As you get ready to start school, here are some thoughts on the first week in AP Calculus. I looked back recently at several of the “first week of school” advice posts I offered in the past. Here’s a summary with some new ideas.
- DON’T REVIEW! Yes, students have forgotten everything they ever learned in mathematics, but if you reteach it now, they will forget it again by the time they need it next week or next January. So, don’t waste the time, rather, plan to review material from kindergarten through pre-calculus when the topics come up during the year. Plan for short reviews. For instance, when you study limits, you will need to simplify rational expressions – that’s when you review rational expressions. Months from now you’ll be looking at inverse functions, that’s when you review inverses.
- Make a copy of the “Mathematical Practices” and the “Course at a Glance” from the Current AP Calculus Course and Exam Description (p. 14 and p. 20 – 23) and give them to your students. Check off the topics as you do them during the year. Or give them the more detailed Unit Guide (e.g., p. 32 – 33 for Unit 1 Limits) as you start each unit. Either way, have your students check off the topics (1) as you teach them and (2) when they understand them.
- In keeping with Unit 1 Topic 1, you may want to start with a brief introduction to calculus. Several years ago, when I first started this blog, Paul A. Foerster, was nice enough to share some preview problems. They give a taste of derivatives and integrals in the first week of school and get the kids into calculus right off the bat. Here is an updated version. Paul, who retired a few years ago after 50 (!) years of teaching, is Teacher Emeritus of Mathematics of Alamo High Heights School in San Antonio, Texas. He is the author of several textbooks including Calculus: Concepts and Applications. More information about the text and accompanying explorations can be found on the first page of the explorations. Thank you, Paul!
- If you are not already a member, I suggest you join the AP Calculus Community. We have over 23,000 members all interested in AP Calculus. The community has an active bulletin board where you can ask and answer questions about the courses. Questions often ask how to better teach a topic – get hints and share your ideas. Teachers and the College Board post resources for you. College Board official announcements are also posted here. I moderate of the community, and I hope to see you there!
- There is a Facebook group called AP Calc Teachers – AB/BC that you also may find helpful.
- If you haven’t done so already, read your Instructional Planning Report (IPR). Especially helpful will by the comparison of your classes mean scores with the state and global mean scores for each question. If yours are higher great; if lower, that’s where you might need to do something different.
This blog has been written with teachers in mind. Students are always welcome to read it. You may give links to any of the posts you think your class may be interested in.
This year I plan to write a new series of posts especially for students. It will be called “The Why Series.” The inspiration was Why Radians, one of my most read posts. The posts will be short pieces introducing units and parts of units that (I hope) will explain why the topic is important, where the topic is leading. The first post “Why Limits” will appear soon.
Teaching Calculus 
Paul, You have been my inspiration for “how” to teache since my very first year of teaching in 1978. I taught out of your little yellow trig book. That book, along with your teacher notes, framed the last 45 years for me. Thank-you. When I began teaching calculus, your exporations were my goto for getting students thinking.
Also, Thank-you Lin for all of your helpful posts which have also been a live saver for me during my 20 years of teaching calculus, along with your book that I reference often as well. You guys are amazing and such a role-model for me.
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Majorie
Thank you!
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Hi Lin and fellow AP Calculus instructors. Thanks for the timely and succinct comments on how to start the calculus course with calculus, rather than with a review of precalculus. This was the topic of my presentation at the international Advanced Placement Annual Conference (APAC) in Seattle two weeks ago (quite an accomplishment for an 88-year-old!). You can access the four Explorations (a non-boring name for “worksheets”) and my PowerPoint presentation elaborating on them using this link to my Dropbox:
https://www.dropbox.com/sh/798k0tmbqnz4wvy/AADZMDbpCDsUNlWr0R2NDYL_a?dl=0
The Dropbox posting contains other relevant information:
(1) The Seattle APAC PowerPoint and handout also have information on how to present volume problems in such a way that all involve the same calculus—integrating (accumulating) dV, the differential of volume (not accumulating a rate of change of volume). The only difference is in geometry and algebra— finding the volume, dV, of a slice—its area times its thickness (dx or dy)—in terms of a sample point (x, y) on the graph, then getting dV in terms of one variable so that you can integrate it. This topic was to have been presented at the 2022 Austin APACs canceled by COVID.
(2) The T-Cubed Dallas PowerPoint in the Dropbox has the presentation (subtitled “Epsilon and Delta Without Tears”) to have been made at the 2020 (canceled) conference. One of the Explorations detailed in that PowerPoint is in the Seattle handout mentioned above. The other Explorations (and 150+ others) can be accessed by going to http://www.flourishkh.com and requesting a free 30-day online trial of Calculus: Concepts and Applications from Kendall Hunt, the current publisher. During that trial, you can download any of the materials you want to use later with your students. You could even invest in an online or hard copy of the text, but the important thing is for you to see how calculus can be approached by having your students get a gut feeling for the four concepts (limits, derivatives, definite integrals, indefinite integrals) before becoming immersed in the myriad details such as, “bring down the exponent and reduce it by 1,” which gives no clue that the answer is an instantaneous rate.
When the time comes for review of a topic, the best way I found to start off is by saying, “You recall … .” Students who didn’t recall think, “Phew! He’s going to tell us, so I’d better learn it now.” Students who did recall are not insulted by having it repeated.
In addition to the reasons Lin mentions for not starting the calculus course with review (consuming time, forgotten again when needed), there are other more subtle reasons for starting calculus on Day 1: (a) If the students do remember precalc topics, their mindset becomes, “This is easy! I won’t have to study.” By the time they realize there is new stuff to learn, they’ve developed bad study habits. (b) If the students don’t remember it, their mindset becomes, “Better drop the course. I don’t even remember precalculus.”
Best wishes to you and your students as you start the new school year!
Paul
Paul A. Foerster
Teacher Emeritus of Mathematics
Alamo Heights High School
San Antonio
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