Wednesday, February 28, 2018

High School -- Geometry Performance Task: Properties of Quadrilaterals, Continued (Day 118)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

P is the circle's center. 9.9sqrt(2) is the length of the hypotenuse. Find the area of the shaded region to the nearest whole #.

(The shaded region is a segment of a circle -- the region between a chord of a circle and the arc subtended by the chord. Angle P is both an angle of the triangle and the central angle that also subtends this same arc. The word "hypotenuse" is a giveaway that this is a right triangle.)

To find the area of the segment, we subtract the area of the triangle from that of the sector. Let's find the area of the triangle first, as this is easier. This is indeed a right triangle -- indeed an isosceles right triangle, since its two legs are congruent radii. Thus it is a special right triangle, a 45-45-90 triangle -- its hypotenuse is 9.9sqrt(2), so its legs, the radius of the circle, are 9.9. Either leg of a right triangle is its base with the other leg the height, and so its area is 9.9^2/2 = 98.01/2 = 49.005 square units.

Now to find the area of the sector, we note that the radius is 9.9, so the area of the entire circle must be 9.9^2 pi, or 98.01pi square units. Since the arc (like the central angle) is 90 degrees, we conclude that the area of the circle is 1/4 that of the circle. So it is 98.01pi/4 = 24.5025pi square units.

So the exact area of the segment is 24.5025pi - 49.005 square units. Of course, we use a calculator to round this off. I obtain about 27.97 square units, which we round off to 28. Therefore the area is 28 square units -- and of course, today's date is the 28th.

It's possible to estimate this answer without a calculator. Notice that the area is (9.9^2/4)(pi - 2), and if we approximate 9.9^2 by 98 and pi by 3.14, we must then multiply (24.5)(1.14). This product works out to be 27.93, which is close enough to the correct answer.

This is a strange question -- we're given an exact length 9.9sqrt(2), yet we must round off the answer in the end. Perhaps we might as well have specified the hypotenuse as an integer, such as 14. (Notice that 9.9sqrt(2) is about 14.0007.)

This problem uses a special 45-45-90 triangle. Special triangles don't appear until Lesson 14-1 of the U of Chicago text. Glencoe students are currently learning about this in Chapter 8. Our students can still use the Pythagorean theorem to find the radius, but it's tricky.

Today I subbed in a high school chemistry class. Since this is not a math class, there will be no "Day in the Life" today.

Meanwhile, this is the second day of a two-day Performance Task on the blog, so there's no worksheet for me to post today. Hopefully our students can figure out the answer. As a hint, students should first try proving that the figure is a rectangle (which requires four slope calculations). Two distance calculations (length and width) are need to confirm that the rectangle is a square. The Midpoint Formula given on this page is a red herring, as it's not needed.

By the way, I hear that PBS is showing a math-related episode of NOVA tonight. Unfortunately, I don't have time to watch it tonight. Whenever I do get around to seeing this episode, I'll report on it on the blog.

And so thus ends this rather short (at least by my standards) post!

Tuesday, February 27, 2018

High School -- Geometry Performance Task: Properties of Quadrilaterals (Day 117)

Today I subbed in another math class -- at a different high school from yesterday. This time I subbed in a Geometry class. I definitely want to do the "Day in the Life" for today, since this is, after all, a Geometry blog.

And so here is the "Day in the Life" for today -- Day 109 in this district:

7:00 -- Rise and shine -- this is yet another "first period" (zero period) class.

The Geometry students are still working on similarity -- Lesson 7-3 of the Glencoe text. In fact, this is what I wrote three years ago about today's lesson (from a tutor's perspective):

Last night I tutored my geometry student again. Section 7-3 of the Glencoe text is on similar triangles, including all three similarity statements: AA, SAS, and SSS.

Well, my student is still slightly confused with solving and setting up proportions. But he was more concerned with a question from Section 7-2 of the Glencoe text, on similar polygons. There is a group of similarity problems in the Glencoe text that are flawed.

Three years ago I ended up writing more about Lesson 7-2 than 7-3. But unfortunately, one of the questions on today's Lesson 7-3 worksheet also contains a (different) flaw, as I'll soon find out.

Most of the first few problems have the students calculate missing lengths in similar triangles. Notice that since one of the side lengths is missing, SSS Similarity goes right out the door. Thus all of the problems on the first side of the worksheet use either AA~ or SAS~.

7:55 -- First period leaves and second period begins. They begin the Lesson 7-3 worksheet.

8:50 -- Second period leaves and third period begins. They begin the Lesson 7-3 worksheet.

9:45 -- This is the same school I subbed at a week ago today -- so now it's tutorial time. Several students arrive for tutorial, but many of them are working on subjects other than math. The two Geometry students I see have a different teacher and are working in a different lesson. I do help these two briefly.

10:25 -- Tutorial ends -- and believe it or not, so does my school day! The teacher I'm subbing for isn't merely a coach -- he's the athletic director. And so in order to allow him to organize and plan all the school sports, he's given only a three-class load that ends at nutrition time.

So this is the shortest "Day in the Life" I've ever done. But I still have plenty to say, about both the curriculum and my New Year's Resolutions.

Yesterday I covered the first resolution, but my only chance to follow it was during sixth period. As this is such an important resolution, let's continue to focus on Resolution #1 for today's half-day:

1. Implement a classroom management system based on how students actually think.

It's often tricky to implement classroom management when students are working on different assignments or finishing a multi-day assignment. Students can always claim "I'm done, so let me do whatever I want."

Today, on the other hand, everyone is working on the Lesson 7-3 worksheet. This means that I can manage the classroom strongly -- just make sure that everyone is working on it.

