Monday, August 31, 2015

Lesson 1-6: The Need for Undefined Terms (Day 4)

This is what I wrote last year about today's lesson. I have updated the post to reflect the number of results of a certain Google search.

Lesson 1-6 of the U of Chicago text is where the study of geometry formally begins. This section states that three important words in geometry -- pointline, and plane -- are undefined. This may seem strange, for mathematics is all about definitions, yet these three important concepts are undefined.

In college-level math, one learns that these undefined terms are called primitives, or primitive notions. Just over a hundred years ago, the German mathematician David Hilbert declared that there are in fact six primitive notions in geometry: point, line, plane, betweenness, lies on, and congruence. But most textbooks list only the first three as undefined terms. This is because texts actually define the last three using concepts from other branches of mathematics. "Lies on" or "containment" -- that is, what it means for, say, a line to contain a point -- is defined using set theory (which is why the very first sentence of this section states that a set is a collection of objects called elements). "Betweenness" of points -- that is, what it means for a point to be between two other points -- is defined later in this chapter in terms of betweenness for real numbers (their coordinates of course). And the definition of "congruence" is the cornerstone of Common Core Geometry -- we use reflections, rotations, and translations to define "congruence." So we're left with only three primitive notions -- points, lines, and planes.

Lesson 1-6 is a fairly light lesson. So pointline, and plane are undefined -- big deal! Of course, we can do things with points, lines, and planes, but that's not until 1-7. So instead, I use this as an opportunity to remind the students the reasons for taking a geometry course.

The students in a geometry course are around the age where thoughts such as "I hate math" become more and more common. This is the age where they wonder whether they'll ever have any use for the math that they're learning. They begin to wonder whether they'll ever use any math beyond what they learned in elementary school and wish that math classes were no longer required beyond elementary school, for can't they live very successful lives not knowing anything higher than fifth grade math?

As of today, a Google search for "I hate math" returns 406,000 results. And we can easily predict the most common reason for hating math -- of course it's because it's hard. We don't hate things that are easy -- we hate things that are hard. And the class that turns so many off from math is algebra. Indeed, if you choose some school and tell me only its standardized test scores in ELA and math, I can very reliably tell you whether it's an elementary or a secondary school. If the math score is higher, it's probably an elementary school -- if the ELA is higher, it's likely a secondary school. And so now we, as geometry teachers, have the students for the math course right after the one that caused them to hate math in the first place.

The number of search results for "I hate math" has increased 10% over last year. But I notice that much of the increase is generated by a number of tutors that have taken the name "I hate math" -- that is, they tutor for students who hate math, as opposed to hating math themselves.

But we do see some results that are obviously from genuine math haters. One girl has posted a YouTube video of about 6 1/2 minutes on why she hates math. The girl in the video is an eighth grader who is struggling with the Quadratic Formula in her Algebra I class. She says that she hates math because without the class, she'd have a 4.0 GPA, but with math she struggles just to get a D+. I don't link to the video, but anyone can find it via a Google/YouTube search.

There are also images that say "I'm still waiting for the day that I will actually use xy + (420) > x - 5y[2 + 9 = 7] in real life." Well, of course we will probably never use a non-linear (because of the xy term) inequality such as that one in the real world. This image would have been much funnier if, instead of that inequality, the image contained the type of equation that traditionalists lament don't appear in Common Core texts, such as "I'm still waiting for the day that I will actually use (a quadratic-in-form equation with radicals) in real life."

So why do we require students to take so much of a class they hate in order to graduate high school? As it turns out, we can answer this question from one of the sections that we've skipped, Lesson 1-1:

"A point is a dot."

And this section gives many examples of dots -- the pixels on a computer screen. The shapes that appear on our screens consists of dots, which can be modeled in geometry by points. We look at images on our TV screens all the time. And one of the most geometry-intensive computer programs that we have are video games -- we must create images consisting of dots that move rapidly.

The point of all this is that we can surely have math without entertainment, but we can't have entertainment -- at least not most modern forms of entertainment -- without math. We can only imagine how much technology would disappear if math were to disappear.

Elementary school math -- at least early elementary arithmetic (before the dreaded fractions) -- is easy. And college majors majoring in STEM know the importance of learning math. The problem is those in-between years in middle and high school. If math were merely an elective in secondary school, many students would avoid it and choose easier classes. Then there wouldn't be enough STEM majors in college because they wouldn't have had the necessary algebra background. The only way to bridge the gap between "math is easy" (early elementary) and "math is important" (college STEM majors) is to require the subject during the intervening middle and high school years. Otherwise we'd have no modern technology or entertainment.

When I give notes in class, I prefer the use of guided notes. This is not just because I think the students always need the extra guidance, but that I, the teacher, need the guidance. In the middle of a lesson, I often forget what to teach, or forget how to explain it, unless I have guided notes in front of me.

And so today's images consist of guided notes. I begin with Lesson 1-6 and its definitions. Here I emphasize the fact that pointline, and plane are undefined by leaving spaces for the students to write in their definitions -- which they are to leave blank (or just write "undefined")! Notice that Lesson 1-6 distinguishes between plane geometry and solid geometry -- a crucial distinction in Common Core Geometry because the reflections, rotations, etc., that we discuss are transformations of the plane.

Then I move on to Lesson 1-1. This is based on an online discussion I had a few years ago on why students should learn math. I also include it as guided notes so that the students are listening when the teacher gives the reasons that they are taking this course. (The answers to the blanks beginning with the conversation are MBApolynomialinvestingdatasupermarket, and -- the object Americans use that has more computing power than the A-bomb -- cell phone!)

In the year since I first posted this lesson, I've been thinking about how to rewrite the lesson so that students are more responsive to it. In particular, I was thinking about last week's bridge puzzle, on which I wrote, "Back then, people spent their Sundays taking walks over bridges." Think about that statement for a moment -- entertainment back then was limited to Sundays. Back then, six days a week were workdays, on which no one expected to be entertained. Even on Sundays, the morning were devoted to church, so only the afternoons were amusing. And when we finally get to Sunday afternoon, all people did was cross bridges -- something that we wouldn't find entertaining today.

What has changed since the 18th century? The answer is technology -- that is, mathematics. Just as I mentioned in the worksheet, one especially widespread form of entertainment is the cell phone. We don't have to wait until Sunday afternoon for entertainment -- with our modern phones, we can be entertained at almost any time. Games and videos can be played anywhere, and if our friends live across the bridge, we don't need to cross it, since we can call or text them. All of this technology is available now because of mathematics.

