# Lesson Plans

A half-dozen decades ago (really!), when I first started out teaching, we were given a large grade book that included pages for lesson plans. For each week of the year there were two facing pages ruled into thirty two-inch squares.

We were expected to write our lesson plans in the squares. A lesson plan consisted of something like “Product and Chain rule” or “Factor perfect square trinomials” or even just “Section 4.7.” Also, you were expected to include the homework assignment. We had to have plans written for at least two weeks in advance. Principals could collect these (although they rarely did) and check up on you.

Lesson planning changed over the years. Even though no one ever checked up on us, I soon started including more in my lesson plans. The suggested structure came to include “behavioral objectives:” brief statements of what the student should be able to do once the lesson was taught. Not a bad idea. Schools started to require the teacher to write these on the board at the beginning of the class, so students would know what they were expected to learn to do. An even better idea.

As time went on and lecturing got a bad name, you were supposed to include activities (other than copying down what you wrote on the board) in your lesson. Also, a good idea.

All this came to mind when I was asked to look at a website that offered FREE lesson plans for AP Calculus AB.

The website is Calc-medic.com. There are 150 daily lesson plans closely following the AB Calculus Course and Exam Description. The lessons are free and available to AP Calculus AB teachers. All you need to do is register. (BC lessons are planned, but since all AB topics are also BC topics, the plans will help BC teachers as well.)

If nothing more, they are a good pacing guide. BUT there is a lot more.

Before you continue, I suggest you read Tips for Lesson Planning from the Calc-Medic Blog. It discusses lesson planning, and I am sure it will be helpful whether you follow their lesson or write your own.

They call their approach “Experience First, Formalize Later” (EFFL). Each of the Calc-Medic EFFL lesson plans is organized like this:

Learning Objectives: A statement of what the lesson will teach.

Success Criteria: One or more succinct first-person statements of what students should be able to do: “I can use …”, “I can determine …”, “I can reason …”, “I can distinguish ….”

Quick Lesson Plan: Each lesson consists of four segments: (1) an activity – 15 minutes, (2) debrief [the activity] – 10 minutes, (3) Important Ideas – 10 minutes, and (4) check your understanding – 20 minutes. More about this shortly.

A brief Overview of the topic.

Teaching Tips – items you should be sure to mention with hints.

Exam Insights – notes on how the topic may appear on the exams and reference to specific AP exam questions.

Student Misconceptions – a discussion of things that may confuse students or that they may overlook.

## The Activity.

Each lesson has a handout in PDF or DOCX (so you can adapt it) and an annotated Answer Key to the activity. This is the heart of the lesson plan.

The entire lesson is in the activity handout. It begins with a set of questions that will lead the students to the topic of the lesson. These are often close to AP format and have questions based on analytic, graphic, tabular and/or a written stem. Regardless of the way the question is presented, student writing is usually included.

The activity includes a box for student notes (summarized in the answer key).

Practice questions are in the “check your understanding” part of the lesson. Teachers can use the annotated answer sheet to help decide what to present to the students and help them make their notes. To help the teacher, answers are in blue, and annotations are in red.

The lessons do not include any homework assignments. Nor are the lessons linked to any textbook. This allows you to adapt them to your textbook and situation. The authors do assign homework. They explain their philosophy in “How Do We Assign Homework?” from the blog.

There is the old generic lesson plan: (1) Tell them what you’re going to tell them, (2) Tell them, and (3) Tell them what you’ve told them. This works for classes that are primarily lectures. The EFFL lessons at Calc-Medic are more of a discovery approach. The “Activity” leads students up to the new concept(s) presented. This is then firmed up in the “Debrief” and “Important Ideas” parts of the plan and practiced in “Check Your Understanding.”

There are occasional suggestions of where to find off-site information related to the lesson, such as College Board Curriculum Modules and this blog. Links to this information are not provided; it would be helpful if they were.

