Maybe this isn't a big problem for you, oh Calculus teacher, but it's a yearly issue for me. What do you do about kids not taking the AP Exam? No one would fault you having them work through the remainder of the curriculum, because not taking the exam isn't an indicator the kid is woefully lost. However, the way I do things, if I advise a kid not to take the exam, or they've chosen not to, there's a lot of evidence behind that decision. It is very likely they need (and have needed) some help for a while.

Last year I didn't have a lot of these students. Coming up a with a list of practice assignments for them to complete was easy, and fielding their questions wasn't a big burden. They did their thing and, hopefully, they got to leave Calculus with at least some fundamental knowledge about the course.

This year the challenge is bigger, I have a significantly larger number of non-testers. My filtering process got stricter (on purpose), so it was a natural result.

To improve these students' ability to be self-sufficient, I generated assignments for them as before. However, I added a little treat. For years and years and years kids have been asking me to make a database of recordings. I struggled with workflows, file sizes, and practicality, ultimately deciding each time it wasn't worth it.

Enter iPad Pro.

I've had Wacom devices for years, and still use a Cintiq daily in my classroom, where working with a full computer makes sense. Here, I have a more focused purpose: short (<5 min) explanations for kids who already heard this once, with material relevant to a specific assignment. Enough to help them answer their own questions so I can focus on other tasks during class time. Generating things like has been tedious with those devices. Having a self-contained slab I can write on works better.

The workflow is dead simple. Attach iPad to a Mac, open QuickTime player, start a New Movie Recording, and select the iPad as the camera:

An external mic is optional, the internal microphone will probably do the job. Export the files as 720p movies, and plop them in Dropbox.

Their titles mirror that of the relevant review assignment. A non-tester grabs a computer, opens the shared folder link, and works at their own pace.

Some MAJOR clarifications about why I don't consider this in the same category as super buzzword-y "Personalized Learning":

  • This is a limited run engagement, kids are using this for 2 weeks while I address other issues with exam takers
  • They got live, in person exploration/explanations from me once before, these videos are not intended for first time learning, the kids using these have some working knowledge of the concepts in play
  • This is intended for use during class, I'm not explicitly requiring they do any of this at home
  • The videos were produced quickly (record video, export, done) with a laser focus objective: offer clarification through worked example if necessary
  • This is an opportunity for kids who have been behind the curve to feel some accomplishment and back track

I'm not replacing myself with videos any time soon. I stand by my belief that having kids work through an exhaustive database of tutorials, machine graded tasks, PDF worksheets, and whatnot is nowhere near a valid replacement for talking face to face, having in person discussions with peers, etc.

If you have access to sufficient devices and are considering some kind of tutorial database, use it sparingly. Find a specific use case and keep it simple.

AuthorJonathan Claydon

The coming school year will be notable for a number of reasons, among them: no more iPads in my classroom.


It sounds like a big deal, but not really.

Use Cases

In 2012 we were given 4 iPads for classroom use by the district. My class sizes were hovering at about 30 and I knew that wouldn't be enough to have much of an impact. Slowly, through donations, personal purchases, and some additional school purchases I peaked at 33 iPads of varying vintages by spring of 2016.

In that four year span, I attempted everything. I look at distributing and collecting things via Dropbox, Google Drive, iCloud photos, you name it. I tried playing around with various apps only to find that the best workflow involved a word processor, Desmos, photos, and a printer. None of those components specifically requires an iPad. It was easy and worked every time, but it could be replicated with other methods if necessary.

Over a year ago I started to realize that there was nothing iPad-y about the way I used them, opening up questions about where my tech use goes from here. In fact, the simple use case I had was beginning to become more tedious as software pushed ever forward but the hardware aged (about 2/3 of the stock had 2011-era internals).


Managing that many iPads yourself is a pain. Mobile Device Management systems require business Apple IDs to use most of the features. Acquiring a business Apple ID requires that you own and operate an actual business, with tax ID numbers and all that. Apple makes an app called Configurator that's allegedly for this purpose, but I'm not linking to it because it never worked. Where does that leave me? Manually updating iOS thirty some odd times, retyping iCloud credentials constantly, retyping the WiFi password when the security certificate expires every week or so, and having to wipe each device at the end of each school year. Plus the cost and time associated with my charging system.

Relatively simple work, but bleh.


