x = (1/4)y^2 - (1/2)y + 7
is a parabola with directrix, y = ?.
Hmm, even though Blaugust is over, one of its participants blogged about this last month:
http://mymathclub.blogspot.com/2018/08/parabola-coordinates.html
I already linked to the blog of Benjamin Leis last week. Here he gives various parabola formulas, including a directrix formula:
Directrix: y = (4ac - b^2 - 1)/(4a)
So what are we waiting for? For this problem we substitute a = 1/4, b = -1/2, and c = 7:
y = (4(1/4)(7) - (-1/2)^2 - 1)/(4(1/4)
y = 7 - 1/4 - 1
y = 23/4 or 5 3/4
And this answer doesn't make sense. The answer to every Pappas problem is the date, yet there obviously isn't a fractional date like 23/4 or 5 3/4.
Maybe the Leis formula is incorrect. So let's try completing the square instead:
x = (1/4)(y^2 - 2y) + 7
x = (1/4)(y^2 - 2y + 1) + 7 - 1/4
x = (1/4)(y - 1)^2 + 6 3/4
Now how do we find the directrix from this? We might recall that it has something to do with 1/(4a), which in this case is 1/(4(1/4)) or 1. The parabola opens up to the right. Then the focus must be 1 unit right of the vertex while the directrix is the opposite -- 1 unit to the left. So the directrix is x = 5 3/4 -- the same value that we found using the Leis formula. Therefore the error lies not with Leis.
Hmm, notice that since the parabola opens to the right, the directrix is a vertical line, x = 5 3/4. Yet Pappas asks for the directrix in the form y = ?. So we've already spotted an error with Pappas. Notice that Leis gives his directrix in the form y = ? as well, except that he's clearly referring to a parabola that opens upward. In short, a parabola of the form y = f (x) has a directrix of the form y = ?, while a parabola of the form x = f (y) has a directrix of the form x = ?.
But that still doesn't explain where the value 5 3/4 comes from. Anyway, let's work backwards from the date to figure out what Pappas originally intended. Saturday's date was the first, and so we look to see what y = 1 has to do with the parabola. We notice how the number 1 appears when we were completing the square -- the vertex is (6 3/4, 1). This suggests that when she wrote "directrix," she really meant axis of symmetry.
For a problem writer to confuse "axis of symmetry" with "directrix" is a tough error. When we complete the square and see (6 3/4, 1) as the vertex, we're able to figure out the mistake. But if we use the Leis formula, we never see the coordinates of the vertex at all -- and why should we? A student in his class would see the word "directrix" in the problem and jump to the directrix formula.
And we're able to figure out the error only because this is a Pappas problem whose answer is meant to be the date. If this were a textbook, then the only hint that "directrix" is supposed to be "axis of symmetry" is the variable -- y = ? instead of x = ?. If I were a student, I'd just find the directrix and assume that y = ? is a typo (or more likely, not even notice the wrong variable). It would never occur to me to find the axis of symmetry.
This is definitely an Algebra II problem -- axes of symmetry are taught in Algebra I, while directrices never are. Oh, and parabolas that open rightward don't appear in Algebra I either.
Meanwhile, yesterday's Pappas problem was an interesting Geometry problem:
What is the y-coordinate of the point (-1, 9) reflected about the line y = x + 4?
Even though I like doing Geometry problems from the Pappas calendar, I don't do the ones that land on weekends or non-school days like Labor Day unless they contain errors. Hey, wait a minute -- in this problem Pappas drew the graph of y = x + 3 instead of y = x + 4. That's an error, which means I get to do the problem! (It's not as egregious an error as "directrix," but I really want to do this problem on the blog.)
Anyway, it's interesting because reflections are so important in Common Core Geometry. But in most reflection problems that appear on the PARCC or SBAC, the mirror passes through the origin. In this problem, the mirror doesn't pass through the origin. So how do we solve it?
By the definition of reflection, a mirror is the perpendicular bisector of the segment whose endpoints are the preimage and image. This means that to solve this problem, we can:
- Find the slope of a line perpendicular to y = x + 4 (easy).
- Find the equation of a line passing through (-1, 9) with the slope we just found.
- Find where this line intersects y = x + 4.
