I've continued to invest in my iPad infrastructure. I now have my own computer lab of sorts, 25 iPads and a color printer. Any sort of digital product has become incredibly convenient to output.

At the heart of the work flow has emerged Desmos, the greatest thing to happen to a math teacher. Part of my crazy Algebra II idea involved a change in how students perceive graphing. For years, kids have told me they hated it. I never understood why. I don't know completely, but from what I've observed it involves the tedious nature of graphing by hand, the limitations of calculator screens, and an isolation of graphing as a diversion from algebra proper. In Pre-Cal it can get especially ugly when attempting to do anything meaningful with polar equations or conics.


Progress I've been able to make thanks to Desmos:

  • graphing by hand is dead in my classroom
  • students can analyze polar equations and conics in a meaningful way
  • students spend their time on the math, not fiddling with window parameters
  • student work that requires graphs is far more professional looking
  • students can manipulate systems of nonlinear equations with ease
  • students in Algebra II have made honest to goodness connections to what graphs mean


Adjusting to a Desmos environment really just takes patience and frequency. In Algebra II the students used it on the third day of school. I kept cycling back to it over and over. It was the heart of our graphing units. It helped me rethink the way I do transformations which had lasting effects on the way I approach a lot of Pre-Cal topics.

The pictures only tell half the story.


Traditional graphing calculators do such a horrible job of showing off the beauty of functions. Intersection points with a tap. Precision zoom with the pinch of a finger. Never having to describe the logic behind a theta step or attempting to square up a pixellated graph. The tedious nature of graphing just disappears. You gain so much more time as an instructor because you aren't dictating a dozen button presses. Or, in some cases, skipping a graphical unit entirely because of all the button pushes.


Words don't do it justice, but my instruction is completely different and significantly worse without this modern tool. As a professional, I have become SO much better with my conceptual algebra understanding because I can generate a graph in 5 seconds. Among other things, Desmos helped my personal understanding of complex numbers; helped me learn the mechanics of polar ellipses, hyperbolas, and parabolas; let finally illustrate vectors decently; and teach growth and decay problems that don't suck.

This is the part where I'd link to a list of activities that are powered by Desmos, but it'd be half of my entries for the school year. I've tagged them all appropriately.

Ok, not everyone has a classroom set of iPads or laptops. You can't get to the computer lab every day. But even if the one change is to start using Desmos as a math professional, you'll be doing yourself a big favor. Students deserve to know that graphing technology didn't stop advancing in 1996. I don't care if they can't use it on the SAT.

AuthorJonathan Claydon

Last time, the Algebra Wall was in the infant stages. It is now a real live thing, and here's the mess that got us here.

In summary, I wanted the students to build a large display piece showing examples from all the topics we have covered this year. I drafted a long list of pieces, assigned responsibilities, and let them work. We discussed locations, took a long time to make sure everything was neat and tidy, and wound up with something cool to show the school.

Here's our chosen location:


Here's a few shots of construction. Students were assigned problems on a Tuesday. Their first task was to draft up a copy of the work necessary to complete their task. Then after conferring with each other or me, they produced panel worthy versions of the work.


We have longer periods on Wednesday, so the 90 some minutes that day saw lots of progress. Friday some items started to appear and the panel organizers started kicking out titles and organizing work. We would've completed it the following Monday but some state testing took kids out here and there, so Tuesday we were done. Tuesday afternoon a couple kids helped me hang it in the hallway. I used some command strip velcro picture hangers and overdid it. No one should be able to casually yank these off the wall.

The final display (click for bigger):

Fun process. It was a big everyone working together moment. Something that helps solidify math class as your math family. A couple kids speculated that working all the problems involved would make one heck of a final exam. In essence, this was like a final review, though it's just now April. Their seven months of iPad and desmos experience really showed here. Only once or twice did I have to reject a graph because it wasn't scaled properly or hard to read. At no point was I diagnosing any technical concerns. I'm open to adding some extra features towards the end of the year. A big long line of circles, ellipses, and hyperbolas underneath could look good.

Kids knocked it out of the park.

AuthorJonathan Claydon

The Algebra II Experiment continues. No major updates to speak of because I ironed out the format in the fall. Now I'm just tinkering with content. How complex do I want to get with rational functions, for example. This semester I sought to improve the credibility of the financial applications that come up during exponential and logarithmic functions.

In my most recent textbook makeover, I looked at more authentic approaches to exponential growth and decay. My main criticism of textbook growth and decay comes with financial applications. Often students are to predict future value given a starting amount of money and interest rate. Sometimes they might mention compound interest.

