Every so often the stars align and we have a real casual experience all day long in my room. Yesterday was the end of a marking period and every kid was in “finish this stuff” mode.

In Calculus we were wrapping up a function analysis project:

BC Calculus was up to something similar, and we talked about a benchmark they took recently.

Over in College Algebra kids had two tasks to accomplish, wrap up a project on polynomials:

…and take a test. After submitting their project, they grabbed their weekly (ish) assessment and completed it on their own time. College Algebra is a very casual environment. Whole group instruction isn’t really a thing in there. In any given week kids have a to do list (new lesson, assignment, assessment) and have to complete it by the end of the week. There are 23 kids in this class and they work at 23 different paces. Often that means a handful are done early. Most of the time they’ll get a bonus activity (via Desmos or something else), or sometimes I’ll bust out the puzzles:

So, in yesterday’s 50 minute period, lots of kids were wrapping up their assessment, a few handed in their projects and started/finished the assessment, and others were finished, all with some Weezer in the background. 23 kids, all hanging out, taking care of whatever they needed to get done.

And it played out that way all day long. Hanging out, doing some math, listening to music. It is my favorite learning environment. To spend the whole day in it is simply sublime.

Posted
AuthorJonathan Claydon

I’m on a mission to fix all kinds of things about Calculus, especially Calculus AB. Next on deck is Curve Sketching. Previously, while working on integration of Intermediate Value Theorem and Mean Value Theorem, I integrated some curve sketching ideas. Students calculated average rates of change and created rudimentary graphs of a function’s first derivative.

A few weeks later, students now have more familiarity with the first and second derivative and we can talk about how those tables help us analyze a function.

Start with an arbitrary function and interval, and create a table of f’(x) and f’’(x) in Desmos.

At the moment we aren’t concerned with the graphs, those this will be useful later on. Have students recreate the table, but we’re going to declutter the results. Rather than worry about all the values generated, let’s look at whether the first and second derivative were positive or negative at the point.

Having discussed the Intermediate Value Theorem, we have a discussion about where values of zero should appear on our table. For the first derivative, we reestablish a connection we made before, that if the slope of a function is positive, it must be increasing. A zero should mark the transition between increasing/decreasing or decreasing/increasing and these points are important enough to have names.

Next we have a discussion about the second derivative, which is a newcomer to the party. Some days before this activity, we plotted tangent lines, computed second derivatives, and looked at whether the tangent line was an overestimate or underestimate. That opened up the idea of concavity, that the concavity of a function plays a role in how accurate a tangent line will be.

Now it’s time to define concavity a little better. We look for points where the second derivative must be zero and what that could mean. At this point I’m talking with the table and graphs in view, so students can see that something is happening to f(x) at the point where there should be a zero on f’’(x).

Going back to their horizontal table, we now annotate the table with our findings. Based solely on sign value, we can quickly determine where a function is increasing, decreasing, concave up, concave down, and the role of the various critical numbers.

The purpose of all this is to improve a HUGE weakness I’ve seen over the years. For whatever reason, while I could get students sketching f, f’, and f’’ like geniuses, there was a disconnect between how they were making their sketches and what they represented. If a non-sketching question said something about the first derivative being positive, I’d get nothing but blank stares. Very few of them were able to determine that corresponded to increasing behavior.

By building this competency with tables AND graphs, I’m hoping things improve quite a bit. By sticking with equations of tangent lines and tables as recurring themes, I’m hoping free response style questions are more comfortable. It’s way too early to tell, but I’ve really liked how this is going.

Posted
AuthorJonathan Claydon

Since becoming a goofy idea I revealed to the public in 2015, it is FINALLY time for you to get involved in the action. I am proud to announce that Varsity Math merchandise is now on sale for a limited time. Grab a t-shirt, sticker, patch, socks, or a full matching set! If you would like to donate to the classroom needs of my students, you have that option as well.

Varsity Math is something I enjoy very very much and I think the kids like it ten times more than I do. Come join the team!

For some more background info, have a look at the highlights from last year’s Varsity Math Day.

Limited to customers in the United States only. TX residents add 8.25% sales tax on all purchases (except donations made through the button). Store open for a limited time.