Tests and Quizzes are not included. However, when the lesson calls for a test or quiz there are detailed suggestions on how to write and grade the assessment. These are generic, but nonetheless useful. There are suggestions of what to include when writing the assessment (e.g., calculator problems), what question topics to include (specific to the topics assessed), grading tips, and reflections.

The last twenty lessons are a 4-week day-by-day review for the AP Exam. Some, but by no means all, of the review lessons are linked to the Calc-medic “Review Course.” Everything mentioned above is free; the full “Review Course” is available for a per student fee.

Also worth your time is the Calc-Medic Blog with posts on topics related to AP Calculus and teaching AP Calculus. There are posts on pedagogy, the AP Exams, original videos by the authors, slide decks, and discussion of individual topics in more detail. These provide helpful insights for the teacher. It would help if these were linked to and from the lesson(s) they discuss, and if they had “tags.”

Mathmedic.com, companion website. contains similar lesson plans for Algebra 1, Geometry, Algebra 2, and Precalculus. The 180 Days of Precalculus lesson plans are not aligned with the upcoming AP Precalculus course, but lessons for this course are planned in time for the 2023 – 2024 school year. BC lesson plans are also in the works.

The sites are the work of Sarah Stecher and Barb Montgomery, teachers at East Kenwood High School in Kenwood, Michigan. They are both experienced AP Calculus Teachers. They have done a fabulous job with the lessons and their Calc-Medic Website. Both new and experienced teachers will find the lesson plans and blog helpful.

I cannot recommend Calc-Medic more highly.

ALSO:Last year I reviewed an iPad app called A Little Calculus. This app demonstrates graphically all the main concepts of AB and BC Calculus. It is quite easy to use and a quick way to prepare and present good visual examples for your class. The app has recently been updated to allow you to save and recall your own examples. Several new topics have been added. If you are not familiar with it, take a look.

P.S. Hope you like the Blog’s new look!

Updated August 15, 18, 22, 2022

# Let ‘um Try!

Last week, I received an email from a mother whose concern really ticked me off.

The mother of a student entering eleventh grade this fall wrote because her son wanted to take AP Calculus next fall. He will not be allowed to take the course because he missed the cut-off grade in his precalculus course by less than two percentage points.

That’s wrong!

The College Board, under prerequisites in the current calculus Course and Exam Description, states, “Before studying calculus, all students should complete the equivalent of four years of secondary mathematics designed for college-bound students.” Her son has done that. There is nothing about achieving a certain arbitrary average.

A percentage grade does not tell you anything useful. (I’ve discussed this before. See here and the opening paragraphs here.)

How can you pass a student in precalculus and then turn around and tell them they aren’t ready for calculus? If they are not ready, fail them in precalculus.

It is certainly reasonable to council a student with an average or below average grade. You can, and probably should, sit down with them and their parents and explain that they may find the AP course difficult, and to do well they will have to commit spending more time and effort than they may be used to. Offer them extra help – it’s you job! If their grade was a D or D– you can be a little more insistent that they think it over carefully. But to flat out deny them the opportunity is just wrong.

Passing Rate.

Often teachers and administrators are concerned about their passing rate in AP courses. I once attended a session at an NCTM meeting where a teacher explained how he achieved a great passing rate each year in AP Calculus. It is easy to do. He explained that he carefully weeded out those students who were not likely to do really well. Once in the course, if they were struggling, he counseled them not to take the exam. (Why waste your money?) So, if passing rate is your concern, that’s how to do it.

He ended with, “I don’t want those students in my denominator.”

Let’s say you carefully select the students who take the course and later the exam. You have 10 students who take the exam, and they all get a qualifying score (3, 4, or 5). Well, great you have a 100% pass rate.

Now let’s say you let another 10 students into the course. Twenty kids take the exam and 15 get qualifying scores. Even better. An additional five students have earned qualifying scores. That’s what counts! You’ve done a better job! And the five students who did not qualify will benefit from having taken a college-level course and will be better prepared for math in college.

Your pass rate has dropped to 75%, but you’ve helped more students – and that’s what it’s all about.

