# Taking Stock

I spent 5 hours today cleaning out my desk, going through files, recycling mountains of paper. In concrete terms, it means school is over. Graduation is tomorrow. And then: I’m on summer vacation [1]. So now a bit of a brain dump as I take stock.

I’ve found this year to be an important transition point:

For the first time, I taught ninth graders, and for the first time, I taught geometry. And in order to do that, I worked an insane number of hours with my partner-in-crime and co-teacher BK in order to write an entire curriculum from scratch, from head to toe. Yup, you read that right. We — in essence — wrote a textbook. We sequenced the course, we wrote materials and designed activities for the course, and we had kids do all the heavy lifting. There are particular moments as a teacher which standout as “big moments.” Moments where we know we’ve developed immensely as a teacher. Transitioning from individual and partner work into total groupwork was one of those moments. Converting my non-AP calculus course into a standards based grading course was one of those moments. And writing a curriculum from scratch, in a single year, with an insanely thoughtful collaborator was the most recent of those moments [2].

The previous two years (before this school year) were two of the hardest years I’ve had as a teacher. We teachers were called on to do a lot in the wake of our school’s five year strategic plan — and it became overwhelming. I had no work-life balance. And  I became a bit curmudgeonly because of those tough years. But this year, things have been better. I still have no work-life balance, but the overwhelming onslaught of initiatives have subsided. One of the things I did to actively try to stay positive this year was to write down every single day one good thing that happened to me — big or small. From the first day of classes to the last. And those things are archived here. This was especially important because at the start of the school year, my mom was diagnosed with cancer (she is doing very well, fyi, no worries).

That being said, I am going to make a goal: that next year, I am going to just let the things that I can’t control go… There’s no point in getting worked up over something that you can’t do anything about. Instead, I’m going to stay loose, and bring back my frivolity and humor, and go off the beaten path in class more. While organizing today, I was looking through a number of old emails and cards from students, and saw so many inside jokes and fun times that they references… and then I thought about this year… and I came up blank. I couldn’t think of a time that I doubled over laughing in class. I couldn’t think of an ongoing joke that I had with a student. I could think of great lessons and a ha moments, but nothing frivolous and fun. So my vow is to make sure that next year involves more joy and laughter. For me, and for my studentsEvery day.

Wow, yes, this braindump led me to something big. With that, I’m out.

[1] That doesn’t mean I’m done with school. I have lucky 13 college recommendations to write. And two summer projects that each will take 25 hours each to complete (revise my multivariable calculus curriculum; plan for our new schedule next year with longer blocks).

[2] I’ve written entire course curricula before. Calculus, for example. But that took a few years to write and get added to. And Adv. Precalculus, which I did in a single year, but lacked the collaboration and innovation that I was able to do this year with BK.

# Some Geogebra Fun

I have an awesome friend and colleague at my school who is a geogebra master. He has started keeping a blog — Geogebrart — posting fairly frequently some stunning, jaw-drapping mathematical art he created using this powerful program.  Check this recent one out — which happens to be one of my favorites! Dualities!

Although I know most of the basics of Geogebra, I have not yet progressed to the stage passed “novice.” However I really want to get there, because this program is so freaking awesome.

When I was at TMC14 this summer, there was a sesh run by John Golden, Audrey McLaren, and Jedidiah Butler. They are like Jedi masters of Geogebra (though I know Audrey will play coy and say she isn’t…). When I was there, I learned about conditional objects, and it was awesome. (The google doc they used to help people out is here.) In about 30 minutes, with the help of John Golden and some kind people near me, I was able to make a rinky-dinky geogebra file which has a triangle on it, and has three points on the three different sides. When you drag each point close to where an altitude of the triangle would hit that side, I had something like “WOW!” or “YOU DID IT!” pop up! And if you got all three points close, something like “ALL THREE?! YOU’RE A SUPERSTAR!” show up.

Okay, okay, I wasn’t going to show you it because it’s sooooo dumb. But heck, whatever, here it is. Click on the image to check it out.

Okay, you and me, we both know that file is totally useless a teaching tool. And it is gross looking. By all accounts, I should not be excited by it. But the weird thing was: I was really proud of it, and I wanted to show everyone around me what I created. Even though I know it was simplistic and useless, I wanted to create a file that did X and I was able to do it! Although it felt dumb to get psyched about it, I was so excited that I could create something that would do what I wanted it to — that I couldn’t do before!

