1. Implement classroom management based on how students actually think.
7:55 -- Second period is the first of three Algebra II classes.
Over the weekend, I prepared work for some more conic section problems. Once again, I name second period as the best class overall of the day, as they worked hard on these problems.
Fortunately, the TA (mentioned in my October 23rd post) is present today. I ask him about the parabola equation. He informs me that yes, the teacher did use the y = a(x - h)^2 + k (or a similar equation with "x =" if the symmetry axis is horizontal) rather than the form used for the QR codes (which were more like (x - h)^2 = 4c(y - k)). I make sure that the students understand that the teacher will be grading their Thursday tests, not the QR codes. So they should write the equations the way the grader wants to see them.
There is also a new student in this class. At his old school, he wasn't studying conic sections -- so I doubt it's fair to expect him to master one of the most difficult units in the text by Thursday's test. I believe that the other students in his group helped him at least understand circles (likely the simplest conic) today.
8:45 -- Second period leaves and third period arrives. This is the second of three Algebra II classes.
While this class is hard-working, it's also slightly more talkative. A few students sit in other seats to form groups of five (though I'm not given official seating charts for Algebra II).
9:40 -- Third period leaves. It is now time for tutorial.
Students are supposed to ask permission to use cell phones during tutorial (and, of course, verify that the phone is being used for work). But one student takes it out without doing any work -- and he's one of the Algebra I students who landed on Friday's bad list.
Since he claimed that he was using the phone for math, I make him complete the first page of last Friday's graphing packet (four graphs) or else I'd report him for using phones in tutorial without doing an assignment. He finally completes the graphs, and so he will begin today with a clean slate.
10:25 -- Tutorial leaves for snack.
10:45 -- Fourth period arrives. This is the last of three Algebra II classes.
This class is slightly more talkative than third period, with even more students moving seats. Once again, I don't have a seating chart, but I know that six students don't belong in a group of four desks.
As the class ends, one student asks a question about ellipse and hyperbola problems where the foci are given rather than either a vertex or co-vertex pattern. She tells me that she has trouble recalling the formulas a^2 = b^2 + c^2 and c^2 = a^2 + b^2 for the ellipse and hyperbola respectively. But unfortunately, I don't have time to tell to give her any tips for Thursday's test.
Here's what I might have told her if I'd had time. Consider graphing, on two separate pairs of axes, an ellipse and a hyperbola centered at the origin. To make it easier to visualize, assuming that the vertices (and hence the foci) land on the x-axis. Then for the ellipse, the foci (c, 0) must lie inside the ellipse (a, 0), and so a > c (and a^2 > c^2). But for the hyperbola, the vertex (a, 0) is closer to the origin than the focus (c, 0), and so c > a (and c^2 > a^2). (Assume that a and c refer to the vertex or focus with positive x-coordinate.)
11:35 -- Fourth period leaves. Fifth period is the teacher's conference period, which leads into lunch.
1:15 -- Sixth period arrives. This is the first of two Algebra I classes.
And here's what I know you've been waiting for -- the lyrics of my new "x's and y's" song. I decided to start with the Brianna and Chase version and keep their first verse. It was difficult for me to figure out the second verse from the YouTube video, and so I replace this with my own lyrics. The replacement lyrics go with today's lesson on graphing using intercepts.
First Verse:
I walked into math class, the math teacher said,
"We're gonna graph our x's and y's."
Today, I just wanna say.
Took out my calculator to add,
Wrote the numbers on the graph.
You know my x's and y's.
Pre-Chorus:
One, two, three, they gonna run back to me,
'Cause I'm the math genius that they never have seen.
One, two, three, they gonna run back to me,
I always want an A but I keep getting a B.
Chorus:
x's and y's, I solve them,
And graph my x's and y's.
I just might try,
x's and y's.
Second Verse:
Where y = 0, that's the x-intercept,
Where x = 0, that's the y-intercept.
My my, how the lines go by.
Plus-y gets high, minus-y gets low,
Plus-x goes right, minus-x goes left.
You know, that's how the graphing goes.
(To Pre-Chorus)
It's interesting to compare Brianna and Chase's version to Elle King's original lyrics. For example, Elle King's second line was "I showed him all the things that he didn't understand" -- which fits almost with Brianna's first line (since that's what math teachers do -- show students things that they don't understand). Yet the young student doesn't keep this line in her version.
Meanwhile, I can tell that Brianna tried to retain King's line about getting high and low (middle of the second verse), but I can't quite tell what the young girl is singing in her video. So instead, I write about getting high and low on the y-axis.
