Lesson 1-3: Ordered Pairs as Points (Day 13)

Today I subbed in a seventh grade science class. This marks my first day of subbing of the new school year.

In my old district, today is Day 13, but in my new district, it's only Day 5. This is the first time that I've ever subbed as early in the year as Day 5 in my district. And it's only the second time that I've ever subbed before Labor Day -- in 2015 I also subbed on the Friday before the holiday, but in that district, it was Day 8. Last year, I started subbing on Day 7. (You can refer to my posts in September 2015 and 2018 to read about those subbing days.)

We return to "A Day in the Life" format for blogging days on which I sub. I'm inconsistent when it comes to "A Day in the Life" -- I always try to blog when subbing a math class, but for other classes it often depends on the situation.

Since my focus is on classroom management, I try to post on middle school days unless there is a special aide who manages most of the classes. Today there's an aide for two of the five classes, which isn't a majority. Therefore I will post "A Day in the Life" today.

But before we start, here is the Blaugust prompt for today -- the thirtieth from Shelli's list:

A Day in the LIfe  (#DITL)

Hey, that's very convenient! I want to post "A Day in the Life" for today's subbing, and today's Blaugust prompt is also "A Day in the Life." And so let's just begin "A Day in the Life":

8:15 -- This is homeroom. Ordinarily there is a video to watch during homeroom, but today the online link isn't ready until it's too late. So instead, I lead the Pledge of Allegiance, and then an eighth grader from ASB gives the announcements.

8:20 -- Homeroom ends. This is the middle school where all classes rotate, and today the rotation begins with fourth period. In case you're wondering why the rotation begins with fourth period on Day 5, it's because the rotation starts with Period 1 on the first two days.

Anyway, fourth period is an honors science class. But since today is only Day 5, it's still the first week of school. And you know what that means -- first week activities! Thus many of the activities suggested by other bloggers actually appear in this class.

First of all, the students do have name tents. These aren't exactly the same as VanDerWerf name tents, where students ask a question or leave a comment everyday during the first week and the teacher must respond. But in addition to the name, students write their goals for science on the back and write something inside too. So it's likely these tents were inspired, directly or indirectly, by VanDerWerf.

Each day this week there's supposed to be some sort of "seating challenge" where students must complete a challenge and then take a seat based on the results of the challenge. I don't have a clue what the seating challenge is supposed to look like. But because it's a sub day, the seating challenge is cancelled, and kids are free to sit wherever (as they don't have assigned seats due to the challenges).

Finally, we reach the main activity of the day. I write the instructions to the activity on the board:

1. Choose one picture of a beaker.
2. Write first name and initial of last name on the beaker.
3. Draw items that represent you, in color with no white or pencil showing.
4. Cut it out after it is completed.
5. Turn it over and write full name and period on the back.

This activity is due on Tuesday. Completed projects will be hung on the walls. Oh, and any names I write on the bad list will receive detention. Of course, this is an honors class, so there's no behavior issues at all.

9:15 -- Fourth period leaves and fifth period arrives. This is the first of two classes with an aide.

10:05 -- A lock down drill is announced. I don't wish to turn this post into politics, but what does it say about recent events that there needs to have a lock down drill during the first week of school?

Some students are talkative during the lock down drill. Notice that this isn't an honors class -- and the special aide had already left for another classroom before the drill began.

10:20 -- The lock down drill ends, and it's time for snack.

10:30 -- Sixth period begins. This is the second of two classes with the aide. This is not an honors class -- in fact, one of the students is absent because she's already been suspended. Notice that not even my seventh grade boys at the old charter school -- some of whom were extremely defiant -- managed to get themselves suspended that fast! What's more unsettling is that this girl has just arrived this week from her (K-6) elementary school.

11:15 -- Sixth period leaves and first period arrives. This is the second of two honors classes. Of course, there are no behavior issues in this class.

12:10 -- First period ends and it's time for lunch. My break extends to second period conference.

2:05 -- The homeroom class returns for third period. This class is labeled as a "sheltered" class, which indicates that there are many English learners in this class. As often happens in such classes, this class is quieter than Periods 5-6 (since students are reluctant to speak due to language issues), but not as quiet as the honors classes. There's also a second girl who's been suspended from this class.

2:55 -- Third period leaves, thus ending my day.

I've been making it a habit to sing some of my old songs in class as an incentive. Today I choose a song that's (somewhat) related to science from this time three years ago -- "The Need for Speed" (aka the mousetrap car song -- see yesterday's post for pictures of the mousetrap cars). In fifth period -- longer due to the lock down drill -- I sing another science song, "Meet Me in Pomona, Mona" (on this opening day of the LA County Fair). This is just before the lock down, not during the drill itself (when everyone needs to be silent to hear instructions over the intercom).

Another habit I wish to develop this year is emphasizing the positive. Of course, it's easy to say positive things about the two honors classes, since those students are so quiet and hardworking.

But I have to push myself to say good things about students in the other three classes. Today, I praise students who are making nice drawings or coloring them in neatly. Also, when the class is noisy, I remind the students that most of them are being quiet, except for those few who are clearly doing something wrong (like speaking loudly or faking knocks at the door). There's still room for me to grow here, but at least I was able to avoid arguments or writing anyone on the detention list.

Normally, whenever I sub in a science class, I lament seeing all the wonderful lessons that I could have taught at the old charter school, but didn't. Today's lesson really wasn't much science-y -- but it does remind me of some opening week activities that might have improved my classes. Indeed, I discussed many of these earlier this month while reading Blaugust posts (including yesterday).

For example, the seating challenges seem awkward during the first week of school, when the teacher is still trying to learn student names. But then again, that's what the name tents are for -- with name tents, the teacher doesn't need a seating chart to learn the names. Thus we can have activities that don't require a seating chart -- and hence avoid arguments about seating during the first week.

Among other Blaugust participants, Denise Gaskins posts what she calls a carnival -- dozens and dozens of links to math teachers of all levels, from elementary to high school:

https://denisegaskins.com/
https://denisegaskins.com/2019/08/30/playful-math-education-carnival-130/

Under "Teaching with Wisdom and Grace" she links to both Sara(h)s as well as a few others who post opening week activities.