Today I was again inspired by my colleague and friend’s geogebra art, so I wanted to create some of my own.

I was quickly able to make this in 10 minutes [click on the picture to go to the file and mess around with the parameters! cool things happen!]

My goal was to define a curve parametrically and then have — at a ton of points on the curve — a circle to be drawn so it would look like a tube. That ended up looking only moderately neat. So I changed it so that as one traveled on the parametric curve drawing the circles, the radius of the circles would change (based on some formula I fed it). The reason this wasn’t so hard for me? I knew all the commands to do this except for the parametric curve command, which was easy to figure out.

But then I wanted to try my hand at something that would take more than 10 minutes and that would challenge me. I wanted to have something “show” a sphere via the animated drawing of “slices” (ellipses). It was inspired by this beautiful gif, but I knew that was going to be too hard for me to start out with. So I decided I would start out with a simple sphere with slices going horizontally and vertically, with no rotation.

After somewhere between 90 and 120 minutes, I did it! (You can click on the gif to go to the file and play around with some of the parameters.)

Although the image isn’t as cool as the one that took me 10 minutes to create, I’m way prouder of this. It is because it took a ton of learning and trial and error in order to figure out how to do this. The set of problems I encountered and somehow figured out:

• I know how to create a single ellipse in the center of the circle, but how do I make another ellipse a certain distance away that still only touches the edge of the circle?
• How do I make the ellipses “width” (minor axis) decrease so that it is fattest near the equator, and almost like a line near the poles of the sphere?
• Without manually typing a zillion ellipses, how do I tell Geogebra to create all the vertical ellipses at once, and all the horizontal ellipses at once?
• The way I was generating the ellipses resulted in a problem… once an ellipse “hit the pole”(became a point), it would turn into a hyperbola. So I needed to find a way to make sure that once an ellipse “hit the pole” it would disappear.

I figured all this stuff out! So even though the sphere doesn’t look nearly as cool as I’d like, I feel so much more accomplished for it than with the super-cool-looking circles of variable radii drawn on a parametrically-defined curve.

***

Note: it’s amazing how “simple” this sphere image is once you figure it out. Once you create three sliders:

t goes from -5 to 5 [incriments of 0.1]
StepSize1 goes from 0.05 to 2 [increments of 0.05]
StepSize2 goes from 0.05 to 2 [increments of 0.05]

and you enter the following two (that’s it!) geogebra commands:

Sequence[If[abs(t – n StepSize) < 5, x² / (25 – (t – n StepSize)²) + (y – t + n StepSize)² / (1 – sgn(t – n StepSize) (t – n StepSize) / 5)² = 1], n, -5 / StepSize 2, 5 / StepSize 2, 1]

Sequence[If[abs(t – k StepSize2) < 5, (x – t + k StepSize2)² / (1 – sgn(t – k StepSize2) (t – k StepSize2) / 5)² + y² / (25 – (t – k StepSize2)²) = 1], k, -5 / StepSize2 (2), 5 / StepSize2 (2), 1]

Then you’re done! Well, you should animate the t-slider to make it cycle through everything without you having to drag the slider!

Seriously, two commands, that’s all it takes. But hopefully from the commands themselves you can understand why it would take me so long to figure out…

# Playing with Math

Sue VanHattum (of Math Mama Writes) is in the finishing stages of editing a rich collage of works that is aptly named Playing with Math: Stories from Math Circles, Homeschoolers & Passionate Teachers.

Truth be told, I tend to eschew reading about math education because most of what I’ve read feels dry and irrelevant to me. I tend to stick with who I trust when it comes to math education: my colleagues, whether they be in-person or virtual. And although I didn’t tell Sue this, because she was so kind to share an advance copy with me, I fretted about falling asleep while slogging to get through 67% of this book because of the subtitle. (I have never led or been to a math circle, nor do I work with homeschoolers.) I’m just an average joe teacher who keeps his sights on his classroom and his kids, and… well… that’s about it.

Now for the punchline: I couldn’t stop reading it. All 100% of it.