After I sing the song today, the students begin the lesson on Chromebooks. But still, I have the students try the following four graphs on the board, in order to establish a good list:
- x + y = 3
- 4x + 3y = 12
- x - 5y = -5
- 5x - y = -5
The guy who almost gets in trouble in tutorial is back in this period. He starts throwing an eraser at another student, so I must place him on the bad list. This time -- after having seen him work hard on the four graphs in tutorial -- I give him the chance to erase his name from the board. It's at the end of class and I don't want him to be late to seventh period, so I just ask him what y = 0 and x = 0 mean in the context of the song. He correctly identifies the x- and y-intercepts.
Overall, sixth period becomes the best Algebra I class of the day, since the class showed much improvement from Friday.
2:05 -- Sixth period leaves and seventh period arrives. This is the second of two Algebra I classes.
This time I must write a girl's name for throwing a paper airplane -- and I don't give her the opportunity to erase her name, because she tries to erase it herself. This is when I'm distracted with making sure that the Chromebooks are put away and plugged in.
By the way, this teacher has numbered the groups 1-9, which the four students within each group labeled as A-D. Her procedure for Chromebooks is to name a letter, and that person either takes out the Chromebooks or puts them away. In her lesson plan, she tells me to have Student B take out the Chromebooks, but not which student to collect them. I ask Student B to collect the laptops, but then some of them complain, so I choose Student C instead. Then while the C's grudgingly plug in the laptops, some of the A's, B's, and D's try to leave the room before the bell rings -- and this is when the girl erases her name from the board.
I've blogged before about procedures and routines -- and certainly, taking out and plugging in Chromebooks each counts as a procedure. But many students find following these procedures a pain, and so on sub days, they make every excuse not to follow them -- and call the sub a pain in the back for enforcing them.
For example, the laptops are numbered 1-36 -- and there's a one-to-one correspondence between seat numbers and laptop numbers (so Chromebooks 1-4 are for students 1A-1D, Chromebooks 5-8 are for students 2A-2D, and so on). But many students in seventh period don't take the correct laptops -- and it doesn't help that sixth period students never put them away properly.
With seating charts available for Algebra I, I try to enforce the seating chart -- so I don't have five or six students sitting at a cluster meant for four. But within each cluster, many students still don't sit in exactly the single seat listed on the seating chart. Normally this doesn't make that much difference (though it makes the ideal silent roll-taking -- glancing at the empty seats to determine who's absent within seconds -- impossible). But in this case, the seat-switches make it impossible to tell which kid is Student B, so I can't be sure whether any of the students getting Chromebooks are actually Student B (nor which Student B's are just lazy).
Some of these procedures are nearly impossible to enforce as a sub. The students won't cooperate with certain procedures, such as sitting in the exact assigned seat. And more often than not, they won't cooperate with laptop storage procedures, even if the laptops and slots are numbered. If a sub tried to enforce these procedures, the students tell the sub that he's wrong to do so, and that the only correct thing to do is to err on the side of letting students do whatever they want.
But I think back to when I was a regular teacher at the old charter school -- in particular, when I handled laptops during IXL time. I mentioned how I could have labeled the slots so that the students had a particular place for their own laptop.
After seeing today's class, I wonder whether I could have labeled my seats 1A-1D, 2A-2D, and so on up to 6A-6D, and then labeled the computers and slots the same way. Notice that I didn't quite have enough laptops for all the students -- only about 18 or 19 of them.
So in this case, I label my laptops as A's, B's, and C's, with no laptops for the D's. Then I seat the students strategically. In the smaller eighth grade class, I leave the D's (and most B's) empty, so that every student get a laptop. In the larger sixth grade class, I strategically place in the D seats either the top students who don't necessarily need extra IXL time, or else certain special ed students who have trouble using computers.
Then comes the tricky part -- enforcing the seating chart and laptop arrangement. If a student sits in the wrong seat (even if it's just one seat off) or places the Chromebook in the wrong slot (even if it's just one slot off), I punish the student, period. I punish the students even if they complain that it's not fair, or it's a pain in the back to follow all the procedures, or it doesn't really make a difference where they sit or where they put the laptop. I punish the students even if they gang on me -- one student breaks the rule, and the others tell me that it's unfair to punish the first student. And a natural punishment for not putting the laptop away properly is that they must do a long written assignment on the next IXL day instead. (If I ask C's to put computers away and one is out of place, then it's Student C who gets punished.)
It was Harry Wong who, in his teaching book, stressed the importance of procedures. But even Wong doesn't anticipate, for example, how difficult it is to get a group of teenagers to all sit in their assigned seats (so that attendance can indeed be taken silently, as he suggests). Even first- and second-graders are much more likely to be obedient regarding the seating chart than high school students, who rather break the rules to sit near their friends.