When first period begins, I do problem #1 on the board. Later on, I do #2 as well. I'm hoping that the students can finish at least the first side of the paper. But many students fail to finish -- and those who do the work skip questions or jump to the other side. I end up having to leave the names of two students who do nothing other than copy #1 and #2 from the board.

In second period, I leave Questions #1 and #2 on the board and start on Question #3. Once I try the problem, it's no longer any wonder why students keep skipping it -- the question is flawed.

Here is the Question #3 from the Glencoe Lesson 7-3 worksheet:

3. Identify the similar triangles. Then find the measure of QR.

In Triangle QSV, point R is between Q and S, while T is between V and S. The following lengths are given on the diagram:

QS = VS = 36
QV = 36sqrt(2)
TS = 24
QR = x

I tell my students another story from my tutoring days, which I've also chronicled on the blog:

I do admit that students might be tricked by this sort of question. I remember once when I was teaching or tutoring a student who was in the similarity chapter of the text. Upon seeing the two triangles in a question much like this one, he immediately started writing a proportion. I told him that a proportion can be set up only if the triangles are similar, and so I asked him, how did he know that the triangles are similar? His response was, of course the triangles were similar because we were in Chapter 7, the similarity chapter of the text!

Yes, my student cleverly figured out that in the similarity chapter, nearly every problem would have a pair of similar triangles. But this problem illustrates what's wrong with his thinking

And here's the problem on the worksheet with today's students -- the given info isn't sufficient to conclude that the triangles are similar!

If you don't believe me, then try to prove that Triangle QSV ~ RST. We've already eliminated SSS~ as a possibility for any question on this worksheet, but what about SAS~? The two triangles have Angle S in common, but one of the adjacent sides, RS, has length 36 - x. Since the unknown value that we;re solving for is part of one of the sides, this rules out SAS~ as a possibility as well.

This leaves only AA~. But no angle information is given, so we cannot conclude that Angles Q and TRS are congruent, nor Angles V and RTS. Notice that in the next three questions, we're given that two of the sides are parallel (which would give us congruent corresponding angles), but no parallel lines are stated in this question.

Notice that if we could conclude that Triangle QSV ~ RST, then x = 12. But what's to stop us from choosing another point on QS, calling it R', and then claiming that QR' = 12 instead of QR?

Ironically, here are the instructions for the first four questions on the other side of the worksheet:

"Determine whether each pair of triangles is similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Explain your reasoning."

And doubly ironically, all four pairs of triangles have enough info to conclude they're similar. (There are two SAS~ questions and one each of SSS~ and AA~.) The only pair of triangles not to contain sufficient info is on the wrong side of the worksheet as #3.

Well, let's answer this question for the wrong #3. Here is what we need to conclude that Triangles QSV and RST are indeed similar. Any one of the following is sufficient:

  • Angle Q = TRS
  • Angle V = RTS
  • QV | | RT
  • RT = 24sqrt(2)
This last one needs some explanation. The side lengths of QSV are 36, 36, 36sqrt(2). This should be recognizable as the sides of a 45-45-90 triangle, with the right angle at S. Then knowing RT will allow us to conclude that the triangles are similar by HL. For some strange reason, HL Similarity never appears in any texts, yet it's obvious that HL~ should work. The same proof for the other similarity theorems works for HL~ as well.

There's one more sufficient statement that will allow us to conclude similarity -- RS = 24, then the triangles are similar by SAS~. But if we had RS, then we could find x = QR by the Betweenness Theorem (Segment Addition) as QR = QS - RS, without using similarity at all.

In second period, I end up spending so much time on Question #3 that I rush through the final problem on the front of the worksheet -- Question #7, a word problem on shadow lengths. Students, of course, try to avoid word problems, and so many don't finish Question #7. (Some students also try to avoid Question #3 because of the square root, which isn't the real problem with it.)

But returning to classroom management, I do catch one student who doesn't even have his worksheet on his desk. I can't catch his name, so I must return to identifying him by backpack. Since it's still wintertime, he's wearing a coat, which I also use for identification. Students wear different clothes everyday, but they often wear the same jacket -- and of course, they don't change their backpack.

In third period, I do Questions #1, #2, and then #4-7 on the board, saving a discussion about Question #3 for last. Students still have some trouble with the word problem in #7, but this time I'm able to get them through it. I don't need to write any names.

During tutorial, another management issue turns up. This is one that has come up with several other teachers I've spoken to during breaks. A student texts someone on a cell phone, then claims that the recipient is her mother about to undergo surgery. Indeed, this is what the other sub I saw in the lounge had to kick out a student for yesterday. That student yelled at the sub that some things, like life and death, are more important than school, and that exceptions to the no phone rule should always, always, be granted during emergencies, no questions asked.

From my perspective, this is a blatant attempt to neuter the no phone rule. Students who claim that they are texting a parent are almost always texting another teenager, and the topic isn't an emergency, but entertainment. Their idea is that since teachers can never be certain that a text isn't an emergency, the only 100% fair thing to do is to let students text whenever they want to whomever they want, even if it interferes with their education. To them, letting kids do what they want is the only fair thing, and everything else is unfair. To them, if you want to be considered a "good" or "fair" teacher, it's very, very important to let them have fun, to the exclusion of everything else.

And returning to the girl in my classroom, the second time she takes out her phone, she and a group of girls started laughing -- odd behavior for someone whose mom is undergoing surgery. This clinches it, of course -- she's texting another teen to discuss entertainment. The problem, of course, is that this is tutorial, so it's impossible to get any identifying information for a student who might not even be in any of his classes. Of course, she claims that she's finished all her written work. In other words, to her, the only fair thing to do is let her have a free 40-minute texting and entertainment period everyday (since she can claim she has no work everyday) -- nothing less is fair.