And Beth Ferguson also adds some activities for Algebra I/II and Statistics, though these aren't necessarily for the first week:

http://algebrasfriend.blogspot.com/
http://algebrasfriend.blogspot.com/2019/08/data-collection-labs-to-model-functions.html

But it's Ferguson's post from yesterday that reminds me of how in some ways, I blew the first week of school almost as much as I blew it in science.

I tried to find a link to a blog that describes what a "seating challenge" might be. I found this old 2014 post describing what it might be. (This link might appear in the Ferguson or Gaskins lists, but I just used Google since searching all of their links might take too long.)

https://mathinate.wordpress.com/2014/08/24/first-week-plan/

Notice that the author of this post just happens to be named Sarah -- but of course, it's not VanDerWerf or Carter:

Tuesday:
To get started, students will find their seat by solving a simple math problem; the seats are numbered 1 – 24. 

Wednesday:
For this first day, students will seat themselves by alphabetical order of first name.

Thursday:
Students will seat themselves by birthday, January to December.

Friday:
Students will seat themselves in order by height, shortest to tallest. They have to do this one silently, which is an extra challenge.

Except for the math problem, I can easily see a science teacher doing something like this. And once again, I could have done this in my own classroom three years ago.

Yesterday, I explained how 2016 was a big year for posting first day activities -- VanDerWerf, Carter, and Ferguson all posted that year. This was the same year that I was in the classroom and thus could have used some first week activities. But for various reasons, I didn't take any of them into account:

• Ferguson posted hers too early for me. During Blaugust, I read all of Ferguson's posts during the month of August. But Ferguson's opening week activity post was dated July 22nd.
• Carter posted hers too late for me. She posted during her first week of school, which was one week after mine. (I did use her "Survival in the Desert" in her first week/my second week.) In 2016, she posted many activities that week since it was the first time that her school stared on Monday rather than Thursday (requiring five days of opening week activities, not just two).
• VanDerWerf posted hers at the perfect time -- nine days before my first day of school. The problem was that in 2016, I had never heard of VanDerWerf. Although she had been teaching for decades, she didn't start blogging until the winter of 2015/6. Thus 2016 was her first opening week since creating her blog. But I didn't know of her blog until 2017 -- long after my first week of school. I haven't been in a classroom during the opening week of school since -- until today, that is.

Indeed, Carter and Ferguson had labeled their posts "Blaugust," while VanDerWerf never did. But then again, Shelli had invited me to join Twitter, which I ultimately turned down. If I had joined, it's likely that I would have seen VanDerWerf's tweets. Indeed, I decided to search some old 2016 tweets and saw that Shelli herself referenced VanDerWerf that August. Thus if I had accepted Shelli's invite to join Twitter, I would have seen VanDerWerf's tweet and likely used one of her activities -- either name tents or her other famous opening activity (known simply as "1-100").

Let's get to the rest of this post. Today on her Mathematics Calendar 2019, Theoni Pappas writes:

x is a rational number. What must x equal for this triplet to represent the sides of a right triangle?
{30.5, x, 5.5}

This is another one of those Pythagorean triple problems where we don't know whether x is the hypotenuse or a leg. Assume that it's the hypotenuse:

a^2 + b^2 = c^2
30.5^2 + 5.5^2 = x^2
930.25 + 30.25 = x^2
960.5 = x^2

And 960.5 isn't the square of a rational number. Thus x must be a leg:

930.25 - 30.25 = x^2
900 = x^2
x = 30

Therefore the missing member of the triple is 30 -- and of course, today's date is the thirtieth. The Pythagorean Theorem is taught in Lesson 8-7 of the U of Chicago text.

This is what I wrote last year about today's lesson:

Lesson 1-3 of the U of Chicago text is called "Ordered Pairs as Points." (It appears as Lesson 1-2 in the modern edition of the text.) The main focus of the lesson is graphing points on the plane. Indeed, we have another description of a point:

Third description of a point:
A point is an ordered pair of numbers.

The idea of graphing points on a coordinate plane is a familiar one. But sometimes I wonder whether we should make students graph points and lines so soon in their Geometry course.

Once again, here's how I think about it -- the students coming to us just finished Algebra I. Some of them struggled just to earn a grade of C- or D- (whatever the lowest allowable Algebra I grade is in your district is so that the students can advance to Geometry). The students who just barely passed Algebra I are tired of seeing algebra. They may look forward to Geometry where they won't have to see so much algebra -- and then one of the first things we show them is more algebra.

Then there's also the issue, first brought up by David Joyce, that students should use similarity to show why the graph of a linear equation is a line. This idea appears in the Common Core standards for eighth grade, but it's awkward in high school. Graphing linear equations is a first semester Algebra I topic while similarity is a second semester Geometry topic -- and it's difficult to justify delaying graphing linear equations by three semesters just to conform to Joyce's wishes.

In the past, I've tried -- and failed -- to teach linear graphs after similarity. (This includes last year, when I tried to follow the eighth grade standards, but I left the class before graphing equations.) This year, my plan is simple -- I will conform to the order of the U of Chicago text. The U of Chicago text introduces linear graphs in Lesson 1-3, and so that's when I'm teaching it.

The bonus question asks about longitude and latitude. I've already located my own coordinates as being near 34N, 118W.

Thus ends another Blaugust. I'm not posting on the 31st since that's a Saturday. Instead, my next post will be on Tuesday, after the Labor Day holiday.

Lesson 1-2: Locations as Points (Day 12)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

Find x.

This is yet another problem where all of the givens appear in an unlabeled diagram. Since this problem involves a circle, we'll label the center O. Five points on the circle are significant, so we can label these in alphabetical order as A, B, C, D, E. Let chords AC and BD intersect in the circle at P. So finally I can provide some angle values: AOB = 20, CED = 19, CPD = x.

Clearly this problem involves the circle theorems of Chapter 15 of the U of Chicago text, especially the measures of central (Lesson 15-1) and inscribed (Lesson 15-3) angles. Let's use them both:

Central Angles: Since AOB = 20 is a central angle, Arc AB = 20.
Inscribed Angles: Since CED = 19 is an inscribed angle, Arc CD = 38.

Finally, we must use the Angle-Chord Theorem of Lesson 15-5:

Angle-Chord Theorem:
The measure of an angle formed by two intersecting chords is one-half the sum of the measures of the arcs intercepted by it and its central angle.

