August and the school year is about to start!

As I did last year, this year my weekly posts will point you to previous posts on topics that will be coming up a week or two later. I try to stay a little ahead of you so you’ll have time to read them and incorporate what you feel is helpful into your plans. I will occasionally write some new posts as ideas come to me. (You could help them come by sending them. Send your questions and suggestions to LinMcMullin2@gmail.com)

Resources

First, here are some suggestions on pacing.

The Course and Exam Description

• The Course and Exam Description  (CED)This is the official course description from the College Board. The individual list of topics that are tested on the exams (the Concept Outline) begins on page 11 and are listed in the Essential Knowledge (EK) column along with its Big Idea (BI), and Learning Outcome (LO) . Also, you will find the Mathematical Practices (MPACs) starting on page 8. These apply to all the topics.

• To help you organize all this see my post on Getting Organized using Trello boards. A board listing all the Essential Knowledge and MPAC items are included.

Exam Questions

AP Calculus teachers should have a collection of the past AP Exams handy. Use them for homework, quizzes, and test through the year. Study them yourself to understand the content and style of the questions. Here are some places to find them:

• Another good reference is Ted Gott’s free-response question index and his MC unsecure Index by topic 1998 to 2018 The indices reference all the released free-response and multiple-choice questions. They are Excel spreadsheets. Each question is referenced to its Key Idea, LO and EK and includes a direct link to the text of the question. Click on the drop-down arrow at the top of each column and choose questions exactly on the EK you want to see. Ted plans to update this after the new multiple-choice questions are released. I will let you know when and where it is available. Thank you again, Ted!

• I have an index of a different sort. It lists the ten Type Problems and which question, multiple-choice and free-response, that are of each type. You can find it here. This will be updated when the 2018 exams become available.

• Past free-response questions that have been released along with commentary, actual student samples, and data can be found at AB FRQ on AP Central and here BC FRQ on AP Central. Be aware that these are available to anyone including your students.

• Multiple-choice questions from actual exams are also available. The 2012 exam in the blue box on the course home pages (see above). This is open to anyone including students. More recent exams can be found at your audit website under “secure document” on the lower left side. This must be kept confidential because teachers use them for practice exams – they may not be posted on-line, on your school website or elsewhere, or even allowed out of your classroom on paper. Unfortunately, some teachers have not obeyed these rules and the exams can be found online by students with very little effort. Be aware that, nevertheless, your students may have access to the secure questions. For my suggestion on how to handle that see A Modest Proposal.

The AP Calculus Community

• Finally, if you are not already a member, I suggest you join the AP Calculus Community. We are fast approaching 17,000 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!

## Have a great year!

PS: Here is a link to some precalculus topics that come up in calculus

# AP Calculus Resources

So, soon time to start another year – both for you and for me. In my over 200 posts to this blog I’ve discussed a lot of calculus topics in hopes that it may help some of my readers teach or learn about calculus. The problem is that I’ve sort of run out of topics and ideas. Nevertheless, I will write more. I would like your ideas, suggestions, and questions about calculus and teaching calculus. You may email me at lnmcmullin@aol.com. Thanks.

My plans for this year are to point you to some of my previous posts. Each week, to stay ahead of you, I will list links to previous posts that you may find helpful in the next week or two.

This post lists links to resources that AP Calculus teachers should have handy for reference during the year.

• The Course and Exam Description  This is the official course description from the College Board. The individual list of topics that are tested on the exams (the Concept Outline) begins on page 11 and are listed in the Essential Knowledge (EK) column along with its Big Idea (BI), and Learning Outcome (LO) . Also, you will find the Mathematical Practices (MPACs) starting on page 8. These apply to all the topics. I have posts on each separately: see Mathematical Practices,  MPAC 1, MPAC 2, MPAC 3, MPAC 4, MPAC 5, and MPAC 6. Also, you will find a list of Instructional Approaches that outlines various ideas for use in your classes – and not just your calculus classes. The book also contains sample free-response and multiple-choice questions that show how the MPACs and Essential Knowledge item are related to each question.

• To help you organize all this see my post on Getting Organized using Trello boards. A board listing all the Essential Knowledge and MPAC items are included.

