Piecewise Functions are a fun twist at the beginning of Pre-Cal. Graphing them however, has always been a bit annoying. The first year I taught Pre-Cal this topic was a disaster. Slowly I found a better way to do it. Start with evaluation. Have students evaluate a function over a wide range of points. Give them a grid and have them plot the results.

Some errors will surface. Overlapping pieces, boundaries drawn in the wrong location. This is natural. But, it gets better.

As a culminating move, I teach them how to graph with restrictions in Desmos. This year, I tried something new. You, dear child, are going to DESIGN a piecewise function. Normally, I suck at open ended stuff.  But, my few weeks of bizzaro world gave me the impression that my new crop of students would be up to the challenge.

Requirements:

Design a function that at a minimum

  • has 3 pieces
  • has 3 different types of behavior: increasing, decreasing, constant, or undefined
  • has 2 different types of functions: linear, constant, radical, quadratic

Present a large version of your function, the equation that generated it, and a description of the intervals that exhibit different behavior.

I set up the project with some practice on describing intervals and how to manipulate various function types (a review of transformation rules), and then set them loose.

Most went with something pretty standard, but I did get a few clever applications. Students primarily built their functions on an iPad and then made a copy. However, since I don't have a enough iPads to go around, they had the option of using their phones if they wanted. At least 3 or 4 kids in each class successful pecked out a function on the web or app version of Desmos. A few shared an iPad with a partner.

Super happy with the result. I expect this open ended thing will continue.

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AuthorJonathan Claydon

Calculus is my giant scary project this year.

For once, I'm doing something super traditional. Relying on published resources. I am not the biggest fan of textbooks, our standard issue Pre-Cal book in particular. For years and years everything I produce has been independently created. Kind of necessary if you do crazy stuff like make your own Algebra II curriculum. I made LOTS of things.

Given the formulaic nature of the AP Exam, wandering off in my own direction may not be advisable. Thankfully, I only had to buy one of the books you see here. The AP workshop I attended over the summer was free book after free book, plus the one adopted by my school.

Breaking the traditional lecture format of AP classes, that's the real goal here. These books? I can work with these books.

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AuthorJonathan Claydon

My major challenge with calculus is confidence. For whatever reason, the AP Exam causes our students to freak out. I have a number of ideas to try and fight this that will trickle out here as the year progresses. One of them is to teach them how to use resources. And second, provide them with resources that are going to mirror the instruction they're given in class. Meaning, sending them to instructional videos by some random guy on YouTube is not involved here.

To start, I want to work any homework assignments I give them. I got a special notebook for it and everything. Then I thought, well, these should be available to them. But borrowing the notebook only helps one kid at a time. Five minutes of fiddling with Dropbox later and I created my Calculus share folder:

I post the solutions around the time I give the assignment. We have several conversations in class about the right and wrong way to use solutions. Homework is such a tiny part of their grade, the lack of understanding that will come out on tests isn't worth it. Most understand that attempting it themselves is the appropriate thing to do. In fact, one kid found an error in MY work, which was cool.

That last one might scare you. Hold on, that's the TEST ANSWER KEY? Are you crazy? No. The AP Exam is the final performance. I gain nothing by giving them assessments that they can't learn from. ALL their skills have to be sharp in 8 months. If I can't get them into an analysis and growth mindset now, they'll never convince themselves they can study for something like an AP Exam next semester. Their tests are short and frequent. Material on them overlaps. Once something appears on a test, it can appear again WHENEVER. I'm asking you limit questions in March, kids. Making old assessment part of their preparation routine is something I see that can fight the confidence problem. I might have messed up, but darn it I'll be ready the next time.

As you read this they'll be taking Test 2. I'm cautiously optimistic that some of them see a value in this opportunity.

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AuthorJonathan Claydon
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Help me out here, Algebra II people, we have a problem.

The answer I'm looking for, is "where does the given parabola intersect the line y = 7?"

One of the first things I decided to cover in Pre-Cal this year was solving quadratic equations. It creeps up on you in Calculus and beyond that I don't want my students to wait until they're 22 to realize its value like I did.

When I introduced the topic, I asked "why do quadratics have two solutions?" In every class someone suggested it had something to do with the x-axis. I pushed and wondered "what's your obsession with the x-axis?"

I tried to confront this problem last year with my method for introducing the quadratic formula. Students saw the quadratic formula only as a means for finding x-intercepts. They are not totally wrong, but they don't know the truth.

Part of this is the fault of textbooks. As I noticed last year, there is an obsession with giving students quadratics that are equal to 0 and equating roots, solutions, and zeros as like terms. When really, what they need to understand, is you can construct a quadratic which in turn has x-intercepts that are equivalent to the intersection (or lack thereof) of the starting expressions.

Compounding the problem is then showing them half a dozen methods for solving, muddying the waters the whole way. Should I factor? Should I complete the square? What about this formula thing? Students have no idea where to start. Every class admitted this to me. All of them could tease out factoring and the quadratic formula as methods, but very few could describe when one is more desirable.