Using this theorem, we obtain:

Angle CPD = (Arc AB + Arc CD)/2
x = (20 + 38)/2
x = 58/2
x = 29

Therefore the desired angle is 29 degrees -- and of course, today's date is the 29th. By the way, the Angle-Chord Theorem rarely appears on state tests such as the SBAC. The last major theorem needed for SBAC is the Inscribed Angle Theorem.

This is what I wrote last year about today's lesson:

Lesson 1-2 of the U of Chicago text is called "Locations as Points." (It appears as Lesson 1-1 in the modern edition of the text.) The main focus of the lesson is graphing points on a number line. Indeed, we have another description of a point:

Second Description of a Point:
A point is an exact location.

Yesterday I made a big deal about the first description of a point -- the dot -- since many of our students are interested in pixel-based technology. Locations as points aren't as exciting -- but still, the second description is something we think about every time we find a distance. The definition of distance is highlighted in the text:

Definition:
The distance between two points on a coordinatized line is the absolute value of the difference of their coordinates.

Other than this, the lesson is straightforward. Students learn about zero- through three-dimensional figures, but of course the emphasis is on one dimension. One of the two "exploration questions," which I included as a bonus, is:

• Physicists sometimes speak of space-time. How many dimensions does space-time have?

The answer, of course, is four -- even though there might be as many as ten dimensions in string theory. We ordinarily only include Einstein's four dimensions and don't consider the extra six dimensions of string theory as part of "space-time."

Here's the other bonus question:

• To the nearest 100 miles, how far do you live from each of the following cities?

a. New York
b. Los Angeles
c. Honolulu
d. Moscow

Well, part b is easy -- I worked in L.A. last year and my daily commute obviously wasn't anywhere near 100 miles, so my distance to L.A. is 0 miles to the nearest 100 miles. The U of Chicago text gives the distance from L.A. to New York as 2451 miles as the crow flies, but 2786 miles by car. I choose to give the air distance in part a, in order to be consistent with parts c and d (for which only air distance is available). We round it up to 2500 miles. My answers are:

a. 2500 miles
b. 0 miles
c. 2600 miles
d. 6100 miles

Hmmm, that's interesting -- I'm only slightly closer to New York than to Honolulu.

Here is the Blaugust prompt for today:


A peek into my classroom - show us your classroom or describe a typical day / hour

Well, yesterday I wrote that I don't have any videos of my teaching. And only once did I take any photos of my classroom (not counting photos submitted to Illinois State). For the sake of this Blaugust post, I'll post those pictures again today.

As for a typical hour, I did write about what I originally wanted a typical 80-minute block to look like, about three weeks before the first day of school:

10 minutes: Warm-Up
10 minutes: Go over homework/previous day's lesson
20 minutes: New lesson (Foldable note taking)
10 minutes: Music break
20 minutes: Guided practice
10 minutes: Closure/Exit Pass

This was set up for a traditional lesson. Of course, soon I learned more about the Illinois State text and its nontraditional lessons. Many parts of this 80-minute plan changed -- but I always kept some form of a Warm-Up, music break, and Exit Pass.

Here are the changes to this plan caused by Illinois State. First of all, the Warm-Up turned into the Illinois State Daily Assessment, which is supposed to take only five minutes, not ten. Going over HW and the previous day's lesson were awkward since there was supposed to be only one traditional lesson per week, and the HW was to be done online. Most of the time, I had the students take notes directly into the Student Journals, which was also where the guided practice was. Thus in the end, the typical 80-minute block became:

5 minutes: Warm-Up (Illinois State Daily Assessment)
10 minutes: Review previous week's lesson (from Illinois State)
20 minutes: New lesson (Illinois State Student Journals)
10 minutes: Music break
25 minutes: Guided practice (Illinois State Student Journals)
10 minutes: Closure/Exit Pass

If I remember correctly, the Illinois State pacing guide assigned one hour to the traditional lesson, and notice that the Illinois State parts of this lesson do add up to one hour. The only non-Illinois State parts of this lesson plan are the music break and Exit Pass.

The traditional lesson, as I wrote earlier, would be one day per week. As I realized much too late, the ideal weekly plan would have been something like this:

Monday: Coding (with coding teacher)
Tuesday: Traditional Lesson
Wednesday: Learning Centers
Thursday: Science
Friday: Weekly Assessment

Once again, there's only repeat posters for Blaugust today. We begin with Benjamin Leis:

http://mymathclub.blogspot.com/
http://mymathclub.blogspot.com/2019/08/15-75-90-alternate-forms.html

It's been a while since I've talked about one of my favorite triangles the 15-75-90. So here's a short post on a new detail  that I realized about them the other day.

We think about special right triangles in Lesson 14-1 of the U of Chicago text. In that lesson, we know of two special right triangles -- 45-45-90 and 30-60-90. But as the Exploration question in that lesson implies:

There are many triangles that could be considered special.

And clearly Leis considers the 15-75-90 right triangle to be special. He writes the following problem:

ABCD is a square. Triangle ABF is an equilateral triangle (with F inside the square).
If CD = sqrt(6) + sqrt(2) then EC = ?

This question almost looks like a Pappas problem -- and indeed since the final answer turns out to be 16, this could appear on the sixteenth day of the month. But Pappas never includes 15-75-90 triangles on her calendar -- and Triangle CDE turns out to be 15-75-90. (Hint: Triangle BCF is isosceles.) On the other hand, she includes 45-45-90 and 30-60-90 triangles all the time.

We can think back to the previous Leis post. OK, so maybe we shouldn't really teach the Cubic Formula in Algebra II classes. But can we teach 15-75-90 triangles in Geometry?

Here's an interesting mini-activity to consider when teaching Chapter 14. Just after introducing the trig functions, we ask students to select three acute angles -- say 6, 37, and 86 degrees. Then the teacher writes the following:

sin(6) = ?
cos(37) = ?
tan(86) = ?

Let's enter these on a calculator:

sin(6) = .1045284633
cos(37) = .79863551
tan(86) = 14.30066626

The teacher writes these values on the board, and lets the class "study" them a little. Then without warning, the teacher erases these values and announces a pop quiz. The students must now fill in the missing values:

sin(6) = ?
cos(37) = ?
tan(86) = ?

accurate to ten digits without a calculator. Of course, the students will protest -- it's impossible to have learned all of those digits that fast! Then the teacher "cancels" the pop quiz, and states that the students should have chosen different angles -- 30, 60, and 45 instead of 6, 37, and 86:

sin(30) = .5
cos(60) = .5
tan(45) = 1

Oh, so for some reason, this pop quiz would have been a lot easier with 30, 60, and 45 instead of 6, 37, and 86 degrees. Then this leads to a class discussion -- why is tan(86) such a complicated value but tan(45) so simple? And of course, this leads to special right triangles.