• Another good reference is MC unsecure Index by topic 1998 to 2018. This lists ALL the free-response questions from 1998 to the present (2016). This is an Excel spreadsheet. Each question is referenced to its Key Idea, LO and EK and includes a direct link to the text of the question. Click on the drop-down arrow at the top of each column and choose questions exactly on the EK you want to see. Ted plans to update this after the new multiple-choice questions are released. I will let you know when and where it is available. Thank you again, Ted!  Updated to include the 2017 FRQs. (August 2, 2017)

• I have an index of a different sort. It lists the ten Type Problems and which question, multiple-choice and free-response, that are of each type. You can find it here. This will be updated when the 2017 exams become available.

• Past free-response questions that have been released along with commentary, actual student samples, and data can be found at AB FRQ on AP Central and here BC FRQ on AP Central. Be aware that these are available to anyone including your students.

• Multiple-choice questions from actual exams are also available. The 2012 exam in the blue box on the course home pages (see above). This is open to anyone including students. More recent exams can be found at your audit website under “secure document” on the lower left side. This must be kept confidential because teachers use them for practice exams – they may not be posted on-line, on your school website or elsewhere, or even allowed out of your classroom on paper. Unfortunately, some teachers have not obeyed these rules and the exams can be found online by students with very little effort. Be aware that your students may have access to them. For my suggestion on how to handle that see A Modest Proposal.

• Finally, if you are not already a member, I suggest you join the AP Calculus Community. A of the end of July, we have 15,210 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!

Have a great year!

# Subtract the Hole from the Whole.

Sometimes I think textbooks are too rigorous. Behind every Riemann sum is a definite integral. So, authors routinely show how to solve an application of integration problem by developing the method starting from the Riemann sum and proceeding to an integral that give the result that is summarized in a “formula.” There is nothing wrong with that except that often the formula is all the students remember and are lost when faced with a similar situation that the formula does not handle. .

The volume of solid figure problems are developed from the idea that if a solid figure has a regular cross-section (that is, when cut perpendicular to a line, each face is similar – in the technical sense – to all the others). They are all squares, or equilateral triangles, or whatever. The last shape considered is usually a “washer”, that is, an annulus or two concentric circles. This is formed by revolving the region between two curves around a line. Authors develop a formula for such volumes: $\displaystyle \pi \int_{a}^{b}{{{\left( R\left( x \right) \right)}^{2}}-{{\left( r\left( x \right) \right)}^{2}}dx}$.

Now there is nothing wrong with that, but I like to give the students their chance to show off. They can usually figure out the answer without Riemann sums. Here is my suggestion. After students have had some practice with circular cross-sections (“Disk” method”) I give them a series of three volumes to find.

Example 1: The curve $f\left( x \right)=\sin \left( \pi x \right)$ on the interval [0, ½] is revolved around the x-axis to form a solid figure. Find the volume of this figure.

Solution: $\displaystyle V=\int_{0}^{1}{\pi {{\left( \sin \left( \pi x \right) \right)}^{2}}dx}=\frac{\pi }{4}$

Example 2: The curve $g\left( x \right)=8{{x}^{3}}$ on the interval [0, ½] is revolved around the x-axis to form a solid figure. Find the volume of this figure.

Solution: $\displaystyle V=\int_{0}^{1/2}{\pi {{\left( 8{{x}^{3}} \right)}^{2}}dx=}\frac{\pi }{14}$

These they find easy. Then, leaving the first two examples in plain view, I give them:

Example 3: The region in the first quadrant between the graphs of $f\left( x \right)=\sin \left( \pi x \right)$ and $g\left( x \right)=8{{x}^{3}}$ is revolved around the x-axis. Find the volume of the resulting figure.

A little thinking and (rarely) a hint and they have it. $\displaystyle V=\frac{\pi }{4}-\frac{\pi }{14}$

What did they do? Easy, they subtracted the hole from the whole. We discuss this and why they think it is correct. We try one or two others. And now they are set to do any “washer” method problem without another formula to memorize.

Extensions:

1. In symbols, when rotation around a horizontal line, if R(x) is the distance from the curve farthest from the line of rotation and r(x) the distance from the closer curve to the line of rotation the result can be summarized in the formula

$\displaystyle V = \int_{a}^{b}{\pi {{\left( R\left( x \right) \right)}^{2}}dx}-\int_{a}^{b}{\pi {{\left( r\left( x \right) \right)}^{2}}dx}$.