VARSITYMATH.NET

Posted
AuthorJonathan Claydon

You spend a lot of time around young people and you can’t help but do a lot of analysis of how you were as a young person. By far the biggest difference I see in myself and my students is how local their field of view seems to be, at least in my observations. When I say local field of view, I mean a lot of their time and energy is spent on the present and some handful of hours ahead of the present. What’s happening next week? month? year? You wouldn’t know they were concerned with such things.

And that’s not a bad thing, kids should get to be kids and only worry about the here and now as much as possible. Planning their next meal with friends or scraping together $10 for an impulse purchase is what youth should be about. Yes, college or post secondary plans, but kids are just so good at living right now.

Somewhere in the last 15 years, my brain got bored with right now. Or perhaps, with enough experience, you can handle the right now so well that it just doesn’t require a lot of effort?

Just off the top of my head, school related things I happen to be processing and their due date:

  • logistics for laser tag (7 months)

  • approximate sales expectations for my upcoming Varsity Math merch line (1 month)

  • is summer camp happening again? how many might go? (9 months)

  • painting a wall in the school visitor entrance (3 months)

  • National Honor Society induction logistics (6 months)

  • approximate AP testers (5 months)

  • AP benchmarking ideas (2 months)

  • TMC 2019 (10 months)

  • TMC 2020 (22 months)

  • does Varsity Math have a logical conclusion (20 years)?

Stuff like what am I teaching tomorrow? what should I eat today? what game/show/movie should I get into? just aren’t interesting questions. Have I just solved those problems too many times?

Is this why I tinker with curriculum so much? To keep my own attention?

I still very much like planning what I’m teaching tomorrow and what I should eat today, but they just aren’t a source of concern. I started noticing this about 3 years ago when I passed an experience milestone.

As friend Rachel put it the other day, it’s like I’m just in a constant state of playing tug of war with the future. As one long term project ends, another steps in to take its place. The never emptying to-do list.

Does this stress me out? No. It’s bizarre really. I just really like working long term problems. I mean, I’m still in the middle of my longest problem to date, improving AP effectiveness. We’re at 4.5 years and counting with only preliminary results.

You might be tempted to call me a workaholic or something. Some have before. I just don’t see it that way. For one, I don’t see it as work. I do “non-work” things, whether you hear me talk about them or not. I haven’t gone to “work” in 10 years. I’m solving interesting problems, something it turns out I really wanted in a career. Teaching offers fascinating logistic problems that I really really really like. It’s probably the biggest thing I liked in engineering school, mapping out plans to get stuff done, forged in a (now demolished) dorm basement as I figured out what it takes to pass exams. Secretly, I don’t really care for summer break anymore (though it is still a nice respite, and I mean, duh, TMC is the greatest), as there aren’t as many problems to solve.

It’s almost a bit of a game I play with myself, “bet you can’t figure out how to start a company” being the latest.

All this to say watching kids is just fascinating stuff, it helps me try to remember what it was like before whatever switch flipped in my head.

Posted
AuthorJonathan Claydon

Not long ago I had this simple idea for introducing the idea of an antiderivative. Turns out it wasn’t long before the idea would pay off again. In College Algebra, we were looking at radical equations. Solving stuff with square roots wasn’t new, but solving stuff with cube roots and fourth roots was. I place a huge emphasis on connecting algebra to graphs in College Algebra. If we’re solving a cube root equation, I want them to be able to graph it.

I cheat a little bit and tell them Desmos “can’t” accept a cube root as written with a radical symbol. It very much can, if you go digging around in the function buttons. However, I use it as an opportunity to introduce root and fractional exponent equivalence. It was a bit of a struggle to get kids to remember, so I thought we’d play a game with some of the extra index cards I had lying around.

As before, groups of students were given a random pile of cards. Each card had 1 match in the deck. I did not tell them how they were supposed to make matches.

Thrown into the mix were some rational expressions and their equivalent form with negative exponents. Students had no exposure to this other than from previous classes. I made the coefficients unique enough to where they were able to put things together using that as a context clue. As groups concluded their pairs, I went around and had a discussion with each group about what rules they had developed to make their matches.