On a Casio calculator that I found in my old classroom, the square root symbol is displayed. This leads to even more simple values for functions:

sin(45) = sqrt(2)/2
cos(30) = sqrt(3)/2
tan(60) = sqrt(3)

The teacher then states that any value that displays an exact value on the screen -- whole number, fraction, or radical -- is fair game for memorization on a pop quiz. Thus it's fair to ask for tan(45) or tan(60), but not tan(86). But then we might stumble upon something:

tan(15) = 2 - sqrt(3)

And students now might wonder, why does tan(15) gives an exact value? Voila -- we just motivated the teaching of 15-75-90 triangles in a Geometry class.

Notice that we can use the addition and subtraction formulas from trig to find tan(15) -- in this case, either tan(60 - 45) or tan(45 - 30) will work. But it's possible to find this values exactly using a purely Euclidean approach, without any use of advanced trig formulas at all.

We begin with Triangle ABC with A = 75, B = 15, C = 90. Let BC = 1 and AC = x. Notice that the tangent of 15 degrees is now AC/BC = x/1, so x = tan 15. Let the hypotenuse AB = y.

Now we reflect this triangle over its leg BC, with D the mirror image of A. Since Angle ABC = 15 -- and its reflection DBC = 15 -- we have Angle ACD = 30. And thus, if we drop a perpendicular from A to DB and label the foot of this perpendicular E, then ABE is a 30-60-90 triangle.

We know one side of this triangle -- the hypotenuse AB = y. Thus its shorter leg is AE = y/2, and thus its longer leg is BE = y sqrt(3)/2. And since DB = y (as DB is the reflection image of AB = y), we can also find DE = y (1 - sqrt(3)/2).

Now ADE turns out to be another 15-75-90 triangle (since there's clearly a right angle at E, and D is the image of A = 75). We can find tan 15 using this triangle:

tan 15 = DE/AE

But we already know that x = tan 15:
x = y(1 - sqrt(3)/2) / (y/2)
x = y(1 - sqrt(3)/2)(2/y)
x = 2(1 - sqrt(3)/2
x = 2 - sqrt(3)

Thus x = tan 15 = 2 - sqrt(3), just as it appears on the calculator.

Notice that y = tan(75), and y is the hypotenuse of a triangle whose legs are now known, and so we can just use the Pythagorean Theorem:

AC^2 + BC^2 = AB^2
1 + (2 - sqrt(3))^2 = y^2
y^2 = 1 + 4 - 4sqrt(3) + 3
y^2 = 8 - 4sqrt(3)
y^2 = 4(2 - sqrt(3))
y = 2sqrt(2 - sqrt(3))

These are the values that appear on the 15-75-90 triangle in the Leis post.

Meanwhile, another Blaugust poster today is Beth Ferguson:

http://algebrasfriend.blogspot.com/
http://algebrasfriend.blogspot.com/2019/08/five-top-posts.html

Just as I do every year, Ferguson decides to link to her five most popular posts -- except it's of all time, not just the past twelve months. As it turns out, her most popular post is from 2016:

http://algebrasfriend.blogspot.com/2016/07/a-collection-of-first-week-activities.html

It's yet another list of activities for the first week to school. Ferguson's list isn't as famous as the ones posted by the two Sara(h)s -- in fact, she acknowledges Sara and Sarah in today's post. In fact, we see that Ferguson posted her list in July 2016, while the Sara(h)s posted the following month. Even though this is her most popular post, her list is still overshadowed by the Sara(h) lists.

But here are those pictures from my classroom three years ago to fulfill first part of the Blaugust prompt, followed by the Lesson 1-2 worksheet.

Lesson 1-1: Dots as Points (Day 11)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

A 112-meter rope is used to mark off a rectangular planter box. What width should the rectangle be to maximize the area?

As it turns out, of all rectangles with the same perimeter, the square has the most area. Thus we are asking for the side length of a square whose perimeter is 112.

Clearly, the answer must be 112/4 = 28. Therefore the desired width is 28 meters -- and of course, today's date is the 28th.

Let's locate this lesson in the text. The area of a rectangle appears in Lesson 8-3. Meanwhile, the Isoperimetric Inequality (i.e., that a circle maximizes area for a given perimeter) appears near the end of the text, in Lesson 15-8. If it's not known that a square maximizes area for a given perimeter, then we must use algebra (or even calculus) to prove this fact.

These lesson numbers refer to the U of Chicago text, of course. And anyway, speaking of that text...

This is what I wrote last year about today's lesson:

Our focus is now the U of Chicago text. Just like the Serra text, it's an old Second Edition (1991), and there are newer editions in which the chapters are ordered differently. Since my plan this year is to follow the order strictly, let's revisit the chapter order in my text:

Table of Contents
1. Points and Lines
2. Definitions and If-then Statements
3. Angles and Lines
4. Reflections
5. Polygons
6. Transformations and Congruence
7. Triangle Congruence
8. Measurement Formulas
9. Three-Dimensional Figures
10. Surface Areas and Volume
11. Coordinate Geometry
12. Similarity
13. Logic and Indirect Reasoning
14. Trigonometry and Vectors
15. Further Work With Circles

Let's compare this to the modern Third Edition of the U of Chicago text. The first thing we notice is that the new text has only 14 chapters, not 15. We observe that the first twelve chapters are more or less the same in each text, and so it's Chapter 13 that is omitted in the new version. Instead, the material from the old Chapter 13 has been distributed among several different chapters.

You might recall that in the past when I used to juggle the lessons around, it was Chapter 13 that I moved around the most. So you could argue that when I was breaking up Chapter 13, I was actually adhering to the order in the new Third Edition -- unwittingly, of course!

Let's look at Chapter 13 in the old text, and I'll give the lesson in the new text to which the old Chapter 13 material has been moved:

• Lessons 13-1 through 13-4 (on indirect proof) are now the first three lessons of Chapter 11, just before coordinate proofs. (Lesson 13-2, "Negations," is no longer a separate lesson in the new text.)
• Lesson 13-5, "Tangents to Circles and Spheres," is now Lesson 14-4, in the circles chapter.
• Lesson 13-6 through 13-8 (on exterior angles of polygons) have been incorporated into Lessons 5-6 and 5-7 (on Triangle Sum).