I'd love to admit that my returning Pivot students nailed this question like they should have, but that wasn't true.

The traditional treatment of quadratics and what the two solutions represent does more harm than you think.

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AuthorJonathan Claydon
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In January 2011, I tried something crazy. It's now a local phenomenon.

I had a mind blowing meeting with a group of middle school teachers a couple weeks ago. A group ready to change just about everything they do regarding student work management and assessment. Not just ready, but determined.

I began to seriously look at my assessment practices almost four years ago. Really, it was a time concern. I spent my first year giving big unit tests and grading homework and all that stuff they say you're supposed to do and found it incompatible with the demands on my time from coaching. By accident, I wandered into SBG, notebooks and all that. Three years ago, I started sharing my successes with my math department. Two years ago, many of them were ready to try it.

And now, years after a secret experiment, SBG and notebooks is the majority practice at all levels of math in my school, Algebra 1 to Calculus. We've all implemented it differently, PreAP is excluded for the most part (though not mine), our notebook management systems are varied, and I will say that not everyone may experience the same altering side effects I did, but we made a change.

Even crazier, in just a handful of meetings, an entire middle school hopped on board because many of my colleagues had experienced such positive results and testified as much to them. At some distant point in the future, there's a kid who will graduate having spent their 6th - 12th math career judged by how they show growth.

Challenges remain:

  • SBG is no bandaid for bad lessons, learning to be interesting is a whole other hurdle
  • this middle school is in uncharted territory, with no one specifically on campus who has implemented this stuff before
  • our Algebra 1 team is trying SBG, notebooks, AND a tweaked curriculum, there will be kinks, but my role this year is their direct line of support
  • new teachers are always a tough sell, never getting the full philosophy behind the move or experiencing the classroom challenges that lead to this stuff in the first place
  • SBG and notebooks can't be forced as a matter of policy, it will guarantee half-hearted implementation, each teacher needs the eureka moment, and that may never come

We aren't perfect. It is not math department heaven on earth. Simple requiring a student to have a notebook does not mean you will have 100% engagement. Implementing SBG will not guarantee you have a 0% failure rate or that kids will achieve that understanding they've seemed to be missing.

The point of this is to say that change is possible. Change at scale is possible. And it happened organically. Our district did not mandate any of these policies. There is a ton of work to do. But the fact that we even have these discussions locally without anyone thinking I'm a crazy person is important.

If you have great practices that are working for you, you don't have to be the lone voice in the wilderness. Student results speak for themselves. Chances are there's a teacher in your district who wish things could be better but doesn't know where to start. Show them what to do. Share everything. And slowly, you can convince a lot of people to step back at their current methods and ask "why?"

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AuthorJonathan Claydon

This is year six. I've taught academic (on level) classes for every year prior. It's what I know, and what I really enjoyed. Last year I took a random assortment of on level kids on a crazy experiment and they blew me away.

The world is different now. In what started as a quest to attempt Pivot Algebra II on a single PreAP class, I'm now knee deep in AP and PreAP. I have 154 students in 5 classes (2 Calc, 3 PreCal). The first week was, strange.

  • PreCal numbers are 34, 36, 36, the largest classes I've ever taught, and yet the size has not been a problem
  • despite the giant size, every PreCal class exhibits laser focus when requested
  • upon my encouragement, 8 of my on level kids from Algebra II signed up for the PreAP experience, they hold their own and are teaching the PreAP lifers around them how to do stuff, I almost cried when one suggested something might be failing due to the presence of complex numbers
  • Calculus students who had me for on level Pre Cal shared in my amazement at how quiet my classroom can be when it's time for business while the lifers looked at them strange "what do you mean, class was loud?"
  • said Calculus students who had me for on level almost disowned me when I told them homework was a regular part of an AP class
  • secretly, I don't care for the whole PreAP distinction and don't use it on tests or my objectives area, I think that designation gets corrupted by teachers who use it as at excuse for bad teaching, excessive homework, and general "this is what it's like in COLLEGE" bullcrap
  • my in class support is far more advanced, WAY fewer students need help getting started on classwork, and anyone with a question starts it with "well this is what I tried and I THINK this the problem but SHE says it could be..." while I get a little wide-eyed and then examine their work
  • in but two sessions of classwork, I've already settled half a dozen PreCal arguments about equation solving methods, with both parties having strong cases
  • in those two sessions of PreCal classwork, out of nowhere someone would ask "so it's homework if we don't finish, right?" as I had to take pause and then say "no, that's not really my thing"
  • the students were totally ok if they ran into something weird on an assignment and there wasn't an exact example to follow

Classroom Management wise, it's like we're closer to Columbus Day than Labor Day. I'm not totally used to this yet. The thing that makes me excited though, is finding ways to step up my own game and challenge them in ways they haven't seen before. Already, applying what I learned from the Pivot project is going well. I helped stamp out the misconception that quadratic solutions ALWAYS involve the x-axis.