Within 10 minutes, we had decoded fractional exponents and negative exponents without any boring lecture about exponent laws.

Posted
AuthorJonathan Claydon

This moment is a little surreal. If you’ve never heard the origin story of Varsity Math, here’s the moment I revealed it at TMC 15.

At TMC 16, I spread the love to everyone in attendance, offering a free sticker to all 200 some-odd attendees. In the years since, Varsity Math has become an enshrined institution at my school. Kids are incredibly all in on the concept.

At recent conferences, I’ll sport my Varsity Math merchandise and people will ask me for some. I am only able to give out a handful of extra stock that’s left over from the school year. And I always feel bad because there’s not really a lot to go around, and I don’t want people to feel left out. Others want to get something going like this with their department but may not have the means to generate merchandise. This year, I am excited to announce that you, teachers at large, can join on the fun as well. After figuring out a lot of random paperwork, Varsity Math is now a legally recognized company in the state of Texas.

Starting in November, an online store will open at varsitymath.net which will redirect to a page on this site. It will be open for a couple weeks as I test out taking and fulfilling orders. Some months later it will probably open up again. You can guarantee I will do a run for TMC 19. On offer will be stickers, t-shirts, socks, and sponsorships. If you purchase a sponsorship, 100% of that money will go towards the various Varsity Math activities at my school. Students are not always able to cover the costs associated with the merchandise and activities I provide. Traditionally I am able to cover their costs, but outside help will allow me to do more and possibly, one day, lower the cost of entry for all students.

I have put out feelers and it seems like there’s a demand, now we’ll see what happens. If you would like official Varsity Math gear, keep an eye on this space and Twitter for the opening of the store. Talk to people in your math department and buy shirts as a group! I would love to see you all sporting the best in math clothing (next to Desmos swag).

Huge thanks to everyone who has shown enthusiasm for my little high school joke and got me to this day. See you soon!

Posted
AuthorJonathan Claydon

It’s about time to start talking about antiderivatives in AB Calculus. We’ve defined a few rules so far: power rule, chain with trig, chain with e^x, and chain with ln x. For several years I’ve wanted to be early about our discussion of integrals. It has been helpful for conceptual understanding later if they know about the relationship between derivatives and integrals early. In fact, knowing about integration makes Curve Sketching a lot easier.

I was just going to do some simple introduction, but then I decided we could use a moment to get up and walk around.

Start with a 53¢ pack of index cards. Write a function on the top of the card and its derivative on the bottom. Repeat about 40 times.

Cut the cards in half. For fun, include several functions that have the same derivative.

Shuffle the cards a bunch until they’re good and scrambled.

The plan is to hand them the stack of cards and tell them that each card has a match. I’m not going to say how the cards are related to each other, only that have a match in the deck. After we spend some time sorting them, we’ll talk about the results.

How are the cards related?

Could some cards match with multiple cards?

Eventually they’ll tease out the idea that one card was the derivative and one was the original. Then bam! we hit them with the idea of antiderivative, the result of working backwards from a derivative. It is highly likely that when finding some matches they will do this, knowing they have the derivative in hand, in search of the original.

The fact that multiple functions can have the same derivative is always an interesting discussion, and just like that we’ve justified the presence of +C.

Posted
AuthorJonathan Claydon

I have traditionally had trouble fitting in Intermediate Value Theorem and Mean Value Theorem into Calculus without them seeming arbitrary. Kind of by accident, I found a nice way to not only talk about both theorems, but introduce early idea for curve sketching. And it was super simple!

Start with a table of a continuous function:

Screen Shot 2018-10-04 at 11.26.51 PM.png

Have students calculate the average rate of change on all the intermediate intervals. In this case, 3 to 7, 7 to 11, etc. In addition, have them calculate the average rate of change for the extreme values of the table, in this case 3 to 26.

We now have a list of slopes: 1.525, 2.975, -8.6, -1.3667, 1.683 with an extreme slope of 0.1227.

Next, have students graph the values from the table.

They did this by hand, but I used Desmos here for demonstration purposes. Have them describe the behaviors of the function. Next, make a graph of the slopes, including the slope we found from the endpoints:

Great moment to talk about why we would graph the slopes this way, and the assumptions involved when taking an average rate of change. Now some questions. How many times is the slope zero? How many times is the slope equal to the overall slope? Could we sketch a function that would output these values?