Yet the greatest paradox is that, while math makes all of this technology possible, students use this technology to justify avoiding the study of mathematics. Traditionalists don't like the fact that students don't study as much now as they did in the past. Nowadays, the idea that one should study for two hours at once -- that is, go two hours without cell phones, TV, or other entertainment -- is unthinkable for many students, yet before modern technology, the idea of being entertained as often as once every two hours was equally unthinkable. The girl in the YouTube video says that she must study two hours per day just to pass her Algebra I class -- and that she's lucky if she can finish her weekend homework by Sunday afternoon. If the hypothetical "Math God" that she mentions in the video could make math disappear, she'd have a 4.0 GPA, and much less time needed to study -- but then the technology that makes YouTube possible would no longer exist, and she'd be spending Sundays crossing bridges to entertain herself.

We don't need to go back to Euler's day, 300 years ago, to find generations of students who were willing to work hard and forego entertainment. But some traditionalists go back to 100-year-old texts because they feel that newer texts have too many pictures. Technology progressed so much that photography, even in the mid-20th century, was inexpensive (going back to "a point is dot,") but that photo technology made texts even as early as then too entertaining, and therefore, not educational enough for the traditionalists.

The phrase Millennial Generation refers of course to the millennium. Strictly speaking a millennial is one who was born in the old millennium and graduated from high school in the new millennium. By this definition, I am not a millennial, since I was born in December 1980 and graduated high school in June 1999 -- still the old millennium. But some authors, such as Mark Bauerlein, consider the Dumbest Generation to be anyone under 30 at the time of its publication (2008). By this definition, I am a member of the "dumbest generation."

Naturally, most traditionalists and members of older generations who criticize millennials blame the problems of our generation on technology. This is why, when I teach this lesson, I want to point out that using technology to justify being a "dren" who can't count change makes us -- including myself as a member of the generation -- look bad. Of course, in a few years, I can't credibly claim to be in the same generation as my students -- some incoming students starting high school this year are already born in the new millennium (and so are no longer "millennials"). The important thing is that all of us, my age and younger, need to avoid being the "dren" who can't solve simple math problems and instead work on becoming the hero whose knowledge of math saves the day. This is what I want my students -- including those like the girl from the video, if she ever scrapes by Algebra I and is placed in a Geometry class like mine -- to realize.

Friday, August 28, 2015

Activity: Number Patterns (Day 3)

Today is the first lesson that I didn't post last year yet I'm including this year. I'm filling up the entire first week of school with activities, but last year I posted the first day of school on a Thursday, so I needed only two activities. With the first day of school on a Wednesday this year, here's a third activity to complete the week.

During the year, I pointed out that many texts begin with a lesson on inductive reasoning, which often entails completing number patterns. I like this sort of lesson at the start of the school year, but the U of Chicago text unfortunately doesn't contain such a lesson. I found today's activity in a different text that I mentioned earlier this week -- Michael Serra's Discovering Geometry.

Lesson 1.2 of the Discovering Geometry text is on Number Patterns. In this lesson, students will use inductive reasoning to find patterns in sequences of numbers, letters, and names.

I decided to create my worksheet from a variety of questions from the text. I began with some simple sequences where students had to find the next two terms. Notice that Exercise 5 is the Fibonacci sequence -- a nod to Fawn Nguyen, who gives her Geometry students a worksheet on that famous sequence on the first day of school.

Speaking of Fawn Nguyen, yes, many of my opening activities are based on Nguyen's. As it turns out, one of her best known lessons is on patterns -- except these are visual, not numerical, patterns:

Of course, many of the visual patterns lead to numerical patterns. For example, pattern #2 -- which looks very much like yesterday's building activity -- lends itself to the sequence 1, 3, 5, 7, ..., 85 as the number of rooms in each building. Nguyen makes her students find the 43rd term -- my worksheet only asks for the next two, which could be as far as the eighth and ninth terms.

Exercises 7 and 8 on my worksheet come from neither Serra nor Nguyen. Instead, they come from the texts used by the students I tutored. Those students enjoyed trying to figure out the patterns in these lists of names. I know that I've spent so recent posts about future presidents, but these lists are all about past presidents. Exercise 7 refers to George Washington, John Adams, Thomas Jefferson, and James Madison, so the correct answer is another James (Monroe), another John (Quincy Adams), and then Andrew (Jackson).

Exercise 8 looks similar, but this time it refers to money -- George Washington ($1), Thomas Jefferson ($2), Abe Lincoln ($5), and Alexander Hamilton ($10). So the correct answer is Andrew (Jackson again, $20), Ulysses (Grant, $50), and then it's all about Benjamin (Franklin, $100). As it turns out, one can actually extend this sequence. I was recently watching old 1960's episodes of the game show Let's Make a Deal on the new BUZZR channel, and often the host Monty Hall would offer contestants $500 bills (William McKinley) and $1000 bills (Grover Cleveland). There was even an episode when Monty showed a contestant an extremely rare $5000 bill that the bank had allowed him to show on that episode only -- had the contestant won it, she would have received a check for $5000 as the bill would have to be returned to the bank. James (Madison, not Monroe) was on the $5000 bill, so the sequence would continue Benjamin, William, Grover, James. It may be a good idea for teachers to give the related number sequence 1, 2, 5, 10, 20, 50, ..., as a hint.

Exercises 9 and 10 come from the Investigations section in Serra. This gives students the opportunity to work with a partner during the first week of school. Finally, Exercises 11 and 12 are two of the four conjectures that are part of this lesson. Notice that I wrote them just as Serra did, with the full equations written as 51 + 85 = 126. Perhaps I should have just written 51 + 85 = and required the students to calculate the sum. This is because I wanted to stick to the way Serra presents them and besides, I wanted the students to have fun during this activity making the conjectures, not having to calculate the sums. Of course, teachers who prefer the students to do the addition can simply white out the answers before copying.

The worksheet was getting long, so I stopped here. but notice that there are still many problems left in this section in the original Serra text. For the benefit of those who don't own the Serra text, let me reproduce Exercises 31 and 32:

Quatros: 4, 108, 60, 52, 36, 144
Not Quatros: 2, 29, 106, 18, 15, 22, 6
Which are Quatros? 86, 737, 42, 72

Semirps: 2, 13, 11, 23, 53, 97, 71, 47
Not Semirps: 15, 25, 209, 21, 190
Which are Semirps? 123. 67, 51, 27

Notice that "Quatros" are simply multiples of four. The word "Quatro" comes from the Latin word for four -- and we'll see that root later on in Geometry when we cover quadrilaterals. As it turns out, the modern Portuguese word for "four" is quatro. A few other Romance languages pronounce the word for "four" identically to the Portuguese, albeit with a slightly different spelling.

As for "Semirps," any nerd -- or even a dren -- can see that "Semirp" is "primes" spelled backwards. I do find it a bit awkward that the text pluralized "prime" to "primes," reversed it as "Semirp," then pluralized it again to "Semirps." Then again, one advantage to calling them "Semirps" rather than "emirps" is that the extra s- may trick readers into thinking about the prefix semi-, which is Latin for one-half -- especially right after seeing the Latin root for "four" in the previous question.

