MPAC 3 Computing

Posted on September 27, 2016 by Lin McMullin

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 3: Speaking 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.

Well, not really. A photo from a schoolroom in Russia, taken on my vacatin this summer.

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).

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MPAC 2 – Connections

Posted on September 20, 2016 by Lin McMullin

-Steve Jobs

Continuing the series on the Mathematical Practices for AP Calculus (MPACs) today we look at MPAC 2.

MPAC 2: Connecting concepts

Students can:

a. relate the concept of a limit to all aspects of calculus;

b. use the connection between concepts (e.g., rate of change and accumulation) or processes (e.g., differentiation and its inverse process, antidifferentiation) to solve problems;

c. connect concepts to their visual representations with and without technology; and

d. identify a common underlying structure in problems involving different contextual situations.

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

While “limit” seems to disappear shortly after the definition of derivative is past and reappears briefly with the definition of the definite integral, in fact all of the calculus depends on limits. Limit seems to be used for other things – continuity, end behavior, asymptotes – but really limit is what makes all of the calculus work and provides the firm foundation for derivatives and integrals and therefore is always in the background of everything calculus. Students need to be made aware of this.

Connecting the concepts in calculus and in previous work in mathematics, seeing the same ideas in different contexts, and using one concept in different ways to solve different type of problems is what makes mathematics in general and the calculus in particular so universal in its application and effectiveness. The ideas in mathematics relate to each other; they are not separate items.

The “Rule of Four” helps students see and understand these connections; technology makes the Rule of Four easy to apply in multiple situations.

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

All the way through the teaching and learning of mathematics these connections exist. Teachers need not only to be aware of them but be sure to point them out to students. Whenever there is an equation, discuss what it means in the context of the problem, see what its graph tells you, and, when a new use comes up, relate it to the previous applications. This is not intended as a way to address different learning styles. The Rule of Four approach is for all students – some will see the idea better on way or the other, but all students will benefit from seeing the connections and the various approaches.

The MPACs overlap with each other. Building notational fluency (MPAC 5), attending to the proper implication of algebraic and computational processes (MPAC 3), connecting multiple representations (The Rule of Four, MPAC 4), proper reasoning (MPAC 1), and communicating the ideas (MPAC 6) all lead to connecting the concepts.

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 40% of the multiple-choice and all six free-response questions on both AB and BC exam reference MPAC 2.

Here are some previous posts om these topics

Limits

Why Limits?
Limits and Local Linearity
Finding Limits
L’Hôpital Rules the Graph Connecting t hew equation and the graph.
Determining the Indeterminate
Fermat’s Penultimate Theorem This theorem is the first step on the way to the Mean Value Theorem and limits makes it so.

Examples of connecting the concepts of graphing functions and linear motion problems

Matching Motion
Motion Problems: Same Thing, Different Context
A Vector’s Derivatives Relating Parametric/Vector derivatives to linear motion derivatives.

MPAC 1 Reasoning

Posted on September 13, 2016 by Lin McMullin

My series on calculator/technology use will continue during the year. Meanwhile, today I am starting a short series about Mathematical Practices for AP Calculus or MPACs as they are called.

Earlier this year I did some work verifying the alignment of several textbooks to the Essential Knowledge (EK), Learning Objectives (LO), and MPACs of the new Course and Exam Description for AP Calculus. The publishers provided a reference (page or exercise number) for each LO and EK in their books and a separate reference for each MPAC. The books I looked at all hit the LOs and EKs pretty well at over 95%. But the MPACs, not so much.

The MPACs are intended to get students thinking and working like mathematicians. The more I look at them, the more I think they can do that.

Each MPAC starts with the statement “Students should ….” It’s easy for authors to do a good job explaining the calculus in the LOs and EKs; that’s what authors do. The writers do what’s listed in the MPACs because they are good mathematical practices and the authors are good mathematicians, but the authors do not often point out what they are doing in this regard.

The MPACs are for students to do. Textbooks need to provide opportunities for students to do the them. This pretty much has to be in the exercises. Some of the exercises provide the opportunity to do some of the things listed in the MPACs, but this is often more accidental than intended.

Little or no opportunity is intentionally provided to learn and practice the MPACs. It is up to the teachers to provide these opportunities.

While they are called Mathematical Practices for AP Calculus, in fact they really apply to all of mathematics. The calculus examples in the MPACs may easily be changed to apply to mathematics teaching and learning earlier in the curriculum. Properly applied they should have an impact on the entire curriculum. If the goal is to help students learn to think and work like mathematicians, then starting in AP Calculus is way too late.

I will use this and the next few post to discuss the MPACs in detail and provide some suggestions as to where and how teachers can help their students to think and work like mathematicians.

MPAC 1: Reasoning with definitions and theorems

Students can:

a. use definitions and theorems to build arguments, to justify conclusions or answers, and to prove results;

b. confirm that hypotheses have been satisfied in order to apply the conclusion of a theorem;

c. apply definitions and theorems in the process of solving a problem;

d. interpret quantifiers in definitions and theorems (e.g., “for all,” “there exists”);

e. develop conjectures based on exploration with technology; and

f. produce examples and counterexamples to clarify understanding of definitions, to investigate whether converses of theorems are true or false, or to test conjectures.

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

While the word logic does not appear here, these six items (with the possible exception of e) are the tools of logic and the basis of mathematical reasoning. The word prove has appeared very rarely on the AP Calculus exams, students have been asked to justify their answers, apply a definition or theorem to a particular function, and show that they know what ideas can be used in a situation and show that they can use them. Conjecturing, producing examples and counterexamples are the basis of mathematical reasoning.