Notice, that I like to keep the $\pi$ inside the integral sign so that each integrand looks like the formula for the area of a circle. What the students need to know is to subtract the volume hole from the outside volume. With that                idea and the disk method they can do any volume by washers problem.

2. You should show the students how this equation above can be rearranged into the formula in their books,

$\displaystyle V = \pi \int_{a}^{b}{{{\left( R\left( x \right) \right)}^{2}}-{{\left( r\left( x \right) \right)}^{2}}dx}$.

This is so that they understand that the formulas are the same, and not think you’ve forgotten to tell them something important. It is also a good exercise in working with the notation. (see MPAC 5 – Notational fluency)

3. Next discuss what ${{\left( \pi R\left( x \right) \right)}^{2}}-\pi {{\left( r\left( x \right) \right)}^{2}}$ is the area of and how it relates to this problem. See if the students can understand what the textbook is doing; what shape the book is using.. Discuss the Riemann sum approach. (MPAC 1 Reasoning with definitions and theorems, and MPAC 5 Notational fluency)

4. With the idea of subtracting the “hole” try a problem like this. Example 4: The region in the first quadrant between x-axis and the graphs of $f\left( x \right)=\sqrt{x}$ and $g\left( x \right)=\sqrt{2x-4}$ is revolved around the x-axis. Find the volume of the resulting figure.

Solution:$\displaystyle V=\int_{0}^{4}{\pi {{\left( \sqrt{x} \right)}^{2}}dx}-\int_{2}^{4}{\pi {{\left( \sqrt{2x-4} \right)}^{2}}dx}=4\pi$

(Notice the limits of integration.)

Traditionally, this is done by the method of cylindrical shells, but you don’t need that. You could divide the region into two parts with a vertical line at x = 2 and use disks on the left and washers on the right, but you don’t need to do that either. Just subtract the hole from the whole.

.

# MPAC 6 Communicating

Saving the best, or perhaps the most important, until last, MPAC 6 is the verbal part of the Rule of Four. Problems and real-life situations are “translated” from ideas or words into symbols, equations, graphs, and tables where they are examined and manipulated to find solutions. Once the solutions are found, they must be communicated along with the reasoning involved. The aspects of good mathematical communications are those listed in this MPAC.

MPAC 6: Communicating

Students can:

a. clearly present methods, reasoning, justifications, and conclusions;

b. use accurate and precise language and notation;

c. explain the meaning of expressions, notation, and results in terms of a context (including units);

d. explain the connections among concepts;

e. critically interpret and accurately report information provided by technology; and

f. analyze, evaluate, and compare the reasoning of others.

AP® Calculus AB and AP® Calculus BC Course and Exam Description Effective Fall 2016, The College Board, New York © 2016. Full text is here.

Justifying answers and explaining reasoning in words has long been required on AP calculus exams. The exams have also required students to explain the meaning of expression involving definite integrals and the value of a derivative in the context of the questions.

How/where can you make sure students use these ideas in your classes.

Since to write mathematics well textbook authors do the things listed under this MPAC, but they rarely require students to write about or explain mathematics. They do not show students how to write good explanations of their work and solutions nor, do they provide exercises requiring explanations. Therefore, teachers must do it.

When you get to the end of the year and start working on old AP calculus exams for review you find many questions requiring students to communicate their methods and reasoning, the meanings of their work and results, the connections among different concepts, interpreting what their technology has shown them.

But waiting until the end of the year is way too late. This kind of work should be included in students’ mathematical work from the beginning, before Algebra 1. It can and should be done at every level. By the time they get to calculus, students should not be at all surprised at being asked to explain verbally and in writing what they are doing and why they chose to do it that way.

Find or provide opportunities for students to consider the reasoning of others (MPAC 6f) as well as explain their reasoning to each other. This can be accomplished with group projects, study groups, checking each other’s work, etc. You can also provide templates hits and tips for writing well.The Course and Exam Description  includes an entire section on “Representative Instructional Strategies” (pp. 33 – 37). Among the suggestions are various ways to have students work together and separately on improving their communication skills. The following section (pp. 37 – 38) discusses what a “quality response will include:

• a logical sequence of steps
• an argument that explains why those steps are appropriate, and
• an accurate interpretation of the solution (with units) in the context of the situation”

Provide less than perfect answers for students to critique and improve. (Hint: Use the sample student responses that are released each year along with the exams to show good and not-so-good answers and reasoning.