As cloud computing became a real, viable thing for student work (Google Docs in 2012 was uh, yeah...and it's still so-so on iOS), and as a 1:1 Chromebook pilot in AP English 4/AP Government proved useful, I started looking at Chromebooks as an alternative. Price wise it's equivalent to a refurbished iPad mini, with none of the maintenance headaches and batteries that last for-e-ver. Also, Summer Camp proved the ideal test market. It was a small group of kids, the librarian had nothing else to do with the Chromebooks for the summer, and in the years since we started tech deployment, every kid has a Google account. In the end? Totally frictionless.

A small technical hurdle was solving the Chromebook printing problem, but that wasn't terribly difficult. Some detail on that later, but the easiest way involves a printer with Ethernet (and boy did I purchase a monster).

Having a class set (which the district is providing, for that I am grateful) combined with teaching kids that will be issued one to take home (through other classes) will open up some opportunities. It was already handy for accessing Calculus solutions, and will make some of the deeper parts of Desmos more accessible.

AuthorJonathan Claydon
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I wrote about proven but dull iPad use roughly a year ago and my opinion hasn't changed.

If you ever walk into my room, you'd think it's a technological wonderland. There's TVs and a printer, and I use a non-conventional input method. I generate all of my assignments in house electronically, etc, etc. All of that helps me do my job efficiently. But if you're one of my students, you'd probably hear more about markers than anything else. My method of note giving is actually quite frustrating for the students. I can manipulate stuff all over the screen, shrinking and moving content as necessary. Until I can equip them all with an iPad Pro or something, there's not much they can do but sigh and erase because they didn't leave enough lines empty. The phrase "you KNOW we can't do that" comes up often as I happily rearrange things.

Anyway, a couple of recent events have prompted me to examine just what a student device enhances. One, there's a (slim) possibility a local company might be augmenting what we have on hand. Two, I've started teaching a lot of kids who are incidentally involved in a 1:1 program through their AP English and Eco/Government classes. Leading to scenes like this:

You see that and think, surely they're Google doc'ing or collaborating or making a presentation or something, right? That sort of thing does happen elsewhere in the building and it works well, just not in my room for my subject. Their primary use for me is flipping through scans of handwritten solutions to AP problem sets.

iPads and Desmos still come in handy:

But the struggle remains. Does it pass the pencil test? Do I save something going the electronic route? In very narrow scenarios, that answer has been yes. But most of the time, pencil is still so much better, especially when it comes to feedback.

At a local EdCamp, there was buzz about Google Classroom. But the end result was a lot of people migrating fill-in-the-blank worksheets and debating ways to have students fill in the blanks electronically. Or yet another way to boil math down into computer friendly multiple-choice sets. When asked (I usually just listen at these things), I said it's the wrong approach entirely. You haven't thought about whether filling in blanks or skimming through multiple choice was an appropriate assignment in the first place. Ask any college kid putting up with MathXL.

In some future scenario where all my students have a Chromebook or something, I think I'd stop making copies of my problem strips as a first move. In Calculus it might lead to me writing the workbook I want but no one wants to sell. As far as having students download slides or fill in forms or what have you, I'm just not convinced. Activity Builder is on the radar, but I still don't know if it's my style. In my particular math classroom, the advantages aren't high enough to merit further investment.

AuthorJonathan Claydon
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This is a continuing series of posts about how I approach topics in Pre-Cal They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students understand the unusual nature of a trigonometric equation. Students understand there is a base set of solutions on the domain 0 to 2pi, but an extension of that domain extends the solution set. Students understand that it is possible to make a statement that summarizes all the terms in the infinite series of solutions. Students understand that the frequency of the trig functions relates to the way solutions repeat along the domain.


We don't start with algebra. We start with the concept of an inverse trig function. For convenience, I stick with angles on the unit circle. My first request is that given a particular x or y coordinate from the unit circle, can you find all the angles associated with that coordinate? We condense that statement into math notation, sin^-1 x or arcsin x, for example. The idea that multiple answers are possible is interesting, and lays the foundation for the algebra later.

Next, we look at graphs. I hand out a set of trig equations.

To keep things simple I have them graph the even set or odd set on a restricted domain of 0 to 4.5pi, recording every intersection they notice with a sketch of what they saw. This continues a big algebra idea, every equation has a graph that validates the algebra. For more complex algebra, I always like to start with the graph.