- Find the distance between (-1, 9) and this point of intersection.
- Find a point on this line the same distance from (-1, 9) as the point of intersection. This ordinarily requires us to find the equation of a circle with the correct center and radius and find the two points of intersection with the line -- one is (-1, 9) and we want the other. It's also possible to skip this step by using vectors instead.
But we can avoid this process entirely by graphing. We begin by graphing both the mirror y = x + 4 and the preimage (-1, 9) on a coordinate plane.
Now since we need a line through (-1, 9) perpendicular to y = x + 4, we begin counting out a slope of -1 (the opposite reciprocal of 1) starting at (-1, 9). So we count down 1, over 1, down 1, over 1, down 1, over 1, and stop as soon as we reach y = x + 4. This is a total of down 3, right 3, and the point of intersection is (2, 6). We now treat this as our vector, and go down 3, right 3 to (5, 3), which is the final image point.
Therefore the y-coordinate of the image is 3 -- and of course, yesterday's date was the third. Most of the time, we consider graphical methods to be more difficult than algebraic methods -- for example, if we're given a system of linear equations, we'd rather solve it algebraically than graph it. But graphing is excellent for reflection problems, since we're not given the equation of one of the lines (provided, of course, that the point of intersection has integer coordinates).
This is what I wrote last year about today's lesson:
Lesson 1-4 of the U of Chicago text is called "Points in Networks." (It is combined with the old Lesson 1-1 to form the new Lesson 1-3 in the modern edition of the text.)
But there is a twist here. I actually taught the Lesson 1-4 activity two years ago as it was my first day of school activity for my middle school classes. And so I'm actually going to reblog the experience of my first day as a teacher from that year. Yes, this means that this will be yet another "crying over spilled milk" post that has dominated my blog these past few weeks. In this post, I'll only repeat what I wrote about my eighth grade class, as well as the commentary I wrote at the end.
11:25 -- My eighth grade class arrives. This is my smallest class, with only 12 students -- but there are only eight students present at the start of class. I begin the class the same way I start all my classes, with a Warm-Up question:
What is 2 * 2 * 2 * 2? (That is, 2 times 2 times 2 times 2.)
Most students answer correctly, although a few tried to add. A student or two is upset that the very first thing we do on the first day of school is multiply. I point out that the answer is 16 -- and that today is the 16th. I always go around to stamp correct papers -- many teachers point out that students enjoy getting stamps, and my students are no exception.
11:35 -- My student support aide arrives -- the English teacher and I are each assigned one. Actually, she arrives with the four missing students, all girls.
We move on to an Opening Activity -- the Konigsberg Bridge Problem. I've written about this problem previously on the blog and even suggested it as a first day of school activity -- well, now I'm finally giving the activity on an actual first day of school. This is a little of what I said about this problem here on the blog:
The Königsberg Bridge Problem is a famous math problem from nearly 300 years ago. Fawn Nguyen, a well-known math blogger and fellow Southern Californian -- she lives in Ventura County -- used this as an activity in her geometry class:
http://fawnnguyen.com/famous-bridge-problem/
As we all know, the Königsberg Bridge Problem is impossible to solve -- it has no solution. But I don't want to start the class with a problem that the students can't solve -- they're already frustrated enough with problems that do have solutions when they just can't find them.
The whole point of this lesson is to point out that students should look for patterns, and that sometimes it's just as important to know why something is impossible as it is to know why something is possible.
Let me complete this with a note on pronunciation. The U of Chicago text points out that the name Euler ends up sounding like "Oiler." But how does one go about pronouncing the name Königsberg? I once read that the o-umlaut ends up sounding like "uh," almost like "ur." A Google search reveals a ten-second video in which this name is pronounced:
But there is a twist here. I actually taught the Lesson 1-4 activity two years ago as it was my first day of school activity for my middle school classes. And so I'm actually going to reblog the experience of my first day as a teacher from that year. Yes, this means that this will be yet another "crying over spilled milk" post that has dominated my blog these past few weeks. In this post, I'll only repeat what I wrote about my eighth grade class, as well as the commentary I wrote at the end.
11:25 -- My eighth grade class arrives. This is my smallest class, with only 12 students -- but there are only eight students present at the start of class. I begin the class the same way I start all my classes, with a Warm-Up question:
What is 2 * 2 * 2 * 2? (That is, 2 times 2 times 2 times 2.)