When I was working for the National Math and Science Initiative (NMSI), part of my job was to vet schools who wanted to join our program. NMSI insisted that there be no artificial barriers or cut-off points for admittance into all AP courses. Students who had passed the prerequisite courses and wanted to take an AP course had to be admitted. And it worked: the schools had more students qualify every year than the year before.

If all your students are earning 3, 4, or 5, you are being too selective.

I gave the woman what advice I could. I hope it helps. I wish I could have done more for her and her son.

# To dx or not dx

As exam time nears, teachers become concerned about exactly what to give credit for and what not to give credit for when grading their students’ work on past AP free-response questions.

Former Chief Reader Stephen Davis recently posted a note on the grading of a fictitious exam question showing how 2 points might have been awarded on a L’Hospital’s Rule question.  The note is interesting because it shows the details that exam leaders consider when deciding what to accept and what not; it shows the details that readers must keep in mind while grading. This type of detail with examples is given to the readers in writing for each part of every question. With hundreds of thousands of exams each year, this level of detail is necessary for fairness and consistency in scoring.

Why show your students the minimum they can get away with? How does that help them? Do your students a favor: score the review problems more stringently than the readers. If their answer is not quite right, take off some credit and help them learn how to do better. It will help them in the long run.

# Sequences and Series (Type 10)

AP Questions Type 10: Sequences and Series (BC Only)

The last BC question on the exams usually concerns sequences and series. The question may ask students to write a Taylor or Maclaurin series and to answer questions about it and its interval of convergence, or about a related series found by differentiating or integrating. The topics may appear in other free-response questions and in multiple-choice questions. Questions about the convergence of sequences may appear as multiple-choice questions. With about 8 multiple-choice questions and a full free-response question this is one of the major topics on the BC exams.

Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section, students may be asked to say if a sequence or series converges or which of several series converge.

The Ratio test is used most often to determine the radius of convergence and the other tests to determine the exact interval of convergence by checking the convergence at the end points. Click here for a convergence test chart students should be familiar with; this list is also on the resource page.

Students should be familiar with and able to write a few terms and the general term of a Taylor or Maclaurin series. They may do this by finding the derivatives and constructing the coefficients from them, or they may produce the series by manipulating a known or given series. They may do this by substituting into a series, differentiating it, or integrating it.

The general form of a Taylor series is $\displaystyle \sum\limits_{{n=0}}^{\infty }{{\frac{{{{f}^{{\left( n \right)}}}\left( a \right)}}{{n!}}{{{\left( {x-a} \right)}}^{n}}}}$; if a = 0, the series is called a Maclaurin series.

What Students Should be Able to Do

• Use the various convergence tests to determine if a series converges. The test to be used is rarely given so students need to know when to use each of the common tests. For a summary of the tests click: Convergence test chart.  and the posts “What Convergence Test Should I use?” Part 1 and Part 2. In 2022 BC 6 (a) students were asked to state the condition (hypotheses) of the convergence test they were asked to use.
• Understand absolute and conditional convergence. If the series of the absolute values of the terms of a series converges, then the original series is said to be absolutely convergent (or converges absolutely). If a series is absolutely convergent, then it is convergent. If the series of absolute values diverges, then the original series may or may not converge; if it converges it is said to be conditionally convergent.
• Write the terms of a Taylor or Maclaurin series by calculating the derivatives and constructing the coefficients of each term.
• Distinguish between the Taylor series for a function and the function. DO NOT say that the Taylor polynomial is equal to the function (this will lose a point); say it is approximately equal.
• Determine a specific coefficient without writing all the previous coefficients.
• Write a series by substituting into a known series, by differentiating or integrating a known series, or by some other algebraic manipulation of a series.
• Know (from memory) the Maclaurin series for sin(x), cos(x), ex and $\displaystyle \frac{1}{{1-x}}$and be able to find other series by substituting into one of these.
• Find the radius and interval of convergence. This is usually done by using the Ratio test to find the radius and then checking the endpoints. for a geometric series, the interval of convergences is the open interval $\displaystyle -1 where r is the common ration of the series.
• Be familiar with geometric series, its radius of convergence, and be able to find the number to which it converges, $\displaystyle {{S}_{\infty }}=\frac{{{{a}_{1}}}}{{1-r}}$. Re-writing a rational expression as the sum of a geometric series and then writing the series has appeared on the exam.
• Be familiar with the harmonic and alternating harmonic series. These are often useful series for comparison.
• Use a few terms of a series to approximate the value of the function at a point in the interval of convergence.
• Determine the error bound for a convergent series (Alternating Series Error Bound or Lagrange error bound). See my posts on Error Bounds and the Lagrange Highway
• Use the coefficients (the derivatives) to determine information about the function (e.g., extreme values).