For some of these equations there are many intersections, for others there is but one. Some comments on how weird tangent is, and questions on how to type sin^2 x.

Now the interesting question. Why so many answers? What happens if we widen the domain restriction? What if we removed it entirely? Many correctly guess that there are far more solutions than we could ever write down.

But is there something to this? Is there a way to condense a small infinity like this into something a little more manageable? We take a closer look at just what numbers appear as solutions. Are they random or regular? How might we predict the next number?

After the graphing exercise I graph another random trig equation. We focus on the first two positive solutions. I name them Solution Zero. We discuss the frequency of the function in question. I take our Solution Zero and add one repetition of the frequency to those numbers. Amazingly, the next number in the solution sequence comes out. Minds blown.

Eventually we discuss that an infinite set of solutions condenses to Solution Zero + n[frequency] both in radians and degrees. Finding Solution 1000 is an easy task now. It's fun to ask about how we would determine solutions to the left of zero, many determining that n = -1 would do the trick.

Lastly I demonstrate the algebra necessary to find these solutions without a graph, validating the need for an inverse trig function.

We build from there and talk about squared functions and how the scope of Solution Zero is much larger (base 4 instead of 2). All of it a lovely dance back and forth between algebra and graphs, avoiding impractical (and graphically dubious) equations used by textbooks to convince somebody (ANYBODY) that kooky trig identities are a thing real people use.


We discuss Trig Equations in the early parts of the second semester. I look for student understanding of both the graphical and algebra aspects of the solution sets. Students validate whether a graph accurately represents a trig equation and determine solutions beyond Solution Zero.

Test 11, Test 12

AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students realize that the hypotenuse of a right triangle, the magnitude of a vector, and the r-value of a polar coordinate are three different names for the same quantity. Students realize that the direction of a vector and the theta-value of a polar coordinate are the same quantity. Students understand how to use polar equations to find r-values and how those r-values are plotted. Students understand how parameters in a polar equation affect the properties of its graph.


A couple years ago I moved straight from vectors to the polar coordinate system because the math is identical. With such recent exposure to the mechanics it just becomes a vocabulary lesson.

We start simple, taking rectangular coordinates and finding their polar equivalent. We draw some right triangle systems and define the hypotenuse as the length of the radius. The theta-value of the coordinate is just the angle relative to zero. It's the same math from vectors.

Next, what makes these things unique. Graph of some high-res polar grids. Have students make a table in 15 degree increments, 0 to 360. Give them some simple polar equations like r = 3 + 3 cos theta and r = 3 + 3 sin theta. Have them complete the table for all the values of thetas. Ignore the groans. Once finished, show them how to plot a point along the circles and then connect the dots.

To better understand the weird shapes, they do an exploration. Given a set of polar graphs and some equation frames (a+/-b cos/sin theta; m sin/cos k theta) students determine the equation associated with a graph and we have a discussion about properties. They have iPads to do the graphing.

Questions to ask: why does 5 + 3 sin theta appear the same as -5 + 3 sin theta? why do graphs have a favored orientation? what determines petal count in a rose curve? What induces a loop? is there anything odd about petal counts? how would you make a rose curve with 10 petals?

Next we get a little more technical. Students are given a second set of polar graphs. This time, no iPads to help. Come up with the equation. Using the equation you came up with, find r-values at 0º, 90º, 180º, and 270º. Use this information to make a connection between negative r-values and the appearance of loops. Discuss the angle separation of rose curve petals. Discuss how far off-axis a sin based rose curve is rotated from its cos counterpart.

One last set of polar equations. Having their fill of cardiods, limaçons, and roses, we discuss ellipses and circles. Pass out the iPads again and send them to a polar ellipse generator. Give them a set of ellipses and have them determine the appropriate parameters. Have a brief discussion about any effects they notice and give them some definitions: semi-major axis, eccentricity, and focus. The key part of this study is to get an idea for eccentricity and how it relates to the dimensions of a shape.

If you really want to keep going (and at this point it's mid-April and probably have the itch to start Calculus), you can explore parabolas and hyperbolas.


Great stuff here. After the initial discussion of equations that match graphs, the students get a chance to design their own. Then we draw them outside on the sidewalk.