Go back to their description of behaviors. When the function was increasing, what type of slope did you have? when the function was increasing? when the slope appeared to be zero?

Before you know, you’ve teased out the concepts behind the relationship between K(t) and K’(t) in addition to a good demonstration of the Mean Value Theorem at work. Finally, with a graph of the original function, a discussion of Intermediate Value Theorem comes naturally. Pick some arbitrary y-values and have students decide if they should exist. How would you know?

I don’t even know what made we think of it, but adding the graph of the slopes to the mix really made this an interesting problem. Everything I need to cover in the next few weeks is sitting right there! And students found it all intuitive, we just needed to add some formality to our justifications. I really enjoyed how this turned out.

Posted
AuthorJonathan Claydon

Based on a Global Math Department Presentation from 9/18

I have a single goal, that within a few days of the start of school, my students are convinced that their time with me is going to be the best part of their day. I even say this on the first day of school. Although once you’re at a school long enough, you earn a reputation. In recent years I’ve appended that statement to say “this will be the best part of your day, I hope it lives up to the hype.”

At the same time, I want it to be the best part of their day for the right reasons. Yes, we are going to have some fun, but we’re going to be productive at the same time. Kids can have weird perceptions of what makes a “great” class. I’m reminded of a conversation I heard between middle school students talking about their really fun social studies class in which they did nothing and watched movies every day. That’s not the sort of thing I’m going for here.

Inspirations

I draw most of my inspiration from two teachers. I loved going to their classes every day. It was exciting to spend time in their room because I knew I was going to have a good time. As a teacher, I really want to have a good time as well, since I’m in the room all day after all. So if I’m having a good time, they should too.

My 6th grade math teacher, Mr. Richardson, was a legend. In my elementary school there was a buzz about him. When you got to 6th grade, you needed to have math with this guy. He had a counterpart in social studies that was equally hyped, but sadly, I didn’t get them both.

What did he do?

First and foremost he organized us into teams. These would change throughout the year. We had jobs. We had to have a logo. There was an in class economy. Each week you’d earn a salary depending on your job, and his currency could be traded for trinkets from a store. How this man kept track of 150 some odd salaries is beyond me, but we got paid every week.

He gave us interesting problems. There was a curriculum to tackle, but he went out of his way to challenge everyone in the room. We did a multi-week travel project, where we had to plan out every detail and track every expense, writing the checks and everything. He was the sneakiest direction writer I ever met. There was ALWAYS some little treat hidden in his directions. It was his way of teaching us to pay attention. I sucked at these.

Most importantly, he respected everyone in the room. It didn’t matter that we were 11. If you came up with a clever method, he’d name it after you. He was constantly grouping and regrouping us based on our needs. He would give different types of assignments to everyone. And he expected you to keep them all organized. I was horrible at keeping them organized.

My 12th grade Calculus teacher was Mrs Westerfeld. Compared to the other guy, her classroom operated a lot simpler. She’d do two things consistently, complain about George W Bush (she was from Crawford, TX where he has a ranch) and let you know she cared.

Twice a year she’d sit everyone down and read you a story. She read us the Polar Express and gave us all a little bell. At the end of the year she made the whole class write one nice thing about everyone else in the class. You’d put your name on a card and pass it around in a circle. When you received a card, you wrote your nice thing, and she’d add hers. She did this for 150 students.

But the other days of the year she got down to business. Plenty of homework and challenging tests, but you loved it.

Classroom

I send several important messages to my students. I never say them out loud, I let the actions speak for themselves. The setup of my classroom is integral to this process.

There are six tables in the room. Students sit in semi-random groups throughout the year. The goal is to build an identity with their table mates. If we do a group activity, kids work with their table. If we have a contest, kids compete with their table. If it’s time for classwork, they do it with their table. If I start flirting with rearranging seats they will give me the biggest STINK EYE if I even THINK about breaking them up. They were complete strangers on day one! Are you serious? Several times a group of kids who were complete strangers at the beginning of the year have asked if they could do Secret Santa for JUST THEIR TABLE.  It’s the most adorable thing I’ve ever seen. I’ve had several students for multiple years, and these repeaters want their table back. Doesn’t matter that the kids are slightly different. That’s THEIR TABLE.