I was also considering including the first two sequences from the "Improving Reasoning Skills" section, which contains some bonus problems:

1. 18, 49, 94, 63, 52, 61, ...
2. O, T, T, F, F, S, S, E, N, ...
3. 4, 8, 61, 221, 244, 884, ...

I was able to figure out the first one, and I'd seen the second one before, but the third question stumped me -- and I suspect that it will stump our students as well. (The second one in the sequence is as easy as One, Two, Three!)

One of my favorite websites when considering number sequences is the On-Line Encyclopedia of Integer Sequences:

The OEIS is one of the oldest sites on the Internet. Notice that it was first created in 1964 -- long before the Internet existed as we know it! Back in the 1960's, users had to submit queries by sending it a primitive form of e-mail. Nowadays, of course, it is web-based like most other sites.

Many of the sequences in Serra's text are entries in the OEIS. Here they are:

180,360,540,720 (Notice the geometrical interpretation -- sum of the angles of an n-gon!)
1,3,6,10,15,21 (triangular numbers)
1,3,4,7,11,18 (Lucas numbers, similar to Fibonacci)
1,3,7,15,31,63 (sometimes called Mersenne numbers)
2,6,15,31,56,92 (given by the polynomial that generates these, (n+2)*(2*n^2-n+3)/6)
3,12,48,192,768 (there are some signed sequences in the database, but here the signs were ignored)

In fact, the only sequences I didn't enter were the ones containing letters or fractions, since this is in fact an integer sequence database.

Here's a sequence related to one of the lists of dead presidents that I entered:


Finally, here are the answers to the bonus questions:

18,46,94,63,52,61 (but I really did figure this one out before entering it into the OEIS)
4,8,61,221,244,884 (too hard for me, no problem for OEIS)
2,3,6,1,8,6,8 (too hard for me, no problem for OEIS)

The last sequence did stump the OEIS, however. Neither one of us figured out the sequence:

6,8,5,10,3,14,1, ...

Maybe you can figure it out, then try submitting it to the OEIS! Unfortunately, the OEIS has been swamped with submissions for months. Still, you can see why I enjoy the OEIS as a handy resource for integer sequences.

Let me wrap up this post by reminding you, the readers, that all of these problems come from my own version of Serra's text -- the first edition. My copy of the U of Chicago text is also the first edition. I only know about the contents of the newer editions by reading other websites such as the following link to a Minnesota classroom:

Assuming that each bullet point corresponds to a chapter, we see that the biggest difference between the old and new versions of the U of Chicago text is that the old version delays translations and rotations to Chapter 6, while the new gives all of the isometries in Chapter 4. I observe that the new Chapter 5 is called "Congruence Proofs" -- but I wonder what figures the text is proving congruent and how, considering that "Congruent Triangles" isn't until Chapter 7 (just as in the old text). Indirect proofs have been moved up from Chapter 13 to Chapter 11, with the new Chapter 13 being about circles -- the old Chapter 15. Trigonometry -- the old Chapter 14 -- has been dropped altogether.

I hope you enjoyed today's activity. Now you have three choices as to what to cover the first week of school -- and one can cover one, two, or all three activities during the first week.

Thursday, August 27, 2015

Lesson 1-5: Drawing in Perspective (Day 2)

We think back to what I posted for the second day of school last year. I stated that we could have an alternate activity for the first week of school. Let's look back at what I posted that day:

Lesson 1-5 of the U of Chicago text is the last section before the real geometry -- by which I mean the geometry taught in most other textbooks -- begins. But like the previous section, Lesson 1-5 contains some material that may be suitable as an Opening Activity.

Now recall that yesterday, I recommended -- following Fawn Nguyen -- that the lesson on Points in Networks should be the Opening Activity spanning the entire first week of school. So we don't really need a second Opening Activity. But still, I present Lesson 1-5 as an alternative to the activity in 1-4, so that one has a choice whether to open the course with 1-4 or 1-5. And as it turns out, Nguyen covered something similar to 1-5 in her class as well:

Now the U of Chicago text focuses on perspective drawings. Many introductory art classes teach the concept of one-point perspective -- and possibly two- and three-point perspective as well. But the text admits that most mathematicians don't draw, say, a cube in perspective.

Nguyen's lesson takes a different approach to drawing three-dimensional figures. For one, the focus on this lesson is on buildings. Her lesson begins by having some buildings already drawn and the students counting the "rooms" and "windows." (As it turns out, one "room" is one cubic unit of volume, and one "window" is one square unit of lateral area.)

I like the way that Nguyen's lesson begins. Unlike the bridge problem, where I wanted to avoid beginning the school year with a problem that's impossible to solve, here we begin with a very solvable problem. The only issue I have is with the second question, because it requires materials. I work from the assumption that most classrooms don't have the blocks and isometric dot paper that Nguyen's classroom has.

(As an aside, notice that cubes drawn on isometric dot paper are definitely not in perspective. This is because, while edges perpendicular on the cube intersect at 120 degrees on the iso dot paper, edges parallel on the cube remain parallel on the paper. Therefore there are no vanishing points.)

Now here's the difference between last year and this year -- in 2014, I didn't post a worksheet for this opening activity. Instead, I suggested that we can just use Fawn Nguyen's worksheet. But this year, I give a worksheet that we can use.

Then again, my worksheet is very similar to Nguyen's. On the front side, I gave the same example as she did and the three buildings for the students also come from the Ventura County teacher. I used two of her easier buildings -- A and B -- and the more challenging Building F.

The back side of my worksheet differs slightly from Nguyen's, though. Her worksheet specified the number of rooms and windows and asked the students to draw the buildings. Mine, on the other hand, simply has the students draw four different buildings with eight rooms and then asks them to count the number of windows in each one. When designing my worksheet, I took in consideration that my students might not necessarily have access to the manipulative interlocking cubes that are apparently available in Nguyen's classroom. Hopefully, this will be easier for the students.

Or one can simply do yesterday's bridge problem instead.

Wednesday, August 26, 2015

Lesson 1-4: Points in Networks (Day 1)

Today is the first day of school -- at least for this blog. I am following the calendar of one of the two districts where I sub, and this is when the students there go back to school.