Last year at my middle school, only three students claimed that emergencies should override the no phone rule. Two of the claims turned out to be genuine -- one was the special scholar (January 6th post) whose mother was making arrangements to pick up her elementary school sister, and the other had a death in her family. (She and her younger brother were eventually picked up from school.) To this day, I still believe the third student was lying about the emergency.

If I were a parent, I would never text my children during the school day, not even if a death in the family happens. If a relative were to die one minute after school starts, I would wait seven full hours, until one minute after the final dismissal bell, to text my children. I'd be too shocked by the news even to consider texting anyone. And besides, I'd assume that my kids would have their phones confiscated if they were to text -- the last thing they need is to lose a relative and a phone the same day. If it's absolutely necessary to contact them, I'd make every effort to do so without texting them -- including contacting or driving up to the school. And if I can't find any way to contact them without a text, I'd tell them not to respond until after the final dismissal bell. And if I were a student, I might not even want to text my parents anyway! (Note: with shootings in the news, here I assume that if an active shooter arrives at my kids' school, all classes are cancelled and students can safely text without phone confiscation.)

OK, that's enough about classroom management for today. I do wish to write a little more about teaching the Glencoe text. If I were a regular teacher with the Glencoe text, how would I pace it.

Since the Glencoe text has a Chapter 0, it's tempting to follow the same digit pattern that I've established for the U of Chicago text. But this is a little too fast -- it forces Chapter 7 on similarity into the first semester. I like the idea of starting the second semester with similarity -- and apparently this is exactly what's done in this district.

Of course, to be still in Chapter 7 five weeks into the semester is a bit slow, especially if the goal is to reach Chapter 13 by the end of the year. A better pace would be to start the Big March with Chapter 8, on trigonometry. (That's right -- trig at the Big March again!) Apparently, the Geometry students I see in tutorial are already in Chapter 8. They are studying special right triangles -- and they are able to recognize the 36-36-36sqrt(2) still written on the board as a special 45-45-90 triangle.

In the Glencoe text, Chapter 4, on congruence, appears before Chapter 9, on transformations. I find an old copy of the first semester final in the classroom, and apparently, Chapter 9 is pushed up into the first semester. This fits the Common Core definition of congruence. And it also appears that Lesson 9-6 is combined with Chapter 7. This makes sense as Lesson 9-6 is on dilations.

I also find a copy of a Performance Task, similar to what students may find on the SBAC. There's a hole in my U of Chicago pacing plan since Chapter 11 has only six sections. And I've never posted a Performance Task before, despite this being a Common Core blog.

And so I post this task as an activity for today and tomorrow. Actually, I create my own version of the problem rather than the district copy. This is to block out the name of the district since I don't post identifying information (and to avoid any issues with SBAC, which might mistake this practice question for a real test question). Also, I changed it from one day to two days. On the actual SBAC, students are typically given a two-hour block to complete the Performance Task, so they should have two days to complete it.

The question is about the coordinates of a square. It fits perfectly with Chapter 11 of the U of Chicago text -- and indeed it's similar to the Lesson 11-1 activity from two weeks ago.

In the Glencoe text, the Performance Task fits with Chapter 6, on quadrilaterals. Glencoe Chapter 6 is similar to Chapter 5 in the U of Chicago text (and Chapter 6 in the Third Edition), except that coordinates appear early. Indeed, I lamented three years ago that the Distance Formula appears as early as Chapter 1 in the Glencoe text!

Monday, February 26, 2018

Lesson 11-6: Three-Dimensional Coordinates (Day 116)

Lesson 11-6 of the U of Chicago text is called "Three-Dimensional Coordinates." In the modern Third Edition of the text, three-dimensional coordinates appear in Lesson 11-9.

This week the Big March starts in earnest. As I've written before, the first week of the Big March isn't terrible, since at least it's a four-day week. The second week starts the string of five-day weeks -- these are what actually make the Big March the Big March.

Today I subbed in a special ed high school math class. Because it's a math class, I will definitely post a "Day in the Life" for today -- Day 108 in this district.

Before 9:15 -- There are several things different with today's schedule. First of all, many of the high schools in this district, including this one, has late days on Mondays. This was also true at one high school in a previous district I subbed at. But I never wrote a "Day in the Life" for any such late Monday, and thus I never mentioned it on the blog.

Late Start Mondays, of course, occur so that teachers can have a morning Common Planning meeting just as many K-8 schools have early out Wednesdays. As it turns out, the reason this special ed teacher called for a sub is so he could hold IEP meetings with parents. Therefore he actually attends the Common Planning meeting, then has enough time to meet me and give me the lesson plans before his first IEP meeting.

It turns out that all classes are doing the infamous holiday graphing worksheets that I've posted many times before on the blog. (Actually, I find out the name of these graphs -- "Cartesian Cartoons.") I inform him that I'm a math teacher. So often, math teachers don't get math subs, and so he's prepared the "Luck O' the Irish" worksheet (for St. Patrick's Day) so that students work independently. Perhaps if he could be assured of a math sub like me, he could've had me help the students more substantially with their math.

This teacher doesn't have a first period (which is really zero period in this district). On Mondays, first period is after the Common Planning meeting. At other districts with both a zero period and late day Mondays, either there is no zero period on Mondays, or zero period actually occurs before the PD meeting (which was the case at the district where I no longer sub).

9:15 -- Second period begins. This is an Algebra I class, but of course all classes are actually working on the "Luck O' the Irish" graphing worksheet.

The official name of this class is Algebra 1B. These students are all sophomores who are enrolled in a two-year Algebra I course. Many schools spread out the content of Algebra I over two years for their struggling students -- especially the special ed students. Last year they took Algebra 1A, and this year they are in Algebra 1B.

10:05 -- Second period leaves and third period arrives. This is the only class that's officially a Geometry course. The students are all juniors who completed Algebra 1A and 1B their first two years of high school.

