Visualizing Unit 9

As you probably realize by now, I think graphs, drawing and other visuals are a great aid in teaching and learning mathematics. Desmos is a free graphing app that many teachers and students use to graph and make other illustrations. Demonstrations can be made in advance and shared with students and other teachers.

Recently, I was looking a some material from Unit 9 Parametric Equation, Polar Coordinates, and Vector-Valued Functions, BC topics, from the current AP Calculus CED. I ended up making three new Desmos illustrations for use in this unit. They will also be useful in a precalculus course introducing these topics. Hope you find them helpful.

Polar Graph Demo


You may replace the polar equation with any polar equation you are interested in. There are directions in the demo. Moving the “a” slider will show a ray rotating around the pole. The “a” value is the angle, \displaystyle \theta , in radians between the ray and the polar axis. On the ray is a segment with a point at its end. This segment’s length is \displaystyle \left| {r\left( \theta \right)} \right|. As you rotate the ray you can see the polar graph drawn. When \displaystyle r(\theta )<0 the segment extend in the opposite direction from the ray.

This demo may be used to introduce or review how polar equation work. An interesting extension is to enter something for the argument of the function that is not an integer muntiple of \displaystyle \theta and extend the domain past \displaystyle 2\pi , for example \displaystyle r=2+4\sin (1.2\theta )

Basic Parametric and Vector Demo


A parametric equation and the vector equation of the same curve differ only in notation. So, this demo works for both. Following the directions in the demo, you may see the graph being drawn using the “a” slider. You may turn on (1) the position vector and its components, (2) add the velocity vector attached to the moving point and “pulling” it to its new position, and (3) the acceleration vector “pulling” the velocity vector.

You may enter any parametric/vector equation. When you do, you will also have to enter its first and second derivative. Follow the directions in the demonstration.

Cycloids and their vectors


This demo shows the path on a rolling wheel called a cycloid. The “a” slider moves the position of the point on the wheel. The point may be on the rim of the wheel (\displaystyle a=r, on the interior of the wheel (\displaystyle a<r), or outside the wheel (\displaystyle a>r  – think the flange on a train wheel). Use the “u” slider to animate the drawing. The velocity and acceleration vectors are shown; they may be turned off. The velocity vector is tangent to the curve (not to the circle) and seems to “pull” the point along the curve. The acceleration vector “pulls” the velocity vector. The equation in this demo should not be changed.

The last two demonstrations give a good idea of how the velocity and acceleration vectors affect the movement of the point.

Hope you find these helpful.



Blog Guide

As I am not doing many new posts these days, I want to call your attention to the “Blog Guide” tab above. This tab will guide you to the information on the blog. It will help sort through the approximately five hundred posts and find those that concern the topic you are interested in.

The “Before Calculus” section discusses things usually taught before calculus.

The “Pedagogy” sections had notes on pacing, teaching, testing, grading, and scoring.

The “Graphing Calculator Use” page contains links to what students should know and be able to do on the AP Calculus Exam with their graphing calculator. There are also links to how to use a graphing calculator to teach some of the topics in the course.

The “AP Exam Review” has links to the ten common type questions on the exams with notes on what students should know about each of them. Good for review and as you teach each topic during the year.

Then there are links to the ten units, different from the type questions, in the current Course and Exam Description for AP Calculus AB and BC.

The “Good Questions” links are to specific questions, mostly from past AP exams, which are discussed in detail. They explore the richness of the question.

“Odds and Ends” has links to, well, odds and ends – other posts you may find interesting and helpful.

Meanwhile: An interesting article “How I Rewired My Brain to Become Fluent in Math”

Today and Tomorrow

Today is this blog’s tenth anniversary!

 My first post was on July 15, 2012. At the time I was working with the Arkansas Advanced Initiative for Math and Science. I was thinking of a series of emails with teaching hints for the calculus teachers I was working with. It occurred to me that a blog format would be more useful to them and to others who stumbled across it. So, that’s how all this all got started.

This is my 492nd post in addition to the 98 pages available from the menu bar. As of this morning, the blog has had, 956,803 visitors and 1,628,857 page views – and counting.

Teaching mathematics is more than just proving the theorems and doing the standard examples.  I certainly have not posted about everything there is to know about calculus – which would be difficult, since I don’t know everything. It was never my intent to write an online calculus book or even cover all the topics in the course description. Textbooks do that well enough. I hoped to provide some insight and ideas to help teachers explain things.

