Friday, August 23, 2019

Lesson 0-8: Perspective (Day 8)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

Segment BE is a median for trapezoid ACDF, whose area is 128 square units. Find the length of AF.

(Here is the given info from the diagram: D, F right angles, CD = 9, DE = 4, with E on DF.)

Let's use the trapezoid area formula as given in Lesson 8-6 of the U of Chicago text:

A = (1/2)h(b_1 + b_2)

The right angles at D and F imply that DF is the height of the trapezoid, and since BE is a median, it means that DE = 4 must be half of the height. So we can plug 4 in directly to (1/2)h. Then CD = 9 must be one of the bases, and our task is to find the other base.

128 = 4(9 + b_2)
32 = 9 + b_2
23 = b_2

Therefore the missing base is 23 -- and of course, today's date is the 23rd.

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

Lesson 0.8 of Serra's Discovering Geometry is called "Perspective." This is the second of two sections appearing in the Second Edition but not in the modern editions. Serra begins:

"Many of the paintings created by European artists during the Middle Ages were commissioned by the Roman Catholic Church. The art was symbolic; that is, people and objects in the paintings were symbols representing religious ideas."

Unlike Lesson 0.5 on mandalas, which we choose to include on the blog even though it's "missing" from the modern editions, Lesson 0.8 can just be left out altogether. This is because we'll be starting the U of Chicago text next week, and that text already has a lesson on perspective (Lesson 1-5), so Day 15 would just be a repeat of Day 8.

Then again, we recall that in my class last year, the students in all grades had trouble drawing cubes even though those were on isometric paper rather than in true perspective). So we might wish to teach perspective on both Day 8 and Day 15. True perspective drawings should most likely be completed on plain unlined white paper, with a straightedge to draw lines toward the vanishing point. Lined notebook paper for one-point perspective drawing may also be acceptable -- but not for two-point perspective (the subject of today's worksheet).

The worksheet below comes from "marcandersonarts" and "Daisuke Motogi."

Here is the Blaugust prompt for today:

Time Capsule - revisit an old post and reflect. If you are new to blogging - find a post on this day from the past on someone else's blog-read, share, reflect.

Well, technically I've already reblogged last year's post for Lesson 0.8. There were some other things I wrote about in that post that I could add to my reblogging that's not directly related to 0.8:

This is what Theoni Pappas writes on page 235 of her Magic of Mathematics:

"This famous drawing by Leonardo da Vinci appeared in the book De Divina Proportione, which Leonardo illustrated for mathematician Luca Paoli in 1509. Leonardo wrote an extensive section on the proportions of the human body in one of his notebooks."

This is the first and only page of the section "Secrets of the Renaissance Man." Of course you can't see what drawing Pappas is referring to here, so let me provide a link:

Pappas explains:

"In his book, he also made reference to the works of Vitruvius, the Roman architect (circa 30 B.C.) who also dealt with the proportions of the human body."

The title, De Divina Proportione, refers to the "divine proportion," which is also known as the golden ratio or Phi. Both Vitruvius and Leonardo believed that Phi = (1 + sqrt(5))/2 appeared in certain ratios of the human body. This is explained at the following link:

  • In the distance from the Da Vinci’s guide line drawn at the hairline to the guide line at the foot, the following are all at golden ratio points:
    •  the navel, which is most often associated with the golden ratio of the total height and not the height of the hairline
    • the guidelines for the pectoral nipples
    • the guidelines for the collar bone
  • In the distance from the Da Vinci’s guide line drawn at the elbow to the guideline at the fingertips
    • the base of the hand is at a golden ratio point.
Pappas concludes:

"Leonardo adds, The length of a man's outspread arms is equal to his height."

This estimation appears in a Square One TV song, "Rule of Thumb" by Kid 'n Play. The rappers are trying to measure the length of the floor. One member knows that his height is about six feet, so he concludes that the length of his outspread arms from fingertip to fingertip is also six feet. There isn't a separate video on YouTube for this song, but it does appear at the start of this YouTube video on Math Talk, a spin off of Square One TV. This was recently posted a few months ago:

How can I connect this back to the science class I taught last year? Well, the actual ratios of the human body isn't part of the curriculum, although the human body itself is.

Under the old California standards, here's how a seventh grade life science class was organized. It began with a little chemistry with an emphasize on the elements required for life (hydrogen, carbon, oxygen, and so on). Then the lessons focus on cell structure, DNA, and genes. Next would be evolution and the history of life on earth. This is usually followed by biodiversity, with lessons first on microbes and fungi, plants, and then animals. Within the animal unit, typically invertebrates are covered first, then the various orders of vertebrates -- fish, amphibians, reptiles, birds, mammals. So humans appear last in this section -- but then this is followed immediately by the human body, also known as anatomy. Since I should have followed the California standards for seventh grade, this meant that the unit on the human body should have appeared at the end of the year.

And if you prefer me to reblog a 2016 post (my year in the classroom) rather than 2018, then let me do so right here. I don't want another August post, so instead I'll choose a September 2016 post, halfway through the first trimester:

Today the students take a test. This is for all three grade levels, now that I've changed my original assessment schedule. The eighth grade test is on rational approximations. So far, many students fare well on the test, since much of it involves approximating a square root on the calculator and rounding it off from zero to three decimal places.

Day 30 marks the midpoint of the trimester. In the past I've referred to half of a trimester by a special name -- the "hexter."

I've written about my plans to give four tests this trimester. Therefore, I ought to have two tests during each hexter. But as it turns out, I ended up giving only one test the first hexter, since I printed up the progress reports before grading the tests. Furthermore, the last major grade before printing the first hexter progress reports was a Dren Quiz, which was easy.