When discussing polar conics, you can have some really fun discussion about the solar system. Polar ellipses are plotted using Kepler's equations, which nicely map the orbits of any celestial body you've heard about. After the exploration of making equations of arbitrary ellipses, send them to Wikipedia and lookup the semi-major axis and eccentricity for Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto, Eris, and Halley's Comet. Sticking those values into a Kepler equation gets you a lovely solar system, complete with Pluto's encroachment of Neptune.

There's a great video to show as a followup that discusses how bodies interact with one another in space.


I assessed the mechanics: coordinate conversion and appropriate equations for a graph. Then wandered into the conceptual with discussion questions about eccentricity and written descriptions of rose curves. It spanned three tests and 4 SBG topics.

Test 16, Test 17, Test 18

AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students understand how values of the unit circle are periodic and are connected with the appearance of a sin and cos graph. Students understand the ideas of amplitude and period. Students understand that it is possible for an identical graph to be produced by a sin and cos function with appropriate use of transformations. Students understand that tan has a different period by default.


There's a lot to do here and I find this to be one of my favorite topics. We move from values on the unit circle to more generic methods for sketching the graph of sin and cos functions. We make some science connections with sound waves.

First grab some trig graph paper and have students get out a unit circle and some markers. Have them systematically plot the values of sin for the whole circle and connect the dots. Do the same for cos and then tan.

Define the terms amplitude and period. Some of this will be familiar from prior science classes. I interchange period with wavelength and frequency. As a further connection, we talk about the "speed" of the function and define this base form as "normal" speed.

Together, I show them how you can plot the graph of modified sin/cos/tan functions if you know something about how amplitude and period frame the picture. We plot a few examples together. Students often find it weird that sin 2x has a shorter period. There is a tendency to start talking about arbitrary opposites a la "oh well, we do the opposite of the modification, therefore 2x implies x / 2" hand waving. I found it really useful to continue that metaphor of speed. The function sin 2x completes its run two times "faster" than the "normal" we defined previously. For something to be faster, the elapsed time must be shorter. Therefore, one wave takes up less distance along the x-axis. This becomes really handy when you want to talk about sin 2x/3, where the whole opposite operation thing gets ugly to describe. Rather, at 2/3 the speed, we should expect waves to be longer. Throwing in stuff like -3 sin x or sin x + 2 doesn't cause much confusion. I don't do horizontal transformations just yet.

Next they work through two activities to get more practice. It's easy to go fast through this topic and wind up with lots of confusion. I made a concerted effort to slow down and offer lots of opportunities for students to discuss this with each other.

After the activities, they completed some individual practice.


First I have to make some connections between pictures and the equations associated with them. I hand a out a set of 16 unidentified graphs (Answer Key). Students are given an iPad and desmos to determine how the pictures were made.

Next students play a little game drawing randomly generated trig functions. Students are given whiteboards, markers, cards, and 1 die. Students flip a card which describes something about the amplitude, period, and/or transformation of a function. Students roll the die. A 1/2 means sin, 3/4 cos, and 5/6 tan. Students must draw the appropriate function that has the properties of the card. I wrote a more extensive description at the time.

Next students are given a chance to design some trig functions of their own, labeling the graphs appropriately. They design 3 sin and 3 cos functions. They must demonstrate amplitude changes, slower periods, and faster periods in whatever combination they choose. Students create the graphs with desmos on an iPad, print them out and annotate.

Last, we make some connections with sound waves. How much I spend with this depends on where I am in the semester. Last year we were right up against the end, so the science stuff was just a couple side discussions. You could flesh it out more as I have in the past, or start with this. I use a tone generator to talk about how we perceive amplitude and period with sound. I also have a set of blinking lights that are fun to talk about too. You can go really deep into harmonics and springs and things if you have time to fill in the physics background.


This traditionally hits at the end of the first semester. Assessment was pretty straightforward. I'm looking at their ability to sketch sin/cos/tan functions by interpreting amplitude and period correctly spread across two SBG topics. I ask some conceptual questions as a third topic.

Test 9, Test 10

AuthorJonathan Claydon

There are rumors big scale Chromebook deployments could come to the district. There were some pilot programs last year in English IV and Government. I was given a Dell Chromebook 11 to see what possibilities there were for high school math.


My primary point of comparison is an iPad. The Chromebook I used sells for $249. You can get refurbished iPad minis for that price, but not with retina screens. Definite price advantage. The keyboard is fine. The case and hinge feel sturdy. Battery lasts for quite a while, with infrequent use would only need a charge once a week. But man, the trackpad is no good. Plugging in a mouse was necessary.


