and AC. Thus the Transitive Property of Inequality tells us that BD is shorter than every other segment in the diagram. So the shortest side in the diagram truly is opposite 20 degrees -- and of course, today's date is the 20th.

Suppose Pappas had asked for the longest side instead. Well, in Triangle ABC, the longest side is AC, but in Triangle BCD, the longest side is CD. So we have both AC > BC and CD > BC, but we can't compare AC and CD to each other. The Transitive Property of Inequality isn't sufficient to determine the longest side -- instead, more sophisticated analysis, such as the Law of Sines, is needed. (Oops, I meant "sine rule" -- never mind, I'm talking about the IB anymore today.) But don't worry -- she usually won't ask that question on her calendar. The problems will nearly always be set up to use only the Transitive Property of Inequality.

Let's get to today's lesson. I've written above that one of the most difficult units always seems to begin right around the start of the Big March. Many students have trouble with graphing throughout Chapter 11, and furthermore, today we learn the Distance Formula, which of course will be difficult for some students. But then again, student who have trouble with formulas would have struggled with the surface area and volume formulas of Chapter 10, even before the Big March began. (Well, at least it's not trig -- my Big March topic of the past and the current Honors Algebra II Big March topic.)

In the past, I combined Lesson 11-2 with Lesson 8-7, on the Pythagorean Theorem (and indeed, this lesson in the Third Edition is titled "The Pythagorean Distance Formula").

I never wrote anything about the Distance Formula two or three years ago, since in those posts I always ended up writing more about the Pythagorean Theorem. But David Joyce has more to say about the Distance Formula:

Also in chapter 1 there is an introduction to plane coordinate geometry. Unfortunately, there is no connection made with plane synthetic geometry. Here in chapter 1, a distance formula is asserted with neither logical nor intuitive justification. Of course, the justification is the Pythagorean theorem, and that's not discussed until chapter 5. In that chapter there is an exercise to prove the distance formula from the Pythagorean theorem. The Pythagorean theorem itself gets proved in yet a later chapter.

Fortunately, the U of Chicago text avoids this problem. Our text makes it clear that the Distance Formula is derived from the Pythagorean Theorem.

Today I post an old worksheet from three years ago. It introduces the Distance Formula -- but of course, it teaches (or reviews) the Pythagorean Theorem as well  -- including its similarity proof, which is mentioned in the Common Core Standards.

(Oh, and speaking of similarity, the Geometry students I help out during Off-Season Basketball practice are still working on similarity. Thus similarity is the Big March topic for Geometry students in this district!)

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