As it turns out, all of my eighth graders are earning a C or better. But there are a few students who were failing until the Dren Quiz raised their grades to a C. This might make the progress reports misleading, since the grades were artificially inflated by a Dren Quiz -- oops! As it turns out, most of my failing students are seventh graders. The first test was difficult, and no Dren Quiz can erase all the 10% and 20% scores received on the test.

Here is the song for today:


If you want to find unit rates,
There's one thing you must know.
To find a unit rate,
All you do is divide!
To see if it's proportional,
All you do is divide!
Write it as a fraction,
Reduce it then you're fine.
Graph it at (0, 0),
Then just draw a line.

If you want to find square roots,
There's one thing you must know.
To find an estimate,
4 and below, round down!
To find an estimate,
5 and above, round up!
1 place for tenths, 2 for hundredths,
3 for thousandths, you're fine.
Graph it between two values,
Right on the number line.

Tomorrow is also the beginning of a new module. Learning Module 3 of the Illinois State text is called "What's the Best Advantage?" In this module, students will finish the mousetrap cars that they started back in Module 1.

For my eighth graders, this will be an excellent opportunity to integrate science in the lesson. As I wrote earlier, the next NGSS science lesson on the computer is on motion and force. I've been delaying it until the students can learn about force and Newton's Laws. Well, as it just so happens, the students are supposed to measure the force used to launch the mousetrap cars -- in Newtons! So the idea is to have the eighth graders use the first hour to use the mousetrap cars and then the second hour to complete the online assignment.

But the problem is that our mixed-up Wednesday schedule might finally be changing. Here's how the old schedule worked: first period I would have sixth grade, then second period I'd have the eighth graders for "science" (the online assignment), and then third period I'd keep the eighth graders for STEM (which I'd use for either math or an Illinois State project). The problem with the old schedule is trying to fit music into the schedule. According to the music teacher's schedule, eighth grade music started near the end of first period and was intended to last into second period. So the eighth graders began the day in the history classroom and switched to music when the music teacher arrived -- only to have it end 15 minutes later when the sixth graders arrived to the history classroom.

The new, more logical schedule has eighth grade music line up with second period. This means, among other things, that the eighth graders won't be in my room for both an online science lesson and a STEM project, since they'll still be in music. I assume that I will begin the day with the sixth graders in my room -- but I can't send them to the history classroom after first period, since the eighth graders will still be in there for the music lesson. So I'd either keep the sixth graders an extra period or have them go to English and have the seventh graders come to my room.

Under the old schedule, the seventh graders came to my room for fourth period -- but then their music lesson took place in my room, and I didn't see them for math or science at all! Frankly, I wouldn't mind seeing the seventh graders tomorrow, and I bet they'll enjoy beginning the project. If they're in my room only for music as usual, then I'll just have them do the project on Thursday -- indeed, I suspect the project will bleed into Thursday for all the grades no matter what.

And if I do lose an hour with my eighth graders, then I'll just do the project today and have them do the online science assignment tomorrow after lunch -- that time on Thursday is usually for online math assignments, but I'll just have them do science instead.

But this is a two-day post, and I won't know what happens until tomorrow. That's right -- we received an email informing us of the new music schedule, but we were never told what to do with the students outside of music time! The English teacher has given up trying to figure out the schedule and says that she'll just give an online English assignment to whatever kids show up in her classroom! So this is what Wednesdays are like at our middle school, even one hexter into the year!

Returning to 2019, the Blaugust prompt is to reflect on these old posts. Note that in 2016 I mentioned a planned science lesson, then in 2018 I wrote about my failure to teach science. Yes, in 2016 I'd made so many plans to teach science properly, but none of them turned out well.

In that old 2016 post, I seemed to give the uncertainty of the Wednesday music lessons as an excuse not to teach science properly. Of course, the decision to make Wednesday the day for science was, in hindsight, a rather poor choice. I probably should have made Thursday the science day -- after all, I was guaranteed to see all three grades on Thursday. So I could have taught science to all three grades that day without any problems.

Today we return to Beth Ferguson, the only Blaugust participant who posted today:

Once again, Ferguson is a former teacher who's now an elementary math coach, so once again this week we turn back to basic arithmetic:

I’ve learned a lot by working with these children.  The math curriculum in our elementary schools is excellent.  The curriculum emphasizes finding solutions using multiple methods.  Typical algorithms are introduced after exploring a variety of other strategies designed to help students internalize place value and the meaning of each operation.

And of course, by "other strategies" she means "not the standard algorithm" -- so there's nothing in this post that traditionalists will like. Indeed, she includes two pictures here -- the "plus-minus" method for addition and "chunk out" for division.

Whatever "number talks" are, they sure don't look like anything traditionalists would approve of.

Instead of dwelling on elementary math again, let's get back to Benjamin Leis and the Geometry problem that he posted yesterday. Even though we arrived at a correct solution, I doubt it's the method that Leis intends for us to use.

Indeed, this reminds me of the tweeter CCSSIMath. Every once in a while, CCSSIMath would tweet a problem and say that Japanese seventh (or eighth, or whatever) graders can solve the problem, but Americans at that age can't -- because Common Core is holding them back. Many of these are Geometry problems that can be solved with some clever insight -- but I can't figure out that clever insight even after staring at this problem for a few hours. And if I can't figure it out, I wonder how seventh (or whatever) graders are supposed to solve it!

Anyway, Leis labels this as a Geometry problem, not a Trig problem. The solution I posted yesterday requires the difference formula for tangent plus advanced algebraic manipulation that most Geometry students haven't mastered yet. Therefore what I posted isn't the intended purely geometric solution.