Google Drive and its companion apps are definitely meant for a desktop browser. If given a set of these, I could have students save a lot more of their work. I'm still not sold on handing things in digitally. I like physical products. If anything being able to save things like custom trig graphs or piecewise functions would let students spend time on them outside the classroom if they don't finish. I could add some more elaborate write up portions perhaps?

I couldn't try out Google Classroom. It requires inviting accounts enabled for it through Google Apps for Education. And I have no children at the moment. It doesn't seem to be much more than a central point for students to grab things from you. The add-on Doctopus appears to offer similar management features. I didn't explore add-ons much.

Printing was among my major questions. Google handles this through Cloud Print, a work in progress. In theory you tie a wireless printer or a printer tethered to a desktop you control to your Google account and any subsequent device you sign into can print to that device. I successfully sent stuff to the printer many times, but some OS X weirdness prevented it from working. I don't blame Google here. Everything on their end seemed to work.


I set out to reproduce things I've done with iPads. Regardless of hardware, when it comes to technology the application has got to really trump the quickness of pencil/marker and paper. Remaking things you can do in Desmos on an iPad was simple. My experience with Desmos classroom activities was no different than with an iPad.

Clockwise from top left: Identify a region between curves; determine linear equations for a street map; import data and perform a regression; prepping Desmos output for print

The Chromebook excelled in two scenarios. For a couple years I had Pre-Cal kids wander around and take pictures of random right triangles, dimension them, and then give me a the set of trig ratios associated with an angle in the triangle. The drawing tools in Drive make for some better results. A student submission from that version is on the left, my recreation is on the right.

The desktop implementation of Drive is what wins here. You have do a lot of import/export steps on the iPad to get it sent to the right place, part of the reason I abandoned this idea.

The second improvement came with Geogebra. I don't have much experience with it, but I've known it to be a little fiddly and it works better with a mouse. Were I geometry teacher, I'd be very happy at the thought of student computers capable of using it. Geogebra installs a Chrome app (nothing but a bookmark really) version of itself.

The 3D graphing features seem worthwhile. It could offer a lot of enhancement to the modest discussion of 3D I'm able to do today.


At first pass, a Chromebook would allow me to do the same things I can do with my iPads. As stated before, the best thing I've found for technology is off loading the heavy lifting parts of graphing. That's roughly 10 to 12 class days (once every 4-5 weeks) throughout the course of the year. Putting a desktop experience in a cheap package gives me a chance to do those activities and a little more. They still wouldn't come out everyday.

There are easy ways to modify some of my established practices. Students could do more writing. Students could submit projects to me instead of printing them out. Students have all the benefits of the internet and we could find some research angles. Students could do a lot more self-paced learning (like watching tutorial videos! just seeing who's still with me...) There's tons of unexplored potential with Geogebra and I just teach the wrong subject to exploit it. But, the whole experience doesn't do much to make the pencil and paper needs of math class better or obsolete.

If the testing conglomerate every modernizes I could see using these things all the time in Calculus. A notebook, pencil, and Chromebook/iPad is not a bad way to do math.

So are 1:1 Chromebooks a good idea for math? As part of a larger deployment where ELA/SS have gone paperless, sure. Exclusively for math? Just give me a cart.

AuthorJonathan Claydon
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A while ago I shared some thoughts about my iPad workflow (singular). Maybe you read it? A non-descript tall man did and then it got retweeted to the moon and back.

I didn't touch on the logistics much. It is not a 1:1 environment. The classes that used these the most had 36, 35, and 34 kids in them. Sharing was required.

Here are the ingredients of the tasks:

  • iPad/iPhone sitting on the same wireless network as my teaching computer
  • Desmos
  • Pages
  • Brother HL-3170CDW
  • handyPrint ($5), makes USB connected printers visible to iOS devices

Printing wasn't always involved, but in the case where the kids were designing something, the idea was the print it out and add some details. Any sort of like on high mandated Chromebook thing would need to answer the printing question for me. Desmos is exponentially better than spending time graphing things by hand. The Pages portion isn't totally necessary, but if you want to be efficient with toner and paper, Pages lets you place multiple photos on a page. Printing from the iOS Photos app gets you just one.