While the students work on their "Luck O' the Irish" worksheets, I decide to take a closer look at the textbook for this class. This is, after all, a Geometry blog, and on previous special ed math subbing days I've written about the different texts that I find in the classroom.

This district apparently uses the Glencoe Geometry text. I've written about the Glencoe texts many times on the blog -- most recently two weeks ago when SteveH, the traditionalist, recommended "proper Glencoe textbooks."

This text is also the one used by the student I was tutoring three years ago. Thus I wrote extensively about this text that year. I'll repeat the table of contents here.

The Glencoe Geometry Text
0. Preparing for Geometry
1. Tools of Geometry
2. Reasoning and Proof
3. Parallel and Perpendicular Lines
4. Congruent Triangles
5. Relationships in Triangles
6. Quadrilaterals
7. Proportions and Similarity
8. Right Triangles and Trigonometry
9. Transformations and Symmetry
10. Circles
11. Areas of Polygons and Circles
12. Extending Surface Area and Volume
13. Probability and Measurement

There are two chapters here that I don't recall from my tutoring days -- Chapters 0 and 13. Let's look at these two chapters in more detail.

0. Preparing for Geometry
0-1. Changing Units of Measure Within Systems
0-2. Changing Units of Measure Between Systems
0-3. Simple Probability
0-4. Algebraic Expressions
0-5. Linear Equations
0-6. Linear Inequalities
0-7. Ordered Pairs
0-8. Systems of Linear Equations
0-9. Square Roots and Simplifying Radicals

13. Probability and Measurement
13-1. Representing Sample Spaces
13-2. Probability with Permutations and Combinations
13-3. Geometric Probability
13-4. Simulations
13-5. Probabilities of Independent and Dependent Events
13-6. Probabilities of Mutually Exclusive Events

Note that the Common Core Standards for high school include a Stats and Probability strand. Here in California, these standards are incorporated into the Algebra I and Geometry courses -- in particular, stats appears in Algebra I, and probability appears in Geometry. This explains why Chapter 13 appears in the Glencoe text. I can't find the word "California" anywhere on the front cover -- so perhaps other states are following this pattern as well.

As the title of Lesson 13-3 implies, Geometry and probability are linked. Lesson 8-9 of the U of Chicago text hints at geometric probability. The only other mention of probability in the U of Chicago text is in Lesson 6-1. Here P(E), the probability of an event E, is given as an example of function notation to prepare the students for transformation notation T(x, y).

I've never posted a probability unit on the blog before. Perhaps I should consider posting such a unit this year. After all, I can't call this an authentic Common Core Geometry blog if I'm leaving out a key part of Common Core Geometry classes -- the probability unit.

10:50 -- Third period leaves for snack.

11:10 -- Fourth period Algebra 1B arrives. Of course, they work on the Cartesian Cartoon page.

12:00 -- It is now fifth period. As we've seen before a few weeks ago, special ed teachers often move to another room to co-teach one period. This teacher is scheduled to co-teach what appears to be a freshman Algebra I (not 1A or 1B, but just 1).

These students are in Chapter 7 of the Glencoe Algebra I text, which is on exponents (as an introduction to polynomials). The main teacher demonstrates two laws of exponents (product of powers and power of a power). She uses key phrases like "multiply the little numbers" and "distribute the outer number to the inner numbers" to help the students remember these laws.

12:45 -- Fifth period leaves for lunch.

1:30 -- Sixth period Algebra 1B arrives. Of course, they work on the Cartesian Cartoon page.

2:15 -- It is now seventh period. This teacher is a football coach -- but football season ended three months ago. And so my day is now over.

Let's look at our focus New Year's Resolutions for today:

6. If there is a project-based curriculum such as Illinois State, then implement all components of it.
7. If there is an official assignment to review for state testing, then implement it fully.

These resolutions are important for me to fulfill if I were a regular teacher, but these are, of course, special situations that aren't always applicable when I'm just a sub. For now, we'll just repeat only the first five resolutions.

1. Implement a classroom management system based on how students actually think.

This resolution is often tricky to fulfill on special ed days, when there's usually an aide to take care of the classroom management. But this aide leaves after fourth period (and I myself am the "aide" for fifth period). This leaves sixth period as the only class to manage -- and before she leaves, the aide warns me that this class might be a difficult one.

I watch the main teacher for fifth period place some papers on the sixth period kids' desks right when lunch begins. And so I do the same -- at lunchtime I place the "Luck O' the Irish" papers so that they are already on the desks when sixth period arrives. This reduces some trouble at the start of class, since the students already know what they're supposed to be doing.

As it turns out, sixth period is already a small class, with only a dozen students -- and almost half the class is missing. Two guys are summoned to other rooms, while three girls are absent. Earlier, during lunch, another sub has told me how she kicked out one girl for talking back instead of putting her phone away. This could very well be the same as one of the girls who's absent from my class. With the biggest troublemakers out of the room, this class is actually the quietest of the day!

Today I should post the Lesson 11-6 worksheet. But two and three years ago, I never made one. In the past, I switched the chapter order and covered Chapter 11 before Chapters 9 and 10. I didn't really want to cover three-dimensional coordinates before the main chapters that describe the Geometry of three dimensions. And of course, once I reached Chapter 10 I never returned to Lesson 11-6.

It's my tradition, on days I sub for Geometry, to post the lesson I sub for instead. In other words, what happens in the live classroom takes priority over the U of Chicago order. And besides -- at least today's "Luck O' the Irish" actually fits in Chapter 11 on coordinate geometry, albeit in only two rather than three dimensions. If you wish, you can pretend parts of the graphs are in different planes and make it into a 3D lesson.