The book isn’t composed of traditional articles-as-chapters. Playing with Math is, rather, a collage. I was treated to bursts of math puzzles, activities, and games (the majority of which were completely new to me) wedged between short and medium-length vignettes from people who are working with kids on math. (There are almost 50 contributors to this book, some of whom I know!) I can see this book being a great present for one of my NYC colleagues, because as I was reading it on my laptop, I kept thinking how perfect this book would be for subway reading because each piece was only a handful of pages. A testament to the book is that as I was reading it, I wanted a zillion post-its and tabs to flag this or that.

Even though I haven’t been to a math circle nor am in any way involved with the homeschooling community, reading the pieces around those topics were interesting precisely because I know so little about them. But moreso, they got me thinking about ways I could differently think about my classroom and my kids. When it came to the math circles, it gave me ideas on how to let go and trust kids to take charge of their own mathematical learning more. And when it came to homeschooling (and unschooling), I wondered how much kids lose their love of learning precisely because of the structure of school. The author of the pieces did this by telling stories. Some were like video cameras, documenting and explaining the “teacher moves” in some particular math circle sessions. Some were powerful and wrenching first person narratives about mothers trying to help their children. And the teacher section was a curation of powerful stories of teachers like me, trying to be a little bit better each year. Some pulled lines to whet your appetite:

We began today’s math circle, the first of six sessions, sitting in an “ogre.” Not a circle, not an oval, but an ogre, the kids’ way of precisely describing the shape we made.

Peter Panov and David Plotkin can barely stay in their seats. They’re firing questions and comments and conjectures and quips at their instructor, Jim Tanton, as fast as he can respond. The whole class of thirteen-year-olds was giggling when I walked in. On the board is a list of some Pythagorean triples and a procedure for generating more. Tanton had just generated the triple (-1,0,1), and a general hilarity about the idea of a triangle with a negative side-length erupted. Now it’s as if he were dangling strings in front of a pack of puppies. They’re all worrying at the problem, tossing out ideas, wiggling in their seats.

Looking back now, I see how far off the mark we were. We should have advocated for our daughter to ensure she received an intellectually, socially, and emotionally appropriate education. But we were overwhelmed by the more-pressing problem of Ryan, so we missed her quiet desperation. I wish I had been more proactive and looked below the surface. I wish I had worked more closely with her teacher. I wish I had trusted my own instincts about my daughter’s needs and abilities.

I waited eagerly for him to arrive the next morning, looking forward to the moment when he would put AAAAAALLLLLL those tiles together in neat rows by category, and he would have to exchange several times (not to mention his surprise at seeing all the units disappear when multiplying by ten). Instead Roland came in, shook my hand, and said: “My dad told me that all I have to do is add a zero to 8,696 and I’ll have my answer, because when you multiply by ten you just add a zero.” My heart sank. Oh no, Dad! You robbed your son of such a cool experience!

Several years ago, my school experienced a shortage of geometry books. There was talk of teachers sharing class sets and photocopying pages for students. I decided to try a different strategy. I took this as a professional challenge to see how long I could teach without a textbook. I knew whatever happened would be a growing experience for me as well as my students. Through no fault of the school library, two or three weeks stretched to seven. By that time, I was well into my “textbook-free” strategy, so I just kept the ball rolling … for the rest of the year.

I like stories, and that’s what this book is. Not disquisitions or pronouncements or shallow research studies. Stories. The authors bring to life their experiences and interactions with kids and their insights and their frustrations, and I started care about these people, their children, their classrooms.

If there is one theme that stood out to me, it is this: we need to work at undermining the constraints that we are confronted with (whether it be textbooks for teachers, or the entire school experience for some parents) to allow us to do what we all know is best for kids… playing and engaging with math in a way that tugs at internal motivation (curiosity, the excitement of discovering something) rather than external motivations (praise, grades). We need to continue to find ways for doing math to be beautiful and creative acts of passion and wonderment and joy. The contributors of Playing with Math are working on this, and I am inspired by their stories.

Sue speaks about the origins of this book here:

And she is having a crowd-funding campaign. “The book has been written, edited, and illustrated. The money raised here will allow us to pay the artists, editors, and page layout folks, and it will pay for the print run.” I contributed so that I could get a paper copy of the book and finally mark it up with all the post-its and flags I want!

# Teaching Award

About a month ago, I received a teaching award at my school. Technically, I suppose it isn’t an award, but a chair (“the William C. Stutt Chair for Math, Science, and Technology”). Fancy, right? I wasn’t going to blog about it, but it is something I want to archive and that’s the biggest (but not the only) reason I blog.

