Blog Guide

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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”

The Hindu – Arabic Series

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At first glance, the topic of series seems to be something encountered late in the year of a BC Calculus course. But everyone uses series any time they use numbers, which is to say very often. Let’s look at this particularly important series.  

The way numerals were written way back when was clumsy. If you don’t believe me, try multiplying or dividing with Roman Numerals. Around AD 1200, Leonardo of Pisa introduce the Hindu-Arabic system for writing numbers to Europe. He learned this system during his travels in the Middle East. Leonardo is also known as Fibonacci although he was not given that name until the 1500s. While he is better known for his famous sequence, I think improving the way numbers are written is a much more important contribution to mathematics (even though it would have caught on eventually). This is the system used world-wide today.

A page of Fibonacci’s Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled with Latin numbers and Roman numerals (black) and the value in Hindu-Arabic numerals (red). (Some numerals are slightly different than those in use today.)

Hindu – Arabic notation is a shorthand for a series, sometimes finite, often infinite.  It is a sophisticated idea when thought of in modern terms. This “new” system is a place-value system: the value of each digit depends on its position relative to the decimal point, defined by a sequence. For example,

\displaystyle \begin{array}{l}456.789=4(100)+5(10)+6+7(\tfrac{1}{{10}})+8(\tfrac{1}{{100}})+9(\tfrac{1}{{1000}})\\\end{array}

This notation has advantages over other methods of denoting numbers. The decimal representation of a number is unique (well almost, as we’ll see below), whereas every Rational number may be written as many different fractions. Certainly, makes computation easier. It also makes finding approximations and arranging numbers in order easier than writing them as fractions.

But other things also happen.

The Hindu-Arabic decimal system revealed that all Rational numbers written in this notation are repeating decimals. A repeating decimal is an expression containing a string of one or more digits that repeated forever. For example, 1/3 = 0.333333… with the “3” repeating forever and \displaystyle \frac{{241}}{{55}}=4.38181818... with “18” string repeating forever. (Some numbers repeat zeros forever; they are a special case called terminating decimals.)

The decimal form of a fraction may be found by using the division algorithm. Since only those numbers less than the divisor may appear as “remainders,” eventually one of them will appear again after which the succeeding digits will repeat.

Conversely, any repeating or terminating decimal can be written as a quotient of integers. This example shows the procedure.                                                              

Let \displaystyle n=4.3818181818...

Then \displaystyle 100n=438.18181818...

Subtracting the first from the second

\displaystyle 99n=433.8

\displaystyle n=\frac{{433.8}}{{99}}=\frac{{4338}}{{990}}=\frac{{241}}{{55}}

So, all the Rational numbers can be written as a repeating or terminating decimal, and conversely all repeating or terminating decimals are Rational numbers. Numbers that cannot be written as repeating or terminating decimals are exactly the Irrational numbers.  

An Irrational Number – the Diagonals of a Square

The length of the diagonals of a square is a non-repeating decimal. That is, the length must be expressed as an infinitely long decimal that contains no string of digits that repeats. \displaystyle \sqrt{2} is an Irrational Number and there are a lot of others like it!

By the Pythagorean Theorem the diagonals of a square with sides of one unit have a length denoted by \displaystyle \sqrt{2} – the number whose square is 2. In a previous post, I showed a way, one of several, to find closer and closer decimal approximations to \displaystyle \sqrt{2}. The table below shows the results.


n = decimal places
Ln < \displaystyle \sqrt{2}Gn > \displaystyle \sqrt{2}
012
11.41.5
21.411.42
31.4141.415
41.41421.4143
51.414211.41422
61.4142131.414214
71.41421351.4142136
81.414213561.41423157
91.4142135621.414213563
101.41421356231.4142135624
111.414213562371.41421356238
121.4142135623731.414213562374
131.41421356237301.4142135623731
141.414213562373091.41421356237310
151.4142135623730951.414213562373096
161.41421356237309501.4142135623730951
171.414213562373095041.41421356237309505
   
Each number in list Ln was produced by taking the preceding number and affixing the digits 0, 1, 2, … , 9 to it, squaring that number, and finding the largest whose square was less than 2. The Gn is the next number, the smallest number with a square greater than 2. The numbers in Ln have squares less than 2; the numbers in Gn have squares greater than 2.

