Posted in Assessment, Calculus, First day

AP Calculus BC: reflection and ideas for the future

I loved teaching AP Calculus BC last year! I was a little intimidated at first… I hadn’t taught or even used Calculus in 15 ish years. It was a great experience and I can’t wait to do it again! But alas… not enough sections this upcoming year. I do want to reflect on what I did and what I would like to do in the future so I don’t forget.

My favorite lesson was actually the first day of school. I showed three linear graphs and asked students to write down everything they knew about them.

First day graphs

After talking about key features – intercepts, end behavior, domain and range – and linear vs. constant functions and a line that’s not even a function, I surprised them: these aren’t lines!

First day graphs 2

Students worked in teams to match each graph with an equation and come up with a convincing reason why. Then teams presented their solutions via gallery walk.    

First day student 2   img_1357

We touched on domain and range, even and odd functions, and other key features of functions; we also began exploring the ideas of limits and local linearity. Students really latched onto the idea of “zooming in” on the graph, which ultimately paid huge dividends throughout the course. I found myself saying weekly, if not daily, “remember the first day of school when we zoomed in on those three graphs?”

Another win last year was practicing derivatives and (some) antiderivatives at the same time. This is one of my favorite teaching strategies in general; if you can do something one way, you should be able to think “backwards.” I only did basics, like the table below, but it really got students thinking both directions.

I like to give a little peek into what’s coming, and since we already knew that the derivative of a constant is zero, it wasn’t too challenging to sneak in conversation around this “+C” business early.

Here’s another one, this time after trig derivatives and chain rule:

In the future I would like to do this even more intentionally and often.

Around the same time, we were sketching derivatives given a graph, so I threw in some antiderivatives in the same way: if this is the derivative graph, what could the original graph look like? Here they could really see the “+C” as teams came up with different vertical translations of the same graph.

As much as I loved exploring and wondering about these different ideas, the biggest struggle I had was the order in which to introduce concepts. Everything is so connected, and I didn’t want to muddle their understanding by doing too much at once. I ended up sticking pretty close to our textbook; especially in my first year, especially since my PLC teammate would also be following the textbook, especially since the instructor of my APSI used the same textbook at his school and provided me with a bunch of already aligned supplemental resources (!!!), it felt important to stay consistent.

Now that I have a little more “big picture” vision, I would do things a little differently.

First, standards-based grading! I started this in second semester last year, when we began studying series. It was a natural break to begin something new, but the content turned out to be perfect for the kind of self-reflection and improvement that I needed from students. (I’ve been implementing various versions of SBG every year I’ve been teaching; this might be my favorite version so far, but still far from perfect.)

Learning Objective 4.1A – Determine whether a series converges or diverges.

This learning objective encompasses A LOT of different tests for convergence, and what I found is that our textbook gave very bare answers to homework problems, so students thought they were doing great when they were actually missing big chunks of explanation that would communicate sound mathematical reasoning (and let’s be honest, would cost them points on the AP exam). By changing my grading strategy, I was able to better communicate my expectations and give feedback for improving.

After our first quiz on the topic, I gave students their highlighted quizzes back (no points marked, my typical M.O.) along with a Google doc for reflection. This included a rubric for them to score their own progress on this learning objective.

Rubric series

I showed an example of a 4 to the whole class, emphasizing three parts to communicating whether a series converges: conditions, evidence, conclusion. Many students were providing the evidence, but not first checking that the conditions for using the convergence test have been met or providing an appropriate conclusion; we looked at two examples of 3s that illustrated these omissions. Then students made corrections to each problem on their quiz and gave an overall score for their understanding with evidence to support their score. Ultimately they were very accurate and I only had a couple of conversations with students whose evaluation of their learning was different from mine.

The best part – every student earned a 4 on their next assessment of the same learning target. After a whirlwind first semester, I finally felt like I was doing something right!

I always give cumulative quizzes/exams, so this quiz also included derivatives from first semester, along with a rubric for self-evaluation similar to the one above. I got some very honest, “I thought I was a 4, but this evidence says that I’m currently at a 2” evaluations. They realized for themselves that they still had some work to do on the “easy” stuff! I really hope to start this from the very beginning of the year when/if I get to teach this course again… I think the potential for continuous learning here is huge.