None of those situations model how money actually grows, since anyone doing it right is going to make regular contributions.

In researching exponential/logarithmic applications, I stumbled upon two good situations: loan payoff time, and mortgage payments.

You can find dozens of websites with calculators that will tell you how many months it will take to pay off a credit card or an estimate of your mortgage payment. But how do these calculators work?

Authentic Scenarios

In 2014, credit card statements started including new information:

Screen Shot 2014-03-29 at 10.25.31 PM.png

In an attempt to save you from yourself, the bill sends the message that the minimum payment is a Bad Idea™. But where did 3 years comes from? And $478? Credit education is probably number two behind "count change" in a national "what should kids learn in math" survey.

Houses are another interesting study. What's the real cost of a house? Are houses the profit center everyone claims? Are you really missing out by "throwing your money away" in a rental?

My students had fantastic conceptions about all these things.

First, the two formulas you need. These were really hard for me to find. It's like banks don't want you to know? That can't be right.

Payoff time, n, for principal, P, given APR, r, and constant payment, x.

Payoff time, n, for principal, P, given APR, r, and constant payment, x.

Required payment, x, to payoff principal, P, given APR, r, and number of payments, n.

Required payment, x, to payoff principal, P, given APR, r, and number of payments, n.

The left formula I used to generate the 3 years and $478 from the credit card bill. The right formula to generate the expected mortgage payment for a certain loan amount.

Talking Points

To open, we discussed credit cards. Specifically, how interest is determined, and how high interest rates can be, even if you're low risk. Compound interest enters by defining the term "APR" which some students knew, but most didn't. I started with a $500 purchase making $10 payments. We discussed the work manually. Deduct $10, calculate interest on the remaining balance. Deduct $10, calculate interest. Making $20 in payments only nets a $7 reduction in principal. That surprised everyone.

Houses was an even better discussion. Interest rates are lower, but the idea of paying for something over 15 or 30 years was a surprise. My students were not aware the loans were that long. You want to drop some jaws, demonstrate that a $200,000 loan at 4% costs you $344,000 to own.


The math here was tricky for them. Conceptually, they understood what we were doing. Mechanically, it was tough. Lots of breaking things down into steps. We juggled intense decimals through a long calculation. Payoff time being expressed as "-n" was confusing. We went slow. From an assessment stand point, I asked a lot of question about the concept. What's the advantage of making larger payments? What do the variables stand for? That kind of thing. Exposure to HOW the calculations are done was enough for me even if they didn't become an expert. Many of them could handle the work after a while. A lot couldn't. But the goal of financial education was achieved.


This unit doesn't happen without a decision 6 months ago to challenge the notion of what an on-level Algebra II student can do. The laundry list of steps to make the calculation, the nasty decimals involved, the multiple instances of log an exponential terms, it doesn't happen. Even last year this discussion disintegrates into a mechanics nightmare.

AuthorJonathan Claydon

I had an idea a couple months ago. Throughout the year I've found that we have time on our side in my Algebra II class. Despite a harder attempt at curriculum we have cruised through seven of the eight major parent functions and it's March.

It's about time for a break in the action. Last semester we spent a couple days on The Estimation Wall, now it's time for something larger and more public. While nice, The Estimation Wall sits right outside my classroom. My classroom isn't on a main drag, foot traffic is low.

Around the corner is plenty of vacant wall space near the entrance to the math department. Somewhere in there will hang our Algebra Wall.


The goal of the activity isn't really practice. It's a display piece (we have many murals on display throughout the school). A showcase of the variety of functions available in Algebra II with some focus on their applications. It starts with foam board panels.

Currently we'll build eight panels: Linear, Quadratic, Absolute Value, Radical, Exponential, Logarithmic, Rational, and Exp/Log Applications. In the future we MAY add Conics and Inverses, depends on how April goes. Why not wait until April for the whole thing? I'd like this to be on display longer than a month.

Hanging the panels isn't something I've figured out yet. Upon asking permission for this project it was requested the panels be removable for the summer.


The panels are pretty big. Each measures 20" x 30" and will be arranged 2 x 4. Each will have the type of function on display, the parent function, and how you would identify if something's linear, quadratic or whatever. Before assigning the project, I mocked it up to get an idea.

A challenge with the idea is getting each student equally invested. I dislike projects that give kids the opportunity to twiddle their thumbs. I heavily avoid assigning roles and my "projects" are really displays where equal effort was required from all. Some finish faster than others, but no student becomes burdened trying to complete an entire poster themselves while two others steal the credit.