Just like last year, I begin with Lesson 1-4 of the U of Chicago text, since it contains an excellent opening activity for the first day of school -- Euler's bridge problem. As I mentioned in my last post, I may make several changes to last year's curriculum for the first semester, but most of those changes won't appear until October or November. Because of this, many of these posts will be identical to what I wrote last year. This is what I wrote on this topic last year -- and of course, I changed the word "Section" to "Lesson":

I begin with Lesson 1-4 from the U of Chicago Text. Notice that the first five lessons of the text consist mainly of descriptions of a point -- these are omitted from most texts and could be considered a waste of time when there are 15 chapters to cover. But the first day or week of school traditionally consists of an opening activity, and it might be possible to make a suitable activity out of these lessons. Now some lessons, like Lesson 1-3 on ordered pairs, are certainly important. But many students find graphing to be a huge turn-off -- certainly not what most teachers want out of an opening activity.

But Lesson 1-4 -- now there's something. The Königsberg Bridge Problem is a famous math problem from nearly 300 years ago. Fawn Nguyen, a well-known math blogger and fellow Southern Californian -- she lives in Ventura County -- used this as an activity in her geometry class:

Notice that Nguyen taught this lesson after her class had "just finished with the State tests" and that this was her "geometry lesson for the entire week!!" Here, I suggest this as an opening activity, but it can still span the entire first week of school. On this blog, though, the first week consists of only Wednesday, Thursday, and Friday.

Nguyen was disappointed when her textbook, published by McDougal Littell, simply gives away the answer to the problem. Fortunately, the U of Chicago text doesn't do this. It instead gives Example 1, with network diagrams, and Example 2, asking which of the networks in Example 1 are traversable.

As we all know, the Königsberg Bridge Problem is impossible to solve -- it has no solution. But I don't want to start the class with a problem that the students can't solve -- they're already frustrated enough with problems that do have solutions when they just can't find them. Because of this, I decided to create the following images. The example is actually Example 1, Network III from the text. Problem 1 is Network I from the same example, and Problem 2 is actually Exercise 12 -- this one's a bit more challenging, but the network is traversable. Problem 3 is actually Königsberg.

The second image below is actually the same as the first, but with the maps drawn as networks. This is designed to be given on the second day. Just as Nguyen suggested in her own class, here is where the teacher begins discussing "vertices" (called nodes in the U of Chicago text), "edges" (arcs), and "valence" (even and odd nodes).

My suggestion is that the teacher have the student count the arcs at each vertex for the pentagon (all are two) and the pentagram (all are four). Both of these are easily traversable. The other diagrams all have some nodes with three arcs, and these are what make the networks harder to transverse. Eventually, the students will see that a network can't have too many "threes." Then the students can try drawing some of their own networks -- possibly with some "ones" and "fives" as well -- to find out that it's the odd nodes and how many of these there are that make the difference.

The whole point of this lesson is to point out that students should look for patterns, and that sometimes it's just as important to know why something is impossible as it is to know why something is possible.

Let me complete this with a note on pronunciation. The U of Chicago text points out that the name Euler ends up sounding like "Oiler." But how does one go about pronouncing the name Königsberg? I once read that the o-umlaut ends up sounding like "uh," almost like "ur." A Google search reveals a ten-second video in which this name is pronounced:

Oh, and since I mentioned Fawn Nguyen's blog in this post, the readers may be wondering what Nguyen has taught in the meantime. Well, she hasn't posted anything for the new school year yet -- her most recent post is dated July. Based on her posts from last year, this week should be the first week at Nguyen's school as well.

Instead, let me comment on what Nguyen did for the first day of school last year. She wrote that she had three preps -- sixth grade math, eighth grade math, and Geometry. No, she didn't give this bridge problem on the first day last year -- here's a link to what she actually did:

Notice that for Geometry, Nguyen gave a lesson to introduce the students to conjecture and proof -- and it's all about the Fibonacci sequence.

Monday, August 24, 2015

How to Fix Common Core: Some Last Words

Today is my final post of the summer -- and as you can see, it is the last post of my "How to Fix Common Core" series.

As we have already seen, these "How to Fix Common Core" posts tend to be slightly political -- if only because the Common Core debate is inherently political. When I reach the first day of school, I want to get back to purely mathematical posts with no politics. Since this will be my last chance to make a political post, I might as well get as political as I possibly can.

In today's post, I begin by discussing which 2016 candidates are pro-Core or anti-Core. Afterwards, I talk about the current president and his daughters' education. Readers who wish to avoid politics can just ignore this post altogether and wait for my next post, which will be purely mathematical.

I've already mentioned the first presidential debate and the comments made by some of the candidates regarding Common Core. In today's post, I decided to visit the well-known website "I Side With" and take the 2016 Presidential Quiz:

Now I was hoping that I could answer only the Common Core question, since the only political issue that I care about on this blog is the Core -- but unfortunately, I had to answer more questions in order to get a list of candidates. So instead I decided just to answer "yes" on all of the questions that appear on the first page, but rate them as "least important." (It appears that some of these "yes" answers favor Democrats and the others favor Republicans, and so hopefully they cancel out.)

Then, of course, I answer the Common Core question and rate it as "most important." Here is the Common Core question given at the link:

Do you support the adoption of Common Core national educational standards? learn more

I will take the quiz several times -- first answering "yes," then answering "no," and so on down this list, to see which candidates appear.

First, I answered "yes" on the Common Core question, and received the following results:

97% Hillary Clinton (D)
92% John Kasich (R)
88% Jeb Bush (R)

There are no surprises here. It is well-known that Hillary is supportive of the standards, while Kasich and Bush are the two GOP candidates who are pro-Core.

Now let's try "no" on the Common Core question:

92% Bernie Sanders (D)
90% Donald Trump (R)
90% Mike Huckabee (R)
90% Rand Paul (R)
90% Bobby Jindal (R)

Now the four Republicans who appear on this list are expected -- we already know that Trump, Huckabee, Paul, and Jindal oppose the Common Core. But unexpected here is the presence of a Democrat at the top of the list -- just as there are pro-Core Republicans, there exists an anti-Core Democrat, Sanders. As there aren't as many Democrats in the race as Republicans, we haven't heard as much from Sanders, but apparently he once said something that was against Common Core and so it made it onto this website.

It's interesting that for both "yes" and "no," a Democrat appeared at the top of the list. It could be that while I tried to be neutral with the other questions and weighted them as "least important," those other questions still ended up leaning slightly Democrat. Notice that within each party, the candidates seen as closest to the center (Clinton for the Democrats, Bush and Kasich for the Republicans) tend to favor the Core, while those furthest from the center on either side tend to oppose the Core.

If I choose "No, the Common Core Standards are a watered down version of my state's current education standards," I get the following list:

90% Donald Trump (R)
90% Mike Huckabee (R)
90% Rand Paul (R)
90% Bobby Jindal (R)

Now Sanders drops down the list, while the four Republicans remain at 90%. This shows us that the GOP Core opponents are much more likely than Sanders to believe that the standards are watered down -- which, as we know, is the dominant criticism from the mathematical traditionalists as well.