The other thing that I would really want to do is address the sequence of content; I still don’t have a good feel for this yet. In my other courses, I try to space out related content so that students have time to practice content before trying to build on it. For example, in EM3, we teach unit circle and trig functions (all no calculator) in unit 2, practice it throughout homework in units 3 and 4, then unit 5 brings in inverse trig functions after they have built up fluency. I don’t know yet what this looks like in Calculus, but I would like to attempt it. Thoughts so far:

  • Do some work with series earlier in the course. Chapter 10 in our textbook is all about series and it is HEFTY. It could easily be two units with all of the convergence tests taught earlier and practiced before needing them to test endpoints for intervals of convergence later.
  • Teach derivatives and integrals simultaneously??? Maybe start with graphical and tabular interpretations of both, then move to analytic approaches? Can this be done??? I have more questions than answers.
  • I felt like I needed more time for teaching and practicing integration techniques, so find a way to do this earlier. My students sucked at u-substitution, honestly because I think I sucked at teaching it. Maybe by teaching chain rule and “backwards chain rule” together, students will become more fluent. Maybe by separating it from other integration techniques and letting them get good at one technique at a time, they can feel more equipped when I ask them to compare and contrast strategies. Maybe Coaching Questions would be a good routine to use practice using integration techniques.
  • Polar area – I feel like I can sneak this in earlier with other area integration problems. Still not sure.
  • The construct of teaching all AB content first, then moving to BC content, seems forced. Is there a reason for doing this, other than my textbook said so? Yes, the AB content is foundational for BC content, but can/should it be better integrated?

And there we have it folks – a lot of partially formed ideas about AP Calculus BC! I really do hope to teach this class again. Shoutout to my department chair – if you’re reading this, in case I haven’t said this enough, I would LOOOOOOOOVE to teach AP Calc again in the future. 😉 😉 😉

Posted in Assessment, Calculus, Instructional Strategies, Intervention

Exit Slips and Intervention

I’ve been trying to figure out the “best” way to write this blog post… finally I decided to just write it in whatever imperfect way it comes out because I think it’s one of the most powerful things that we can do as teachers and I want to spread the gospel far and wide!


But it’s not just the exit slip, it’s what you do with it. You’re getting some information about your students’ understanding, and then following up in a very strategic and targeted manner.

Here’s what I did recently with an exit slip on limits and continuity.

Exit slip 1

Intervention #1: Pulled small groups

I gave this exit slip at the end of a class period on Tuesday. On Tuesday afternoon, I quickly sorted exit slips into “yes” and “no” piles based on their answers to #1(a)-(c). Then I sorted the “no” pile into subcategories based on the mistake that was made. I ended up with these categories:

  • On #1(a) – students who answered 2, which is the value of f(-2) not the limit as x approaches -2;
  • On #1(b) – students who answered 3, which is the limit from the right not the limit from the left;
  • On #1(c) – students who answered 2, which is the limit from the left and the value of f(3);
  • On #1(c) – students who answered 1, which is the limit from the right.

I made new student teams based on like mistakes. On Wednesday, I projected a PowerPoint slide with their new teams; as they worked on some practice problems together, I visited each group to address the misconceptions in their exit slips and answer any remaining questions. (In the past I have pulled small groups to work with me at a side table, but this year I’ve decided to regroup teams temporarily and visit each team.)

Intervention #2: Targeted whole class intervention

When I looked at students’ responses to #1(d), I noticed that all students had said that the function was not continuous at x=1, but most did not reference the definition of continuity in their explanations. This told me that I needed some whole group instruction around the definition of continuity, specifically in how to justify their reasoning. So after I addressed each team’s individual needs on #1(a)-(c), I asked students to discuss the prompt below in their teams.


We had a discussion of the three conditions required for a function to be continuous at a point, and students were able to specifically identify the reasons for which each function was not continuous at x=0.

Intervention #3: Peer teaching

Finally, on problem #2, the mistakes were all algebraic/computational and had nothing to do with limits: students either forgot how to factor a difference of two cubes or made a mistake in dividing. In every team, there was at least one student who had correctly factored, simplified, and evaluated the limit, so those students supported their teammates in finding their mistakes and making corrections.