There's one thing I could have done at the end of class today -- entertain the A's, B's, and D's at the end of class by playing the original Brianna song on YouTube (which I'd set up during conference period so I could practice singing -- the speakers were already on). This might keep them from trying to leave early, so I can focus more on the C's and their Chromebook charging, as well as the girl who erases her name.
3:00 -- Seventh period finally leaves, thus ending my day.
Let's look back at the focus resolution again:
1. Implement classroom management based on how students actually think.
As it turns out, our queen has spoken:
http://fawnnguyen.com/serenity-prayer-and-teaching/
On Third Pi Day, Nguyen posts about what teachers can and can't change in the classroom. The following section is intriguing:
- How we treat our students.
I failed and failed at this. The same way I’d failed at times as a parent to my own three children. I yelled, sent the kid out, made sure I got the last word because I needed everyone to know I was in charge. The side effects of my behavior always included shame, regret, guilt. Mostly shame. To give myself some grace, most of these incidents involved my believing the child had lied or demeaned another person.
I read what a student had written about another teacher, fresh from a recent incident. He didn’t want to give me the paper, and I only asked for it because he was supposed to be writing an assignment on that paper. As I was reading, he said, “I didn’t mean to… I was mad…” I finished reading and looked up, “Do you feel better now that you’d written this?” Tears brimmed his big brown eyes, he nodded, “Yes.” I crumbled up the paper and tossed it into the garbage can, “I’m glad. No one else needs to see that note. I love you. [The teacher whom he’d written about] loves you too. We care about you.” He straightened up, wiped his eyes, and thanked me, and off he went to lunch. Not until he was out the door that I thought, he still owes me the assignment. But then I thought that no one else needed to know that he wrote on a different topic instead. Full credit.
- First, a group of girls wrote the letter, not just a single student.
- It was the director (principal) who asked the girls to write a letter about their concerns. So if I had asked them for the letter, they would have refused and said it was for the director only.
- I didn't have any written essay assignments (as Nguyen's pretense for reading the letter was that it was for an essay assignment). Notice that even though Nguyen was, like me, a math teacher, she mentions earlier in this post that she was a science teacher. (Who knows -- if I had taught science properly, there might have been a written science assignment to collect that day, so that I could read the letter.)
But most importantly, I figure that those eighth grade girls hated me almost as much as they did their English teacher. On the other hand, Nguyen's student trusted her enough to give her the letter -- and accept her statement "I love you...We care about you" as genuine. And that's the point Nguyen is making here -- if I had treated my students better, they might have confided in me, and maybe there's no need for any smear letter at all. I can't control how the girls treated me, but I can always control how I treated them.
Other things that Nguyen mentions are:
- How we react and respond to the adults in our building.
I don’t think I said two words about anything during my first 900 days as a teacher, and I was a science teacher at a middle school. I did what I was supposed to do: show up, teach, return the classroom key to the office before I go home, rinse, repeat. I was the quiet type anyway, so quiet that when I announced my decision to become a teacher, this person laughed, “How can you be a teacher? Ha!! You can’t be a teacher, you’re so quiet!” I didn’t say anything, I just smiled, a quiet smile that betrayed my suspicion that she might be right.
I found my voice on Day 901, on staff dress-up day, maybe it was Halloween. I walked down the hallway in scrubs borrowed from my husband at the time. A male teacher who also donned scrubs said pleasantly, “You can be my nurse.” Equally pleasant, I said, “I’m dressed as a doctor today.”
- Make the curriculum come alive.
There are a lot of good resources and people out there to help us with this. Teach in a way that no software or Khan can replace or replicate what we do. To make math come alive, we need to come alive. Make up for our shortcomings with all that we are passionate about, and hopefully topping that list is building a relationship with our kids. Even if math is not their favorite subject or dividing fractions is a big zit, they still enjoy coming to your class and think you’re cool for coming to their games and wearing that stupid costume, for the third year in a row.
There are several reasons that the students who wrote that smear letter against me didn't necessarily like me either. One might have been my inability to teach science -- sometimes they would complain and ask when the real science lesson was going to be.
But another of the eighth graders' complaints was that I spent too much time telling them about the importance of grades. This often happened when the whole class is talking, and that they should be quiet, otherwise they won't learn anything and can't get an A.
This seems to indicate that it's a bad idea to overemphasize grades in my classes. Yet today, I just sang my second song that actually mentions grades. Today's song contains the line "I always want an A but I keep getting a B," and that "All About That Base" parody earlier this calendar year also mentions grades -- "She said, 'Teachers like effort, so you better get all A's.'"