Finally, if I choose "I support the concept but not the implementation":

63% Hillary Clinton (D)
58% Bernie Sanders (D)
53% Bobby Jindal (R)
53% Martin O'Malley (D)

I like this list the best because "I support the concept but not the implementation" best reflects my own personal feelings about the Core. After all, that's what this series "How to Fix Common Core" is all about -- my desire to fix Common Core indeed indicates my support for the concept but not the implementation of the Core. Also, I am still undecided about which candidate to support in the election -- and of course there are other issues to consider besides Common Core. So I definitely prefer for my own list to have no candidate from either party above 90%. (Of course, when I do decide upon a candidate, I won't post it on the blog, since that would be off-topic politics.)

Now that's enough about the possible future president, but what about the current president? One frequent criticism of the standards and their implementation is that President Obama, as well as Secretary of Education Arne Duncan, send their children to private schools where Common Core is not implemented. For example, here's an article from the Washington Post:

It is understandable that that the president would send his daughters Malia and Sasha to a private school, since there may be security issues at a private school. But the point being made by Core opponents is that the president and Secretary of State are hypocrites -- that is, they impose standards that are knowingly bad on the population, then protect their own children from these bad standards by placing them in private schools as far away from the Common Core as possible.

Of course, I don't actually agree with this assertion. But I've come up with a way to implement national standards without worrying about presidential hypocrisy -- because of this, let's call the following proposal the (Presidential) Consistency Core.

Here's how the Consistency Core would work -- we look at the curriculum at whatever school the president sends his (or her) own children to, and then simply define that curriculum to be the Consistency Core Standards! This means that the curriculum could change every 4-8 years, whenever there is a new president.

Under this proposal, if the president has no school-aged children, then we go to the vice president, then the education secretary, then other members of the Cabinet, and finally other high-ranking members of the party in control of the White House, until we reach one with school-age children. Notice that of the declared GOP candidates who participated in the debates, Cruz, Jindal, Rubio, and possibly Walker all have children who will still be school-aged in 2017, so the Consistency Core would apply to them. Because the new president would take office in January 2017, the old standards would stay in effect through the 2016-17 school year, and then the standards reflecting the new president's school would be implemented for the 2017-18 school year.

It is well-known that Malia and Sasha attend Sidwell Friends School in Washington, DC. As it turns out, Vice President Biden's grandchildren attend this school, and indeed presidents going all the way back to Teddy Roosevelt have been sending their children there. So there's a chance that the next president's children would also go there, meaning that the Consistency Core Standards wouldn't change in 2017 anyway.

Now let's see exactly what these Consistency Core Standards would look like. As we look at these, we keep in mind that Malia started attending the school in the fifth grade and will begin her junior year in September -- she will end up graduating from the school. Sasha, meanwhile, began in the second grade, is about to start the eighth grade, and will still be in her sophomore year when her father's term is over.

Here is a link to Sidwell Friends:

We see that at Sidwell, middle school is defined as grades 5-8 -- a configuration that is common in other states, but not California. Below this is not "elementary school" but the lower school, and above this is not "high school" but the upper school. We also notice that Sidwell is one of the few remaining schools that starts after Labor Day, and not in August when many other schools around the nation are beginning school.

As this is a math blog, I obviously want to focus on the math curriculum. But it's interesting to see what the curricula for the other subjects look like at Sidwell. Let's look at English for eighth grade -- the grade that Sasha is about to enter:

In the eighth grade students currently read Brave New World, The Tempest, Things Fall Apart, Lord of the Flies, and The Odyssey, among other works fiction, non-fiction and poetry.
In both years there is a particular emphasis on writing, both analytical and creative.  Students learn to plan, draft, and revise paragraphs, essays, and imaginative pieces.  They develop grammatical understanding in relation to their writing, and their vocabulary grows with respect to their reading. Efforts are also made to establish interdisciplinary connections with other subject areas in lessons, units, and/or assignments.

Some people interpret the Common Core standards as having too much emphasis on nonfiction and not enough on literature, so they would be glad to see the above listed works included in the Consistency Core curriculum. Furthermore, many of these works are introduced much earlier at Sidwell than at many public schools, even before the Core. At my own school, I read The Odyssey as a freshman, Lord of the Flies as a sophomore, and Brave New World as a senior!

There are many differences between the history curricula at most public schools and at Sidwell -- especially in the lower school.

First and Second Grade

First and second graders explore subjects in greater depth while actively engaged in discussions, field trips, and hands-on projects. Building a model longhouse, creating an island mural, growing silkworms, and mapping the route of the Mississippi River are activities which necessarily draw upon and connect many disciplines. Longer term studies of selected topics emphasize the integrated nature of learning and help children compare and reflect on their world and their heritage as citizens. Class trips along with other primary experiences, assemblies, interviews, parent stories, internet resources, shared readings of fiction and non-fiction, videos, and displays of objects all help bring historical and social themes to life.
During the U.S. history year, first and second graders focus on Native American cultures and experiences prior to the arrival of Europeans. During the world cultures alternating year, studies vary by class and by year. Recently classes have investigated life in Mexico, Australia, Kenya, and the Scandinavian countries. Regardless of the region under consideration, important themes include daily life, cultural practices and traditions, geography and environmental issues, rights and responsibilities, similarities and differences with life in 21st century America.

Third and Fourth Grade

Third and fourth graders might explore the burial practices of the ancient Egyptians one year and the pathways of the underground railroad the next. Our alternating year schedule allows students to investigate both American history and an ancient culture during their last two years in Lower School. The selected ancient culture may vary from year to year or class to class. In recent years classes have focused on eras such as ancient Greece, ancient Egypt, ancient and traditional China, Japan, or India, and the Inca and Mayan civilizations.
During the American history year the third and fourth grade classes explore the peopling of the United States. While the amount of time spent on a specific topic may vary by classroom, children learn about the experiences of early European settlers meeting indigenous peoples, colonial life and the struggle for independence, the westward expansion, and the waves of immigration, both voluntary and involuntary (including slavery) that helped build the United States.
Geography, science, drama, the arts, fiction, and non-fiction writings are integrated into each year’s study. Students are involved in learning through active, hands-on projects, simulations, discussions, readings, research investigations, videos, field trips, special visitors, and computer and library resources.

There are actually many similarities between social studies at Sidwell and history at my own elementary school under the path plan. The Preparatory Path corresponded to grades 5-6, so did students on that path learn fifth grade U.S. history or sixth grade world history? The answer is that they learned both, in alternating years. So one year all Preparatory Path classes would focus on American history, and the next year would be spent on ancient world history. We see that Sidwell follows the same pattern, even though the school doesn't combine grades into paths as my own elementary school did.