Most students answer correctly, although a few tried to add. A student or two is upset that the very first thing we do on the first day of school is multiply. I point out that the answer is 16 -- and that today is the 16th. I always go around to stamp correct papers -- many teachers point out that students enjoy getting stamps, and my students are no exception.
11:35 -- My student support aide arrives -- the English teacher and I are each assigned one. Actually, she arrives with the four missing students, all girls.
We move on to an Opening Activity -- the Konigsberg Bridge Problem. I've written about this problem previously on the blog and even suggested it as a first day of school activity -- well, now I'm finally giving the activity on an actual first day of school. This is a little of what I said about this problem here on the blog:
The Königsberg Bridge Problem is a famous math problem from nearly 300 years ago. Fawn Nguyen, a well-known math blogger and fellow Southern Californian -- she lives in Ventura County -- used this as an activity in her geometry class:
http://fawnnguyen.com/famous-bridge-problem/
As we all know, the Königsberg Bridge Problem is impossible to solve -- it has no solution. But I don't want to start the class with a problem that the students can't solve -- they're already frustrated enough with problems that do have solutions when they just can't find them.
The whole point of this lesson is to point out that students should look for patterns, and that sometimes it's just as important to know why something is impossible as it is to know why something is possible.
Let me complete this with a note on pronunciation. The U of Chicago text points out that the name Euler ends up sounding like "Oiler." But how does one go about pronouncing the name Königsberg? I once read that the o-umlaut ends up sounding like "uh," almost like "ur." A Google search reveals a ten-second video in which this name is pronounced:
12:05 -- Because I know how tough the 80-minute block schedule can be on middle school students, I provide a music break. I get out my guitar and I play the following inspirational song:
The Dren Song -- by Mr. Walker
[OK, I'll skip the song since I already reblogged it in a recent post.]
Along the way, I explain that a "dren" is a reverse-nerd -- a nerd is someone who's good at math, and a "dren" is someone who doesn't understand the basics of arithmetic. As it turns out, the student who complained about 2 * 2 * 2 * 2 enjoys this song and looks forward to my next song.
I show my students the September 2015 Boys' Life article about the mathematicians and scientists who work for NASA and the possible future of people traveling to and living on the moon. But as it turns out, eight of the 12 students in my class are girls, so I don't expect Boys' Life to motivate them.
Instead, I tell them about the movie trailer that was released just yesterday -- Hidden Figures, about the scientist Katherine Johnson who worked for NASA and the Apollo projects in the 1960's. For those of you who have read my blog before, it goes without saying that I plan on watching this movie, and I highly recommend that my students watch it in January as well.
12:15 -- I proceed with my next Opening Day activity -- Personality Coordinates. This activity comes from the King of the MTBoS, Dan Meyer:
http://blog.mrmeyer.com/2013/personality-coordinates-icebreaker/
Each person in a group picks a dot and writes her name next to it.
Now the group’s job is to label the axes. Physical attributes don’t require all that much thought and don’t reveal all that much, so don’t allow them.
That’s it. It requires a surprising amount of creativity and conversation. Happy first day of school, teachers.
12:30 -- My support aide leaves, and this is a good time to end the period with an Exit Pass:
If you don't know the answer, ..
The answer is "at least know where to find it," which is posted in a corner of the room. (I mentioned this in an earlier blog post.) Some wrong answers are "ask the teacher" and "you're a dren."
12:45 -- My eighth grade class goes out to lunch.If there's anything I could change about the way I ran the class today, it would be to teach the entire class in reverse order. That way, the Exit Pass becomes a Warm-Up, a scavenger hunt to find the rest of the quote, Personality Coordinates occur earlier in the class, and the 2 * 2 * 2 * 2 question doesn't turn off students right at the start of the period.
I would also rewrite the Konigsberg worksheet. I'd already changed the worksheet to add more bridge problems, including some trivial ones. But now I'd number those trivial problems #1 and #2 (rather than #3 and #4, as they were numbered today).