I'm liking it here in bizarro world.

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AuthorJonathan Claydon

At TMC14 the entertaining Steve Leinwand gave a talk about some of the trappings of math teaching. One section of the talk dealt with asking the simple question "about how much?" with a expression containing some oddball square roots.

At random around 11pm one night, that point resurfaced. A weakness I've noticed for a while is that students lack an understanding for the value of things like sqrt(2) or how you could use what you know to approximate sqrt(27). Or the value of 3/2 or how to estimate 1/3 of something. I had a conversation with another student once about the final price for a car, and they had no idea about how to approximate tax/title/license on top. After explaining it, they ask the poignant question "why does no one mention this in school?"

This year I think I'll do something about it. It's like a really focused version of Estimation 180 without all the pictures. I'm thinking I pose four situations a day and have a discussion about how these could be approximated to build their "oddball" number sense.

I probably wouldn't type them all in advance, this was an exploration exercise. Probably after a few months I could compile them here for those of you that would like a home version.

This could be fun. If you too have noticed a deficiency in number sense, don't complain about it or blame calculators, try to do something.

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AuthorJonathan Claydon
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My classroom is my eternal playground. Some highlights of this year's version.

The teacher space is tiny. There are 36 kids in here at certain points of the day, and they need the space more than I do. My first year my teacher desk was 1000% bigger than this. A class of 25 was also mind blowingly huge at the time. The device cabinet has moved over here and the landing zone for kid things like pencil sharpeners and tissues.

There's seating for six at every group now. I developed some emergency plans in case that's not enough. The back of the room is a lot more efficient. One of those TVs used to float out in the middle of the room which was a pain to walk around. I'm hoping it's not too cramped in the little rows between the tables.

I had notebook in two spots last year because there used to be some other things on this table. Whatever junk it was is gone so all the books should fit over here without an issue.

I organized the supply table a little more. The glue sticks are in a container for dishwasher pods. Every table has some scissors is former pineapple containers. The bonus was all the pineapple I got to eat to acquire these containers. No more staplers, they all got broken. RIP all the staplers.

I am extraordinarily pumped for this school year.

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AuthorJonathan Claydon

Prior to the start of school, I stumbled on this sign (not on my campus):

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Further up there is a diagram showing exactly how the cabinet is supposed to appear. Now, look, this isn't intended to call this person out and tell them they're doing it wrong or anything like that. And I'll be the first to tell you that I'm not strict about much. Chances are there were a series of incidents that lead to this sign and intervention was necessary. I have a serious face too. But, how you react to this sign could be an interesting indicator of your classroom culture.

To me, a sign like this is an immediate sign of mistrust. You, dear child, are GOING to screw up and I WILL catch you.

I have a lot of devices in my room. Being the control lover that I am, I too appreciate a nicely organized cabinet. When I ask for them back, I get them back. The kids make a stack on a table. We use them often enough they know how this works. Then I stick them back in the cabinet. No boxed threats required.

Students are not by nature against you. You can make a lot of progress if you recognize that they can be trusted individuals with something to offer.

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AuthorJonathan Claydon
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As my iPad set grew, charging them started to become a problem. We have a cart to store them in that has a few power strips and some dedicated USB chargers. There are only so many outlets to go around, so I couldn't use all the stock chargers. I had a set of 10 and had to do charging in shifts. It made the cabinet a mess. The dedicated USB chargers are underpowered for their task (10W, 0.5A shared among 4 ports, a full size iPad draws 12W on its own), and when the cart is loaded up, they can't sustain enough juice for an iPad (a string of ding-ding-ding noises as the iPad gains and loses power).

Enter this bad boy.

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These are 5-port, 40W, 8A chargers by Anker. They cost $25 each and I bought 5. These have the juice necessary to power a ton of devices and let me get rid of the mountain of power chargers cluttering up the cabinet.

After some sharpie labeling and cable snaking, here they are installed:

The body is coated in rubber, so mounting them to the side with a velcro patch didn't really work out, but this is ok. When outputting full power they get very warm (VERY warm), so they need some air to breathe. I plugged in 20 devices and heard none of the ding-ding-ding problems from the underpowered chargers. Plus, look at all these adapters I don't have to use:

As a bonus, I used the fifth one to solve a nagging problem I had last year. With the prevalence of cell phones (I can hear you now OMG CELLPHONES HATE HATE HATE, but calm down a bit), my outlets would get commandeered whether I granted permission or not. Often it was at the expense of the pencil sharpener or some other critical thing.

Sitting on the top of the cart is my solution, a phone charging station. Charge it here, stop touching my outlets. 

The reduction of cabinet clutter is fantastic. There were three power strips in that thing, all plugged into each other (super safe guys). Now it's only one.

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AuthorJonathan Claydon
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