Ordinarily I don't post copyrighted material. But Cartesian Cartoons are so easy to find online that I see no harm in posting yet another copy of it. (And besides, I've posted some of them before during years past.) Some students today do remark that this is a bit early to be doing a St. Patrick's Day assignment, but oh well!

One girl has trouble understanding how to do the graphs. I try to help her a little, but her points are still a bit off. I tell her that she is "almost smart"and that she should work harder to make the graphs look right. I hope my words can motivate her.

Friday, February 23, 2018

Lesson 11-5: The Midpoint Connector Theorem (Day 115)

Lesson 11-5 of the U of Chicago text is called "The Midpoint Connector Theorem." In the modern Third Edition of the text, the Midpoint Connector Theorem appears in Lesson 11-8.

Today I subbed in a high school ROP multimedia class. (For those who aren't familiar, ROP, or Regional Occupational Program, is a California career technical education program.)

Since it's not a math class, there is no "Day in the Life" today. But I do wish to point out that depending on the class, students are directed to create their own graphics and videos -- including a short one-minute trailer of a hypothetical movie. Footage from actual films as well as one new scene of their own is used to create the trailer.

Last year, the coding teacher wanted to do a similar project with our eighth graders. It was more successful at this school -- partly because these are high school students not eighth graders. But another factor is student behavior. Even though last year's eighth graders generally behaved better for other teachers (including the coding teacher) than for me, they still misbehaved enough to discourage him from proceeding with the project. I suspect that if I had been a stronger classroom manager last year, their improved behavior would have carried into coding -- and then they would have been allowed to do a video project.

One student I meet in today's class is from Puerto Rico. He tells me that his home has been damaged by last year's hurricanes, but he is attending school here in California, funded by a "scholarship" in his favorite sport, volleyball. I ask him about his family back on the island, and he informs me that they are doing well, even though they're still without electricity. It's one thing to read about the problems Puerto Ricans are facing -- it's another to meet a displaced victim face to face.

Unlike the rest of Chapter 11, this is a lesson I covered better two years ago than three years ago. And so this is what I wrote two years ago about today's topic:

Lesson 11-5 of the U of Chicago text is on the Midpoint Connector Theorem -- a result that is used to prove both the Glide Reflection Theorem and the Centroid Concurrency Theorem. Last year I only briefly mentioned the Midpoint Connector Theorem on the way to those higher theorems, and then when we actually reached Chapter 11 I only covered it up to Lesson 11-4, as I knew that I'd already incorporated 11-5 into the other lessons. But this year, I'm giving 11-5 its own worksheet.

Midpoint Connector Theorem:
The segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.

As I mentioned last week, our discussion of Lesson 11-5 varies greatly from the way that it's given in the U of Chicago text. The text places this in Chapter 11 -- the chapter on coordinate proof -- and so students are expected to prove Midpoint Connector using coordinates.

It also appears that one could use similar triangles to prove Midpoint Connector -- to that end, this theorem appears to be related to both the Side-Splitting Theorem and its converse. Yet we're going to prove it a third way -- using parallelograms instead. Why is that?

It's because in Dr. Hung-Hsi Wu's lessons, the Midpoint Connector Theorem is used to prove the Fundamental Theorem of Similarity and the properties of coordinates, so in order to avoid circularity, the Midpoint Connector Theorem must be proved first. In many ways, the Midpoint Connector Theorem is case of the Fundamental Theorem of Similarity when the scale factor k = 2. Induction -- just like the induction that we saw last week -- can be used to prove the case k = n for every natural number n, and then Dr. Wu uses other tricks to extend this first to rational k and ultimately to real k.

I've decided that I won't use Wu's Fundamental Theorem of Similarity this year because the proof that he gives is much too complex. Instead, we'll have an extra postulate -- either a Dilation Postulate that gives the properties of dilations, or just one of the main similarity postulates like SAS. I won't make a decision on that until the second semester.

Nonetheless, I still want to give this parallelogram-based proof of the Midpoint Connector Theorem, since this is a proof that students can understand, and we haven't taught them yet about coordinate proofs or similarity.

Today is an activity day. The old Lesson 11-5 worksheet from two years ago wasn't officially an activity page, but I included an extra questions which serve to make this an interesting activity:

First, notice that the first thing we see in the U of Chicago text is a picture of Sierpinksi's Triangle -- one of the fractals that we learned about two months ago when reading Benoit Mandelbrot's book. It's included here since it's closely related to midpoints. Then there is a problem from the text about Midpoint Quadrilaterals. Wu's proof is added as the last bonus question.

Thursday, February 22, 2018

Lesson 11-4: The Midpoint Formula (Day 114)

Lesson 11-4 of the U of Chicago text is called "The Midpoint Formula." In the modern Third Edition of the text, the midpoint formula appears in Lesson 11-7.

Like Lesson 11-1, Lesson 11-4 wasn't truly covered on the blog two years ago. I actually did a better job covering Chapter 11 three years ago than two years ago.

And so this is what I wrote three years ago about today's lesson:

Lesson 11-4 of the U of Chicago text covers the other important formula of coordinate geometry -- the Midpoint Formula. As the text states, this is one of the more difficult theorems to prove.

In fact, the way we prove the Midpoint Formula is to use the Distance Formula to prove that, if M is the proposed midpoint of PQ, then both PM and MQ are equal to half of PQ. The rest of the proof is just messy algebra to find the three distances. The U of Chicago proof uses slope to prove that Mactually lies on PQ. Since we don't cover slope until next week, instead I just use the Distance Formula again, to show that PM + MQ = PQ, so that M is between P and Q. The algebraic manipulation here is one that's not usually used -- notice that instead of taking out the four in the square root of 4x^2 to get 2x (as is done in the last exercise, the review question), but instead we take the 2 backwards inside the radical to get 4, and then distribute that 4 so that it cancels the 2 squared in the denominator.