When AP exam questions are written the writers reference them to the LOs, EKs and MPACs. The released 2016 Practice Exam given out at summer institutes this summer is in the new format and contains very detailed solutions for both the multiple-choice and free-response questions that include these references. (This version is not available online as far as I know.)  None of the multiple-choice question, but all six free-response questions on both AB and BC exam reference MPAC 6 (although see 2014 AB 18 for an idea of how MPAC 6f may be tested).

Here are some previous posts on these subjects:

Writing on the AP Calculus Exams

The Opposite of Negative

What’s a Mean Old Average Anyway?

Others

PLEASE NOTE: I have no control over the advertising that appears on this blog. It is provided by WordPress and I would have to pay a great deal to not have advertising. I do not endorse anything advertised here. I noticed that ads for one of the presidential candidates occasionally appears; I certainly do not endorse him.

# MPAC 5 Notational Fluency

MPAC 5: Building notational fluency

Students can:

a. know and use a variety of notations (e.g., ${f}'\left( x \right),{y}',\frac{dy}{dx}$);

b. connect notation to definitions (e.g., relating the notation for the definite integral to that of the limit of a Riemann sum);

c. connect notation to different representations (graphical, numerical, analytical, and verbal); and

d. assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts.

AP® Calculus AB and AP® Calculus BC Course and Exam Description Effective Fall 2016, The College Board, New York © 2016. Full text is here.

The use of symbols is, not only what everyone thinks of when they think of mathematics, but quite rightly it’s a great tool. Notation has made the abstraction of diverse mathematical concepts possible and revealed the connections between disparate parts of mathematics. Each notation is defined somewhere; the new notations of the calculus are defined during the course (MPAC 5b). Students often do not realize that notation is simply shorthand. Symbols seem to have a magical quality and do things on their own. It is up to the teacher to demystify all this by making the connections listed in this MPAC for the students and making sure students use the notation properly.

How/where can you make sure students use these ideas in your classes.

The variety of notations and often their redundancy are confusing to students and therefore need to be carefully explained and properly used. This does not begin in calculus, but rather from the first days of students’ mathematical life: the plus sign, +, is notation. Even earlier, 1, 2, 3, are notations. We hope that by the time students get to the calculus they have had a lot of experience with notation and that their teachers have insisted on using notation correctly. The fact that there is often more than one notation for the same thing is recognized in MPAC 5a.

Notation often has meaning related to graphs. For instance, a horizontal asymptote at y = 3 is the graphical manifestation of the expression$\underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=3$.

Notation speeds up communicating (MPAC 6) about what students are doing. For example, given the velocity expression of a moving object and asked to find the acceleration at t = 5.432, all student need to write is a(5.432) = v’(5.432) =  their answer. This not only identifies the answer, but also explains (justifies) what they are doing.

Notation sometimes serves as directions on how to do some process. The Product rule, the Quotient rule and the Chain rule all help us remember what to do when finding derivatives.

But student often misuse notation. A common misuse of notation is to string their computations together with equal signs where that is neither appropriate nor true. They will calculate the integral needed to find the average value over [0,8] and get a decimal answer, say 1034, and then write 1034 = 1034/8 is the average value – correct answer, poor notation, a point lost. Another common mistake is to calculate an area by unwittingly subtracting the upper curve from the lower and get an answer, say –10 and then write –10 = 10. This loses one point for the wrong integrand and another point for the lie –10 = 10. Likewise, saying this integral = |-10| is not correct.

Both examples are incorrect use of the equal sign. Probably the best way to avoid this is to do computation vertically, one line at a time and not connect them with the equal signs. In the first case, had they written

•     Correct integral = 1034
•     Average value = 1034/8

They earn full credit. In the second example if they write

•      Integral lower minus upper = –10     <loses one point>
•      Area = 10
•     They not only earn the answer point, but regain (recoup in “reader talk”) the point they lost for the wrong integrand, and earn full credit.

The accurate and precise use of notation is also mentioned in MPAC 6.