Of course, both of those lines were written by some other YouTube parodist (like Brianna). I've now since decided that it's OK to mention grades if it's in a song -- or if grading is directly relevant to what I'm trying to tell them (such as in Algebra II when I remind them that they should write the parabola equations not the QR way, but the teacher's way, since she's the one who will grade their tests on Thursday).
But if all I'm trying to do is convince the students to follow rules in general (such as be quiet when I'm teaching new material), then it's better to say "Because I said so" and then punish them.
By the way, earlier tonight, Fawn Nguyen posted again:
http://fawnnguyen.com/six-ways-to-see-visual-pattern-324/
As the title implies, this post is all about a visual pattern (one of 324 of them). Since it's visual, you really need to click on the link to learn more, since anything I write here is meaningless unless you see the actual pattern.
Chapter 19 of Ian Stewart's The Story of Mathematics is called "Number Crunching: Calculating Machines and Computational Mathematics." Here's how it begins:
"Mathematicians have always dreamed of building machines to reduce the drudgery of routine calculations."
This chapter is all about the rise of calculators and computers. The story of computers must begin with one of those dreamers, Charles Babbage (who, if you recall, was mentioned in Douglas Hofstadter's book as well). Quoting Babbage, Stewart writes:
"'Another member, coming into the room, and seeing me half asleep, called out, "Well, Babbage, what are you dreaming about?" to which I replied, "I am thinking that all these tables" (pointing to the logarithms) "might be calculated by machinery".'"
And his dream was known as the difference engine -- and even though Babbage never built it, a version was finally completely recently for a London museum.
Much of this chapter is about numerical analysis -- how computers actually calculate values of complicated functions, and how accurate those calculations are. (When I completed my Masters Degree at UCLA, I actually considered pursuing a Ph.D. in numerical analysis -- and I was even assigned a dissertation adviser! But then that C+ in a graduate analysis course scared me away.)
Stewart actually mentions one important method of numerical analysis: Newton's method:
"The point x_(n+1) is where the tangent to the curve at x_n crosses the x-axis. As the diagram shows, this is closer to x than the original point."
Newton's method is often illustrated by using the square root function. If we wish to find the square root of some number A, we let x_1 be our initial guess. Then for each successive guess x_(n+1), we let this be the number exactly halfway between x_n and A/x_n. (If x_n really were sqrt(A), then note that x_n and A/x_n would be equal. So the number halfway in between is closer to sqrt(A) than the previous guess.)
The author also mentions Euler's method, used to solve differential equations. (Euler's method is mentioned in the Hidden Figures movie -- Katherine uses it to find the numerical solution of an ODE that has no closed-form solution.) Other methods are more accurate than Euler's method:
"One of these, the so-called fourth order Runge-Kutta method, is very widely used in engineering, science and theoretical mathematics."
Stewart describes how these symplectic integrators are used to measure in outer space:
"Symplectic integrators are especially important in celestial mechanics, where -- for example -- astronomers may wish to follow the movements of the planets in the solar system over billions of years."
Stewart moves on to algorithms -- procedures for solving a problem. His first example is Euclid's algorithm used to find the GCD (or GCF) of two whole numbers. (I blogged about this algorithm a few years ago when reading Ogilvy's book.)
"It can be shown that if n has d decimal digits (a measure of the size of the input data to the algorithm) then the algorithm stops after at most 5d steps."
This means that the algorithm runs in linear running time. The class of all algorithms that run in polynomial time is called P:
"So, although we know that many problems can be solved by an algorithm in class P, we have no idea whether any sensible problem is not-P."
The author also mentions the class NP of all "non-deterministic" polynomial algorithms, which he defines as ones for which an answer can be checked (rather than found) in polynomial time:
"And now we come to the deepest and most difficult unsolved problem in this area, the solution of which will win a million-dollar prize from the Clay Mathematics Institute."
And this is the Millennium problem P = NP?
"The most plausible answer is no, because P = NP means that a lot of apparently very hard computations are actually easy -- there exists some short cut which we've not yet thought of."
Stewart concludes the chapter by asking the question, if a computer discovers a proof that no human can verify, can we really call it a proof?
"This assertion is controversial, but even if it is true, the result of the change is to make mathematics into an even more powerful aid to human thought."
The sidebars in this chapter are a biography of Augusta Ada King, what numerical analysis did for them, and what numerical analysis does for us.
Lesson 6-2 of the U of Chicago text is called "Translations." In the modern Third Edition, we must backtrack to Lesson 4-4 to learn about translations.