This list is quite long, but only a few of these items can be asked in any given year. The series question on the free-response section is usually quite straightforward. Topics and convergence tests may appear on the multiple-choice section. As I have suggested before, look at and work as many past exam questions to get an idea of what is asked, how it is a sked, and the difficulty of the questions. Click on Power Series in the “Posts by Topic” list on the right side of the screen to see previous posts on Power Series or any other topic you are interested in.

Free-response questions:

• 2004 BC 6 (An alternate approach, not tried by anyone, is to start with $\displaystyle \sin \left( {5x+\tfrac{\pi }{4}} \right)=\sin \left( {5x} \right)\cos \left( {\tfrac{\pi }{4}} \right)+\cos \left( {5x} \right)\sin \left( {\tfrac{\pi }{4}} \right)$). See Good Question 16
• 2011 BC 6 (Lagrange error bound)
• 2016 BC 6
• 2017 BC 6
• 2019 BC 6
• 2021 BC 5 (a)
• 2021 BC 6 – note that in (a) students were required to state the conditions of the convergence test they were asked to use.
• 2022 BC 6 – Ratio test, interval of conversion with endpoint analysis, Alternating series error bound, series for derivative, geometric series.

Multiple-choice questions from non-secure exams:

• 2008 BC 4, 12, 16, 20, 23, 79, 82, 84
• 2012 BC 5, 9, 13, 17, 22, 27, 79, 90

These questions come from Unit 10 of the CED.

Revised March 12, 2021, April 12, 16, and May 14, 2022

# Polar Equation Questions (Type 9)

### AP Questions Type 9: Polar Equations (BC Only)

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned only with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose curves, limaçons, etc.). The graphs are usually given in the stem of the problem; students are expected to be able to determine which is which if more than one is given. Students should know how to graph polar curves on their calculator, and the simplest by hand. Intersection(s) of two graphs may be given or easy to find.

What students should know how to do:

• Calculate the coordinates of a point on the graph,
• Find the intersection of two graphs (e.g. to use as limits of integration).
• Find the area enclosed by a graph or graphs: $\displaystyle A=\frac{1}{2}\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{{{{\left( {r\left( \theta \right)} \right)}}^{2}}d\theta }}$
• Use the formulas $\displaystyle x\left( \theta \right)=r\left( \theta \right)\cos \left( \theta \right)\text{ and }y\left( \theta \right)=r\left( \theta \right)\sin \left( \theta \right)$  to convert from polar to parametric form,
• Calculate $\displaystyle \frac{{dy}}{{d\theta }}$ and $\displaystyle \frac{{dx}}{{d\theta }}$ (Hint: use the product rule on the equations in the previous bullet).
• Discuss the motion of a particle moving on the graph by discussing the meaning of $\displaystyle \frac{{dr}}{{d\theta }}$ (motion towards or away from the pole), $\displaystyle \frac{{dr}}{{d\theta }}$ (motion in the vertical direction), and/or $\displaystyle \frac{{dx}}{{d\theta }}$ (motion in the horizontal direction).
• Find the slope at a point on the graph, $\displaystyle \frac{{dx}}{{dx}}=\frac{{dy/d\theta }}{{dx/d\theta }}$

When this topic appears on the free-response section of the exam there is no Parametric/vector motion question and vice versa. When not on the free-response section there are one or more multiple-choice questions on polar equations.

