Virge Cornelius Mathematical Circuit Training 2015 Limits Answers

This week was mostly test review and testing for both Precalculus and AP Calculus AB. Below, I share my methods for test review in both classes. My AP Calculus students did not do well on our Unit 4 Exam. Some reflection is necessary for moving forward and how to approach this unit in the future. In Precalculus, at the end of the week we began our journey into Trigonometry, I enjoy teaching the unit circle so it was a good end to the week. In AP Calculus AB, we moved on to the Mean Value Theorem. After our rough mid week test, the students were really happy to be successful with a topic, the lesson went very well.

Precalculus Test Review

In my classes, I like to give practice exams. I always felt in school that I knew the material and what to expect, but inevitably, the exams always felt slightly different than what I was used to. Obviously, this points to a lack of deep understanding of the topics. However, our students are all in a different place cognitively and sometimes these topics only really unravel in the brain later on. I provide a practice exam/quiz so that students will understand the expectations. This is much harder for me to do in AP Calculus because it takes a long time to write questions that are varied enough and rigorous enough to match an entire AP Calculus Exam.

When I'm making a test, I generally just get the formatting correct on the practice exam. I then make 3-4 quick copies in Google Docs and then start to edit them all at once, one question at a time. This helps me create a set of exams that are at the same difficulty level. Once I figure out how to generate one question, it doesn't generally take too long to create a couple variations. All in all, just thinking of random numbers is the hard part some time.

Precalculus Unit 3 – Transformations of FunctionsPractice Test
Precalculus Unit 3 – Transformations of Functions Test:Version A,Version B,Version C

AP Calculus Test Review

In the AP Calculus Teacher's group I am in, I have seen several "circuit training" worksheets from others. I believe these have been inspired mostly byVirge Cornelius and my students really enjoy them (as much as they can enjoy a rigorous math exercise). I wanted to try my hand at a few. My students caught a few errors but I have since corrected them and I present to you my AP Calculus Circuit Training Exercises for L'Hospital's Rule, Linear Approximation, and Motion on a Line. As a bonus, I created a short sheet to practice interpreting the context of a derivative. Note, for the circuit training, the first two pages are the "key", this is essentially how I start to build the question base. After this, I randomly choose a block using Google Sheets to randomly assign which block they will go to.

L'Hospital's Rule Circuit Training
Linear Approximation Circuit Training
Motion on a Line Circuit Training
Interpreting the Context of a Derivative Review

Reflections on AP Calculus AB Unit 4 Test Performance

Every teacher I believe has had that moment. You went through a unit of study and saw even your high flyers fail to show appropriate progress towards the standards. It's a tough time for everyone, students are stressed, parents are not happy with the students and their grades, and as an educator you turn an eye inward to ask what went wrong. I have in previous years had students not take things seriously until an assessment was looming. This caused the students to "freak out" before big exams and try to catch up on all of the learning throughout the unit at the last minute. Our school has academic support time so I would always get slammed during this period of time, trying to reteach students who essentially checked out until it was "important". This year, I decided to run daily checks for understanding in AP Calc to keep students in the game. This has had mixed success.

At the beginning of the year, I used AP style questions to keep the rigor up. This simply resulted in students struggling to grapple with the harder questions as they were only exposed to the basics and then completed homework on it. I then went to an easier form of checkpoint quiz but then the content did not appropriately ramp up to the difficulty of the AP Exam (and by extension, the difficulty of my end-of-unit exams). I felt that throwing AP Classroom and Albert.io questions at them during review time would cover the gap, it did not. It was too much material too late and my student failed to cover the gap in that time.

At this point I am thinking of scrapping the checkpoint quizzes, although I will miss the daily checks for understanding. If anyone out there has advice on what they do I'd love to here it. Perhaps quizzing every few classes will be the answer moving forward.

I have started gathering questions on a per-lesson basis from AP Classroom (ones that are not teacher-access only) and creating daily handouts. They will be an optional part of the homework assignments and we can cover questions at the beginning of each class. We'll see if things go better during Unit 5!

AP Calculus Unit 5 Start (Analytical Applications of Differentiation)

It may seem silly, I have been using AP Classroom since the start of the year, but I only now noticed that they have lesson handouts for students and teachers for each topic. The amount of effort I have put in to each topic to figure our exactly what College Board's intention was, and I could have simply looked at these handouts. Live and learn…

We started this week with the Mean Value Theorem. I felt that AP Classroom's resources for this were very good, we discussed what Existence Theorems were a bit further and since IVT continues to rear its ugly head on every assessment, we compared and contrasted MVT and IVT. Next class we will introduce the Extreme Value Theorem as our next existence theorem and then move into applications of the first derivative. By the end of the week, students will be pros at identifying key information of functions using the first and second derivatives.

Precalculus Unit 4 Start (Radians and the Unit Circle)

I absolutely love this part of the precalculus curriculum. We finally move past our review of Algebra II topics and begin to really dig into trigonometry and high level topics. I created a guided notes handout for my students. If I can't make a rich, inquiry based exploration of a topic, I try to create a guided notes handout for students to take important notes and work through example problems. I'm torn as to whether this is best note-taking practice, I know that having students copy material is supposed to be better for retention but I get so impatient waiting for those same two slow-copiers to finish copying. What do you do in your class as far as notes go? Do you feel that same frustrations?

Anyway, it turned out to be a great lesson, I feel the students really enjoyed the topic as we formalized what a radian was and why exactly the unit circle was such an important tool. My guided notes handout is included below, feel free to give feedback!

Unit 4: Section 1 – Radians and the Unit Circle Guided Notes

"Mathematician" Instead of "Name"

After looking through some ofSam J. Shah's handouts, I really liked that he started to put "Mathematician" to indicate where a student should start writing their name. I started to do the same in my class and I love the impact that it has had. Most students have reacted in some ways and it has opened up some excellent conversations. Some students like the honor and fanciness it brings, it elicits some chuckles. However, some students have exclaimed to me "I am definitely not a mathematician." This has opened up some excellent conversations about what it means to be a mathematician. We have discussed how ability to do something is not what necessarily what creates a label. Anyway, I thought it was a neat little thing and I've had fun with the class discussions that have come up as a result. I encourage any other math teacher to do the same. For other subjects, I feel we should write "Scientist:", "Historian:", "Philosopher:", etc.