But as I said, it's the grades below the fifth grade where the difference lies. We see that Sidwell alternates between U.S. and world history throughout the Lower School. So Sasha, during her father's first term, would have learned history much earlier than I did. This is mostly due to the California
Social Studies standards:

We only need to look at the titles of the elementary school courses:

Kindergarten: Learning and Working Now and Long Ago ........................................................... Grade One: A Child’s Place in Time and Space ................................................................................ Grade Two: People Who Make a Difference ..................................................................................... Grade Three: Continuity and Change ................................................................................................ Grade Four: California: A Changing State ...................................................................................... Grade Five: United States History and Geography: Making a New Nation ..............................

We see that California's K-3 standards are mostly covered in kindergarten at Sidwell (or maybe first grade, to be generous), while Sidwell's 1-4 standards aren't covered in California until fifth or sixth grade (maybe fourth grade, to be generous, if we count "an American Indian culture" as part of California history). So we see that Sidwell's lower school standards are about 2-4 years ahead of those in California. Many people wish that more actual history would be taught in Grades 1-3 than usually occurs under the standards of most states.

Of course, history is not a Common Core subject -- it is left to the states, while science will be made national under the new Next Generation Standards. We notice that just as with my path plan, certain subjects are taught in homeroom at Sidwell -- English, math, and history -- while science is taught in a lab by a specialized teacher. An argument can be made that if we really want our Consistency Core to line up with Sidwell's core curriculum, history should be covered under Consistency Core while science is left to the states, rather than vice versa.

But a national history curriculum may be dangerous. We've already seen with the AP U.S. History debate that the curriculum is highly partisan. This is seen most noticeably with the way various curricula treat Thomas Jefferson, who sadly gets the short shrift in curricula penned by both Democrats and Republicans. In particular, Democrats condemn Jefferson as a slave owner (as in the original proposed AP U.S. History standards) while Republicans condemn him as radical (for example, in the current Texas standards). So one may avoid such controversy simply by not having national history standards.

Of course, it's the math standards that I want to discuss in more detail. In the lower school, the standards are divided up into PK-K, 1-2, and 3-4 -- once again evoking the path plan. There are some topics that traditionalists like and others that progressives like.

For example, under grades 1-2, we see:

Number and Operations 
Estimation skills
Count by 2s,5s,10s
Learn basic addition and subtraction facts
Develop an understanding of place value
Learn about odd and even numbers
Count to 100; learn to read and write numbers through 1000
Develop fractional concepts
Balance an equation
Introduction to multiplication and division
Explore use of calculator

Surely traditionalists don't want first or second graders to "explore use of calculator." Indeed, many traditionalists would prefer that calculators were banned until middle school, if not high school.

On the other hand, under grades 3-4, we see:

Number and Operations 
Read and write numbers through millions, billions
Understand and apply basic place value concepts
Learn basic facts multiplication and division 
Add and subtract multi-digit numbers
Create models for multiplication and division; understand remainders
Understand prime and composite numbers
Solve using factors and multiples
Multiply and divide two digit and three digit numbers by single and double digit numbers
Learn algorithms for multiplication and division 
Represent fractions and decimals with appropriate models
Recognize equivalent fractions; compare fractions and decimals
Estimate, judge, and order fractions and percents
Combine fractions; simple addition and subtraction
Explore positive and negative integers and powers of 10

Traditionalists, of course, would like "learn algorithms for multiplication and division." Of course, nowhere does it say "learn standard algorithms for multiplication and division." But I can almost hear the traditionalists point out that before the Common Core, it never occurred to anyone that "learn algorithms" could refer to anything other than the standard algorithms. Given this assumption, Sidwell teaches standard algorithms in grades 3-4 than the Common Core teaches in grades 5-6, and so they'd embrace including this in the Consistency Core.

Let's move on to the middle school. Here's a link to the middle school curriculum:

Under grades 5-6, we see:

The fifth grade math program continues instruction in computation of whole numbers, fractions, and decimals.  Problem solving and work with mathematical tools pervade the entire curriculum as students practice estimation and geometry skills, apply principles of ratio and proportion, and write simple equations.
The sixth grade math program develops a greater mastery of computation and estimation through applications in various problem-solving activities. Among the topics that students cover are ratios, percents, fractions, variable expressions and simple equations, operations with integers, solid figures and plane geometry. Students demonstrate their understanding through a range of activities including projects and written assessments.

Traditionalists might not like that last sentence, about "activities including projects." Other than that, they'd find the curriculum quite solid, including integers in sixth grade -- a year earlier than required under Common Core.

Here's an important point about the middle school math curriculum:

A part of every student’s day in fifth and sixth grade is Math.  The Middle School utilizes grouping in all four grades.  However, the school remains committed to the goal of flexibility as students move from one year of Math to the next, and parents who are anxious about their child’s placement in a Math group should be assured that choices remain open to students throughout the four-year program.  Grouping, after all, is a tool for best serving the needs of individual students.

We see that Sidwell employs ability grouping -- something that traditionalists favor. Of course, we know that just as with tracking, students are often divided into ability groups based on race rather than their actual ability. Then again, if someone as high-profile as the First Daughters had been placed in a low-ability group due solely to their race, it would be widely reported and we'd know about it by now.

We proceed with the seventh grade standards:

Seventh graders cover topics selected from the following areas: number patterns; properties of groups; matrices; network theory; geometry; transformations; ratio, proportion, and rates; percent; relations and functions; probability; algebraic notation; circuits; symbolic logic; and math history.  Students use technology in a variety of ways.  Problem solving is emphasized throughout. 

Some of these seventh grade topics seem out of place in a middle school classroom. I wouldn't be surprised if for "properties of groups," only finite cyclic groups (clock arithmetic) and the more important infinite groups (integers, rationals, reals) are actually taught. For "matrices," I'd reckon that it's only addition and scalar multiplication -- I can't imagine that Sasha would have been required to multiply two matrices last year. "Circuits" and "symbolic logic" go together -- notice that there exist both parallel (OR) and series (AND) circuits. I wonder how much "math history" the seventh graders would actually have to learn.

Now let's move on to the eighth grade -- a controversial year under the Common Core Standards:

Eighth graders are placed in introductory, regular, or accelerated algebra.  In both regular and accelerated algebra, students develop algebraic concepts and skills through exploration, rigorous analysis, and problem solving.  In introductory algebra, students cover fundamentals in depth at a pace that prepares them for a formal algebra class in ninth grade. 

So we see that eighth graders are expected to take some sort of algebra class. Both the "regular" and "accelerated" algebra courses appear to be Algebra I classes, while since the "introductory" class prepares the students for Algebra I in freshman year, it would be considered a pre-algebra class.