This question typically covers topics from Unit 9 of the CED.

Free-response questions:

• 2013 BC 2
• 2014 BC 2
• 2017 BC 2
• 2018 BC 5
• 2019 AB 2

Multiple-choice questions from non-secure exams:

• 2008 BC 26
• 2012 BC 26, 91

Other posts on Polar Equations

Polar Basics

Polar Equations for AP Calculus

Extreme Polar Conditions

Polar Equations (Review 2018)

Revised March 12, 2021, April 8, 2022

# Parametric and Vector Equations (Type 8)

### AP Questions Type 8: Parametric and Vector Equations (BC Only)

The parametric/vector equation questions only concern motion in a plane. Other topics, such as dot product and cross product, are not tested.

In the plane, the position of a moving object as a function of time, t, can be specified by a pair of parametric equations $\displaystyle x=x\left( t \right)\text{ and }y=y\left( t \right)$ or the equivalent vector $\displaystyle \left\langle {x\left( t \right),y\left( t \right)} \right\rangle$. The path is the curve traced by the parametric equations or the tips of the position vector. .

The velocity of the movement in the x- and y-direction is given by the vector $\displaystyle \left\langle {{x}'\left( t \right),{y}'\left( t \right)} \right\rangle$. The vector sum of the components gives the direction of motion. Attached to the tip of the position vector this vector is tangent to the path pointing in the direction of motion.

The length of this vector is the speed of the moving object. Speed = $\displaystyle \sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}$. (Notice that this is the same as the speed of a particle moving on the number line with one less parameter: On the number line speed $\displaystyle =\left| {v\left( t \right)} \right|=\sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}}}$.)

The acceleration is given by the vector $\displaystyle \left\langle {{x}''\left( t \right),{y}''\left( t \right)} \right\rangle$.

What students should know how to do:

• Vectors may be written using parentheses, ( ), or pointed brackets, $\displaystyle \left\langle {} \right\rangle$, or even $\displaystyle \vec{i},\vec{j}$ form. The pointed brackets seem to be the most popular right now, but all common notations are allowed and will be recognized by readers.
• Find the speed at time t: Speed = $\displaystyle \sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}$.
• Use the definite integral for arc length to find the distance traveled $\displaystyle \int_{a}^{b}{{\sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}}}$. Notice that this is the integral of the speed (rate times time = distance).
• The slope of the path is $\displaystyle \frac{{dy}}{{dx}}=\frac{{{y}'\left( t \right)}}{{{x}'\left( t \right)}}$. See this post for more on finding the first and second derivatives with respect to x.
• Determine when the particle is moving left or right,
• Determine when the particle is moving up or down,
• Find the extreme position (farthest left, right, up, down, or distance from the origin).
• Given the position find the velocity by differentiating.
• Given the velocity, find the acceleration by differentiating.
• Given the acceleration and the velocity at some point find the velocity by integrating.
• Given the velocity and the position at some point find the position by integrating. These are just initial value differential equation problems (IVP).
• Dot product and cross product are not tested on the BC exam, nor are other aspects.

When this topic appears on the free-response section of the exam there is no polar equation free-response question and vice versa. When not on the free-response section there are one or more multiple-choice questions on parametric equations.

Free-response questions:

• 2012 BC 2
• 2016 BC 2
• 2021 BC 2
• 2022 BC2 – slope of tangent line, speed, position, total distance traveled

Multiple-choice questions from non-secure exams

• 2003 BC 4, 7, 17, 84
• 2008 BC 1, 5, 28
• 2012 BC 2

This question typically covers topics from Unit 9 of the CED.

Revised March 12, 2021, April 5, and May 14, 2022

# Other Problems (Type 7)

### AP Questions Type 7: Other topics

Any topic in the Course and Exam Description may be the subject of a free-response or multiple-choice question. The topics discussed here are not asked often enough to be classified as a type of their own. The topics listed here have been the subject of full free-response questions or major parts of them. Other topics occasionally asked are mentioned in the question list at the end of the post.