Another problem I have has to do with explaining my directions clearly. I was hoping to create a seating chart directly from the Personality Coordinates worksheet (since the students are already seated in groups of four), but I couldn't because some groups randomly labeled the dots rather than place the student sitting northwest in the upper-left corner of the page. Also, some students wrote the Exit Pass on a separate sheet of paper rather than the back of the Warm-Up.
I remember explaining my directions to the students -- but I could be remembering my explanations to the 6th and 7th grade classes, not the 8th grade class. Anyway, I know that I don't always explain instructions clearly to my students from my days as a sub, so I must give the students the benefit of the doubt whenever I see them misinterpreting instructions.
OK, let's return to 2018. There are a few things that I want to say about my reflection -- from the perspective of it being a year later and I not having returned to that classroom.
First of all, that year I wrote that maybe I should have reversed the order of the first day. Then the opening Warm-Up question, 2 * 2 * 2 * 2, becomes an Exit Pass.
The answer to that question is 16, and the first day of school that year was August 16th. This was an idea that came from Pappas -- making the answer to the Warm-Up question be the date. But, as we already know, this fell apart because the Illinois State Daily Assessment took over the Warm-Up.
Thus it would have been better for me to establish the Pappas tradition by making the date be the answer to the Exit Pass, not the Warm-Up. In other words, I should have reversed my Exit Pass and Warm-Up not only that first day, but everyday. This also serves another purpose -- after Exit Passes, I often never gave the correct answer because some students are already walking out the door while others are still trying to correct their mistakes. By making the date be the answer to the Exit Pass, the students already know the answer, so never finding out the answer isn't an issue. Again, this then frees Warm-Ups for the Illinois State Daily Assessment, which we would then work out on the board.
That year, I wrote that I often had trouble explaining my instructions clearly. But in hindsight, I think this was part of my overarching management issue -- the students never stopped talking. I now believe that I often spoke quickly because I knew I needed to talk before the students did. And this applied not just to giving clear directions -- my math explanations suffered as well. I know that I could have taught math much better if I could count on the students being quiet during the lesson.
Obviously, at some point on the first day of school, I need to tell the students of the importance of being quiet. At the time, I actually didn't mind the students talking during the Warm-Up, but the problem is that they don't stop talking when I'm giving instructions, lessons, or tests. Therefore it's worth it to keep the kids quiet and in their seats during the Warm-Up as well.
By reversing the Warm-Up and Exit Pass, the new first day Warm-Up is the scavenger hunt to complete the phrase, "If you don't know the answer...." Unlike most Warm-Ups -- especially the Illinois State Warm-Ups (which would begin after that is set up online) -- this Warm-Up actually involves students moving around and talking -- so it's a bit awkward to show the importance of being quiet and sitting down at this point.
Of course, I could tell the students to be quiet before starting -- and then afterward, I tell the students to be quiet again. It's important not to accept any excuses for not being silent, including:
- You're mean!
- You're the only teacher who makes us sit down and be quiet.
- You're unfair!
- You're unreasonable!
- You're weird!
- Making us be quiet is juvenile.
- I wasn't talking.
- Why do we have to be quiet?
I know that Blaugust is over, but I can't help but mention the following Blaugust poster. You see, if the Blaugust challenge had a winner, here's who the undisputed champion of Blaugust would be:
Elissa Miller is a high school math teacher. She does not give her home state, and she teaches multiple classes.
Anyway, she actually posted all 31 days in August. She was also the champion of another blogging challenge, MTBoS30, when she posted 30 times in May 2016. Yes, I know that May actually has 31 days, but the name of the challenge was MTBoS30, so only 30 posts were required. But with 31 posts in August, Miller has beaten her own challenge record!
Here are some interesting Miller posts during her winning month of blogging:
In this post, Miller writes about how she doesn't like making phone calls. But this year, her principal is making her call parents, and so she prepares a spreadsheet full of information.
Here are more spreadsheets -- except this time the students are the ones creating them. It's a project where students keep track of data over a long period of time. Like "Personality Coordinates" from earlier in today's post, the creator of this project is the Former King of the MTBoS, Dan Meyer.
Miller writes about how her summer plans for the new school year have gone awry. She reminds us that everyone, especially teachers, make mistakes. Her post reminds me that despite all the mistakes I made in my first year of teaching, I can and will find a way to improve.
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