This is what I wrote last year about this lesson:
To emphasize the coordinate plane, I've titled this lesson "Translations on the Coordinate Plane" rather than just "Translations." Yesterday I warned you that another translation lesson was coming up soon, and as it turns out, that day is today.
I said that there are several ways to prove that the transformation mapping (x, y) to (x + h, y + k) really is a translation. First is a coordinate proof using the Slope and Distance Formulas, but I mentioned that we can't use those until after dilations in the second semester. Second is to manipulate the transformations until various mirrors cancel, but that requires arcane symbols such as:
T = r_U(n) o r_U(m)
and I said that it would only confuse students. Fortunately, there's a third way to complete the proof that I alluded to a few weeks back -- and I've decided to use this form of the proof in today's post.
We begin by noting that the two transformations (x, y) -> (x + h, y) and (x, y) -> (x, y + k) are already proved to be translations. The first translation slides points along the x-axis h units, and the second slides points along the y-axis k units. But how are points not on either axis transformed?
Now that we're in Euclidean geometry, we can prove that translations slide every point the same distance -- which is what we expect translations to do. We consider the points (0, 0), (a, 0), (a, b), and (0, b), and these points are the vertices of some quadrilateral. We know by definition, the side along the x-axis has length a and also the side along y-axis has length b. We also know that three of the angles are right angles -- the angle at (0, 0) since the axes are perpendicular, the angle at (a, 0) because x = a is perpendicular to the x-axis, and the angle at (0, b) because y = b is perpendicular to the y-axis. In Euclidean geometry, we know that the sum of the angles of a quadrilateral is 360, so if three of the angles are right angles, so is the fourth, and the quadrilateral is a rectangle.
But now we know the lengths of two sides of this rectangle, a and b, and we want to find the lengths of the other two sides. Of course, the other two sides must also have length a and b as opposite sides of a rectangle are congruent. Yet how do we know this? Most Geometry texts would now say that every rectangle is a parallelogram and the opposite sides of a parallelogram are congruent, therefore the opposite sides of a rectangle are congruent. But unfortunately, the U of Chicago text doesn't give the Parallelogram Tests and Consequences until the last part of Chapter 7.
But let's consider the Quadrilateral Hierarchy yet again. Now only is every rectangle a parallelogram, but every rectangle is an isosceles trapezoid in two different ways -- and we've indeed proved that the opposite sides (legs) of an isosceles trapezoid are congruent. Therefore, we really do know that the opposite sides of a rectangle are congruent.
What we've shown is that the distance from (a, 0) to (a, b) is b units, and that the distance from (0, b) to (a, b) is a units. So horizontal and vertical distance work exactly as we expect them too -- that is, we now know that the distance from (x, y) to (x + h, y) is h units, and the distance from (x + h, y) to (x + h, y + k) is k units.
So (x, y) -> (x + h, y + k) moves points h units horizontally and then k units vertically. But that still doesn't tell us that the composite of the horizontal and vertical translations is itself a translation. Let's instead try the following -- let P(a, b) and Q(c, d) be two points, P' and Q' be the images of P and Q under the first translation, and then P" and Q" be the images of P' and Q' under the second one.
Then as the first translation slides h units, PP' = QQ' = h, and as the second translation slides k units, P'P" = Q'Q" = k, and as all horizontal and vertical lines are perpendicular (by the same rectangle argument given earlier), both angles PP'P" and QQ'Q" have measure 90. Thus triangles PP'P" and QQ'Q" are congruent by SAS.
So by CPCTC, PP" = QQ" without requiring the Distance Formula. Also, we have that the angles P'PP" and Q'QQ" -- the angles PP" and QQ" make with the horizontal -- are congruent, meaning that PP" and QQ" are in the same direction. So the transformation maps every point the same distance in the same direction. Therefore the transformation is a translation. QED
Notice that this proof essentially assumes a sort of "Converse to the Two Reflections Theorem for Translations" -- the forward theorem asserts that if a transformation is a translation, then it moves every point the same distance and direction, and so the converse would say that if a transformation moves every point the same distance and direction, then it's a translation. But this converse is trivial to prove -- as soon as we have a point P and its image P' and say that every point is moved the same distance and direction as P, then it's easy to find two mirrors to set up the translation. One of the mirrors can be placed at P and the other at the midpoint of
Also, notice we assume that just because the two angles are equal, P and Q must be moved in the same direction. (And technically, we assume that any horizontal line must be perpendicular to any vertical line.) Both of these can be proved using corresponding angles.
Still, this proof should be more intuitive than the other versions which require symbolic manipulation or formulas we haven't covered yet to prove.
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