Teaching Ap Calculus For The First Time

An write-up detailing activities to usage in your calculus classroom. Although this article does not reflect the 2019-20 updates, it stays a valuable reresource. The Instructional Approaches section of the CED includes many type of research-based instructional tactics aligned to course abilities.

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This is an edited version of a lecture yielded at the AP Calculus Reading Professional Night in June 2001.


Teaching is composed of much more than standing in front of a course and spouting wisdom while our students furiously take notes (or at leastern appear to take notes: personally, I’m convinced that doodling is the task that occurs many often). Thinking about what you desire your students to have the ability to do, devising a strategy for accomplishing this, implementing your strategy, and also then assessing if you also came close is what renders teaching both amazing and also difficult. Students might not realize it, however our objectives go much beyond having actually them memorize the derivative of sin x or learning the formula for finding the volume of a solid of radvancement. Producing students that are important thinkers, self-learners, great communicators, and also persistent employees is simply ordinary hard—specifically when we are contending via the first car, the first significant girlfriend/boyfrifinish, the initially time living amethod from house, the first time doing laundry! Active discovering techniques are one way of getting our students’ attention and also helping them become independent learners. Active discovering is additionally one method of obtaining students to spfinish time doing mathematics—the true trick to effective learning!

My goal is to offer several teaching tactics. Many of these are strategies that I’ve learned from various other civilization and also modified to usage in my own classroom. I’m assuming that some of these techniques may be familiar, and that some of these concepts will be new. I definitely execute not advocate making use of every one of these approaches in one class period, or also in one year! However, I discover that shaking up a course by trying something new currently and also then is refreshing for both my students and me. Hopecompletely you’ll discover one or 2 ideregarding implement, the next time you teach calculus.

In-Class Activities

Matching Games

Find some cheap stick-on nametags and also make "matching pairs." Randomly pass out the nametags when students pertained to class and have them find their "match." For instance, on one nametag write a function (written in blue) and also on a 2nd nametag write its derivative (written in red). Tell the blue tags to discover their derivative and tell the red tags to find their antiderivative. This works well for doing one-step (or few-step) difficulties after a concept has been presented and also typically takes around five minutes.


Create a thematic placemat (equivalent to those you see at restaurants through activities for kids). Include some difficulties, graphs, interpretations, and throw in a joke just for fun. Distribute these before students get to class. Students need to start working on the placemat as soon as they come in. (Tell them it’s to save them entertained until the “meal” is ready.) This commonly takes about five minutes and also works well for presenting principles.

Multiple-Choice Questions

Write a multiple-choice question that deserve to be done conveniently. Ideally, ask a theoretical question soon after a idea is introduced. Have students choose a solution and either ask for a display of hands or have students mark their answer. Then offer them one to 2 minutes to talk about the question through their neighbor. After the discussion, ask students to aget answer the question. Generally, almost every one of the class will certainly have actually the correct answer after the conversation. Follow this up by making certain the students understand also which answer is correct and why. This typically takes about 5 minutes.

For instance, intend we had actually talked about the relationship in between a function, its first derivative, and its second derivative on the previous day. I can desire to use a problem equivalent to AB-3 to examine on their understanding of these principles.


Act out the adhering to script with a partner, being certain to encompass the suggested gestures and also expressions. Before founding, determine that is Alpha and also who is Beta.

Alpha: Hey, Beta, I missed class yesterday. Did we do anything important?Beta: You better not let Dr. Andersen hear you say that. She assumes we do somepoint necessary eextremely day.Alpha: Yeah, yet some days are even more vital that others. So what did we do?Beta: We stupassed away differential equations.Alpha: So what"s a differential equation?Beta: What carry out you think it is?Alpha: Uh, something via an equals sign?Beta: (Nods head encouragingly.) That"s ideal. What else?Alpha: Uh, a differential?Beta: Cshed. You require a derivative.Alpha: So it"s just an equation via a derivative in it. Sounds basic enough. Guess you didn"t carry out anypoint necessary yesterday after all.Beta: I don"t know about basic. Dr. Andersen shelp that, in general, solving differential equations is really hard. She sassist that the majority of of them can not be resolved by anybody, even if they have actually a Ph.D. in math. So what good is acquiring a Ph.D. in math if you can not execute difficult math problems?Alpha: Well, I don"t think it sounds also tough. Here"s our course outline for this day. The first problem says:

Consider y" = x, y(0) = 2

Why is this a differential equation?What is intended by the y(0) = 2?Solve the differential equation.