I'm going to have 22 students build this thing, that's part of the reason the panels are so big. You can fit eight examples and graphs on these things. Factor organizing everything and making title blocks and we have a TON of work.

Divide and Conquer

I debated itemizing the 80 some odd things needed and having a sign up process. But I didn't want anyone crying foul because whoever signed up first snagged all the linear. Since these are big, I do want certain students in charge of organizing all the disparate parts. For example, if I was making a linear piece, I would give it to the person organizing the linear panel when I finish. Having single students in charge of a panel will allow each panel to look consistent. These students would also be in charge of the title block material for their panel.

Next I drafted up all the equations I wanted them to present. On the back are the items for the application panel (some population comparison, financial situations, etc).


I put their names in a spreadsheet to hand out work. I wanted it to balance. I hand picked the panel organizers. Recognizing the extra effort, this counted as part of their work load. I wanted 3-4 work items per student. I gave extra work to my super fast students who didn't have a panel to organize. Each student will be all over the place in terms of problem type. No one gets all the linear.

Click for zoom.

The Estimation Wall took about two and half hours. I want to spend about the same here. It may take a little longer. Being a display piece, I think there will be a proof reading phase. Also higher tolerance on the final product. Nothing glues to a panel until I give it the ok or something like that. Could that be the organizer's job? Maybe? Or they gather all the material for a single approval per panel?

The Algebra Wall will be hung by the time you read this. Part Two will document the chaos that gets us there.

AuthorJonathan Claydon
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Classroom iPads are cool, but there are some headaches associated with managing them.

I use a tiny set of apps, but software moves fast. Since I started building my iPad horde, there have been two major iOS releases, and who knows how many dozen app updates.

For iOS revisions, I wait a few months and then bring the lot home for a big update party:

This has never been a big deal, other than having a lot of devices to monitor as they progress at different rates.

App updates though, can be a source of mild irritation. Their release dates are random and frequent. Keeping pace on a lot of devices is tough. It is complicated by a restriction, I disable installing, updating, or deleting apps on my student devices.

Though they can rearrange icons on the home screen, no kid has the power to wipe the whole thing clean and put a dozen Flappy Bird clones on there. However, as the administrator, I would have to disable this restriction (and type the pin code) before I could attempt to update the individual apps. Upon completion, I'd have to set the restriction again (and type the pin code). For one device, no big deal. As I now have 21 devices, the process is tedious.

I no longer have to manage that part manually. With iOS 7, the App Store can monitor updates in the background. As updates pile up, it will wait for the device to idle and apply all currently available updates for you with no intervention required.

This works with all those App Store restrictions enabled. Thus, a student device can receive maintenance without your intervention, and no one has to manually allow and disallow App Store access for it to happen. Unless the student catches the device applying an update (indicated by an icon animation), they'll never know it's happening.

If you are concerned about a major app update coming along that screws up your workflow, you may keep this disabled. My uses are so simple that I don't feel like this will be a problem. The extra couple hours it saves in maintenance are worth it.

AuthorJonathan Claydon

As seen previously, I used polar equations of conics as an opportunity to talk about the solar system. Now it's time to see if I get the students to take something home about the idea, rather than just point to their pretty poster.

A few days after the completion of their solar systems, it was time for a test. I included a section the related to the discussion we had:

Results were mixed. I figured this might happen. Projects can be tricky. Students don't always see them as part of the curriculum and won't attempt to absorb the learning portion, focusing only on the product. Many saw our discussions as just random asides, not an answer to why someone might care about a polar conic.

I liked this question because it surfaced a big misconception. Student equated "eccentricity" with "size" and very frequently described ellipse B as least eccentric citing variations of "it's the smallest of the four." This was great to learn because it helped shaped the next day.

Which by the way, if an assessment isn't influencing your lesson plan, why are you giving it?

Another Puzzle

First, I explicitly called out the misconception. Size has nothing to do with an object's eccentricity. Size is a function of semi-major axis. They were given these four ellipses:

Screen Shot 2014-03-17 at 4.53.07 PM.png

I used this tactic with cardiods and rose curves about a week before. My question to them is the same: "how did I make these?"


To structure the activity, they set this up in desmos:

I used variables because it would easier to adjust the two numbers involved. I instructed them to manually change a (semi-major axis) and k (eccentricity). To prevent endless trial and error, I gave them the scope of a and k. The semi-major axis is an integer between 5 and 20, eccentricity a decimal between 0.2 and 0.9 with a step size of 0.1.