I don't have nearly as much to say about the Midpoint Formula as the Pythagorean Theorem and its corollary, the Distance Formula. To me, it's a shame that I had to bury the Pythagorean Theorem in the middle of this Coordinate Geometry unit. The main theorem named for a mathematician really deserves its own lesson, but due to time constraints I had to combine it with the Distance Formula the way I just did it in yesterday's lesson.

Returning to 2018, this year I gave the Pythagorean Theorem and the Distance Formulas two separate lessons, but other than that I don't have anything else to say about Lesson 11-5. And I have nothing else to say (not even any excuse to bring up the IB or some other contrived topic again), and so this is a rare short post.

Wednesday, February 21, 2018

Lesson 11-3: Equations for Circles (Day 113)

Lesson 11-3 of the U of Chicago text is called "Equations for Circles." In the modern Third Edition of the text, equations for circles appear in Lesson 11-6.

This is what I wrote two years ago about today's lesson:

The first circle lesson is on Lesson 11-3 of the U of Chicago text, on Equations of Circles. I mentioned that I wanted to skip this because I considered equations of circles to be more like Algebra II than Geometry. Yet equations of circles appear on the PARCC EOY exam.

Furthermore, I see that there are some circle equations on the PARCC exam that actually require the student to complete the square! For example, in Example 1 of the U of Chicago text, we have the equation x^2 + (y + 4)^2 = 49 for a circle centered at (0, -4) of radius 7. But this equation could also be written as x^2 + y^2 + 8y = 33. We have to complete the square before we can identify the center and radius of this circle.

In theory, the students already learned how to complete the square to solve quadratic equations the previous year, in Algebra I. But among the three algebraic methods of solving quadratic equations -- factoring, completing the square, and using the quadratic formula -- I believe that completing the square is the one that students are least likely to remember. In fact, back when I was student teaching, my Algebra I class had fallen behind and we ended up skipping completing the square -- covering only factoring and the quadratic formula to solve equations. And yet PARCC expects the students to complete the square on the Geometry test!

I also wonder whether it's desirable, in Algebra I, to teach factoring and completing the square, but possibly save the Quadratic Formula for Algebra II. This way, the students would have at least seen completing the square in Algebra I before applying it to today's Geometry lesson. [2018 Update: At this point two years ago I got into a long discussion about the PARCC and SAT. But due to my subbing from yesterday, the test on my mind right now is the IB exam. Two years ago, I looked at the PARCC and SAT from a traditionalist perspective -- and this year, I'm in the mood to write about the IB from a traditionalist point of view as well. So even though we just had a traditionalists post last week, today's post gets the "traditionalists" label as well.]

But what about the PARCC test for Algebra I -- does the Quadratic Formula appear there? I took a quick look at the EOY test for Algebra I, and at least one question that asks a student to convert a quadratic equation from standard into vertex form, which is often done using completing the square (but this could also be done by using x = -b / 2a, plugging it into the original equation to find y, and then letting these values be h and k in the vertex formula). I also saw a few problems that appeared to be inappropriate for an Algebra I test and looked more suitable for a higher-level class.

So this goes right back to the Common Core debate. What level of math should students be expected to master at each level? There is a poster who goes by the username SteveH, who posts at the traditionalist website Kitchen Table Math. Here's a detailed discussion of this issue by SteveH:

They could have found schools that produce good numbers of Calc AB and BC students with scores of 3 or higher and detailed their high school math curricula in terms of specific textbooks and syllabi. They could show the number of students who got 3’s or higher on the AP Calc or AP Stat tests who did NOT take algebra in 8th grade. Then they could ask the parents of the successful students what specific support they had to provide at home or with tutors to even get their kids to algebra in 8th grade. This is the hidden tracking and mapping that educational pedagogues specifically overlook with weasel word mappings. They just point to successful students and claim them for their own. My son must be his old school’s poster boy for Everyday Math.

Like many other traditionalists, SteveH wants to make sure that students are able to reach AP Calculus in senior year. One problem with these current PARCC tests, by including some of these harder problems on the Algebra I test, is that schools then say that the Common Core Algebra I test is too difficult for eighth graders, so they wait until ninth grade to let them take Algebra I. Then the students can never reach AP Calculus.

So we see SteveH's proposal here -- he writes that there should be a survey of students, who not only took AP Calculus but passed with with a 3 or higher, that asks them what they math they took prior to Calculus to attain that goal. Then one should have written the math standards to reflect the levels of math given by students in the survey. SteveH's mention of "specific support they had to provide at home of with tutors" refers to students whose elementary schools offer progressive math curricula, such as the U of Chicago's elementary texts, so that parents would have to supplement this with traditionalist (instructivist) math lessons at home. The idea, of course, is that the elementary standards should be rewritten to support more strongly a traditionalist pedagogy.

SteveH's idea, on one hand, is appealing. One criticism of the conversion to Common Core is that parents feel that their students are being treated like guinea pigs. Of course, whether we have Common Core or another set of standards, some class of students has to be the first to use the standards, and the parents of the first class will feel that their students are "guinea pigs" for being the first to use such untested standards -- so there could be no innovation without guinea pigs. But suppose we were to replace Common Core with a SteveH Core based on the survey mentioned in the paragraph that SteveH wrote. Since the SteveH Core Standards would be based on what actual students said they took in the survey, they wouldn't be untested standards -- so the first class of students who learned them would not be guinea pigs!

On the other hand, here are a few things I have to say about the SteveH proposal:

-- SteveH mentioned AP Statistics in his post. Is it possible for students to take Algebra I in ninth grade and still make it to AP Stat? Of course, that's what the survey would find out.
-- Would Integrated Math still exist under the SteveH Core? I bet it's possible for a homeschooled student to make it to AP Calculus, yet learned under the Singapore or Saxon math curricula, which favor the integrated pathway.
-- Why does SteveH find it so important for students to reach AP Calculus, anyway? He writes:

The low expectations start in Kindergarten and that creates adults who will never have that opportunity. By seventh grade it’s all over for most students.

