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

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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

Second semester can be rough. It is a busy time of year for me being "on" for 12 hours most days, and 16 hours on others. By Feburary I'm a little spent. And that's when I'm not trying to teach something for the first time. This February I was a little hashtag overworked, for real. I've always found student opinion to be valuable feedback. I figured prior to Spring Break it was time to take the temperature of the room. I asked my Pre Cal kids if there's anything in particular they liked, whether they found the class difficult, if it's different than prior classes and whether or not they get attention. The pile was large:

Call it fishing for complements, but these are some nice validation after a 10 week grind. It's also a chance to see what the quiet kids have to say. I am always curious about what quiet kids have to say about the hot mess noisy environments I've been known to cultivate.

Here's a selection (click for bigger):

Try it next time you're feeling a little down or if you're worried that something may not be working as intended. I don't spend a lot of time with these, I scan them kind of quickly. I'm looking for a couple of indicators: the student feels comfortable and supported. If most of them tell me that, things are going ok no matter how strung out I may feel.

AuthorJonathan Claydon

For several years I have students explore polar equations with a little puzzle solving. Eventually, students stumble upon some observations about how to build the graphs. As an example:

For something like this, students will point out that there is something awfully coincidental about the presence of 3 and 5 in the equation and the values of the various intercepts on the graph. That leads to a discussion where I have ignorantly offered them the shorthand that "the two parameters add together or subtract" to get you various places. It doesn't tell the entire truth. The presence of the + in the equation confuses a few. Why would you subtract these? Same thing happens if it was r = 3 - 5 cos theta. It still reaches out to 8, and 3 + 5 still equals 8 but that's weird mister, there's no + sign! And in the end you're memorizing a lot of rules about one integer being larger than the other, etc.

What I should have done is given a proper explanation, something I didn't figure out until a week too late last year. Now I was prepared.

Started with the same activity, students made the same observation about the weird additive property thing going on. But rather than let them hold onto a shortcut, we looked at why.

After the initial analysis, I gave them a new set of graphs with the same goal: determine the equation. After they succeeded using whatever patterns they had internalized, I had them do a breakdown on the corners (0, 90, 180, and 270). If a loop is present, there will be a negative value at one of the corners. The largest dimension happens even in the presence of a - because the value of the sin or cos is -1 there. This had an added benefit of helping students understand loops better. What does it mean to go -3 in the direction of 90? It helps sell the idea that polar coordinates have a different concept of positive and negative. It also gives them a more universal idea about how to graph a polar equation. The corners will tell me what I need to know.

I feel much better about this unit now.

AuthorJonathan Claydon
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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

I've had iPads in the classroom since the fall of 2012. It started with 4 and now I have a pile of 25. What world changing things are we doing these days?

Honestly, no one would build an ad campaign around it. My primary workflow is rather dull.

I've played around with exporting student work and sharing photos. It served a purpose but at my current scale isn't practical. I tried collecting data in spreadsheets and sharing it, too many kids had nothing to do. I tried have them analyze a video. I tried having them model equations to pictures. In each case the time investment ratio was just out of whack. There were simpler ways to accomplish the same thing. Despite how easy it was to gather stuff via Google Drive, I could never find other uses for it. Despite how easy it was to have them try and teach themselves, it was just glorified handouts. I like for them to keep records of everything we do. I still love paper and pencil, marker and glue. Can we make that better?


It began last year with an Algebra II experiment. About midway through the first semester I wanted to spend a long time on graphing, and verifying that algebra you do with your pencil can be proven with a graph. It became a recurring theme in that class to take the most efficient graphing utility available and produce great work without spending forever plotting points by hand.

It was so easy. Graph it in Desmos. Arrange screenshots in Pages. Print. Glue. Repeat.

Could this enhance Pre-Cal projects? Also definitely. And nowadays they don't always need my iPads.

Entrusted with a different type of student, I wanted to give them more control of their display work. Let them experiment inside some defined parameters rather than follow a script. It has worked really well. When they don't need to graph and print, Desmos makes the best tool for trying to answer the question "how do you reproduce this?"

So no, we aren't taking photos (for math purposes anyway #hundredsofselfies), we aren't making movies, we aren't watching videos, and we aren't collaborating in Google Docs. But we are graphing stuff. We are adjusting parameters on the fly, plotting polar equations with ease, making our custom piecewise functions just so, and pretty soon, designing an absolutely giant collection of sidewalk drawings.

It's simple. It's versatile. It's out of the way. And that's cool with me.

AuthorJonathan Claydon


How do you recover from apparent disaster? First, I recognized that some of this was generated by me. Usually an across the board problem is evident of a teaching problem. And throwing standard integrals in combination with u-substitution integrals can cause some headaches, especially for amateurs. It was classic overthinking, lots of students trying to make substitutions that weren't there. I didn't yell or anything. Current class grades were posted and I just put on the serious face and said "you be quiet for a minute."

We reviewed the test, they sighed at how easy it all was. I made a peace offering and held a retake on the skills portion the following Wednesday. This is actually not different than my normal policy. Skills portions are always eligible for retake. The only qualifier is that you had to have done the previously two homeworks. As noted last week, a lot of kids couldn't be bothered.

Normally the day prior to the test I'll put some bullet points up for study purposes. This week I tried to make a point:

I'm cautiously optimistic that the message was clear. You never know.

They were also given a series of assignments to do in the class days that followed. I grouped them together as a project grade that took the place of their normal mock AP exam (which is happening later so we can cover more).

Deep breaths and time.

AuthorJonathan Claydon
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I like what I do. I like writing about what I do. It's not all sunshine and rainbows though. Understand that about anyone who documents their practice. Today I share an historic bad day.