But I seem to have little more to add. I have found little new to write about recently. For the past few years, as you’ve probably noticed, many of my post were lists of links to past posts of actual calculus content.

So, I’ve spent some time this month looking at all my past posts and sorting out the ones with real content from those linking to the content posts. I’ve added a new drop-down menu to the navigation bar at the top of the screen called Blog Guide. Here you will find all the content posts organized in a way that I hope you will find useful. (The “link” posts are not there but are still available if you’ve bookmarked any of them.)

Please take a minute to look at the Blog Guide. I hope its organization will help you find your way around. (The “Search, “Posts by Topics,” and the “archives” on the sidebar will also help.)  

From now on, the blog will be on autopilot, so-to-speak. There will be few new posts. If I get an interesting idea, I will share it, but will not be posting regularly.

Some of my best inspiration comes from readers. So, if you have a calculus topic you would like me to discuss or expand on, please email me here and I’ll see what I can do. (The address is also on the navigation bar under “About.”), Also, I would appreciate you letting me know of any typos or broken links.

If you click on the “Follow” link in the sidebar, you will receive an email whenever a new post appears.

I hope to have helped you at least a little and hope to continue to do so. Thanks for reading and supporting

Enjoy your summer and have a good school year.

Starting the Year

As you get ready to start school, here are some thoughts on the first week in AP Calculus. I looked back recently at some of the “first week of school” advice I offered in the past. Here’s a quick (actually, a bit longer then I planned) summary with some new ideas.

  1. The last time I taught AP Calculus during review time a student asked if there was a list of what’s on the exam. Duh! Why didn’t I think of that? So, I made copies of the list (from the old Acorn Book) and gave it to everyone. I should have done that on Day 1. So, my first suggestion is to make a copy of the “Mathematical Practices” and the “Course at a Glance” from the 2019 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. 
  1. 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 thru pre-calculus when the topics come up during the year. Include short reviews in your lesson plans. For instance, when you study limits you will need to simplify rational expressions – that’s when you review rational expressions. When you look at the graphs of the trigonometric functions, that’s when to review the graphs of the parent functions, a lot of the terminology related to graphs, discontinuities, asymptotes, and even the values of the trigonometric functions of the special angles. Months from now you’ll be looking at inverse functions, that’s when you review inverses.
  1. 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!
  1. If you are not already a member, I suggest you join the AP Calculus Community. We have over 18,000 members all interested in AP Calculus. The community has an active bulletin board where you can ask and answer questions about the courses. Teachers and the College Board also post resources for you to use. College Board official announcements are also posted here. I am the moderator of the community and I hope to see you there!
  1.  Here are some links to places on this blog that you may find helpful:
    1. Pacing– organizing your year.
    2. Check the Resource page from this blog.
    3. Calculator information:
    4. Miscellany: These posts discuss basic ideas that I always hoped students knew about mathematics before starting calculus

Adapting 2021 BC 6 the last in the series on adapting questions from the 2021 exam will appear in two weeks on Auguest 31, 2021

Revised August 16, 2021

Summer … At last!

Summer … At last!

I hope you have all either completed your year or are close to it. Take some time to relax.

I am working on a series of nine summer post which will be begin on Tuesdays starting June 22. Each will look at one of the nine 2021 free-response questions. I will not be presenting their solutions; you can find them online. Rather, I will try to suggest ways that you can adapt the questions for use during the year. This may include ways to slightly change the question, ask additional questions from the same stem, and use the question to explore the topic further and deeper. I hope you’ll find them useful.

Please join me then and enjoy your summer!

2021 Review Notes

About this time of year, I have been posting notes on reviewing and on the ten types of problems that usually appear on the AP Calculus Exams AB and BC. Since the types do not change, I am posting all the links below. They are only slightly revised from last year. You can also find them under “AP Exam Review” on the black navigation bar above.  

Each link provides a list of “What students should know” and links to other post and questions from past exams related to the type under consideration.

Note that the 10 Types are not the same as the 10 Units in the Fall 2020 Course and Exam Description. This is because many of the exam questions have parts from different units.

Here are the links to the various review posts:

When assigning past exams questions for review (and you should assign past exam question), keep in mind that students can find the scoring standards online. Even though the AP program forbids this and makes every effort to prevent them from being posted, they are there. Students can “research” the solution. Keep this in mind when assigning questions from past exams. Here is a suggestion Practice Exams – A Modest Proposal