Some suggestions about how and where you can make sure students work with these ideas in your classes.

These items reflect the structure of mathematics. None of the points are specific to the calculus; they can and should be used and developed in all the classes leading up to calculus. Definitions and theorems come into students’ mathematical education before the first year of algebra. The form and structure of axioms, definitions, and theorems, in addition to their meaning, should be made clear to students. So this is something that should start long before calculus and be included every year.

One way you can help students learn how the items in MPAC 1 work is to use True or False (TF) questions; better yet are Always, Sometimes or Never (ASN) questions. These are similar to TF questions except that the students have a middle choice. These questions are an excellent place to hone ones’ skills using the fine points of theorems and definitions. With either TF or ASN questions students should not just answer with a word, but rather be required to explain how they know their answer is correct. They can do this by citing some theorem or definition or producing an example or counterexample. Students can also be asked to discuss, defend, and compare and contrast their answers with other students.

We all know that AP exam questions often require students to “Justify your answer” or “Explain your reasoning.” Here, too, is a good place to practice with the skills of MPAC 1, since justifications and explanations are based on the theorems and definitions

When AP exam questions are written the writers reference them to the LOs, EKs and MPACs. The released 2016 Practice Exam that is in the new format contains very detailed solutions for both the multiple-choice and free-response questions that include these references. About 1/3 of the multiple-choice and all six free-response questions on both AB and BC exam reference MPAC 1.

Here are some previous posts on these subjects:

I have discussed definitions and theorems in previous posts. Here are links to some of them:

Comparing the Graph of a Function and its Derivative

Posted on September 6, 2016 by Lin McMullin

The fourth in the Graphing Calculator / Technology series

Comparing the graph of a function and its derivative is instructive and necessary in beginning calculus. Today I will show you how you can do this first with Desmos a free online graphing program and then on a graphing calculator. Desmos does this a lot better than graphing calculators, because of the easy use of sliders. CAS calculators also have sliders but they are not as easy to use as Desmos.

Let’s get started. Instead of presenting you with a completed Desmos graph, I will show you how to make you own. One of the things I have found over the years is that it takes some mathematical knowledge to make good demonstration graph and that in itself if useful and instructive. Hopefully, you and your students will soon be able to make your own to show exactly what you want.

Open Desmos and sign into your account; if you don’t have one then register – its free and you can keep your results and even share them with others.

In the first entry line on the left, enter the equation of the function whose graph you want to explore. Call it f(x); that is enter f(x) = your function. Later you will be able to change this to other functions and investigate them, without changing anything else.

On the second line enter the symmetric difference quotient as

$\displaystyle s\left( x \right)=\frac{f\left( x+0.001 \right)-f\left( x-0.001 \right)}{2\left( 0.001 \right)}$

Instead of a variable h, as we did in our last post in this series, enter 0.001. This will graph the derivative without having to calculate the derivative. Of course, you could enter the derivative here if your class has learned how to calculate derivatives. If so, you will have to change this line each time you change the function.

In order to closely compare the function and its derivative, on the next line enter the equation of a vertical segment from a point on the function (a, f(a)) to a point on the derivative (a, s(a)). Desmos does not have a segment operation, but here is how you graph a segment. In general, a segment from (a, b) to (c, d) is entered as the parametric/vector function

$\left( a\cdot t+c\cdot \left( 1-t \right),b\cdot t+d\cdot \left( 1-t \right) \right),\ 0\le t\le 1$

The a, b, c, and d may be numbers or functions. Since our segment is vertical the first coordinate will have a = c and will reduce to a. Here’s what to enter on the third line:

$\left( a,f\left( a \right)\cdot t+s\left( a \right)\cdot \left( 1-t \right) \right)$

(Notice that there is no x in this expression; t is the variable. Also, the f(a) and s(a) may be interchanged.)

When you push enter, you will be prompted to add a slider for a: click to add the slider. A line will appear under the expression which will allow you to set the domain for t: click the endpoints and enter 0 on the left and 1 on the right, if necessary.

That’s it. You’re done. Use the slider to move around the graphs.

Using the graphs

Discuss with your class, or better yet divide them into groups and let them discuss, what they see. Since at this point they are probably new to this provide some hints such as “What happens on the graph of f when s is 0?” or “What is true on s when f is increasing?” or “What happens to the function at the extreme values of the derivative?” Prompt the students to look for increasing and decreasing, concavity, points of inflection, and extreme values. All the usual stuff. Work from the function to the derivative and from the derivative to the function.

Have your students formulate their results as (tentative) theorems. You actually want them to make some mistakes here, so you can help them improve their thinking and wording. For example, one result might be: If the function is increasing, then the derivative is positive. By changing the first function to an example like f(x) = x³ or f(x) = x + sin (x). Help them see that non-negative might be a better choice.

You might try giving different groups different functions and let them compare and contrast their results.

This is very much in line with MPACs 1, 2, 4, and 6.

You can do the same kind of thing with graphing calculators. That is, you can graph the function and its derivative or a difference quotient. The difference is that graphing calculators do not have sliders.

Extra feature: Desmos will graph a point if you enter the coordinates just like you write them: (a, b). The coordinates may be numbers or functions or a combination of both. Try adding two points to your graph one at each the end of the segment between the graphs that will move with the same slider.

f(x) = x + 2sin(x) and its derivative.

Teaching Calculus

The pleasure lies not in discovering the truth, but in searching for it.

Monthly Archives: September 2016

MPAC 3 Computing

MPAC 2 – Connections

MPAC 1 Reasoning

Comparing the Graph of a Function and its Derivative

Using the graphs

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Using the graphs

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