That is, math standards that don't lead to Calculus end up closing doors for students, since it's unlikely for a student to get into a competitive college and attain a STEM major, and thus a STEM career, without having had Calculus senior year.

But a counterargument could be that forcing students to take Algebra I in eighth grade, Algebra II in tenth grade, and so on, actually closes doors for students. For example, a student who plans on having a non-STEM job that requires no math higher than arithmetic may wish to participate in sports or other extracurricular activities, but can't because the low Algebra II grade in sophomore year is pushing the GPA below 2.0. Or the student may want an after school job, but the parents won't let their child get one after they see the "D" or "F" in math on the report card.

I have no problem with wanting to get students to Calculus, but I wonder whether it's possible to keep the doors leading to STEM open without closing any non-STEM door.

And suddenly this has turned into yet another unexpected post about traditionalists. This often happens on the blog -- I want to reuse material from last year to set up this year's curriculum, only to find that I wrote about traditionalists. I suppose that this week's material lends itself to traditionalist criticism -- first yesterday's lesson on Cavalieri's Principle and then today's where we're pushing Algebra II material down into Geometry.

First of all, SteveH does still occasionally post at the Kitchen Table Math website. His most recent post there was in a long heated debate spanning from November to January:

The original post was about the phrase "sit and get," a phrase that progressives sometimes use to denigrate traditionalist pedagogy, along with the more common "drill and kill," and "guide on the side" as a phrase progressives use to describe themselves. On the other hand, traditionalists use the phrases "drill and practice" and "sage on the stage" to refer to their own pedagogy.

The progressive Michael Goldenberg writes:

Further, how would teachers who never heard of or considered any other sort of classroom develop slogans about such classrooms? Only those of us who experienced the traditional approach and saw that it was wanting would have motivation for summarizing in a few words what it was, with the implication of what was wrong with it implicit in those words. I guess you can call those "slogans," but I think that's being overly polite. 

On the other hand, no one awake in the 1990s who was interested in mathematics education is likely to have missed all the demeaning and dismissive slogans that groups like Mathematically Correct and NYC-HOLD[both traditionalist -- dw] came up with in regards to "reform" math education. Indeed, the old 2+2=4 website of Mathematically Correct had an entire page of such epithets, as I believe you know perfectly well.

And then SteveH takes up the traditionalist side of the debate:

But they were entirely correct considering curricula like MathLand, which was so bad that it disappeared with nobody claiming responsibility for it only to be replaced by curricula like Everyday Math [U of Chicago elementary texts -- dw] that tells teachers to "trust the spiral" and allows kids to get to fifth grade without knowing the times table. And now we have CCSS implementations like PARCC which officially declare that K-6 is a NO-STEM zone and their top level ("distinguished") only means that students are likely to pass a college algebra course - all the while talking about "problem solving" and "understanding" as if they've figured out some royal road to math. All of this just means that only affluent or educated parents have a chance to prepare their kids for a STEM career. Is that "polite?" My "math brain" son had to have a lot of help at home from his parents to survive math in K-6 math and now his schools claim that he is an exemplar of Everyday math. Nobody asked his parents. They are not interested in the truth.

The problem is that parents and students have no choice in the matter. It's not like future teachers in ed school all "discover" this sort of educational philosophy. No. They are directly taught the pedagogy by rote. It now defines their turf, which isn't about mastery of anything close to STEM-level content and skills in K-6 math.

All of this would be a non-issue if people had choice. Is it "polite" to force all students to accept one approach to math?" It wasn't very polite many of the comments made to my wife and I about education. They were specifically designed to get us to go away. They were personally demeaning.

Your attempt to position modern reform math as unappreciated and repressed by others is a complete failure. This has never been just a war of slogans. There have been very many specific arguments against the math curricula in schools, but they have been ignored. Schools can do whatever they want. Is that "polite?"

After this the debate ends up degenerating into name-calling, politics, and racial comments, so I won't quote them any further. Other than that, it covers some of the same material that I mentioned back in my April Fool's Day post. In particular, SteveH agrees with the traditionalists who claim that tracking already exists -- except that the tracking occurs outside the schools, via parents and outside tutors.

I'm not exactly sure what SteveH would want to see in a K-6 curriculum to make it pro-STEM. I've driven past elementary schools that declare themselves to be STEM schools, but I'm not quite familiar with their STEM curriculum.

As you keep in mind that I agree with traditionalists like SteveH when it comes to strengthening the elementary math curriculum (but disagree with him regarding Calculus), I must point out that "choice" mentioned in these debates is a red herring. SteveH only favors "choice" because he wants parents and students to choose a traditionalist curriculum. If most schools had a strong traditionalist curriculum, it would be Goldberg arguing to allow parents and students to "choose" the progressive curriculum and SteveH defending the status quo.

2018 Update: OK, let's cut this old discussion off and start talking about the IB. Wow, so back then SteveH used to post on other websites besides Barry Garelick's blog. Of course, I'm not sure whether Garelick even had a blog back then.

Here is a post from last October on Barry Garelick's blog. SteveH comments there as usual, and here he mentions the IB exam. (There's no need to discuss what Garelick writes in this post, since this entire post is about SteveH's comments.)

The fundamental issue they never address is that integrated (or whatever) math tried and lost in high school. IB (which leads to taking the AP test) and the AP Calculus tracks have won. That’s what colleges want to see for any STEM program and many non-STEM degree programs. This track starts at the 7th grade math track split to a proper algebra class in 8th grade. Vocational schools and community colleges place students using the Accuplacer test, not CCSS.