I did not make any mention of using the slider that pops up once you give a and k a value. I suggested it would be easier to delete and retype values. Some chose to use the slider. I have nothing against the slider. But I want to make sure they gave some conscience input to what they were typing, rather than wiggle a slider and hope to get lucky.


Like before, it got quiet. Joy after solving one. Heartbreak and frustration when it's OH SO CLOSE. But after 20-25 minutes, an extremely accessible activity. When we reviewed answers, there was little to no disagreement. Everyone had dialed in the same values.

All of this to confront the misconception that surfaced on the test: eccentricity has nothing to do with size. Now that they had played with for a while, it was much more obvious WHY number 3 in this problem set is the most eccentric. And though number 2 and 4 have the greatest size (in fact they have the same a value of 15), it has nothing to do with how eccentric we might consider them.

AuthorJonathan Claydon

Last year I had some time to fill, and discovered that modeling planetary orbits with polar equations isn't too hard. If you can find the semi-major axis and eccentricity value of a planet or dwarf planet or comet or whatever, you can model its orbit. Naturally, I chronicled the results.

Rarely do I like the first result from a project.


  • not enough work, students twiddled thumbs
  • not enough devices to go around
  • my write up tried to do too much, as always
  • no connection to prior content
  • no discussion post project
  • no attempt to assess their understanding of what they made

At the time, I was looking for something unique to do, and this project did the job nicely. However, unless you were the student typing all the equations, you may not have gotten the math takeaways I wanted. Plenty of kids ooh'd and aah'd about science stuff throughout, but it was the "do you realize that we're modeling PLANETS? Like, big giants PLANETS?" that didn't work. I also made a request for them to define "eccentricity." Hint: never asks students to define stuff without guidelines. The result is some bullcrap from the first Google hit on their phone.


  • made the project scale, groups would have to model the Jupiter and Saturn systems if they had a large enough group
  • had more devices to go around, time could be spent finding resources for planet cards while others found equation info
  • scrapped the instructions and made it a landing page for researched information and a checklist of final products
  • integrated within the flow of polar equations
  • discussed planets and their behavior extensively prior to starting
  • held a post project discussion about what was observed
  • used the project to segue to polar equations of conics in general
  • included questions about the project on an assessment


The products turned out better. A field trip sort of sabotaged one class (students 1 & 2 missed day 1 so 3 & 4 started it, 1 & 2 then return to find 3 & 4 absent and have no clue what's going on), but it worked. After an initial 20-30 minutes of growing pains, everyone understood what was required.


Good discussions in the post game. On purpose I included some things they weren't used to: aforementioned Jupiter/Saturn moon systems, Halley's Comet, and the dwarf planet Eris. Every class had a rousing discussion about the fate of Pluto, when Halley's Comet will return ("man I'm probably gonna be dead"), and how an object could wind up with such a wobble.

I have versions of the intended graphs saved.

Solar System, Jupiter, Saturn


Fewer students were idle, more got an understanding of what we were doing, and all of them picked up at least something about space. Now, the math takeaway did not improve as much as I wanted, BUT with the project now an integral part of my trek through polar, that part would be solved in Part Two.

AuthorJonathan Claydon

In August, my podium received a major upgrade.

Similar in cost to the low-resolution projector and interactive whiteboard (probably less if you consider installation), this Wacom Cintiq is an amazing tool. It combines the tablet and separate screen I've used before and gives me a chance to see the room better. It also acknowledges that dragging one student up to "interact" with the board is an antiquated practice that should stop forever. I don't care how advanced your presentation file is, one kid at the board is just one kid at the board. Notes you have hidden behind a draggable box are still just notes.

This technology is for my benefit. Swap to desmos. Swap back to some notes. Swap to an image search. Swap to a movie. Ok cool, but isn't that kind of selfish? I thought education technology was all about the children.

Well yeah, but if the lesson is nothing but working problems off a PDF you haven't done the student any favors.

With or without technology integration, the magic does not happen because you teach the same content with a fancier delivery system. My high dollar Cintiq does not guarantee the students will care about what I have to say or ensure they will be 20% more "engaged" as a result of watching me.

Engagement is all about how you plan, not how many draggable objects are in your slides (or the clever questions you ask about some episode of Mythbusters you find "engaging"). Do your students talk to each other regularly? Is it more important to let them interact with one problem on the board or with their peers? What opportunities will they have to drive the conversation? When will they be puzzled? How do we entertain their tangent questions? Will your plan survive if one kid wants to talk about the space shuttle for 10 minutes?

How often does the place explode into a mess of colored paper?