Some of these changes are those I once made by myself -- for example, including tangents to circles with the other circle lessons.

Besides the breakup of Chapter 13, here are the other major changes made in the Third Edition:

• Chapters 4 through 6 exhibit many changes. In my old version, reflections appear in Chapter 4, while the other isometries don't appear until Chapter 6. In the new version, all isometries are defined in Chapter 4. With this, the definition of congruence (and some of its basic properties) have now moved up from Chapter 6 to Chapter 5. Only Triangle Sum remains in Chapter 5 -- the properties of isosceles triangles and quadrilaterals have been pushed back to Chapter 6.
• With this, Chapter 3 has a few new sections. Two transformations are actually introduced in this chapter, namely rotations and dilations. This may seem strange, since rotations are still defined as Chapter 4 as a composite of reflections in intersecting lines -- and reflections themselves don't appear until Chapter 4. It appears that the purpose of rotations in the new Lesson 3-2 is to introduce rotations informally, as well as tie them more strongly to the angles of Lesson 3-1. (Rotations appear before reflections in Hung-Hsi Wu, but Wu does for different reasons.) Arcs also now appear in Lesson 3-1 instead of having to wait until 8-8. Meanwhile, the new Lesson 3-7 on dilations (which are still called "size transformations") is essentially the old Lesson 12-1 and 12-2. Again this is only an intro -- dilations are still studied in earnest only in Chapter 12.
• Chapter 7 is basically the same as the old text, especially the first five sections (except that SsA in Lesson 7-5 now has an actual proof). The new Lesson 7-6 is the old Lesson 8-2 on tessellations. I see two new lessons in this chapter, Lesson 7-9 on diagonals of quadrilaterals and Lesson 7-10 on the validity of constructions. (David Joyce would approve of this -- but he'd take it a step forward and not even introduce the constructions until this lesson.) Meanwhile, the old Lesson 7-8 on the SAS Inequality (or "Hinge Theorem") no longer appears in the new text.
• Chapter 8 has only one new section -- Lesson 8-7, "Special Right Triangles," is the old 14-1. This is so that special right triangles are closely connected to the Pythagorean Theorem.
• Chapter 9 was always a flimsy chapter in the old book -- it's on 3D figures, yet most of the important info on 3D figures (surface area and volume) don't appear until Chapter 10. Now surface area has moved up to Chapter 9, reserving Chapter 10 for volume (except for the surface area of a sphere, which remains in Chapter 10). The old Lesson 9-8 on the Four-Color Theorem has been dropped, but that was always a lesson that was "just for fun."
• The last section of the old Chapter 12 (side-splitter) is now the first section of Chapter 13, which is the new trig chapter. Lesson 13-2 is a new lesson on the Angle Bisector Theorem, and Lesson 13-4 is a new lesson on the golden ratio. I've actually seen these ideas used before -- including on the Pappas Mathematical Calendar -- but this is the first time I've seen them in a text as separate lessons. This is followed by lessons on the three trig ratios. Vectors, meanwhile, have moved up to Lesson 4-6, so that they can be closely connected to translations.
• Chapter 14 should be like the old Chapter 15, but there are a few changes here as well. Ironically, I, like the text, moved tangents to circles to this chapter (Lesson 14-4) so that it would be closer to the other important circle theorem, the Inscribed Angle Theorem. But inscribed angles have been moved up in the new text to Lesson 6-3. This places that lesson closer to the Isosceles Triangle Theorem, which is used in the proof of the theorem. Meanwhile, Lesson 14-6 technically corresponds to 15-4 ("Locating the Center of a Circle") of the old text, but it has been beefed up. Instead of just the circumcenter, it discusses the other three concurrency theorems (important for Common Core) as well as the nine-point circle of a triangle.

Meanwhile, of immediate concern are Chapters 1 and 2 of the new text. Unlike the others, these chapters haven't changed much from the old text. The only difference in Chapter 2 is that Lesson 2-3, on if-then statements in BASIC, has been dropped. (After all, who uses BASIC anymore, except on the Mocha computer emulator for music?) In its place is a new lesson on making conjectures.

Two of the lessons of Chapter 1 have been dropped. One of them is actually today's Lesson 1-1, as its material has been combined with the old Lesson 1-4. Meanwhile, Lesson 1-5, on perspective, has been delayed to Chapter 9 (which makes sense as perspective is definitely related to 3D). The last lesson in Chapter 1 is on technology -- a "dynamic geometry system," or DGS. (That's right -- goodbye BASIC, hello DGS!) Officially, it still corresponds to the last lesson of the old Chapter 1, since this lesson still introduces the Triangle Inequality Postulate (but now students can test out this postulate for themselves on the DGS).

On the blog, I'll continue to follow the old Second Edition of the U of Chicago text. But if I ever get to sub in a classroom again, the classroom has priority over the U of Chicago order. In this case, if an important lesson is skipped, I could sneak the lesson in by following the Third Edition order instead.

Okay, without further ado, let's finally start the U of Chicago text!

Lesson 1-1 of the U of Chicago text is called "Dots as Points." This lesson has the first description of a point:

First description of a point:
A point is a dot.

This is the start of a new school year. Many students enter Geometry having struggled throughout their Algebra I class. Now they come to us in Geometry, and after all the frustration they experienced last year, the first question they ask is, "Why do we have to study Geometry?" Well, the answer is:

A point is a dot.

The old U of Chicago text writes about dot-matrix printers. This isn't relevant to the 21st century, and indeed they don't appear in the modern edition. But here's another question to ask students -- if you didn't have to take math, what would you do at home instead of math homework? And if the answer is "play video games," then guess what -- video game graphics consists of millions of dots. Or, more accurately, they consist of millions of points, since:

A point is a dot.

Images on video games don't come out of nowhere -- someone had to program in the millions of dots, treating them as points -- therefore using Geometry. So without Geometry, video games don't exist. If you want to answer that question -- "What would you do if there was no math?" -- then next time choose something that doesn't require math to build.

In the modern version of the text, there is a brief mention of pixels as part of both computer images and digital camera images. Again, it's not emphasized as much, since "dots as points" must share the new Lesson 1-3 with "network nodes as points."