See, it can not be also bad if she expects us to fix it.Beta: Well, the first question is simple. It"s a differential equation because it has actually an amounts to authorize and also a derivative.Alpha: Hey, I kbrand-new the answer to that one. What"s this second question all about? It looks choose math gibberish to me!Beta: You need to have actually been here yesterday. That"s somepoint dubbed an initial problem.Alpha: You just reput symbol gibberish with word gibberish. It"s still nonfeeling as far as I"m involved. (Shakes head in disgust.)Beta: Well, the solution to a differential equation is a family of functions and...Alpha: (Interrupts Beta.) Hold it. I assumed a solution was a number. You know, find a number, circle it, you"re done via your math homejob-related.Beta: Well, differential equations are different. Your unrecognized is the function whose derivative satisfies the relationship in the equation. Because a derivative tells you around the form of the curve, tbelow isn"t a unique answer. Yesterday we looked at some pretty wild graphs, dubbed direction fields, that demonstrate this. Anyway, an initial problem offers you a allude on your graph so that you have actually a distinctive solution.Alpha: (Holds up his hand to stop Beta from saying anypoint else.) Hold it. That"s enough. Let"s go to question 3. This one looks straightforward. To discover the function, I simply have to integrate. So, I have ∫y" = ∫x. Ummm, that indicates that y = 1/2 x2. See, I said it was easy!Beta: But what are you doing to do with the initial condition? You are expected to have actually y(0) = 2 and for your answer y(0) = 0.

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Alpha: Well, probably the problem is simply simple wrong. You shelp the solution to a differential equation is a family of functions. Seems to me that the solution is a single function—just prefer I believed it would certainly be.Beta: Hey, I understand. (Jumps up in excitement.) You need the +c.Alpha: (Puts head in hands and also groans.) No, no, not the dreaded +c!! I constantly lose points for that on the exams. I believed that was just Dr. Andersen being also picky.Beta: (Shelp via excitement.) No, simply look. You gain y = 1/2 x2. Now you can use y(0) = 2 to settle for c. So we obtain c = _____. (Working together, Alpha and Beta resolve for c.)Alpha: Let"s see if we got it best. I think Dr. Andersen is about prepared to talk around it. (Both Alpha and Beta look forward expectantly.)

Write and Critique

Placed a difficulty on the board and also offer students adequate time to occupational on it. When the majority of are done, have actually them exchange papers through their neighbor. At this allude, they are to correct the various other person’s work-related. This functions finest through two- or three-step problems—something that’s not trivial however likewise not as well challenging. This is equivalent to having students job-related problems at the board or attempt a difficulty throughout course through the included feature of instrumental analysis and review. It might also assist some students realize the value of functioning together. This generally takes 5–10 minutes, depending on the difficulty.2

Group Writing

Placed a trouble on the board and tell students to begin working on it. After around one minute (before they have had time to finish the problem), tell them to switch papers through their neighbor. They must correct anypoint that is wrong and also then keep working. After a second one to 2 minutes, have actually them switch aacquire. Continue having actually them switch periodically till the trouble is complete. Students should turn in both papers through both names (each pair turns in two papers). This works finest with multi-phase problems. Just like Write and also Critique, it emphasizes instrumental evaluation: students are required to think around what they’re doing by having to end up someone else’s work-related. It typically takes around 10 minutes to execute the difficulty and also then discuss it.

Involuntary Discussion

Pass out index cards on the initially day of class and have actually students write down their name and also whatever before various other indevelopment you’d favor to have. Use these cards the remainder of the semester to randomly call on students to answer inquiries. Make certain students realize that you are not grading them on the quality of their answers. When explaining this strategy, I tell students that each of them will offer a brilliant answer and a stupid answer at some point during the semester. I also enable students to "pass" as lengthy as they don’t do so continually. I generally ask 30–40 concerns in a 50-minute course period so everyone is forced to become involved. This strategy offers instant feedback as to just how well the students understand also the principles, and can be provided properly for the majority of of the course period.

For instance, intend we had actually finished talking about how to uncover neighborhood extremes and I wanted to watch how well the class can "put all the information together" by going with an instance similar to AB-5. My part of the discussion could go as follows:Let p(x) = 4x3 + x2 + ax + k. We know that p(0) = 2 and also that p(x) has actually a neighborhood maximum at x = 1. We want to uncover a and also k.