Why in the world would a kid care so much about a seat? Let me explain.

You Are Welcome

I greet everyone at the door and I greet them by name. I have been the quiet kid in the back of the class. I never felt like those teachers knew who I was. I’m not doing that to my kids. I want them to know I notice them. At the end of every class period I give them all a high five as they leave. I’ve been doing that since 2015 thanks to a presentation by Glenn Waddell. Since then I’ve given about 75,000 high fives and have washed my hands about 3,000 times. Seniors are legitimately depressed when it’s time for the last high five. They’re excited to steal extras at graduation.

You Can Relax

If I have to talk in front of the whole group, I treat it like a discussion. Kids can interrupt me at any point with a question. Initially they’re all very polite and will raise hands or whatever, but they figure out pretty quick that I want them to just jump in. If I ask them to think about something and prepare an answer, they’re not allowed to shout their answer. We want everyone to have time to think. If I have them make a calculation, or estimate a quantity, I write down anything I hear. I never confirm anything as right or wrong unless it seems like that’s the consensus. If the answer is 5, and one kid says 5, I’m not moving on until the group agrees. If a kid said 3, I never disparage them for saying 3. I ask them why they thought 3. Some less daring kid might also be wondering why it wasn’t 3. If I make a mistake, I announce my mistake. Sometimes I’ll explain how I make the mistake. You should never be afraid to be incorrect in front of the group, and that starts with me.

You are also welcome to ask an off topic question. Often I will acknowledge the question and deal with it later. Often I’m the one with the off topic remark. I am easily distracted.

If a kid wants to know where I went to high school, I’ll take a minute and show them. If they want to know where I live, I show them a house that could be mine. I might have a question about how the football team did. That might inspire someone in band to tell a funny story from the football game. I welcome the story. The other day kids were asking about hurricanes, so we spent some time discussing the current hurricanes and what it takes to track to us. I want them to be curious about things, and it doesn’t always have to be about the current discussion. I also want them to know that I care what they’re up to and that their stories are interesting to me. We’ll entertain the idea and go right back to our main discussion. Honestly, I think this helps absorb the day’s content better. It offers a natural opportunity for me to repeat something, and it’s given their subconscious a minute or two to process what we’re doing, even if we were talking about something random. Our curriculum is not so sacred that we can’t take a second to deviate.

You Will Be Productive

I love student work time. All the stories and distractions are great, but students know that in my room, when it’s work time, it’s work time. As much as possible I give students their class time to get their work done. This is also when I secretly get to know them the most. As they work, I wander. I’ll look over someone’s shoulder for a second, or I’ll listen to kids discussing a problem. I rarely intervene. I hop slowly from table to table. If a table needs me, they’ll stop me. The whole table will perk up and listen to the question, and I’ll move on. If I pass a table that hasn’t said much in a while, I’ll stop in to see how they’re doing. If they happen to be off topic, this is fine. 99% of the time kids are naturally back on topic within a minute or two. If they take longer, I’ll politely suggest they get back to it.

Every student has a notebook and every assignment we do is in that notebook. I started doing it a long time ago and it’s a proven winner. At the end of the year a majority of kids can’t believe how much stuff wound up in there. It’s their physical artifact that ok, yeah, I did a lot of work in here.

If a kid is really struggling, I’ll either take a seat next to them or kneel down and talk them through it. Often they’ll want to quit.

I’ll remind them that they do know what they’re doing, and that there’s something they know that will help us here. If they need to go slow, we’ll go slow. I wait patiently for them to rewrite a problem, or correct a mistake, or fiddle with the calculator. We are never in a rush. Last year I had a student who struggled with just about everything. I helped her individually all the time. But this girl was determined to keep at it. Throughout the year I’d have kids show me their classwork and this girl would BEAM when she showed me everything she did by herself. She was in 12th grade and had never felt that way in a math class before.