Let me restate the problem from yesterday along with what we demonstrated:

ABCD is a square. P is on side CDQ is on side BC, Angle PAQ = 45, and AB = 1 cm. What is the perimeter of Triangle PCQ?

Solution: The desired perimeter is 2 cm.

The fact that the perimeter works out to be such a simple value suggests that a purely geometric solution is possible. And in fact, the perimeter of the triangle is exactly half that of the square -- that is, the triangle perimeter is exactly twice the side length of the square. This is significant because two of the triangle sides overlap parts of the square sides.

Working backward, we wish to prove that CP + CQ + PQ = 2, which we know equals BC + CD. And so we subtract out the sides that overlap. So we're now left to prove that BQ + DP = PQ.

Continuing to work backward, we notice that there might exist a point R on PQ such that BQ = QR, as well as DP = PQ. If such a point exists, then it would be easy to show that BQ + DP = PQ.

And in fact, if we then drew in AR, we form triangles that just might be congruent -- Triangle ABQ to be congruent to ARQ, and Triangle ADP to ARP. This seems to be the way to go. So let's try writing out a two-column proof:

Given: information from Leis as stated above
Prove: CP + CQ + PQ = 2

Statements                         Reasons
1. info from Leis               1. Given
2. AQ = AQ, AP = AP       2. Reflexive Property of Congruence
3. Draw R such that ???    3. ???
4. ???                                 4. ???
5. Triangle ABQ = ARQ,   5. ???
    Triangle ADP = ARP
6. BQ = RQ, DP = RP       6. CPCTC
7. RP + RQ = DP + BQ     7. Addition Property of Equality
8. CP + CQ + RQ + RP =  8. Addition Property of Equality
    CP + DP + CQ + BQ
9. CP + CQ + PQ =           9. Segment Addition/Betweenness Theorem
    CD + BC
10. CP + CQ + PQ = 2      10. Given Properties of a Square

In Step 5, we must prove that both pairs of triangles are congruent. All of our congruence theorems require knowing three pairs of congruent parts (ASA, SAS, and so on) -- and these are to be filled in to Steps 2-4. One congruence is already known -- Step 2 from the Reflexive Property.

Now R is to be chosen at a specific location so that a second congruence is guaranteed. But my problem is that, after hours of looking at this problem, I found no way to get the third congruence.

One possibility is to let R be the foot of a perpendicular from A to PQ. This makes two right angles at both ARQ and ARP, so that Angle B = ARQ and Angle D = APQ (as ABCD is a square). But I found no way to get a third congruence out of this.

Another possibility is to draw the ray AR such that Angle BAQ = RAQ and DAP = RAP. Such a ray exists from the given fact that Angle PAQ = 45. Since Angle BAD is 90, it means that the sum of the two angles RAQ and RAP (which is 45) equals the sum of the angles (BAQ and DAP), which makes BAQ = RAQ and DAP = RAP possible. But once again, I found no way to get a third congruence out of this.

A third possibility comes from transformations. Let lines AQ and AP be mirrors, and consider the composite of this two reflections. Since the mirrors intersect at A, the composite must be a rotation centered at A. Since the angle between the mirrors is 45, the magnitude of the rotation must be twice this, or 90. The rotation maps ray AB to ray AD -- and since AB = AD, the image point of B must be exactly the point D.

Now let R be the image of B under the first reflection. (Therefore D, of course, is the image of R via the second reflection.) Then we automatically have the congruent triangles -- ABQ = ARQ as well as ADP = ARP, because in each case, a reflection maps one triangle to the other. The problem here is that there's no reason for this point R to be anywhere near PQ, much less on PQ. (That R lies on that segment is needed for the Segment Addition of Step 9.)

It's possible to use trig to show that R lies on PQ. (Since trig is used, this can't be the original proof intended by Leis.)

To do so, let's set up coordinate axes. We'll place A at the origin and AB on the x-axis, so AD must be on the y-axis. (From the picture from Leis, AD is horizontal. But it's actually more convenient for me to place AB on the x-axis, which keeps everything in Quadrant I.)

Now from yesterday's work, we find the coordinate of Q to be (1, tan t) where t is theta, the angle we defined yesterday as the measure of BAQ. Then P becomes (tan (45 - t), 1).

We want to find the equation of line PQ. Since we have two points, we use Point-Slope. The slope of this line is easily found to be -CQ/CP, which we worked to find yesterday. Simplifying gives us:

m = (tan^2 t - 1)/(2 tan t)

So by Point-Slope we obtain the equation of the line:

y - tan t = (tan^2 t - 1)/(2 tan t) * (x - 1)

Now we wish to show that R must lie on this line. The coordinates of R are easy to find -- because reflections preserve distance, AR = AB = 1, so R lies on the unit circle. Its argument is Angle BAR, and since BAR = BAQ + RAQ and BAQ = t and BAQ = RAQ (reflections preserve angles), we conclude that BAR = 2t. Thus the coordinates of R must be (cos 2t, sin 2t).

Plugging this into our equation, we obtain:

sin 2t - tan t = (tan^2 t - 1)/(2 tan t) * (cos 2t - 1)

Our goal now is to show that R always lies on PQ -- that is, that the above equation is a trig identity:

sin 2t - tan t = (tan^2 t - 1)/(2 tan t) * (cos 2t - 1)
                    = (sin^2 t - cos^2 t)/(2 sin t cos t) * (cos 2t - 1) (multiplying fraction by cos^2 t/cos^2 t)
                    = -cos 2t/sin 2t * (cos 2t - 1)
                    = (-cos^2 2t + cos 2t)/sin 2t
                    = (sin^2 2t - 1 + cos 2t)/sin 2t
                    = sin 2t + (cos 2t - 1)/sin 2t
                    = sin 2t + (1 - 2 sin^2 t - 1)/(2 sin t cos t) (using alternate double-angle formula for cos)
                    = sin 2t - 2 sin^2 t/(2 sin t cos t)
                    = sin 2t - sin t/cos t
                    = sin 2t - tan t
identity proved

Thus R lies on PQ. This means that the triangles ABQ = ARQ and ADP = ARP are always congruent, and the above proof works to show that the perimeter of PCQ is 2. QED

But once again, we needed trig to save the day. Once again, this can't be intended by Leis.-- but at least I know that the triangles are supposed to be congruent (so that we're not wasting all our effort only to find out that the triangles aren't really congruent).