There are several issues here. First of all, SteveH states that IB leads to taking the AP test. OK, my subbing from yesterday verifies his claim here. The students appear to be taking Honors Algebra II as sophomores, IB Math Studies as juniors, and then AP Calculus AB as seniors.

SteveH also tells us that the "integrated math" track lost, while the "IB and AP Calc tracks" won. In a way, yesterday's subbing supports him as well. The IB sophomores are taking Honors Algebra II, not Integrated Math (which clearly doesn't exist in this district).

But hold on a minute here. IB, again, stands for International Baccalaureate. As its name implies, students around the world take the IB exam. And students outside the United States (and Vietnam) take Integrated Math, not Algebra I, Geometry, or Algebra II. Thus, contrary to SteveH's remark, plenty of students take Integrated Math to prepare them for IB courses.

I decided to perform a Google search for the math program at British IB schools. Most schools just simply call the class "math" (or "maths," I guess, since that's the international term). I did find one Scottish school that explicitly describes the math classes:

We notice that there are two levels of classes in Grades 8-10 -- C (core) and E (extended). I assume that these paths lead to the two levels of IB exam, SL (standard level) and HL (higher level). Of course, HL is recommended for prospective STEM majors.

There is one class listed here that SteveH would definitely like:

Maths 8E (Extended - Algebra 1)

This course is designed to provide a foundation for IB and IGCSE Math courses and to teach students to be successful at solving mathematical problems.  Emphasis is placed on problem-solving in the context of real-life situations as well as integrating technology into everyday life and using it as a problem-solving tool.  Some of the topics covered include: properties of the real numbers; graphing and solving linear equations and inequalities; the concept of a function; quadratic and exponential expressions and radicals; and connections with geometry.  Students passing Math 8 Extended earn a High School Math credit for this course.
We see that this class mentions Algebra I, but not Geometry. Thus it's not an Integrated Math class, but more like our Algebra I classes that SteveH wants eighth graders to take. And even though the Maths 8C course is more like our Common Core 8, it's Maths 8E that leads to HL and STEM. And so this is what SteveH wants to see -- eighth grade Algebra at the start of a rigorous track.

But even with this Maths 8E course corresponding to Algebra I, we notice that the classes for both freshmen and sophomores include Algebra and Geometry. Thus even Maths 9E and 10E are truly integrated courses. This counters SteveH's claim that the "integrated track" and "IB track" are two separate tracks, with the former "losing" and the latter "winning":

Maths 9E* (Algebra & Geometry)

* Extended
This course is the first of two in a sequence designed for students that enjoy the challenge of a more rigorous math course, whose foundational math skill are in place, and whose interests may lie more in business administration, chemistry, physics, biology, engineering and mathematics.  Students must be willing to accept the challenge of a more rigorous math course and the rapid pace that comes along with it.  Students will acquire and further develop mathematical skills and learn to apply them to other subjects and to real world problems.  Students should have, and will be expected to use a graphing calculator on a regular basis.
The first portion of the IGCSE International Mathematics 0607 Extended syllabus will be studied.  Topics of coverage will include, but not necessarily limited to, number, algebra, functions, geometry, two-dimensional transformations, mensuration and coordinate geometry.

Then again, I can't help but think back to the local STEM magnet high school. A few years ago, students were required to take eighth grade Algebra I in order to be admitted to the school. But the high school itself used to offer IMP, an Integrated Math program from Grades 9-11. Then seniors would take Calculus. At this school, IMP no longer exists -- ironically, this school used the Common Core transition to return to traditional Geometry and Algebra II, at the same time that other schools were introducing Integrated Math. (The eighth grade Algebra I requirement still exists.)

Nonetheless, both the old magnet requirements and the new Scottish IB classes suggest a compromise between the traditional and integrated pathways -- teach Algebra I in eighth grade and then Integrated Math in Grades 9-10. Hopefully SteveH would accept this compromise, since it at least encourages K-6 math to prepare students for eighth grade Algebra I.

I don't link to the above school website since the Integrated Math program no longer exists there (and it was never an IB school anyway). I did search on Google for a school right here in Southern California that offers both Integrated Math and IB. The school I found is Corona Centennial High:

We see that this school teaches three sections of "Integrated Math I Enhanced/IB." Hence it's another counterexample to SteveH's claim that Integrated Math doesn't lead to IB. According to the following link from this district, eighth graders can take Integrated Math I, so a path to Calculus is possible:

This link also lists several college acceptances, including the Ivy League. Otherwise, SteveH could counter that Integrated Math students are taking IB, but aren't doing well enough to pass the exams or get into elite universities. Indeed, in that same October comment, he writes:

Many of these pedagogues now talk about “math zombies” who get good grades, but still really don’t understand what they are doing, which is very normal because there are different levels of understanding. OK, so where are their other ideal students? What exact curriculum gets them there? Show how those students do on SAT II Math and AP Calculus tests. I might be their biggest fan. No. Their dream students don’t exist, so now they’re left promoting the benefits of engagement and conceptual understanding which are agnostic to pedagogy and curriculum. However, to promote their individual products (not curricula), they have to trash “rote” traditional math using rote, and wrong, justifications. It’s quite ironic.

Unfortunately that last link doesn't list actual SAT II and AP Calc test scores (or IB test scores, for that matter), the final piece of evidence SteveH needs to prove that Integrated Math works.

Back when I was a student teacher, my school had an IB program -- and indeed, I taught an Algebra I class for IB freshmen. Presumably, some IB freshmen also took Geometry, but I only taught the Algebra I class. But if I'm not mistaken, the IB program at this school no longer exists -- it was replaced by an AP-based magnet.

Here are the worksheets for today:

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!)