We've discussed many times here on the blog that the Common Core eighth grade class is not equivalent to Algebra I, as the Core expects students to take Algebra I as freshmen. I'm not sure whether even the introductory algebra course would be considered equivalent to Common Core 8 -- instead "introductory algebra" sounds more like the first part of Algebra I at a slower pace. In particular, there doesn't appear to be any geometry included in the introductory algebra class -- as opposed to both Common Core 8 and Saxon Algebra 1/2 which include geometry. The geometry included in both of these appears to be part of the seventh grade course at Sidwell. I wonder on which of the three tracks -- introductory, regular, accelerated -- will Sasha be placed this September.

Now let's look at the upper school curriculum. We begin with the following link:

I know what the traditionalists are hoping -- they want to see classes like Algebra I, Geometry, and Algebra II, and not Integrated Math I, II, or III. They wish to prove that integrated math is so bad that of course the president would send his daughters to a school with traditionalist Algebra I, Geometry, and Algebra II, and avoid integrated math like the plague.

But as we see on the chart, there are pathways for both traditionalist and integrated math. Moreover, the pathway with Math I, II, and III is actually the higher track! Notice that the integrated pathway leads to both Calculus BC and Linear Algebra (a college-level class beyond Calculus) while the traditionalist pathway maxes out at Calculus AB. So this disproves the traditionalist claim that integrated math is less rigorous than traditionalist Algebra I, Geometry, and Algebra II -- as if the fact that other countries use integrated math isn't already a counterexample.

To see what's going on, we notice that this chart shows only the upper school classes -- it doesn't show which eighth grade classes lead to the various ninth grade classes. Recall that there are three levels of eighth grade algebra, and these readily correspond to the three tracks on the chart. In particular, accelerated algebra leads to Math I, regular algebra leads to Geometry, and introductory algebra leads to Algebra I.

We notice that Math I, II, and III aren't the ordinary integrated classes that many school districts in California and other states are adopting. Indeed, since the students entering Math I have already completed Algebra I in eighth grade, Math I is actually geometry and trigonometry. Math II consists of some topics from Algebra II and Pre-calculus, and Math III wraps up Pre-Calculus and introduces a few basic topics from calculus in preparation for Calculus BC the following year.

This reminds me of what a special magnet school here -- the California Academy of Math and Science -- used to do. Students were required to complete Algebra I at their home middle schools in eighth grade in order to be admitted to the school. Then from grades 9-11, students completed three years of integrated math -- with the freshman math class being more like Math II at other schools than Math I. Then the students were considered to be ready for AP Calculus as seniors.

As it turned out, just as other schools in California used Common Core to justify converting to integrated math, the California Academy of Math and Science used the Core to justify converting back to traditionalist math. (I believe the same happened in the state of Georgia.) Here is the link to the school website. We see that students are still expected to have completed Algebra I in the eighth grade, but now the freshman class is Geometry, not integrated math:

Still, both CAMS in the past and Sidwell in the present implement a sort of integrated "sandwich" -- Algebra I in eighth grade, three years of integrated math in between, and AP Calculus as a senior.

We know that Malia is about to start her junior year. If she's on the left track, she'd be taking Intermediate Algebra, if she's on the middle track, she'd be taking Pre-calculus, and if she's on the
right track, she'd be taking Math III. Of course, I don't know what track she's on.

Now suppose we were writing some Consistency Core standards. Is it possible to have a single set of standards cover all three tracks -- the left, middle and right tracks?

One way to do so could be to start out writing some units for each class on the traditionalist track. We assume that there are ten units in each class, so we have something like:

1. First Algebra I topic
2. Second Algebra I topic
3. Third Algebra I topic
10. Last Algebra I topic
11. First Geometry topic
12. Second Geometry topic
13. Third Geometry topic
20. Last Geometry topic
21. First Algebra II topic
22. Second Algebra II topic
23. Third Algebra II topic
30. Last Algebra II topic
31. First Pre-calculus topic
32. Second Pre-calculus topic
33. Third Pre-calculus topic

And then we simply state that on the middle track, eighth grade is standards 1-10, ninth grade is standards 11-20, tenth grade is standards 21-30, and so on. But on the right track, we say that eighth grade is, say, standards 1-12, ninth grade is standards 13-24, tenth grade is standards 25-36, etc., while on the left track, eighth grade may be only standards 1-8, ninth grade only standards 9-16, and so on.

In fact, we notice that with this plan, it now becomes obvious why there is any integrated math in the first place. A freshman on the right track finishes Geometry early and is able to begin some Algebra II topics early -- so the class integrates Geometry and Algebra II. And a freshman on the left track still has to finish some Algebra I topics, but right afterward can start Geometry -- so the class also integrates Algebra I and Geometry. Only a student on the middle track has traditionalist classes, since the standards were ordered into courses to suit the middle track. One can argue that the student I tutored last year is following a similar plan -- he started Algebra I in seventh grade, then finished Algebra I in eighth grade before moving on the Geometry. Therefore the class he took this year was actually integrated math, as it included both Algebra I and Geometry.

But of course, this list of standards, while elegant, doesn't accurately describe the actual courses that exist at Sidwell. For example, as this is a geometry blog, let's compare the three geometry classes listed on the chart -- Geometry: An Inductive Approach, (regular) Geometry, and Math I:

GEOMETRY—1 credit; year course Open to: 9, 10 Meets 5 times per week Prerequisites: Algebra 1 or equivalent Geometry covers Euclidean plane and solid geometry. Emphasis is on orderly and logical thinking, on the ability to develop a sound, precise, logical argument, and on the theoretical derivation and practical application of theorems and propositions. Proof is an integral part of the course. Specific topics in geometry include line segments, lines, rays, planes, congruence, triangles, quadrilaterals, regular polygons, inequalities, perpendicular and parallel lines, similarity, and circles including tangent and secant lines and chords. Throughout the year, algebra review is a regular aspect of class work. Coordinate geometry is used both as a way to introduce and provide a different perspective on geometric topics and also as one way in which topics of Algebra 1 will be thoroughly reviewed. Additionally, basic constructions are introduced and right triangle trigonometry is covered extensively (through Law of Sines and Law of Cosines), along with basic probability and data interpretation. 

MATH I —1 credit; year course Open to: 9 Meets 5 times per week Prerequisites: Algebra 1 and departmental approval Math I is an intensive and accelerated course in geometry recommended for very able and independent math students. The topics of Geometry are covered with greater attention to rigorous proof and the deduction of results from a small number of postulates. Additional topics include advanced constructions, loci, proof by contradiction, a more intensive study of trigonometry, and probability and data interpretation. This course is student-driven and inquiry based, and students must be prepared to take responsibility for their own progress.

GEOMETRY, AN INDUCTIVE APPROACH—1 credit; year course Open to: 10, 11 Meets 5 times per week Prerequisites: Algebra 1 Students in Geometry, An Inductive Approach study both plane and solid geometry. The inductive approach of the class requires students to explore problems by hand and using Geometer’s Sketchpad. On the basis of that work, students make generalizations which are formalized into the standard postulates and theorems encountered in Geometry. Throughout the year, algebra review is a regular aspect of class work.