St 1, how many kind of pieces of information perform we mean to need to answer this question? Why?St 2, what does it mean to say that p(0) = 2? What does this mean in regards to the graph of p?St 3, just how can we use the indevelopment that p(0) = 2 to discover the unrecognized coefficients?St 4, perform the calculation suggested by Student 3. What perform you acquire as soon as you substitute 0 for x? St 5, utilizing p(0) = 1 allowed us to uncover that k = 1. Can we usage this exact same piece of information to uncover a? Why not?St 6, how are we going to usage the truth that p(x) has actually a local maximum at x = 1? What do we recognize about regional maximums?St 7, Student 6 argued that regional maximums offer us indevelopment about the derivative. Find the derivative of p(x).St 8, now that we discovered the derivative, what perform we perform next? Why?St 9, Student 8 said that "a regional maximum at x = 1 tells us that the initially derivative of p(x) is zero when x = 1." Student 7 uncovered the derivative of p(x). What execute you gain as soon as you plug in x = 1 and collection it equal to zero?St 10, what is our answer to this problem?St 11, what if the original question was p(x) = 4 x3 + ax2 + x + k. Could we have actually solved this with the very same pieces of information? Why or why not?

Lecture Outlines

Write an outline for what you want to cover in course. This consists of interpretations, principle questions, and troubles. Hand also out the outline in development and also have the students use it as a method of taking notes. It assures that at least the difficulties are replicated dvery own correctly! Students can usage these as a means of organizing the material. This functions particularly well with the Involuntary Discussion strategy. I offer students my outlines a chapter at a time. This strategy is typically supplied for the whole course duration. Below is a sample lecture outline for a area introducing the concept of series.

Questions to be answered throughout class:

What does it expect to say that a collection is convergent?

Find the sum of the geometric series


Exsimple why each of the complying with series are divergent.







Find the amount for each of the following:



(b) 1 - 3/2 + 9/4 -27/8 + ...

A specific sphere has the property that each time it falls from a height h onto a difficult, level surface, it rebounds to a height rh where 0 r H meters.

(a) Assuming that the round proceeds to bounce incertainly, find the complete distance that it travels.

(b) Calculate the complete time that the round travels.


is a convergent series (wright here an10). Explain why
is divergent.

Out-of-Class Activities

Reading Quizzes

Tell the students that you truly intend them to read the textbook. Make certain they understand exactly what you want them to check out and what they are supposed to understand prior to coming to class. For example, you could ask them to write down all of the definitions in the area, offering an instance of each. Or you might ask them to give the one or 2 primary concepts of the area plus a question they have actually after analysis the area. I give electronic analysis quizzes. The students have actually the inquiries ahead of time (as part of the lecture reading outlines) and send me the answers by means of email. Below is an example of the analysis quiz I provided for the initially section on series (the same area I used for the sample lecture outline above).

Questions to be answered before class

Please submit answers to these questions via email by 8:30 of the day the section is spanned in course.

What is the difference in between a sequence and also a series?Find
, what perform we know about

Electronic Discussion Board

An digital discussion board is an reliable means for students to exadjust principles around homework troubles and mathematical ideas exterior of class. The software application that we use at Hope College is Discus (emerged by a chemisattempt professor at Hope). The instance I"ve attached is from a course referred to as Mathematics for Elementary Teachers. Students were in teams of 3 and had actually to alternate doing the 3 assigned jobs. These were:

pose a problemsolve the problemverify the solution.

This software has additionally been used as a way of having actually students ask concerns on homework difficulties. Many professors require student to write-up to the board a offered number of times per semester.

Grading the Exam

After offering an exam, randomly choose students’ documents and copy their answers onto one more copy of the exam. The next day, give out duplicates of the randomly graded exam and ask students to grade it as a homework-related exercise. Encourage them to occupational together. Tell them to provide each component of each problem a score of zero to three, with zero being totally wrong and also 3 being completely ideal. Insist that they give comments for each trouble and also a allude worth. The grade for this assignment is just how accurately they grade the test. Similar to the Write and Critique activity, this pressures students to evaluate various other people"s work-related. This also cuts dvery own on the time and concerns you get once you hand also back the exam, because students have actually believed around the problems after taking the test.

Early Turn-In Policy

Tell students that they have actually the option of turning in homework, computer system labs, team problems, tasks, etc. at an early stage. Generally, "early" indicates at leastern 24 hrs before the assignment is due. When they revolve it in, go over it easily and also make comments on one more sheet of paper. These comments need to be brief, such as "number one is wrong," "display more job-related for the last component of question five," "you overlooked one situation in your answer to the last component." Students have actually the opportunity to make these corrections before turning in the assignment for a grade. I typically have 10–25% of the class take benefit of this policy at leastern as soon as. It motivates them to begin their work early on and also is more than likely the only time that students actually pay attention to my comments on their papers! They learn to evaluate their occupational as they respond to comments. Students likewise appreciate having some manage over the question: "What does my teacher want?" Time-wise, I think I come out also through this policy because this outcomes in better, easier-to-grade papers.


1This task originated from a talk by Don Cronkite; he calls his scripts "Bing and Bong."

2This activity is a slight change of the peer review composing that is done by my colleagues in the English department.