When AP exam questions are written the writers reference them to the LOs, EKs and MPACs. The released 2016 Practice Exam given out at summer institutes this summer is in the new format and contains very detailed solutions for both the multiple-choice and free-response questions that include these references. (This version is not available online as far as I know.)  A little more than 1/3 of the multiple-choice and all six free-response questions on both AB and BC exam reference MPAC 5.

Here are some previous posts on these subjects:

A Note on Notation

Definition of the Definite Integral

What is a Solution?

PLEASE NOTE: I have no control over the advertising that appears on this blog. It is provided by WordPress and I would have to pay a great deal to not have advertising. I do not endorse anything advertised here. I noticed that ads for one of the presidential candidates occasionally appears; I certainly do not endorse him.

# MPAC 4: Multiple-representations

We used to call it the “Rule of Four.” Maybe that’s why its MPAC 4.

MPAC 4: Connecting multiple representations

Students can:

a. associate tables, graphs, and symbolic representations of functions;

b. develop concepts using graphical, symbolical, verbal, or numerical representations with and without technology;

c. identify how mathematical characteristics of functions are related in different representations;

d. extract and interpret mathematical content from any presentation of a function (e.g., utilize information from a table of values);

e. construct one representational form from another (e.g., a table from a graph or a graph from given information); and

f. consider multiple representations (graphical, numerical, analytical, and verbal) .of a function to select or construct a useful representation for solving a problem.

AP® Calculus AB and AP® Calculus BC Course and Exam Description Effective Fall 2016, The College Board, New York © 2016. Full text is here.

This is another concept not just of use in the calculus. Students should be using symbols, geometric representations (not just graphs), and numerical ideas along with reading and writing about mathematics from their first days in school.

The symbolic (analytic) aspect of the Rule of Four is perhaps a bit more important in doing mathematics. Things have to be proved analytically. The proper use of symbols in mathematics is the subject of MPAC 5.

The verbal part of the Rule of Four also includes writing and explaining mathematics. This is the subject of MPAC 6.

Technology makes using graphs and table of values very easy. Back in ancient times (that is, in BC – before calculators) when I was in high school getting a graph or a table of values required a lot of work. Now these things are easy and quick when using a calculator; now we can spend our time on what the graphs and numbers mean and what they tell us about the situation we’re investigating.

How/where can you make sure students use these ideas in your classes.

The Rule of Four is definitely not restricted to calculus. Using and relating the parts of the Rule of Four should start way back to the students’ earliest work in mathematics long before Algebra 1. “Graphically” should be expanded to “geometrically;” students should be using drawings and pictures and the like before they learn graphing; and continue to use non-graph representations where appropriate after they learn graphing.

While symbolic or analytic work (working with equations, matrices, etc.) is still where you go when you want to be sure something is true (i.e. to prove things), the others have their place in investigations, in helping to form conjectures, and helping to understanding meaning. By the time they get to the calculus, students should be familiar with looking at functions and other mathematical objects from all four perspectives.

Many problems lend themselves to working with only one or two of the Four. This is natural. While you do not have to force all four aspects into every problem, always consider the others. It is not unusual that one of the other might make things clearer. Students who are required to explain verbally or in writing what they are doing (MPAC 6) will benefit even if that is not strictly required.

When AP exam questions are written the writers reference them to the LOs, EKs and MPACs. The released 2016 Practice Exam given out at summer institutes this summer is in the new format and contains very detailed solutions for both the multiple-choice and free-response questions that include these references. (This version is not available online as far as I know.)  About 1/4 of the multiple-choice and about ½ of the free-response questions on both AB and BC exam reference MPAC 4.

PLEASE NOTE: I have no control over the advertising that appears on this blog. It is provided by WordPress and I would have to pay a great deal to not have advertising. I do not endorse anything advertised here. I noticed that ads for one of the presidential candidates occasionally appears; I certainly do not endorse him.

# MPAC 3 Computing

Continuing our look at the Mathematical Practices today we consider computations. We require students to do computations so that they will learn how to do computations; the answer and the check are just the last steps.

MPAC 3: Implementing algebraic/computational processes

Students can:

a. select appropriate mathematical strategies;

b. sequence algebraic/computational procedures logically;

c. complete algebraic/computational processes correctly;

d. apply technology strategically to solve problems;

e. attend to precision graphically, numerically, analytically, and verbally and specify units of measure; and

f. connect the results of algebraic/computational processes to the question asked.