Implicitly defined relations and implicit differentiation

These questions may ask students to find the first and/or second derivative of an implicitly defined relation. Often the derivative is given, and students are required to show that it is correct. (This is because without the correct derivative the rest of the question cannot be done.) The follow-up is to answer questions about the function such as finding an extreme value, second derivative test, or find where the tangent is horizontal or vertical.

What students should know how to do

• Know how to find the first derivative of an implicit relation using the product rule, quotient rule, chain rule, etc.
• Know how to find the second derivative, including substituting for the first derivative.
• Know how to evaluate the first and second derivative by substituting both coordinates of a given point. (Note: If all that is needed is the numerical value of the derivative then the substitution is often easier done before solving for dy/dx or d2y/dx2, and as usual the arithmetic need not be done.)
• Analyze the derivative to determine where the relation has horizontal and/or vertical tangents.
• Write and work with lines tangent to the relation.
• Find extreme values. It may also be necessary to show that the point where the derivative is zero is actually on the graph and to justify the answer.

Simpler questions about implicit differentiation may appear on the multiple-choice sections of the exam.

Example:

Good Question 17

2004 AB 4

2016 BC 4

2012 AB 27 (implicit differentiation), Multiple-choice

2022 AB 5 (a) Implicit differentiation,

BC classes see Implicit differentiation of parametric equations, and A Vector’s Derivative

Related Rates

Derivatives are rates and when more than one variable is changing over time the relationships among the rates can be found by differentiating with respect to time. The time variable may not appear in the equations. These questions appear occasionally on the free-response sections; if not there, then a simpler version may appear in the multiple-choice sections. In the free-response sections they may be an entire problem, but more often appear as one or two parts of a longer question.

What students should know how to do

• Set up and solve related rate problems.
• Be familiar with the standard type of related rate situations, but also be able to adapt to different contexts.
• Know how to differentiate with respect to time. That is, find dy/dt even if there is no time variable in the given equations using any of the differentiation techniques.
• Interpret the answer in the context of the problem.
• Unit analysis.

Shorter questions on this concept also appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

For previous posts on related rates see Related Rate Problems I and Related Rate Problems II.

Examples

2014 AB4/BC4,

2016 AB5/BC5

2019 AB 4 Related Rate

2019 AB 6

2022 AB2 (d), AB4/BC4 (d) Good example that requires using product and evaluation of an expression that include dr/dt and dh/dt.

Good Question 9

Family of Functions

A “family of functions” is defined by an equation with a parameter (sort of an extra variable). Changing the parameter gives a different but similar curve. Questions should be answered in general, that is, in terms of the parameter not a specific value of the parameter. These questions appeared on some exams long ago, may be making a comeback.

Examples:

1995 BC 5

1996 AB4/BC4

Good Question 5: 1998 AB2/BC2

2019 BC 5

Other Topics

Free response questions (many of the BC questions are suitable for AB)

• Finding derivatives using the chain rule, the quotient rule, etc. from tables of values: 2016 AB 6 and 2015 AB 6
• L’Hospital’s Rule 2016 BC 4, 2019 AB 3 (Don’t be fooled), 2019 AB 4(c)
• Continuity and piecewise defined functions: 2012 AB 4, 2011 AB 6 and 2014 BC 5
• Arc length (BC Topic) 2014 BC 5
• Partial fractions (BC Topic) 2015 BC 5
• Improper integrals (BC topic): 2017 BC 5, 2022 BC5 (c)

Multiple-choice questions from non-secure exams:

• 2012 AB 27 (implicit differentiation), 77 (IVT), 88 (related rate)
• 2012 BC 4 (Curve length), 7 (Implicit differentiation), 11 (continuity/differentiability), 12 (Implicit differentiation), 77 (dominance), 82 (average value), 85 (related rate) , 92 (compositions)

These questions may come from any of the Units in the CED.

Revised March 12, 2021, April 1, and May 14, 2022