I just feel that there must be some way to complete the two-column proof using pure geometry without any trig. Since we should use the fact that PAQ = 45 somewhere, the second method by selecting the ray that makes Angles ABQ = ARQ and ADP = ARP is most likely the correct Step 3 in the proof. But what eludes me is Step 4 -- what other congruence can be made so that the triangles are indeed congruent. So far, all we have is AS, and we need an additional A or S to complete the proof.

Thursday, August 22, 2019

Lesson 0-7: Islamic Art (Day 7)

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

Lesson 0.7 of Michael Serra's Discovering Geometry is called "Islamic Art." This is in the Second Edition -- in the modern editions, "Islamic Art" is Lesson 0.6. Serra begins:

"Islamic art is rich in geometric forms. Islamic artists were familiar with geometry through the works of Euclid, Pythagoras, and other mathematicians of antiquity, and they used geometric patterns extensively in their art and architecture.

"Many of [Muhammad's] followers interpreted his words to mean that the representation of humans or animals in art was forbidden. Therefore, instead of using human or animal forms for decorations, Islamic artists used intricate geometric patterns."

As usual, the questions I derive from Serra's text instruct the student to create Islamic-style art. This art is based on tessellations.

There are a few interesting things in this lesson. First, Serra includes a sidebar called "Improving Reasoning Skills -- Bagels I." As it turns out, Bagels is an old 1980's computer game. I never played it on my old computer, but as a young child, I actually had an old toy (Speak & Math) which included a version of Bagels (called "Number Stumper"). Here's a link to a modern version of Bagels:

During the Responsive Classroom training at my old school, the presenter actually suggested Bagels as an opening week activity. In her version of the game, the word "Bagels" was replaced with "Nada," but the words "Pico" and "Fermi" were retained (so she called the game "Pico, Fermi, Nada"). Again, I don't post any version of "Pico, Fermi, Bagels/Nada," but if you want, you can use it in your own classroom instead of the "Islamic Art" lesson.

I do however include Serra's project for this lesson, "Geometry in Sculpture." This isn't directly related to Islamic art, though. Instead, he writes about Umbilic Torus, a sculpture. It was created by Helaman Ferguson and used as a trophy for the Jaime Escalante award -- named, of course, for the world's most famous math teacher.

Here is the Blaugust prompt for today:

Shoutouts!  Give a shout-out to a former teacher, a colleague, or someone in your school or community who is a difference maker.

Well, I don't have any colleagues, unless you count my fellow subs. But as for a former teacher -- actually, for Blaugust three years ago, I wrote about my favorite teachers. So let me cut-and-paste in from that post:

  • My favorite elementary teacher was my second grade teacher -- who later became my fifth grade teacher as well. She was one of the first to notice that I was good at math, and so she came up with the idea of having a Pre-Algebra teacher from the high school (which went from Grades 7-12 in my district) send me a textbook. As a second-grader I would work on the assignments independently, then my teacher would send my work to the high school before I worked on the next assignment. By the time I reached the fifth grade and was in her class again, she had convinced the high school teacher to send me the textbook for "APA," or Advanced Pre-Algebra.
  • Incidentally, my favorite math teacher was that teacher who sent me the advanced work. I finally met her when I was placed in her Algebra I class in the seventh grade. I was the only seventh grader in a class full of eighth graders, but she made me feel welcome in her class.
  • Just like Fawn Nguyen, I had my favorite history teacher when I was an eighth grader. He was also in charge of the Thespian Club at our school, and so he decided to teach history in a unique way -- he would dress up as a historical figure and lecture as if he were that character. Therefore his lectures were more memorable to the students. A few years ago, he retired from teaching, and many of my classmates held a big party for him.
  • My favorite science teacher was my junior-year teacher. I was an up-and-down student when it came to science -- the first two years of Integrated Science were more biology-leaning and I struggled a little, but the third year had more emphasis on physical science, which is more closely related to my strongest subject, math (as we spent over a month discussing with Kline's book). And so I did very well in this teacher's class -- indeed, she told me that I would finish the whole test in a few minutes and spend the rest of the time making my writing neat, and of course my answers were correct. She wondered why I wasn't enrolled in the magnet program, and I replied that I had moved to my new district as a freshman, while magnet students are recruited in the eighth grade. And so my science teacher convinced the school to admit me to the magnet program as a junior. Even though I was no longer in her class, she was still my most memorable science teacher for this reason.
  • My favorite English teacher was my senior-year teacher -- or to be precise, one of two English teachers I had that year. You see, the magnet program I'd entered a year earlier was a year ahead in English -- that is, junior-level English for neighborhood students was equivalent to sophomore English within the magnet. This meant that I would have to double up on English my senior year in order to graduate from the magnet -- and I didn't look forward to this, since my strongest subject was math, not English. So even though I was the only senior in a class full of juniors, I enjoyed this English teacher's class the most. This teacher allowed us to be creative in our writing -- I remember that for extra-credit, I wrote parodies of the literature we were reading, except with my friends and me as the characters. There was also an essay contest for seniors in which we were to write about a journey we had taken -- I wasn't going to participate, except that the junior English teacher whose class I had to take decided to assign the same topic for an in-class grade! I was in the unique position of writing an essay for class and submitting the same essay to the contest.  So I wrote about my journey through my education (much of which I just wrote about in this post) -- and won $200.