Sometimes I’ll jump into their conversations. Throughout the course of a school year, I develop an individual relationship with each table. If there’s six tables in the room, I’ll have six inside jokes going amongst them. I don’t care how big a class is, I want kids to know that they can have my attention whenever they need it.

We Are A Family

I was really attached to my Calculus class. There were 12 of us and I dubbed us the dirty dozen. At the end of the year our teacher let us decorate a ceiling tile. I went back to visit her and our tile was still sitting up there. This was us.

So let’s talk about Varsity Math.

I took over our Calculus program 5 years ago. Kids were fairly apathetic about the course. They heard it was difficult, so they’d try it for like 3 weeks and drop. I wanted to change that attitude. Kids were going to be excited to join an AP Math class.

Varsity Math is an initiative for AP Stats and Calculus students at my school. It has become a signature institution. Younger students ask for it by name.

We have t-shirts. We have stickers. We have sunglasses. We have snapchat filters. We have a big end of the year party. And we have a monument. We also adopted that baby shark song as our anthem. We played it at prom, twice.

The main idea here is that the 100 or so AP Math students are in this together. They are proud to be in a challenging class. Because they should be proud to be in a challenging class. Since starting this initiative, the drop rate in our AP classes plummeted. AP Exam performance has been ticking up too. And for crying out loud they had a PARADE during lunch last year. The kids are so into their math class that they want everyone else to be jealous of their math class. Kids are ECSTATIC the day I hand out the merchandise. Their time has come. They’re part of the legend.

Take Aways

First and foremost, I’m admittedly a little ridiculous and some of this stuff take a LOT of time. Do not feel like you have to drop everything and start your own clothing line to gets kids hyped about math. Though I hear that works for Desmos.

Even the most difficult student wants to learn something. Kids are very aware of what their classes demand of them, and they adjust appropriately. If they know English is a joke, they’ll treat it like a joke. The biggest compliment a kid can ever pay me is when they say “we really do a lot in here” or “wow class goes fast.”

If you were going to pick something to change, the biggest move you can make is to yield as much class time to students as you can. My lessons are very condensed. If they’re longer, there’s almost always a day of classwork that follows. In College Algebra last year I stopped whole group instruction entirely. Kids would come in, and I’d give them a series of structured tasks. They came to expect it. Within a few minutes of class starting even my most difficult child would ask when she could get the work for the day.

Sit down with your kids. Chit chat with your kids. It doesn’t have to be about math. They will become the hardest workers for you if they know you see them as an equal.

Posted
AuthorJonathan Claydon
2 CommentsPost a comment

College Algebra is an interesting course to teach because for the kids involved, the topics aren’t really new, but there are certainly new things they can discover within them, or get better insight hearing about something a second time. This last week we were starting an introduction to transformations. That prompted this bit of lesson planning:

Opening Acts

I used a set of three pre-built Desmos Activities with the group. I intended to use these last year but just never forced myself to do so.

Opener: Transformation Golf

Middle Innings: Translations with Coordinates

Closer: Practice with Symbols

For the ability level of the kids involved, these went really quickly. They are simple, straightforward activities but do present some interesting challenges. The kids really enjoyed transformation golf in particular.

It prompted a lot of good discussion and offered just enough challenge for everyone. We completed that activity in one 50 minute class period (about 40 minutes or actual working time). The other two (coordinates and symbols) were done in a single 50 minute class period. The combination of these three activities were just to job some memories and reacquaint with transformation vocabulary.

Proving Activity

A longer version came later, but using the polygon() tool in Desmos, we did a short proving behavior. We built polygons using a table, and applied some coordinate rules to those polygons. Students had to modify their polygon in 4 ways, writing down what they did. Then they submitted a link to their graph (my subtle way of teaching them how to sign-in with Desmos and save things). This took about 25-30 minutes of real class time:

I really liked the progression. Kids got to take a familiar skill and learn something new about the calculator. A few days later they did a more involved polygon transformation and applied what they learned to transformations of various parent functions (quadratic, absolute value, radical, natural log). The best part? These three days worth of lessons only took 15 minutes to map out thanks to the great resources in the Desmos Activity Library and the incredibly slick polygon command (launched only a few months ago).

Really happy with how all this came together.

Posted
AuthorJonathan Claydon