It’s given out every three years, and the last person to get it is one of my best friends at the school (who is also the person I look up to as a teacher).

When I was called up, there was a standing ovation from the faculty. Of course, let’s put the cards on the table here: there always is a standing ovation from the faculty when anyone gets an award. But I can’t help but admit I got a real glow-y feeling. I was overcome when I saw my parents there, a surprise! They popped out of the curtain and hugged me. I didn’t quite know what to say, so I babbled. All I remember saying is my teaching motto: “Try to suck a little bit less each day.” I posted this on facebook, me feeling babble-y, and a friend said: “You are amazing. Your comment to the faculty about trying to suck less everyday was perfect and came up again a number of times over the remainder of the meeting. I hope you and your parents had fun celebrating your awesomeness this afternoon. Also, please take that standing ovation personally. We could have gone on clapping forever. There was nothing perfunctory about it. Congratulations!” So yes, me all feeling warm and fuzzy.

I also posted this on facebook: “Although I’m not one who basks in honors and awards (I even skipped out on going to my college Phi Beta Kappa induction and a writing award in college), I do feel like teaching is a profession where you don’t get a lot of positive reinforcement for the emotional struggle that you carry with you every day. A few kind words from students occasionally, or a nice email from a parent, if that. 99% of what we do goes unseen and unacknowledged. It’s isolating and exhausting. So this award was a nice thing, something I can turn to when I feel like I’m emotionally drained and a failure. (Which is more often than not.) But more than that, it reminds me how important it is that we teachers give accolades and kudos to each other in a million unofficial ways, *everyday.* Because most all the teachers (especially the math and science teachers) at my school are pretty awesome. And every one of us are working to do right by our kids. And more than awards that get handed out once in a blue moon, we need to pay attention of the good that everyone else is doing around you, and acknowledging and huzzah!-ing those things. Yes, that’s what I see from this. Let’s prop each other up.”

The little news blurb on our school website is here. Archived.

# Experimentum Crucis: A Symposium Course

This January, for seven days, I taught a seven day course with a friend and fellow teacher. Our school eliminated midterms and instead instituted different programs for different grades. Juniors and seniors were given the opportunity to sign up for full-day courses designed and taught by faculty on topics of interest. Faculty were given the opportunity to design courses which got kids to think about topics in a different way.

My co-teacher and I developed a course that was designed to be interdisciplinary (we were working at the intersections of history, science, and philosophy), hands-on (students would be working in the laboratory), and rigorous (meaning kids would be expected to think and work at a high level).

Designing and teaching this class was one of the hardest things I’ve ever done as a teacher. And I don’t know — honestly, I don’t know — if we were successful or not. Even with the feedback we received. Thus even though it was challenging, I’m not sure I felt it was rewarding. In fact, the reason I’m writing this blogpost now, months after this, is because I was so exhausted with the whole thing I couldn’t bring myself to even think about it in a reflective or objective way.

The origins of the class go back to the previous year, when my co-teacher and I started trying to envision precisely what the big picture ideas were, and how we were going to get kids to go from point A to point B in their thinking. This also was coupled with the question: how the heck do you design seven days with the same group of kids, from 8:3o to 3:15. Seriously put yourself into our shoes for a second. Initially, it’s pretty exciting! All this time! Do what you want! But then you realize: you are going to have 12 to 16 kids in your charge, and you need to fill up that time with multiple activities! Quickly this went from exciting to daunting and anxiety-filling. For months, the co-teacher and I would have meetings, read books and articles, come up with ideas, refine our ideas, and throw out our ideas. Coming up with a lesson plan for a single day took weeks of work. The agony, the hours, the frustration… I don’t wish that upon my worst enemy. But we finished.

Our course abstract:

Can you imagine building a battery without the concept of electrons?  What would it be like to describe chemical reactions without discussing atoms?  Would you believe Einstein’s theory of relativity if no text book told you to and there were no way to test it?

In this course, you will have the opportunity to put yourself in the shoes of scientists who (in retrospect) revolutionized the way people viewed and understood the natural world.  By carrying out famous historic experiments, you will explore the process of creating “scientific models” and “scientific facts,” many of which we now take for granted as self evident. This course will be hands-on and interdisciplinary. In addition to lab work, we will read primary and secondary sources that will allow you to place science in historical context and understand scientific knowledge making as a process and a product of its time.