When I reflect upon my favorite teachers, I notice that they have some traits in common. Two of my teachers taught subjects I didn't enjoy, English and history -- and made them enjoyable by presenting them in a unique way. The other teachers taught my stronger subjects, math and science -- and they recognized that I was talented enough in those subjects to move me up to the next level.

Some traditionalists lament the fact that the Common Core accountability movement encourages teachers to focus on the weaker students at the expense of the stronger students. They say that some strong students want to move ahead in their classes, but the teachers, who claim their hands are tied by Common Core, won't let them.

I'm torn whether I should focus on my stronger or weaker students as I get ready to teach in the middle school classroom this year. On one hand, neglecting the weaker students is why many people spurn tracking, so I want to help my weaker students get ahead. But on the other hand, I myself am the beneficiary of certain teachers noticing my special talents and allowing me to succeed in more challenging classes. Therefore I owe it to my stronger students to support them and celebrate their talents just as my own teachers celebrated my own talents.

This is so important that it bears repeating. I owe it to my stronger students to support them and celebrate their talents just as my own teachers celebrated my own talents.

Recall back on Square Root Day the story I told about teaching my second grade friend the square roots of 0, 1, and 144. I admit that this incident, along with my admiration of my second grade teacher, formed the foundation of my desire to become a teacher. At first I didn't know that Grades 7 and higher even existed -- I knew that my elementary school was K-6, and I'd always believed that students went directly from sixth grade to college. I remember that as a kindergartner, to me the sixth graders looked like grown-ups, and so I expected that they were nearly college students.

Naturally, it was the arrival of my Pre-Algebra text that alerted me to existence of 7th grade. I wasn't sure whether I wanted to be a teacher because I wasn't sure I'd be good enough at any subject other than math, but the benefactor who gave me the Pre-Algebra text was a single-subject teacher who taught math and nothing else. And so I knew at that moment that I wanted to become a single-subject math teacher -- which meant that I'd most likely teach in a high school.

As it happens, I was checking the website of my old elementary school and -- believe it or not -- my second/fifth grade teacher still teaches there, 31 years after I was a student in her classes! (At least, she taught there last year, when the site was last updated.) And according to the website, she is now a special ed & intervention teacher.

For Blaugust today, we revisit the blog of Jenna Laib:

Recall that Laib isn't a math teacher -- she's an elementary math coach. And in this post, she quotes Cornelius Minor, who isn't a math teacher either:

Cornelius isn’t a math teacher. His background is middle school ELA, and and he authored the recent We Got This:  Equity, Access, and the Quest to Be Who Our Students Need Us to Be (Heinemann, 2019). After hearing him speak at the annual Heinemann Teacher Tour in July, my head was swirling with thoughts about the math classroom.

In this post, Laib and Minor discuss the idea that students should feel that they belong in the ELA or math classroom. This is of course a noble goal -- and again, the reason that I selected the five teachers (including both math and ELA teachers) is that each made me feel that I belonged in their classes:

…and while I think my math-specific knowledge (content, pedagogical, PCK/content knowledge for teaching, etc.) is critical to this work, some of the work is the collective work of all educators. How do we make students feel like they belong? Like they are knowers and doers of our content area?

But then Laib's post quickly delves into politics:

As an educator — a white educator — how do I take what I have learned about society and our cruel systems and turn this into a productive force?

And here we go with a mention of race -- once again, the idea is to ensure that students of all races (and genders) feel a sense of belonging.

Once again, my goal is to keep these school year posts race-neutral and gender-neutral. Except to mention the races of the authors of this post (Laib states that she's white, and Minor -- assuming that's a picture of him on his book -- appears to be black), that's all I'm saying about race here. (Of course, if you don't mind reading a race-based post, nothing's stopping you from clicking the links above.)

But this means that when Laib writes:

There were times when I punished 10-year-old Nicole, even though she was doing the same annoying behaviors as her friend Haylee, because Nicole had annoyed me by talking through our entire literacy block — four hours earlier. There were times when I gave Bryant an easy worksheet on addition instead of the engaging, open ended task Richelle got to work on, because Bryant had “too many gaps” in his math understanding.

her underlying fear is, is she treating Nicole and Haylee differently because of their race? And the comment about Bryant and Richelle echoes Chakravarty's post from yesterday.

Of course, I want to make sure that I'm treating all my own students equally. (My biggest fear is that I've mistreated students due to gender rather than race, but the same applies.) But once again, during these school-year posts, my focus should be on math-specific knowledge.

(Some people might find it ironic that I'm struggling to avoid mentioning gender and race in a post whose title mentions a religion -- Islam. Of course, we must be sure not to mistreat students due to their religion as well.)

Let's keep the rest of this post gender- and race-neutral by leaving Laib's blog. (Hmm, "Part 1" implies that she'll be posting a "Part 2" soon.) Instead, we return to the blog of Benjamin Leis:

Its been about a year since I last wrote about MathsJam:  So it feels like time to post an update.  I've been hosting a meeting once a month now since last August.  For the last several months we've been hovering at five attendees. They're never quite the same set of people and I've been growing the mailing list so it seems like we're on the cusp of getting a little bigger. I have yet to figure out the way to aggressively publicize something as quirky as this so the growth remains a slow burn. However, my long term vision is about two tables of folks per night and there are enough people already to make the get together fun so I'm not in a hurry.

Recall that Leis isn't a math teacher either -- he's a middle school club leader. Apparently, the other members of this "MathJam" are other teachers/club leaders.