The Ln list is a sequence of numbers that has two important properties easily seen from how it was developed: (1) it is non-decreasing – each number is greater than or occasionally equal to the preceding number because each time we append an extra digit we get a greater number, and (2) the list is bounded above – the numbers never exceed 100, or 15, or \displaystyle \pi , or 2, or in fact any number from the Gn list. The smallest number they never exceed is \displaystyle \sqrt{2}. We know this because this is how the list was developed.

The number that’s between the two lists is the number we’re looking for is \displaystyle \sqrt{2}, but we can never find an “exact” decimal representation. The two lists give better and better approximations to \displaystyle \sqrt{2}. They close in on it. But neither gets there.

So, how do we know that number exists?

The Axiom of Completeness

All the decimal numbers, the Rational Numbers, and the Irrational Numbers (and no others), make up a set called the Real Numbers.

If this list above can be continued forever, it will never get to \displaystyle \sqrt{2}. To handle this kind of situation a new rule (called an axiom) was imposed.

The Axiom of Completeness: Every non-decreasing sequence of Real numbers that is bounded above converges to – gets closer and closer to – its least upper bound.

Since this is an axiom, it is not proved; it is just accepted as fact.

The axiom says that even though there is no decimal to represent it, the number nevertheless exists.

Someone made the axiom up. This is very different than observing and naming a property of numbers like the commutative property or the associative property. It doesn’t have to be true, but it seems very reasonable, and no one has ever found a counterexample. [1]

In the example, the least upper bound is \displaystyle \sqrt{2}. How do we know that? Because that’s what we made the sequence to do. Any other method of “finding” \displaystyle \sqrt{2}, and there are many, gives us, not just the same kind of thing, but the exact same list! Creepy, isn’t it?

Thus, \displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\left\{ {{{L}_{n}}} \right\}=\sqrt{2}

Not only are all Irrational Numbers handled the same way, but all repeating decimals are also. They never repeat digits and, however you find their decimal approximations, the same thing happens: you have a non-increasing sequence of numbers that is bounded above and, by the Axiom of Completeness, converges to the fraction.

I mentioned above that the Hindu-Arabic system gives a unique expression for every number. Not quite. Consider 1/3 = 0.3333333 ….  Since three times one-third is one, it must be that 3 times this decimal which is 0.9999999…. = 1.

At first, I disliked decimals because they were not “exact.” I got over that. For the cost of the Axiom of Completeness, we have a system for writing numbers that makes computation easy (if, often, only to a very good approximation). It’s worth the cost.

But does \displaystyle \sqrt{2} exist? Is it really there?

[1] As a corollary to the axiom, there is a theorem that says a non-increasing sequence that is bounded below, such as Gn, converges to its greatest lower bound, again \displaystyle \sqrt{2}.

Quick Notes

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Two quick notes:

First, I’ve added a new page with activities and explorations I have collected and used with students and in my summer institutes. Some are formatted as handouts (a/k/a worksheets) and others are black line masters for game type activities. You can find them under the “Resources” tab on the top line menu (direct link). I will be looking through my files for more, so check back now and then. Hope you find them helpful.

Second: last year I reviewed an iPad app called A Little Calculus. This app demonstrates graphically nearly all the 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. Your students may also use it to explore on their own or with directions from you. The app has recently been updated to allow you to save and recall your own examples. This is a helpful improvement allowing you to prepare things in advance or reuse set-ups in later classes. Also, two new topics have been added – Logistic Growth and the Derivative of Exponential Functions. If you are not familiar with it, take a look.

Lesson Plans

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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

Today and Tomorrow

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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 TeachingCalculus.com.

Enjoy your summer and have a good school year.

Let ‘um Try!

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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.”

Sorry, it’s not about you!

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

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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.

BUT as teachers preparing your students for the exam you really don’t need to be concerned about all these fine points as readers do. Encourage your students to answer the question correctly and show the required work using correct notation. This is shown on the scoring standard for each question (on Stephen’s sample it is in the ruled area directly below the question). Don’t worry about the fine points – what if I say this, instead of that. If your students try to answer and show their work but miss or overlook something, the readers will do their best to follow the student’s work and give her or him the points they have earned.

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.