Students are begging to interact with one another. My goal every week is to give them an opportunity to speak math to each other as much as possible. They can verify their assumptions. They can teach a peer who's confused. They can get clarification as I wander around. And yes, chatter on and be off task, but integrate this practice frequently enough and they're so thankful to be DOING something even if it's math.

My tablet is a means to convey information. It happens to make it easy to do and internet-enabled. It offers me the convenience of saving things for later. But if a student's primary means of interaction is demonstrating a problem to 25 silent (and uninterested) bodies, you don't get what it means to interact.

Quality lessons stand on their own whether they're taught in chalk, dry erase, or pixels. The medium is the least important item in the chain.

AuthorJonathan Claydon

For a history on this activity, you might enjoy the more thorough write ups from the first two years:

Sidewalk Chalk Adventures
Return of Sidewalk Chalk

Premise is simple: polar graphs are neat, and I wanted a better way to present them.

The technology for doing this low tech activity has gotten better since the inception. Year 1 the kids squinted at TI screens and did their best. Year 2 we had a few iPads and upon assigning their graphs I required them to sketch a duplicate. Year 3 with an army of iPads and a printer, every student printed the necessary graph and used that hard copy.

Given two graphs and two chunks of sidewalk, present your graph.

The scale of the project always blows me away. A few of the kids will take a step back when they see how much ground a single class covers. Were you to own a small airplane, you'd be impressed at the coverage.

99 students participated, and we covered close to 1000 feet of sidewalk and there it stays until the rain washes it away.

AuthorJonathan Claydon

I made a minor tweak to my Pre-Cal curriculum. A discussion at TMC prompted a change where polar equations are taught right after vectors. All the math mechanics are the same, so it made sense. It also bumps polar to a time of year where the seniors haven't checked out yet and may remember something.

I did a detailed explanation of the unit last year. The part in the middle is a tad different now and man, it was the best way I've seen to dissect polar equation behavior.

The idea came from something I tried in Algebra II. That is, make equation discovery more of a puzzle. Instead of fiddling with sliders and recording observations, I would provide equation frameworks and leave the rest up to them.


Handed this out (screenshots generated from the answer key):

Put this up on the screen:

a+bsin(theta), a-bsin(theta), a+bcos(theta), a-bcos(theta), acos(theta), asin(theta), acos(b*theta), asin(b*theta)

Provided these instructions:

How did I make these graphs? Terms a and b are integers no larger than 10.

Detective Work

The old way, I gave too much in the instructions about things like a = b, a > b, and a < b. I wrote out all these discussion questions and wanted them to write down answers. It was nothing but confusion while they were working. This time the task was simpler: solve my puzzle, write down what you think works.

For the most part, silence. Students plucking away at desmos trying to solve the mystery. Sheer joy when they crack one. Frustration when the x-intercept matches but the y-intercept is OH SO CLOSE. And then, pure euphoria when they crack the system and decipher what the intercepts give away about the value of a and b.


Finally, the discussion. I first solicited answers from the crowd for the eight functions provided. This step is intended to see if there's disagreement. I starred any that weren't unanimous. I pulled up the answer key and typed in their suggestions to check for matches.

We used our results to classify the equation types into three groups: cardioids, circles, and roses.

First, the rose curve discussion. Lots of fascinating theories about what determines the petal count. Often two students would take opposing views, thanks to my accidental genius inclusion of 8cos(4*theta). It has 8 petals. In almost every class, a student would argue the leading "8" is why. Another would counter argue it's the 4 and that petal count is that term multiplied by 2. In one class, a student agreed with the multiplied by 2 theory, but that it only worked with cos. The looks on their faces when "8" became "7," "4" became "5," and their prior assumptions broke.

I just, I mean, you guys, it was THE BEST moment.

Amazing bonuses:

MANY students figured out how limaçons are created (b>a) and acted like it was such a "duh" discovery. MANY worked out that the value of "a" tells you the x/y intercept (depending on orientation) and that combining the values of "a" and "b" via addition and subtraction yield minimum and maximum radii. MANY felt empowered that they even if they didn't notice both those properties, they were proud to have caught on to one of them. More importantly, EVERY kid found this task achievable. Two finished early and on their own just started experimenting.

Lastly, I animated a couple rose curves on top of each other as the children reach for their phones to record it:


Look, maybe you read about 3ACTS and think, ok cool, but I don't teach middle school, these lessons are too simple. EVERY subject in math benefits from the idea behind those movements. Create a puzzle. Ask a question. Make them beg. The theory of 3ACTS enabled a group of seniors to argue with each other about polar equations and hash out the mechanics entirely on their own. There's no way a worksheet produces the same results.

AuthorJonathan Claydon
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