It's second semester in Calculus, we are plodding along through integration. Practice is going fine, there is some confusion about u-substitution, but nothing out of the ordinary. Homework completion rate is a little low, but hey, second semester seniors are super like that.

For Calculus I hacked together a weird SBG kinda thing. Tests have three sections, two skills sections rated on a 0-6 scale, and a concepts section rated 0-8. Rather than have a built in two attempt system, they can redo skills but not concepts after school. I gave 8 tests and 3 mini AP exams last semester. This year we've had 3 tests (with 3 remaining) and they will take 1 mini AP exam and 1 full length exam prior to the real one.

First two tests of the semester were fine, higher than average, I was feeling great about our progress. And then, last Friday:

Complete and utter devastation. The first column is an average out of 12, the second out of 8. I didn't do anything different. I even discussed the test with another teacher the night before.

It has been a long time since something like this happened. I was mad, I was screaming at them mentally. There was a bad energy in the room the day of, I could feel something was wrong, it put me in a bad mood. I delayed grading them as long as possible to avoid any anger bias.

A lot of coping via loud music later and I had calmed a little.

The story of how we moved on coming soon.

AuthorJonathan Claydon
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Frank had a few thoughts to consider a couple weeks ago.

I got to thinking, do I overtest? Would the tests I give be considered high stakes? Having implemented a standards based system for several years, I have tons of data on the idea. I think the answer to the first question is "maybe" and the second is "no." Though I think my students disagree.

Raw Numbers

Some notes: SBG was introduced in Algebra II as a mid-year experiment; in 2011 and 2012 I gave a one time test of basic skills at the beginning of the year in Pre Cal before the SBG style tests began. In 2012 I gave fewer Pre Cal tests because, ironically, Texas toyed with the notion of increased standardized testing: 15 exams required to graduate high school. We administered benchmarks and real versions of all those tests that school year, in addition to the previous slate of tests still required for the class of 2013. That summer they retracted the plan after public outcry.

I allow about 40 minutes for each test. They happen every 7-10 calendar days. What I think is missing from Frank's numbers is a considering of that old stand by the quiz. Usually at least two quizzes accompany those three term tests, and take about 20 minutes each. Also, it is more common for schools here to have six terms per year. Estimate 15 tests and 12 quizzes per school year and you get about 14 hours, excluding the final.

My Opinion vs Student Opinion

If you ask my students, a lot of them would tell you we test a lot. In the case of testing successive Fridays, that's when many will moan "didn't we just have one?" If you ask the right followup questions, you can get them to see beyond the gut reaction. Do we have quizzes? No. Do you have more grades? Yes. Do you know more about how you're doing? Yes. Do you have homework on top of all of this? No.

I have had them interviewed before, year after year, lots of feel like they have a more specific idea about how they're doing. They can misfire on one part of an assessment but celebrate success on another and come out feeling like they learned. They can share heartbreak over coming oh so close to that elusive 4.

The Stakes

Are these test high stakes? I don't think so. My students get nervous about them, sure. They THINK it's life or death (thanks grade culture). Some of them really push themselves to get double 4s as much as possible. A lot are ready to have a shot at improving something that didn't go well previously. How do I know they aren't high stakes? I have watched the data for years. The standard 15 unit tests allows 15 attempts at 15 test grades. My students have over 80 attempts at 40 test grades. One misunderstood topic in the mix doesn't make a dent.

At the end of some grading terms, topic 7 or 8 might make a 1 point difference. Nobody fails a grading term for an isolated problem, it takes a series of miscues. At I've noticed the problem long before it's ever actually a problem.

I think a better question is: what are you getting from your 14 hours?

AuthorJonathan Claydon
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Last semester I gave a nifty test question. Students had to reason through whether something I presented them is valid using a combination of things we have talked about. Some of them followed the lines of thinking I intended and a few went beyond my expectations.

A similar thing happened recently. Last year in my Algebra II revamp I wanted to put more emphasis on the purpose of graphs. My current Pre Cal group didn't necessarily get that connection during their Algebra II experience. Early early on I brought it up as we began with quadratics. Now that it was time for trig equations I brought it up again. Why would trig equations have a series of repeatable solutions?

I had them hack it out via Desmos first:

I gave them some equations and showed them how to plot it. They jotted down a ton of the solutions and I asked if they noticed anything. Hoping they caught on to the presence of a pattern, even if they didn't necessarily know what was causing it.

Anyway, we spent some time solving trig equations algebraically, representing the solutions as a multiple of the period. Some of that was on the test. But then I threw this at them.

The expected process would involve taking the equation given, solving it algebraically, and seeing if the intersection points in the picture appear. Simple.

A few demonstrated some great understanding of trig functions. In this particular example, the student solved the equation, noted the presence of an intersection point and agreed. But, BUT. Do you see the part where they verified that the GRAPH itself makes sense, that the presence of an intersection point might not tell the whole story? I mean, I had to stop for a second. This was so great.

AuthorJonathan Claydon

This summer I come up on three years maintaining this site. Has the mission changed?

When people discuss online collaboration and contribution, some will mention that you have to be a little selfish. And primarily, I'm the primary reader. I am thrilled that other people draw benefits from it. But if anything, I have benefited the most.

The three biggest points I will cite to anyone thinking about archiving their teaching online:

  • You can assemble supplemental materials for your appraiser in a snap. Every year I have a dozen or more blog links as evidence that I met school year goals.
  • Even though no one really looks at them, I love my photo collection. It provides source material for great classroom conversation pieces. And again, valuable evidence.
  • Should you ever be nominated for an award, or field a question from a curious teacher, or lead professional development, it is the first place you can turn to show people how you do you. Nothing makes you look more prepared than a massive archive of everything you've done and the ability to say "here."

I love how this has helped me improve my practices. And thank you to everyone that motivates me to keep it fresh.

AuthorJonathan Claydon