Here is the Blaugust prompt for today:

Observe yourself!  Record your lesson using your phone in your pocket and use it to reflect

Well, you already know that I have very few photos or videos of my classroom to post. Anyway, today's a great day to sit out Blaugust anyway. I already spent much of this post discussing the structure of the U of Chicago text.

Meanwhile, today we return to the blog of Jenna Laib:

https://jennalaib.wordpress.com/
https://jennalaib.wordpress.com/2019/08/28/choosing-a-story-to-tell-examining-a-lesson-close-in-1st-grade/

I had planned to spend my first time in Room 12 observing the classroom teacher and getting to know her students. Instead, the teacher texted me the day before to let me know that she had come down with pneumonia. I immediately offered to teach the class.

…not that I had any ideas for a lesson. First Grade. I didn’t know the students. The classroom teacher, Natalie, would probably be out all week.

Once again, Laib is an elementary math coach. Well, apparently she's a substitute teacher this week -- just as I am. Then again, I don't sub in first grade classrooms.

So as usual, the only Blaugust poster is unrepresentative of what I hope to teach someday. But it's interesting to view Laib's post in light of today's Blaugust topic -- photos and videos of lessons.

No, Laib doesn't post any videos, but she does post photos. And I notice that all photos in both of her Blaugust posts are of student work. And so even if I never take photos with a phone in class, there's nothing preventing me from scanning and posting student work.

Well, there's one thing -- copyright. I usually avoid posting anything with a copyright symbol -- usually worksheets published by a textbook company. So if I teach or even sub in a class and wish to post student work, it's likely that I'd be violating the copyright of some company.

Laib's photos contains copyrighted worksheets -- but then again, it's a photo, not a copy. It seems so much worse to scan a worksheet than it is to take a picture of it.

What I could do someday is take a worksheet that I created here on the blog, have students work on it, and then post their student work here. This is tricky -- it would require a multi-day assignment (which Laib is in fact doing this week). The first day is for finding out what lesson the students are on (to determine which worksheet to use), the second for actually assigning the worksheet, and the third is for taking the worksheet home, posting it, and returning it to the student. (The third day is not needed if I have access to a photocopier.) Even then, it only works if it fits into the regular teacher's lesson plan (for example, if we happen to exhaust everything on the lesson plan).

I may never be able to have a video of one of my lessons. But classroom photos are common -- and indeed, most teacher blogs or Twitter accounts contain many such photos. I believe that my cellphone (low-level, not a smartphone) has a camera, but I almost never use it.

If I ever get my own classroom, I could adopt the following phone policy -- cellphones are almost always forbidden in my class. Here's one exception for the "almost" -- if the student offers to take a bloggable (or tweetable) photo of the classroom for me. In other words, if I see a phone out, the student must take such a photo and send it to me to avoid confiscation. Here "bloggable"/"tweetable" means that it contains no student faces, since these can't be posted without parental permission. For example, Laib's photo of the girl P. is bloggable because it contains only her hand and her work, not her face.

That's all I really have to say about Laib's post. Oh, I guess I will point out that Laib's first graders, in the process of learning addition, already know their "doubles" (4 + 4 = 8, 5 + 5 = 10, and so on). In learning the arithmetic tables, some facts are known before others.

Here is the Lesson 1-1 worksheet:

Benchmark Tests (Day 10)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

The exterior angle of a regular polygon is 13 1/3 degrees. How many sides does the polygon have?

To solve this problem, we use a theorem from Lesson 13-8 of the U of Chicago text:

Exterior Angles of a Polygon Sum Theorem:
In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360.

Thus we must write and solve the following equation:

360/n = 13 1/3
n = 360/(13 1/3)
n = 27

Therefore the desired polygon is a 27-gon -- and of course, today's date is the 27th.

Today I'm posting the Benchmark Tests. Angles of a polygon do appear on this benchmark, and so today's Pappas question might help us.

Here is the Blaugust prompt for today:

How do you support struggling students?  What intervention strategies have you used?

To respond to today's Blaugust prompt, as usual, I think back to three years ago. Let's look at old some posts from around that time.

January 12th, 2017:
I need to mention my eighth grade class today, especially since they're learning about transformations on this Common Core Geometry blog. In the end, I decided to delay the science lesson to tomorrow and teach transformations today.

This means that this week I had three full days to cover the three transformations. On the first two days, the translations and reflections went well, and most students appeared to understand. But I worried as today's lesson approached, because rotations are probably the most difficult of the three transformations for students to understand.

Now keep in mind that I'm using the Student Journals that are part of the Illinois State text. We know that rotations can be centered either at the origin or away from the origin. Rotations centered at the origin have easier formulas -- for example, the rotation of 180 degrees centered at the origin maps the point (x, y) to the point (-x, -y).

But none of the rotations mentioned in the Illinois State text are centered at the origin. Most of the questions direct a student to rotate a line segment around one of its endpoints. This at least makes it a little easier, since every rotation maps its center to itself.

And so here's what I did today -- on the first page, the students are asked to rotate AB 90 degrees clockwise about point A, The coordinates are A(2, 3) and B(7, 3). I had the students change A to "the origin," and then I show them the 90-degree rotation about the origin. To do this, I had the students the paper 90 degrees counterclockwise -- that is, the opposite direction from the rotation. Then they drew the image A' by going 2 units on the new x-axis and 3 units along the new y-axis. They did the same to find B', and then they restored the paper to its original position. The new segment A'B' now appears to be the clockwise rotation image of AB.

Of course the students are confused by this at first, but in the end, I believe that they're starting to get the hang of this. I like teaching rotations this way because it sets them up nicely to learn the slopes of perpendicular lines later on. By the end of class, I think the most confusion came from changing all the questions in the Illinois State text, which were geared towards the rotation centered at A rather than the origin.

So that's one possible answer to the Blaugust prompt -- I help struggling students by making the questions in the text easier.

January 23rd, 2017:
Meanwhile, today is a coding Monday. In case you're curious, sixth graders create logos for an imaginary company, while seventh graders learn about spreadsheets. The students learn about various Excel functions, including mean, median, and mode. I don't normally have music break on coding Mondays, but I couldn't help singing the Measures of Center song from last month to jog the students' memory.

I notice that often when I wrote that my class "struggled" on something, that something was actually the Monday coding assignment, not math! But in this case, the coding lesson was math-related -- and my solution was to sing a song to remind them of what they had learned.