Our course objectives:

Through this course, students will explore:

• science in historical context
• how science is influenced by and a product of its time
• that the process of science involves models changing over time
• that what we take for granted is often messy, weird and sometimes illogical
• that science is a human endeavor
• that the making of science is a process
• how scientific “facts” get accepted/discarded –  that ideas are nothing without the acceptance of many people

• What is an experiment?
• What is a scientific fact?

Anchor Texts:

Thomas Kuhn’s The Structure of Scientific Revolutions
Original papers by Robert Boyle and Alessandro Volta
Secondary texts

Experiments:

Originally, we planned to have a number of experiments: Proust, Boyle, Volta, Oersted, Einstein. However because we had a snowday (there went Einstein and the discussion of thought experiments), and because some of the experimentation took much longer than expected, we had to eliminate more (Proust and Oersted). Thus, we only ended up working extensively on Boyle and Volta.

Content:

One day was spent on a field trip to the Chemical Heritage Foundation in Philadelphia, but the rest of the days were spent having deep class discussions and carrying out two in-depth experiments in the labs. We did Boyle’s Law experiment, and they had to bend glass to make their own J-tube, and play carefully with mercury. (We inducted all our kids into the Royal Society, after reading bits of the original charter, and administering the oath that the initial founders took.) Our kids saw that our modern instantiation of Boyle’s Law (PV=k) was nothing like the original formulation (they only were given Boyle’s original paper to guide their research and help them figure out how to reproduce the original experiment), and they started to get at the idea that Boyle was looking at his experiment through a totally different lens (“the springiness of air”). My favorite part was when kids saw how their little sidebar about Boyle in their chemistry textbooks was just a black box for so much! And how it wasn’t just “one crucial experiment” that suddenly worked and changed our understanding. Mwahaha, the title of our course is precisely the thing we aimed to get our kids to debunk.

Our second experiment was building (well, improving upon) the first voltaic pile. Again they only had Volta’s original paper to work from, they were given many materials that Volta mentioned in his paper to play around with and test (e.g. lye, silver, zinc, tin, coins, leather, cardboard, salt water, etc.), and they were working to win le Prix Volta (a real prize Napoleon and the French Academy of Science offered for research in electricity, after Napoleon saw Volta’s original battery demonstrated). This contest was good to talk about collaboration and competition in science, but my favorite part was having kids read a challenging history of science article about what actually was behind the creation of the battery (a torpedo fish!) and what sorts of things had to have happen for there to be the physical and intellectual space for Volta to even have the conditions for him to come up with his Voltaic Pile. That the battery is historically situated, and tools, ideas, and people had to come together in a specific way for the battery to emerge and look the way it did. I also really liked that students could understand that there could be an explanation of electricity that didn’t center around electrons.

That dovetailed really nicely into how we were talking about Thomas Kuhn. We used Kuhn’s Structure of Scientific Revolutions as our core text that they were reading extensive bits here and there each night, and although I was worried it would be too abstract for them, they grappled with it and came out victors. And I think (hope) it was a real mind-blowing experience when they realized that “old” theories weren’t “bad” because those scientific practitioners who adhered to them were dumb (or at least, weren’t smart enough to see the Truth with a capital T). And listening to them discuss Kuhn, grapple with the idea of Normal Science, and start to see glimpses that (1) science isn’t accumulative in the simplistic way that textbooks tend to say it is, and that (2) we always are looking at data, theories, experiments, observations through specific eyes, and what we see is dictated by the paradigms we accept.

Images: Here are images from the Symposium, without student faces in them. (Hence, we don’t have the majority of my favorite pictures.)

# My Wunderkammer: A Visual Resume

About 6 years ago, I remember receiving a stack of resumes for a math teaching job. We were looking to hire someone to join our department, and there were so many resumes and cover letters to go through. Over 50, maybe around 100. And my eyes started glazing over. The resumes looked similar, and the cover letters were banal. And then: one applicant stuck out.

It was a cover letter that gave a link to a really simple website, and on that website was an educational philosophy, a few sample tests, and some student work. Although it was pretty basic, what I liked was that on that simple site I got a much better sense of who this candidate was. I loved the idea. And I decided then and there that I would create my own teaching portfolio online that would capture who I was as a teacher.

This past summer, I did it.