The main topic of this meeting is paper folding something called a "hexahedroflex" -- which sounds and looks similar to the "hexaflexagons" mentioned previously on the blog. Since my blog is a Geometry blog, I couldn't help but notice his mention of our subject:

Finally: I added on my favorite small geometry piece from this month:

So what are we waiting for? Since Leis posted this problem, let me try to solve it:

ABCD is a square. P is on side CDQ is on side BC, Angle PAQ = 45, and AB = 1 cm. What is the perimeter of Triangle PCQ?

At first I noticed that not enough information is given to find any side length. In particular, we see that Angle BAQ can be anything between 0 and 45 degrees. So we let BAQ = theta, which then implies that DAP = 45 - theta. Now we can use the tangent function:

BQ = tan t (I'll write t so I won't have to keep rewriting "theta.")
DP = tan (45 - t)

There exists a subtraction formula for tangent, so let's use it:

DP = tan (45 - t)
       = (tan 45 - tan t)/(1 + tan 45 tan t)
       = (1 - tan t)/(1 + tan t)

Since ABCD is a square, all four sides are 1, including BC and CD:

CP = CD - DP
      = 1 - (1 - tan t)/(1 + tan t)
      = (1 + tan t - 1 + tan t)/(1 + tan t)
      = 2 tan t/(1 + tan t)

CQ = BC - CQ
      = 1 - tan t

Since we're eventually going to add the sides (to find the perimeter), let's rewrite CQ so that it already shares a common denominator with CP:

CQ = 1 - tan t
       = (1 - tan^2 t)/(1 + tan t)

Angle C is obviously a right angle since it's an angle of square ABCD. Thus we can use the Pythagorean Theorem to find the length of hypotenuse PQ:

PQ^2 = CP^2 + CQ^2
          = (2 tan t)^2/(1 + tan t)^2 + (1 - tan t^2)^2/(1 + tan t)^2
          = (4 tan^2 t + 1 - 2 tan^2 t + tan^4 t)/(1 + tan t)^2
          = (1 + 2 tan^2 t + tan^4 t)/(1 + tan t)^2
PQ^2 = (1 + tan^2 t)^2/(1 + tan t)^2
PQ = (1 + tan^2 t)/(1 + tan t)

And so the desired perimeter is:

Perimeter = CP + CQ + PQ
          = (2 tan t)/(1 + tan t) + (1 - tan^2 t)/(1 + tan t) + (1 + tan^2 t)/(1 + tan t)
          = (2 tan t + 1 - tan^2 t + 1 + tan^2 t)/(1 + tan t)
          = (2 + 2 tan t)/(1 + tan t)
          = 2

Sure enough, all references to tan t cancel out, so the perimeter is independent of theta. The desired perimeter is 2 cm.

Wednesday, August 21, 2019

Lesson 0-6: Knot Designs (Day 6)

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

Lesson 0.6 of Michael Serra's Discovering Geometry is called "Knot Designs." Knot theory is a very recent field of topology.  Two figures are topologically equivalent if one can be bent, stretched, tied, or untied to form the other.

Of course, knot theory isn't a suitable topic for middle school science. But Lord Kelvin -- of temperature fame -- once believed that atoms were knots in the ether:

"Although his theory was not true, the mathematical study of knots is a very current topic today."

...and so it may ultimately have a link to physics after all.  Meanwhile, let's see what Serra has to say about knots. This is Lesson 0.6 in my old Second Edition, while it's Lesson 0.5 in the modern editions, as those editions omit my 0.5.

Serra begins:

"Knots have played very important roles in cultures all over the world. Before the Chinese use ideograms, they recorded events by using a system of knots."

The book depicts a Celtic knot. As Serra explains, the ancient Celts carved various knot designs in stone. He writes:

"Knot designs are geometric designs that appear to weave in or interlace like a knot."

By Serra's definition, the Olympic rings form a knot. Today's worksheet is based on Serra's definition, so there is much emphasis on rings. Of the three questions I selected, one of them has the students draw a knot using a compass (so the shapes will end up being rings). The other two are puzzles involving interlocking rings.

One of the puzzle questions asks students to sketch five rings linked together such that all five can be separated by cutting open one ring. This is easy -- just link four rings to a center ring. The other question is a classic -- the Borromean rings are three rings such that all three are linked, and yet no two of them are linked.

Here is a link to a solution to the Borromean ring puzzle:

According to the link above, the Borromean rings are physically impossible -- unless the rings deviate slightly from perfect circularity. Hence the link labels this as an "impossible figure" not unlike the op art from Monday's lesson. Of course, we can draw op art, including perfectly circular Borromean rings, on paper with a compass.

Actually, I found a few interesting links involving Borromean rings. Here is Evelyn Lamb, who writes a math column for Scientific American once or twice per month ("Roots of Unity"):

Recall that knots and science are related. Here is a Borromean ring consisting of three atoms:

And here's a link to Borromean onion rings, courtesy one of my favorite mathematicians, Vi Hart:

Actually, Serra mentions the Gordian knot as well, even though I didn't include this question on my worksheet.

The recreational math website Cut the Knot is actually named after the Gordian knot. Here is a link to Alexander (Bogomolny) the Great, who explains why he chose that name:

Here is the Blaugust prompt for today:

Tell us about a favorite activity/lesson that makes you jump for joy when you get to use it.

Well, I guess that today's knot design activity might make me jump for joy. It's an opening activity that the students can enjoy as it allows them to exhibit their creativity.

Of course, that's probably not what Shelli meant when she posed this question. She obviously wants us to mention an activity or lesson that we actually taught in the classroom. This requires me to think back three years to my charter middle school classroom.