The sharing thing went better than I had hoped. For graph matching tasks like the one in this photo, usually two per kid was fine and I wouldn't pass out all the devices. Some kids elected to use their phone, which is totally cool with me. Sharing in this context offered a chance for more discussion. I've done this in academic and it lead to a lot of one kid just waiting for the other to figure it out. My PreAP group seems to be more eager and less likely to let that happen.

In instances where a printout was required, I would pass out all 25 and there was less sharing, but it still happened. Students working together were allowed to design whatever it was together and share output, so long as they printed two copies and annotated whatever additional information was required. Each kid had to hand in something. Often they'd sit there and share the device but create enough unique work so that they were NOT the same, which was cool.

I never felt constrained by having fewer devices than students. And I'm having a hard time convincing myself it's time to purchase more. I had several pairs who preferred sharing to working on their own. I dare say this is almost the perfect ratio. As always, I will continue to see if there's something better, but this was hard to beat.

AuthorJonathan Claydon

There's not much to explain that hasn't been said already:

Sidewalk Chalk Adventures
Return of Sidewalk Chalk
Sidewalk Chalk Three

A few tweaks this year. Previously the equations were scripted. I handed out a list and kids made the picture and then drew it outside. This year they designed their own: one limaçon and one rose curve with appropriate equation or table. I showed them how to do all sorts of odd things with rose curves. The scope expanded as well, enrolling my co-worker and her troupe. In total it was about 230 students who produced two graphs each. No joke, all of that put together covers about 2000 linear feet. Powered by Desmos. Because, duh.

Selected images:

There were a lot of pokéballs. Tons of these were fantastic. The size of this installation is ridiculous. Unlike previous years, it's not scheduled to rain right away so this will stay put for a while.

Three years ago I had a stupid idea walking to my car. You never know, man.

AuthorJonathan Claydon

It's polar season. You have some options: squint at poor TI reproductions or find yourself a highly efficient graphing tool.

Last week I mentioned dull workflows. Here's a great example. We spent a lot of time on limacons, cardioids, and roses, your usual suspects. Where polar gets really interesting is conic sections and the idea of eccentricity. Manipulating these kind of equations is tough with pencil and paper. Normally I introduce the idea by having students build a solar system first. It's a fun introduction but there's a major problem. Students understand much about what they're doing if a bunch of planetary research is their introduction. They produce nice work and everything, but I should have let them play around with general ellipses first.

This year I almost didn't learn my lesson but some weirdly timed field trips changed my mind. Basically, there wasn't enough time to get the solar system thing in before Spring Break. I had a day to play with so I figured, why don't we play around.


  • access a pre-built desmos array
  • reproduce some shapes
  • make connections between the physical size of the shapes and the parameters
  • attempt to verbalize the possible effects
  • discuss eccentricity with little idea what that word might mean

The Slider Issue

Previously I've had a desmos set up on the board and had them copy it. This time I was like, duh, make it in advance and so I had them access this:

Screen Shot 2015-03-11 at 1.39.08 PM.png

Normally, I wouldn't really go the slider route. There's a lot of room for them to just kind of fiddle at random until something worked. The problem with polar conics is that you have to make adjustments to two variables and the relationship between those variables in the equation is not trivial. I want success to require some effort, but to come quickly. The primary point of this is to have a discussion about what they did and how to classify these shapes.


I gave very few instructions. I pointed them to the ellipse generator and told them the first idea was to reproduce the graphs and record how it happened. Nearly everyone got them all within 10 minutes. After that, we were done with the iPads. Told you, dull.

I put the 8 ellipses up on the screen. I had them tell me what a and k would reproduce them. I asked them if they could tell me what they thought the two parameters did. Main theory was a controlled size and k controls how "squishy" it is. We talked about how wide or tall they thought each shape was. I stood back for a sec and let them look at those numbers versus the a values (a is half the value of the longest dimension).

Then we talked about k. What is this weird number? Why is it a decimal? Why did the shape disappear with values of 1 and 0?

And now the big question. Look at the set of shapes. Which would classify as the most eccentric? Least eccentric? At no point did I ever hint at what "eccentric" might even mean in this case.

That's the part of the workflow I care about. The iPads were the most efficient way to play with the idea. We barely used them. But it let me have a discussion with them and hammer out theories about eccentricity without ever having to tell them what that word means.

At the end of discussion I teased the solar system idea. I think we are better prepared for the concepts involved there. The thought that their elementary school teacher might have mislead them was extremely concerning.

AuthorJonathan Claydon