I admit that in both of these cases, the students simply shut down. They didn't realize why learning math was worth the effort. I used the field trip to inspire them -- I reminded them (especially the girls) that if they studied hard in math, they could become the next Katherine Johnson.

Today is a test day, which makes this our first traditionalists' post of the new school year. Actually, it's been about a month since our main traditionalists (Barry Garelick and SteveH) have posted. So let's seek out some older traditionalist posts.

In fact, I'm still thinking about yesterday's Benjamin Leis post about quadratics and cubics. One question that came up yesterday is, if we're not going to teach the Cubic Formula to high school students, then why are we still teaching the Quadratic Formula? Well, let's see what the traditionalists have to say about the Quadratic Formula.

We might as well start with Barry Garelick again. His most recent mention of the Quadratic Formula was back in January 2018:

https://traditionalmath.wordpress.com/2018/01/15/count-the-tropes-dept-4/

Here Garelick was actually quoting a non-traditionalist (reform math) supporter, Brett Berry:

Brett Berry:
We forget. All the time. Especially when we don’t understand why the algorithm works or if it has been a long time since we last used it. I can guarantee if I took a survey at Starbucks right now, and asked people how to perform long division, convert a mixed number to a fraction, recite the quadratic formula, factor a binomial, and complete the square — most would fail. Not because math is too hard or people are bad at it, but because memorization and algorithms are not the best ways to retain information. We remember through context, understanding and application.

Interestingly enough, the comments read:

BL:

I never did memorize the quadratic formula. But my math teacher showed us how it’s derived, and I quickly did that before each test in the margin of the test paper.

Problem solved. (And, as a result, I never forgot about “completing the square”!)

Hmm, do the initials "BL" stand for "Benjamin Leis"? Indeed, Leis does seem to echo what BL wrote last year:

http://mymathclub.blogspot.com/2019/08/cardanos-method.html

The quadratic formula song troubles me too because I worry it hides a lack of conceptual understanding. I have other memories of dragging kids through a problem that required completing the square where it was clear their mastery was incomplete.

And by extension, even memorizing the Quadratic Formula can be seen as requiring less "conceptual understanding" (those words much reviled by traditionalists) than completing the square. So it makes sense for BL to be Benjamin Leis (as both appear to find completing the square to be superior to just memorizing the Quadratic Formula).

But then Garelick mentioned the Quadratic Formula in an earlier post, dated June 2016:

https://traditionalmath.wordpress.com/2016/06/21/ps-youre-an-idiot-dept/

My grandmother didn’t take Algebra 1, but I took it in the 60’s and I suppose it’s people like me that Ryan is referring to. I have a bunch of textbooks from that era. I’ll be teaching 8th grade algebra at a school in California, in which the school district doesn’t sit on the high horse that SFUSD likes to occupy. In looking through the Common Core-aligned algebra book I’m forced to teach from, I’m aghast at the dearth of good solid word problems, the short shrift given to exponentials, to rational expressions, not to mention the omission of solving quadratic equations by factoring–I guess the quadratic formula saves a lot of time and there’s no value in teaching that approach. There is a chapter on statistics (as if that’s needed in an algebra class), and a superficial look at exponential functions, which I suppose allows people like Ryan to say “Look how deep this course is. Not your grandmother’s algebra 1”.

Here Jim Ryan is also anti-traditionalist. It all goes back to the district in San Francisco which no longer offers eighth grade Algebra I -- the traditionalists' preferred eighth grade math class. Ryan was trying to argue that Common Core Algebra I is more rigorous that traditional pre-Core Algebra I, so it makes sense to wait until freshman year to take it. Garelick countered that Common Core Algebra I isn't more rigorous, since while it contains stats and exponentials, it lacks factoring.

We know that Common Core emphasizes understanding -- and so we must conclude that, unlike completing the square, factoring does not lead to greater understanding of quadratic equations (or otherwise the Core would emphasize it).

Here's a May 2017 post from our other traditionalist website, Joanne Jacobs, where quadratic equations are mentioned:

https://www.joannejacobs.com/2017/05/how-your-brain-beats-google/

Jonathan Rochelle, the director of Google’s education apps group,  “cannot answer” why his children should learn the quadratic equation, according to a recent New York Times story. After all, they can just “ask Google.”

(By context, I interpret "the quadratic equation" to mean the Quadratic Formula.) Of course, there's clearly a conflict of interest here -- it's the director of Google's ed app groups who advocates that Google makes learning the Quadratic Formula archaic.

Tying this back to Leis and the Cubic Formula, though, we see that anyone who needs to solve a cubic formula (that can't easily be factored) just turns to Google or technology to solve it. If we rely on Google to solve cubic equations, then why not quadratic equations?

Then again, I don't care what the employee of Google thinks about Google. I care about what actual students sitting in our classes think about whether technology makes learning obsolete. For that, we scroll down to one of the commenters in the thread, SC Math Teacher:

SC Math Teacher:
I once had a student incorrectly graph a slope-intercept equation on a quiz. She protested and showed me the graph on her TI-83 calculator. She sure knew how to enter the equation and press the graph button, but she knew nothing about slope and intercept. She copied the graph incorrectly onto her quiz sheet. Knowledge of the subject matter is necessary before one uses a calculator. Or Google.

This is a tricky one. I assume that the girl here was arguing that since she copied the answer from the calculator -- and calculators are a priori always correct -- then she must be correct. But it's possible she could be making the larger argument that the existence of graphing calculators make the need to learn how to graph linear equations obsolete.

SC Math Teacher stated that the girl knew how to enter the equation correctly. I suspect her error was in not understanding how to interpret the calculator window -- so that she was unable to convert the line on the screen to the correct coordinates on her paper.

Another commenter asked of SC Math Teacher:

KateC:
[W]hy do you let your students use a calculator?

Indeed, why would they -- presumably in Algebra I -- be allowed to use a graphing calculator?

SC Math Teacher:
That ship has sailed long before the students reach me. I can stand against the tide, I suppose, but with low-level students struggling to multiply single-digit numbers, the benefits would be little, if any.

Of course, graphing calculators are excellent tools in the hands of the adept.

There are several issues here. By "long before the students reach me," does this imply that the students were using graphing calculators in middle school? If not, then the graphing calculator "ship" hasn't really sailed long before Algebra I.