Now "Geometry: An Inductive Approach" refers to Michael Serra's well-known text, which I've discussed here on the blog before (and I own his first edition). In his text, he delays proofs until the end of the book, so that he can introduce the concepts without the students having to worry about proving anything. I've learned that Serra has published a third edition of his text this year. Two differences between his first and third editions is that Chapter 4 of my edition, "Line and Angle Properties," doesn't appear in the third edition, and the three chapters in my edition where Serra finally shows proofs, Chapters 14-16, appear as a single Chapter 13 in the third edition.

So if my list of standards above were accurate, the regular Geometry and Math I classes would cover the same topics as the Serra class, but only faster. But this is incorrect. The difference between Serra and regular Geometry is that in that latter, students actually have to prove the theorems. They wouldn't simply cover the Serra text faster, or even the same chapters in a different order (such as Chapter 13 on proofs before any of the conjectures), but would use a different text altogether.

Indeed, Dr. Franklin Mason's online Geometry text would work well for regular Geometry. For the Math I course, one could use Dr. M's Geometry text and focus on the "honors" questions -- perhaps even adding in the Triangle Exterior Angle Inequality (TEAI) proofs from his old text, since after all, this course focuses on proving results "from a small number of postulates." Since Math I includes trigonometry, one could cover Dr. M's Chapter 8 last and follow it up with more advanced trig.

So we see that simply numbering the standards and assigning different ranges to the three different levels doesn't fully encapsulate the difference among the three tracks. One could still write Consistency Core Standards corresponding to the classes, but it will be more difficult. Still, the naive numbering of the standards fits well with my test scoring system where 800 = ready for Algebra I, 900 = ready for Geometry, 1000 = ready for Algebra II, and so on.

Oh, that's right -- what would happen to the tests under the Consistency Core? One of the biggest complaints is that the First Daughters at Sidwell don't have to take Common Core tests -- after all, the Valerie Strauss article to which I linked earlier is more about the tests than the standards. Under the Consistency Core, it would make sense just to make all the standardized tests optional. For example, parents might wish to take the tests and then use the results to challenge the placement of their children into an ability group or on a track if they feel that the placement is too low. (The Raenbo CAP tests mentioned in my last "How to Fix Common Core" post might also fit here, but these may be too radical since they were designed to replace grades, not just standardized testing. Sidwell clearly still has letter grades.) Only tests that the First Daughters are required to take would be required for students under the Consistency Core.

Note that the Consistency Core is not necessarily my preferred approach for fixing Common Core. I only mention this proposal for those whose biggest complaint against Common Core is that the president's children aren't subject to it. In particular, if I were to set something like this up myself, I might wait until ninth grade for Algebra I, and then accelerate to Calculus from there. Only one of my three tracks would lead to college -- but then again, based on the California master plan, only about 33%, or one out of three, students needs to go to college. But under Consistency Core I must offer Algebra I in eighth grade because that is when Sidwell offers it.

By the way, before Malia and Sasha attended Sidwell, they were students at the University of Chicago Laboratory School. That's right -- as in the U of Chicago text. The phrase "laboratory school" may bring up notions of students being treated like guinea pigs -- but notice that the U of Chicago Lab School is a private school, so the parents choose to send their children there. The complaints about students being treated like guinea pigs mainly referred to public schools where the schools had no choice but to implement Common Core. (Of course, Malia and Sasha would have read from the U of Chicago elementary math texts that are extremely unpopular with traditionalists.)

Speaking of the U of Chicago text, it's time for me to prepare for the school year, since my next post will be on the first day of school in one of the districts where I work. At the end of the old school year, I mentioned the outline of what the first semester of the school year will look like:

Start of 1st quarter through Labor Day: Introduction
Labor Day through Columbus Day: Reflections
Columbus Day through end of 1st quarter: Rotations
Start of 2nd quarter through Thanksgiving: Translations
Thanksgiving through winter break: Glide Reflections
Winter break through end of 2nd quarter: Conclusion

Recall that this district follows what I call a Middle Start calendar -- the school year begins in August, but not quite early enough to end the first semester before Christmas. Now if I take all of my lessons from last year and naively shift them two weeks later to fit this calendar, the following correlation to the U of Chicago text appears:

August 26-September 4: Chapter 1
September 8-October 9: Chapter 2, Chapter 3 (up to Lesson 3-3), Chapter 4
October 13-30: Mostly Dr. Hung-Hsi Wu's Lessons
November 2-20: Chapter 5, Chapter 6 (up to Lesson 6-2)
November 30-December 18: Finish Chapter 6, Chapter 7 (up to Lesson 7-4)
January 4-21: Finish Chapter 7, Review

But there are a few changes that I said I wanted to make. First of all, I am considering moving the first half of Chapter 7 up to the Reflections unit. This is because the Triangle Congruence Theorems SSS, SAS, and ASA don't appear until Chapter 7 in the U of Chicago text. Some people dislike Common Core because they prefer proofs based on SSS, SAS, and ASA to those based on rotations, reflections, and translations. We found out by looking at the PARCC released test questions that proofs based on SSS, SAS, and ASA do appear on the PARCC -- but squeezing the congruence theorems in the week before Christmas doesn't really do them justice.

The other change I wanted to make was to use translations to prove the properties of parallel lines. I find that this will be more intuitive for the students than Dr. Wu's proof based on 180-degree rotations, but the problem is that I must be careful to avoid circularity.

Provided that I figure out how to prove the properties of parallel lines correctly, I may be able to rearrange the lessons to solve both problems at once. My current Rotations unit focuses so much on Dr. Wu's 180-degree rotations, but if I use translations to prove the properties of parallel lines instead, I won't need to spend so much time on 180-degree rotations. This would free up time in October to derive SSS, SAS, and ASA instead.

Well, I still have plenty of time left to figure out my parallel line proofs. For now, I will be returning to Chapter 1 of the U of Chicago text. By the way, notice that I'm no longer referring to Section 1-4 or whatever, but instead I'll call it what the text calls it -- Lesson 1-4. We're not merely covering "sections" of a book -- we're teaching our students "lessons."

By the way, we may ask, which Sidwell Geometry class does the course that I'm posting to the blog resemble the most? Well, since I'm proving theorems, it's surely not An Inductive Approach. I suspect that my class is somewhere between regular Geometry and Math I. In particular, some of the proofs I post may be on the level of Math I, but based on the work I expect the students to produce, it's probably closer to regular Geometry.

I hope you enjoyed your summer. My next post will be on the first day of school in my district, which is Wednesday, August 26th.