To be clear: this isn’t a reflective teacher portfolio.  It’s a descriptive teacher portfolio. It is something that I put together — a mishmash of snippets — that together hopefully gives a solid sense of who I am, what I do, what I believe in. I think calling it a visual teaching resume or a wunderkammer best describes it. (Click on the image to go to the site.)

There are a few missing things that I would like to add to this site at some oint:

• I would like to add everyday samples of student work. Not projects. Just everyday stuffs.
• I would like to add a section about the two week history of science course I designed and implemented with another teacher this year. (See Days 80-87 on my 180 blog for more.)
• I would like to add a section about the “Explore Math” project (more info here and here) I did in Precalculus this year.
• I would like to finish the student quotation page. I actually have quotations typed for a number of previous years, but I do not have more recent years ready.

It was pretty simple to make (I used the free website creator weebly) and I hope if I ever were to go on the job market, it would catch the eyes of whoever had the giant stack of cover letters and resumes in front of them. I wasn’t really going to make a post about my visual resume, or share it with anyone, because I thought: who would care?

But heck: maybe someone out there is going on the job market and thinks the idea is worth replicating? So I decided to post.

# Doodling in Math

A few years ago, I blogged about this fun little doodle that students often make — and how another teacher and I found out the equation that “bounds” the figure. I honestly can’t remember if I ever posted how I got the answer. If I did and this is a repeat, apologies.

Tonight I wanted to see if I could re-derive it like I did before — and lo and behold I did. I’m curious if any of you have done it the way I did it, or if there are other ways you’ve learned to approach this problem. (There is a student who I had last year who created this amazing 3-d version of this using the edges of a cube and some string. I love the idea of asking — for this 3-d figure — what surface is generated by the intersections of these strings.)

We start out by having these lines which form a family of curves. But of course we’re not graphing all the lines. If we were, we’d get something more dense like this.

The main idea of what I’m going to do to find that curve… I’m going to pick two of those lines which are infinitely close to each other and find their point of intersection. That point of intersection will lie on the curve. (That’s the big insight in this solution.) But I’m not going to pick two specific lines — but instead keep things as general as possible. Thus when I find that point of intersection for those two lines, it will give me all the points of intersection for all the lines.

Watch.

First we pick two arbitrary lines.

We’ll have one line move down on the y-axis $k$ units (and thus over on the x-axis $k$ units). And the second line will be moved down on the y-axis just a tiny bit more (down an additional $e$ units). Yes, we are going to have that tiny bit, that $e$, eventually go to zero.

The two lines we have are:

$y=\frac{k-1}{k}(x-k)=\frac{k-1}{k}x-(k-1)$

$y=\frac{k+e-1}{k+e}(x-(k+e))=\frac{k+e-1}{k+e}x-(k+e-1)$

A little bit of algebra is needed to find the point of intersection. Setting the y-values equal:

$\frac{k-1}{k}x-(k-1)=\frac{k+e-1}{k+e}x-(k+e-1)$

And then doing some basic algebra:

$k^2+ke=x$

Now solving for $y$ we get:

$y=\frac{k-1}{k}(k)(k+e)-(k-1)$

$y=k^2+ke-2k-e+1$

So the point of intersection is:

$(k^2+ke, k^2+ke-2k-e+1)$

Here’s the kicker… Remember we wanted the two lines to be infinitely close together, right? So that means that we want $e$ to go to zero. Thus, our point of intersection of these infinitely close lines will be:

$(k^2, k^2-2k+1)$ or $(k^2,(k-1)^2)$.

Beautiful! And recall that we picked the lines arbitrarily. By varying $0\leq k \leq 1$ and plotting $(k^2,(k-1)^2)$, we can get any two lines on our doodle.

But I want an equation.

Simple. We know that $x=k^2$. Thus $x=\sqrt{k}$.*

Since $y=(k-1)^2$, we have $y=(\sqrt{x}-1)^2$

Let’s graph it to check.

Huzzah!!! And we’re done!

I wonder if I can do something similar with this cardioid:

I think I must (for funsies) do some investigation of “envelopes” this summer. I mean, Tina at Drawing on Math even introduces conics with these envelopes!

An extension for you. Do something with this 3d string-art.

*Of course you might be wondering why I don’t say $x=\pm \sqrt{k}$. Since $k$ is between 0 and 1, we know that $x$ must be positive.