Of course, this comment is all about calculators in general -- especially since drens (that is, "students struggling to multiply") are mentioned later on. The idea being made in this post is that these students (the graphing calculator girl and the drens) all believe that actually knowing math is obsolete. Once again, that's the belief of the students, not just Google execs.

So here's an interesting question -- suppose I were the teacher in this Algebra I class. So I see the girl who graphs lines on her calculator and the drens who multiply on the calculator. Suppose these students make it clear that they believe that actually learning math without a calculator is archaic. So what should I do?

I'm still searching for that proverbial line between what students really need to know and what they can just use technology for. We agree that the Cubic Formula is on the technology side. And I agree with the traditionalists that multiplication is on the "need to know" side -- hence my use of the word "dren" to refer to such students.

Well, I'd start by calling such students "drens," of course. Recall that the use of the word "dren" is not to insult or embarrass a student, but to make a point. So if I see a student try to use a calculator to solve a simple multiplication problem, then I subtly approach him (while the others are working independently) and remind him that what he's doing is "drenny." (Does "drennish" sound better as an adjective than "drenny"?)

If the student continues to argue that he shouldn't have to learn how to multiply because of modern technology, I remind him that most modern technology was developed by students who worked hard and earned A's in their math classes. (This includes the Google exec mentioned above -- he may be arguing that no one needs to learn the Quadratic Formula, but he needed to earn A's in his math classes in order to be hired by Google.)

Thus a "dren" is someone who is more afraid of learning too much ("How dare you try to teach me something I don't need to know?") than too little. And if this is because he's unwilling to work hard and make the sacrifices necessary because he'd rather spend that time doing other things, we must ask whether those other things involve playing video games (developed by A-students), such as Fortnite (developed by A-students), on a cell phone (developed by A-students). Thus a "dren" is also someone who wishes that math class would disappear so that more time can be spent using technology developed by people who earned A's in math. (If math were to disappear, so would that technology!)

OK, so that's what I'd tell the drens, but what about the graphing calculator girl? I typically don't call someone who can't graph lines a "dren." On the other hand, linear equations probably approach the proverbial line between "need to know" and "just use technology."

In previous posts, I argue that the end of Common Core 8 (or Integrated Math I) is a good place to draw the line. This includes some algebra and geometry on the "need to know" side. Quadratic equations would fall on the "just use technology" side, but graphing linear equations would be one of the last things on the "need to know" side. (The SAT's "Heart of Algebra" also represents the approximate position of my line, with "Heart of Algebra" on the "need to know" side.)

But as we see with this girl, it's often one of the first Algebra I lessons where students struggle. So what would I tell her?

Well, I might begin by comparing her test to someone who earned full credit for this question and ask her for the differences between the graphs, then show her why the other student's graph better matches the graph on her calculator. Then I can point out that the students who earned full credit generally did so without a graphing calculator.

All the while, I remind her why it's important for her to learn how to graph lines. Once again, if she enjoys playing games on her phone, I point out that the graphics in the game don't work unless someone codes in where to draw the lines -- and that requires knowing how to graph lines as studied in Algebra I.

Here's one more thing about SC Math Teacher's proverbial "ship" that has sailed -- where all "sailors" on this ship want to use calculators for everything. When I taught at the old charter school, very few sixth graders reached for a calculator, while most eighth graders did. This suggests that the "ship" sails in between -- in seventh grade.

(Let me stop mixing metaphors -- my "line" with SC Math Teacher's "ship." Instead, call my line "a river" for that ship to cross. On one side of the river is simple math that students should know without a calculator, and the other side is advanced math for which a calculator is OK.)

Throughout K-6, students are learning arithmetic, and the teachers show them how to do arithmetic one step at a time. They might use methods that traditionalists disapprove of (such as the lattice method), but they don't reach for a calculator. In seventh grade, the study of arithmetic is complete, and instead the four operations are used to solve problems. Teachers no longer show step-by-step arithmetic, and so they just reach for a calculator. This habit becomes ingrained in their minds as they progress through Grades 8-12.

This is why, in some earlier posts, I wondered whether it's a good idea to delay teaching some operations of arithmetic into seventh and eighth grade -- such as complex multiplication and division of decimals or fractions -- without a calculator. This forces the students to avoid calculators for an extra year or two, so that they can reach Algebra I before SC Math Teacher's ship has sailed.

Today, only one Blaugust participant has posted. So we return to the blog of Sue Jones:

https://resourceroomblog.wordpress.com/
https://resourceroomblog.wordpress.com/2019/08/27/geogebra-anybody-want-to-try/

Whereas, right now I don’t have any official projects with deadlines that keep me motivated to keep learning and doing when I’m not helping students, and

Whereas, I really REALLY want to know Geogebra better, and

Whereas, I work a whole lot better when it’s with other folks,

Therefore I hereby invite anybody out there to spend … hmmm…. a week … dabbling in Geogebra with a *specific* kind of activity to learn to build.

Here Jones is writing about the math software called Geogebra. She's creating a challenge for her readers -- creating an actual quiz in Geogebra. She considers her challenge to be part of Blaugust, even though it actually extends into September.

Hmm, so we jump directly from traditionalists complaining about technology to a Blaugust challenge encouraging the use of technology.

Notice that Jones wants to use Geogebra to make a quiz that the software can grade. She even mentions displaying a "thermometer" to show the students' progress so far. Thus this quiz would be similar to the IXL software that I used at the old charter school. In other words, Jones wishes to use Geogebra because it makes things easier for the teacher.

But what effect will Geogebra have on the students? Suppose, for example, SC Math Teacher were to give that linear equations quiz on Geogebra. I've never used Geogebra, but I assume that it graphs lines just as easily as a TI-83. So the entire class would be just like that girl -- graphing lines without any real effort. And of course, it's possible for drens simply to open another tab and ask Google to multiply small numbers. So I can see why traditionalists might oppose Geogebra.

In the end, I won't join the Geogebra challenge because I don't have the software. It might be interesting to see what sorts of quizzes the other participants can come up with.

OK, let's finally post the Benchmark Tests. These are based on old finals posted to the blog. I admit that the tricky thing about Benchmark Tests in Geometry is that the students are coming off of a year of Algebra I, when they've thought little about Geometry at all. This is different from Benchmark Tests in middle school or Integrated Math, where there should be some continuity from year to year.