AP® Calculus AB and AP® Calculus BC Course and Exam Description Effective Fall 2016, The College Board, New York © 2016. Full text is here.

Pretty much all calculus involves computations. This MPAC says that students should be able to plan and carry out the computations necessary to solve problems. This includes selecting the right processes to use and using them correctly. There may be more than one way to do a problem. It includes the use of technology when appropriate as well as the Rule of Four (MPAC 3e). The results should apply to the question asked.

How/where can you make sure students use these ideas in your classes.

Of course you are going to have you students solve problems and investigate mathematical situations, so in some ways this MPAC is “boiler plate.” Students are supposed to learn what to do, in what order to do it, do it correctly, and check or apply their results in the context of the problem.

This applies to the calculus, but starts much earlier. Teachers should be sure that students do this from before day one of Algebra 1. For the teacher it also means checking their work not just for the correct answer, but for the correct thinking and best procedure.

Even many multiple-choice questions involve do a computation. In your classroom exams and quizzes it is a good idea to have students show their work and reasoning on multiple-choice questions. I regularly gave partial credit for good work on multiple-choice questions that required a computation, even if the answer was correct.

CAS calculators and computer programs are great at doing computations, but they still have to be told what to do and in what order to do it. Problems with long or tricky computations are a place to use this technology. For this reason, choosing what to do is, I think, more important than the actual doing it. Still students need to know how to do basic algebra and trigonometry.

CAS calculators can be used to teach basic computation. If a student enters a linear equation and types the operation to solve the equation (such as -4x, or +2) the CAS will perform the operation on both sides of the equation and give the resulting equation. If a student chooses the wrong operation, the CAS does it anyway and presents the result; the student will not see what he or she expected to see and know he or she made a mistake.See the figure in which the fourth line shows a “mistake” followed by a recovery; the last two lines are the check.

Step-by-step solving with a CAS calculator. The fourth line is an intentional mistake. The user not seeing what he expects on the right recovers nicely in the next line. The last two lines are the check.

Aside 1: I once had a student in a pre-algebra course who did division by subtracting the divisor from the dividend until he got down to zero. Then he counted the times he subtracted and presented this as the quotient. After all, division is just repeated subtraction. Correct procedure? Yes. Good way to divide? No. His previous teachers were not checking what he did; they loved his correct answers. Alas, I was unable to break him of the habit, and he was not able to go much farther in mathematics.

Aside 2: When scoring the AP exam, every year we see students finding the area of a region by integrating the difference of the upper function subtracted from the lower function and taking the absolute value when they came up with a negative answer. Correct algorithm? Yes. Good way to do the problem? I think not. (They earn full credit for this, if done correctly.)

Aside 3Speaking of computing, I recently learned that my youngest son, who just turned 31 never learned his multiplication tables! Yet, he never had any trouble and could do multiplication as quickly as anyone. So I asked him how he did it. He explained that he worked off the perfect squares. If he had to multiply seven times eight, he thought: seven squared is 49 plus another 7 is 56. I suspect his teacher never asked him to explain how he multiplied. On the other hand, if I were his teacher would I consider this a good way or would I make him memorize the tables? I don’t know; what would you have done?

When AP exam questions are written the writers reference them to the LOs, EKs and MPACs. The released 2016 Practice Exam given out at summer institutes this summer is in the new format and contains very detailed solutions for both the multiple-choice and free-response questions that include these references. (This version is not available online as far as I know.)  About 2/3 of the multiple-choice and all six free-response questions on both AB and BC exam reference MPAC 3.

Three out of four – could be better.  A photo of a poster in a math schoolroom in Russia, taken on my vacation this summer.

Here is a previous post on this subjects:

While many posts include computations, I do not seem to have any posts on just the idea of doing computations. I offer my euphonious theorem as an example of choosing an unusual computational path through a problem (and leaving the actual computations to the CAS).

PLEASE NOTE: I have no control over the advertising that appears on this blog. It is provided by WordPress and I would have to pay a great deal to not have advertising. I do not endorse anything advertised here. I noticed that ads for one of the presidential candidates occasionally appears; I certainly do not endorse him.

.