One day in November that year, I wrote about Benchmark Testing Week -- a very stress-filled week, since there were both print and online Benchmarks to take in both math and English, and they took hours to complete. But I was able to complete some activities during that week. I suppose that these activities made the students and me jump for joy -- if only because anything that wasn't a Benchmark Test made us jump for joy that week. (This week at my old school is probably the early-year Benchmarks, but here I'm referring to the end-of-first-trimester Benchmarks in November.)

Of the sixth grade class, here's what I wrote about a lesson I gave that week:

Yesterday I had a Bruin Corps member present during the sixth grade block. So here's what I did -- I began the class with a Warm-Up division problem from Illinois State -- 2400 / 51. Then I divided the class into two groups -- those who got the right answer and those who didn't. I gave the higher group to my Bruin Corps member, so he could help them move on, just as he and his fellow Bruin have helped with the higher grades this week.

But this led to problems. First of all, some students decided not to answer the Warm-Up. I remember one boy who chose not to answer, so I seated him with the students who didn't know division. Then when I assigned a division question, he answered it quickly -- meaning that he already knew all the steps and was just too lazy to do the Warm-Up. He complained that he had to sit with the students who didn't know division when he already knew it.

There were probably also some students who cheated and started copying the answers when they saw who was getting them right. I reckon that some teachers get around this by simply handing out colored cards rather than telling them where to sit. Then the students can't tell as easily who's getting the right answer, making it harder to cheat. Many of these problems persisted into the homework, where some students either skipped completely or wrote in nonsensical answers, such as "55 divided 8 is 55 remainder 3." And remember -- this is all despite the history teacher giving them time to do the math assignment in his class!

I tried to give the inspiration example I've mentioned earlier -- the Cubs. The North Siders have failed to win the World Series for 108 years before they won it all this week. They didn't let their past failure hinder their present success. Yet the students who can't divide aren't thinking like Cubs -- that is, like champions. They fail to divide, and so they keep coming up ways to avoid division rather than think, I don't know to divide now, but if I work hard (like the Cubs), I will.

Today during the IXL time, the struggling sixth graders continued to struggle. Indeed, I can't say that I've taught a single student to divide -- the stronger students already knowing division and the weaker students coming up with excuses. It doesn't help that I'm trying to crack down on students who don't remember their IXL passwords by telling them they can't use the computers. Weaker students who wish to avoid division just claim that they don't know their passwords!

As I reflect on this day two years later, this was probably one of the few times that I was able to implement Learning Centers that year. For about half the class, this was a good day, but for others, it didn't work. There were too many opportunities for students to avoid the work -- from the one boy who knew how to divide but didn't attempt the Warm-Up to the kids who tried to cheat during the Warm-Up to the ones who claim their computers didn't work. The whole idea of Learning Centers would have worked much better if I'd had tighter classroom management to make sure that students didn't attempt those tricks.

The seventh graders probably has the most enjoyable lesson of the week:

In fact, I realize that the seventh grade lesson is so easy that today, I actually go back and have the students work on the Orienteering STEM project from the Illinois State text. This decision is easy to make, since my support staff member and Bruin Corps member are both present during the seventh grade math block. So I take one group outside, give them compasses, and have them create a map to hide the "treasure" (the textbook), and then another group takes the compasses and map and uses them to find the treasure.

While all of this is going on, my Bruin Corps member watches the rest of the class. She sees that they've already mastered the concept of opposites, so she has them do some general addition of integers that are not opposites. She has them play a game for points similar to the "Who Am I?" games that I played as a sub last year, and buys pizza for the winning group. The winners turn out to be the same group that hides the first treasure -- that is, they are outside for part of the game, yet they still come back to win. And this is more amazing because the smartest student in that group is in fact absent today, so it's not as if he's doing all the work.

Yes, this project from two years ago came from the Illinois State text. But since only some of the students are working on the project, it ended up turning into Learning Centers again.

As for the eighth graders, I finally attempted to teach them some science that week:

Well, I tried to start online Benchmarks yesterday during the math intervention time usually devoted to the other online program IXL. But the problem was that there was a power outage! I'd already charged the laptops before the blackout, but it was impossible to access WiFi during the outage, so the students couldn't access the online Benchmarks. The blackout began at the start of lunch and ended right at -- you guessed it -- 2:25 (that is, P.E. time).

The English teacher suggested that I have the eighth graders finish the written Benchmarks for her own class, since the students couldn't finish them during English class. Well, the students refused to work on them, and when I told them that they were supposed to be working on the essay, one girl called out, "Well, you're supposed to teach us science and you're not doing that!"

"Okay, then," I replied, "let's start the science assignment now."

It's a good thing that I purchased that Common Core Science book last month, since yesterday was my first opportunity to use it -- after all, science was the last thing on my mind with all of this Benchmark stuff. (Notice that there is no science Benchmark, even though eighth graders are supposed to take the NGSS science test.) So I just jumped into Chapter 7 of that book, which as I wrote in my October 10th post, is on Matter and Its Interactions. I just had the students start writing the definitions of some vocabulary from that chapter, but they only got through the first four terms after all the arguing about the essay and writing. Still, at least I got some science in at a time when I thought I'd have very little time for science.

Yes, I admit three years later that this science lesson didn't work as well as I wanted. But then again, this was a spur-of-the-moment science lesson caused when a power outage prevented me from giving the online Benchmarks. The class didn't have a science text because copies of the Illinois State science texts were still a few weeks away from arriving. In the meantime, we had to use the online science text -- which we couldn't because of the blackout.

I could have made that lesson more enjoyable by changing it into a game, similar to the impromptu game played by the seventh graders the following day. But as I wrote earlier, even writing words and definitions was enjoyable compared to taking the Benchmarks.