So how much time did this take?

Well… it has taken me longer to write this blog post! 🙂  It took me about 5 minutes to sort the exit slips into piles for intervention #1, maybe another 10 minutes to read the responses for #1(d) and create intervention #2, and about another 5 minutes to look at the work for #2 and make sure that each team had at least one student who had completed the problem correctly.

It took me about 25 minutes of class time to check in with every small group (and would probably have been less if I remembered my timer… oops!). And this is happening as the rest of the class is working on another task, so I’m not losing any time; I’m actually gaining time because I’m not reteaching to the whole class.

Is it worth it?

Absolutely! On their quiz last week, every student correctly identified a two-sided limit, a one-sided limit, and a limit that did not exist; nearly every student was able to correctly show that a function was continuous at a point using all three parts of the definition.

In all honesty, I think this is one of the simplest changes a teacher can make to improve student learning. And although this example is from my Calculus class, I have used this strategy in every class, at every level, with every student population. IT WORKS!

Shoutout to Cassandra Erkens of Solution Tree, who has been a consultant in my district for many years now, and who has helped me to refine my assessment and intervention strategies every year!

Posted in Calculus, Instructional Strategies, Precalculus

Resurrecting the blog

Haven’t blogged in almost two years… how the time flies! I keep thinking of things I want to write about, but there’s never enough time in the day.

My original intent with my blog was to help me become more reflective in my own practice. Although that is still a goal for me, between grading papers, creating a new course, and trying to actually have a life with my own family, I’m not sure that reflecting daily in blog form is in the cards for me this year. So instead, I’m going to try to focus on the results of trying something new, strategies and resources that have been particularly effective, and slight tweaks that we teachers can make in our classrooms that pay big dividends for students.

This year I’m teaching AP Calculus BC for the first time and Enhanced Math 3 for the second time. I’m also reviving our Math Lab course, which I originated at our school about 7 years ago; after I moved to the District Office for a few years, the course fizzled out, but I’m determined to make it a success this year! In addition, our school has adopted an 8-period alternating block schedule with anchor day after a 6-period traditional schedule for nearly 40 years. Life is never dull!

One thing I’m trying to be very intentional about this year is instructional strategies that get students out of their seats during our now 85-minute classes; I’m encouraging all of my colleagues to do the same!

Strategy: Stand and Talk 

Thanks to Sara VanDerWerf (@saravdwerf) for this great strategy. I have students stand up and put their hands up, find a partner from a different team and high five; any student with their hand still raised is still looking for a partner. Then I give them a discussion prompt to work on together. Sara recommends giving each pair a problem/prompt written on a half-sheet of cardstock so it will hold up for multiple classes.

In Calculus, we sketched a graph of a function that meets all of the criteria described by limits. I laminated this half-sheet so that students could write on them with dry erase markers and I could still reuse them in the next class.

stand and talk 1stand and talk 2  stand and talk 3  stand and talk 4

Students drew their graphs, then paired up with another pair to check each other’s work. Even though most pairs turned into “kneel and talk” or “lean over a desk and talk” so they could write on the cards, students got out of their seats, worked with other students outside of their teams, battled misconceptions (they hadn’t experienced a function with more than one horizontal asymptote), and practiced describing function behavior with limits.

In Enhanced Math 3 (Integrated Math 3 + Honors Precalculus accelerated class) we’re working on composition of functions and inverses. Since I only have one section, I wasn’t concerned about reusing the prompt, so I just printed on regular paper and collected this as their exit ticket for the day. Stand and talk inversesI left this pretty open to see what they would do with it, but I had several student pairs who did this very quickly (maybe too quickly???), so I reminded them of our learning target: “Mathematicians understand and apply inverse functions in multiple representations” emphasis on multiple representations. One team said, “at first we said yes, then when we sketched the graphs, we realized we needed to restrict the domain of the quadratic function to make it match the reflection of the square root function.”

When we analyzed the results of the exit ticket together the next day, we noticed that the range of the quadratic function was the domain of the square root function; when we found the range of the square root function, we realized that was the appropriately restricted domain of the quadratic function!

So far just two stand and talks have convinced me this is a must do, ESPECIALLY for block classes! 🙂