If you prefer that I answer Shelli's prompt with a Geometry lesson based on the U of Chicago text (outside of my one year of teaching), then one of my favorites is on the area of a circle, which I first posted in 2018:

If G can be divided into A unit squares, then G' can be divided into A squares each of length k. And the area of a square of length k is clearly k^2, so the area of G' must be Ak^2. For the circle problem k is the radius r, and A is the area of the unit circle or pi, so the area of a circle is pi * r^2. We can do this right on the same worksheet -- there's already a circle drawn of radius 10 times the length of a square, so instead of the length of each square being 1/10, let it be 1 instead. Then the area of the circle of radius 10 is equal to the number of shaded boxes, or 314, since the old unit square has been divided into 100 unit squares.

I wrote about several activities related to the circle area in that post -- which includes activities created by other teachers. Unfortunately, I've never actually taught this lesson in the classroom -- I was planning on giving my seventh graders a pi lesson for Pi Day, but I didn't quite make it to March 14th that year.

Today's Blaugust poster is -- um, nobody. So far, no Blaugust participant has made any post dated August 21st yet. Of course, that's probably because most real Blaugust participants are too busy teaching today to post.

The most recent Blaugust post is from Sue Jones, who works at a community college in Illinois:

Mirabile dictu, I have student workers and managed to get my “first half fours” video made and up. It’s about how to figure out 4 x a number, you double it twice. I decided I’d stick to pictures on this one and on the “bigger ones and how to double anything,” show how x 2 is “doubling” so … x 2 x 2 is … x 4….

Clearly, Jones works with students who are far below college-level. It's a point I've made before on the blog -- students don't learn all their times tables at once, but learn the easier ones first. Here we see that a student who has mastered the 2's times tables (doubling) can learn the 4's simply by doubling twice.

Jones admits that this post isn't necessarily the best Blaugust post:

Resolving tomorrow to do a Blaugust #mtbos worthy blog…. and/or … the geogebra sharing idea. Whoops, that needs a video, too. 16 steps…

And by now "tomorrow" actually refers to yesterday, so I assume that the Blaugust-worthy post isn't really coming.

Another recent poster who appears on Shelli's Blaugust list is Elissa Miller. She was a major participant in previous challenges:

First of all, when we click on the link, we first see a link to another blog, Sam Shah. I notice that Shah has signed his name on Shelli's Blaugust list. But at that link, we see no actual posts written by Shah himself, but rather an invitation to yet another challenge (that is, separate from Blaugust).

And then we see that Miller didn't even bother to participate in Shah's challenge herself. Instead, the post is written by yet another teacher, Leila Chakravarty.

We begin with Miller/Shah's prompt:

Prompt: How do you express your identity as a doer of mathematics to, and share your “why” for doing mathematics with, kids?

And the following response is written by Chakravarty. Notice that Chakravarty describes herself as being both "fast" and "slow" at math:

It started in first grade when two boys who I consistently outperformed were placed in second grade math, but I remained in first, getting sent to the back of the room for finishing my work too fast and bothering people who weren’t done yet. I would like to point out that boys who bother people because they are bored are “being boys” and “need a challenge.” Girls who do so are non-compliant. It continued in seventh grade when I was enrolled in a lower math class despite a strong placement score because my elementary school was notorious for lackluster math preparation.

Math was not for freshman girls who worked too slow and had too many questions, who needed to study for hours and go to the tutoring center. Math was not for queer brown girls who were homesick, navigating identity, or too distracted by music classes. Math was not for someone with her high school math teacher’s voice echoing in her head.

Clearly both gender and race are key issues in Chakravarty's post. Since this is a school-year post, I wish to keep this post as gender-free and race-free as possible. (I will point out that Chakravarty is a South Asian name, so this explains her use of "brown." I avoid interpreting her use of "queer" here.)

But clearly speed is a key issue in this post. Chakravarty's problem here seems to be the fact that many teachers equate "good at math" with "fast at math." Her counterargument is that she is very good at math, but not very fast at math.

If we think back to the traditionalists, they mention speed at math mainly when it comes to basic math, such as the arithmetic of single-digit numbers. They suggest that students who are fast at basic arithmetic have more time to spend on doing higher math, such as Algebra I. (For example, it's easier to factor a quadratic polynomial if one can add and multiply quickly.) Students who haven't mastered single-digit arithmetic tend to get frustrated when trying to solve an Algebra I problem.

In Chakravarty's case, notice that she was fast as a young first grader (when, in her pre-Core class, she probably learned how to add single-digit numbers). She only describes herself as "slow" only when referring to post-elementary school math. Thus the reason that she's slow at higher math most likely has nothing to do with the failure to master arithmetic, as the traditionalists suggest.

Was I guilty at equating "good at math" with "fast at math" as a teacher at the old charter school? It's possible to argue that any sort of multiplication quiz, such as a Dren Quiz, rewards speed. But my Dren Quizzes didn't have a strict time limit (such as 50 problems in one minute). Yes, I admit that I did cover one day for another teacher who gave such a timed quiz -- but that was another teacher.

There was one day when I was guilty of equating "good" with "fast." It actually happened about a week after covering for that other teacher. (Did she corrupt me that day?) My seventh graders were working on a quiz on opposites (the same lesson that I described earlier in this post), but everyone was talking loudly. After I collected a few quizzes, I then declared that everyone who hadn't turned one in would get a zero for talking during the quiz. Thus I inadvertently rewarded speed -- only those who were fast enough to turn it in would get a grade.

I remember one girl who was a strong yet slow student -- she became upset. I must admit that I had treated this girl just as Chakravarty's teachers had treated her, because I rewarded speed for no reason.