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?


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The Opposite of Negative

Next year, for the first time in 15 years, I am going to be teaching high school full-time. While I have enjoyed writing and working primarily with teachers for the last 15 years, I’m looking forward to “going back to the classroom” as they say. It looks like I’ll be teaching BC calculus and Algebra 1 – two of my favorite classes. I’m very positive about that.

With that in mind I have been thinking of some of the things I want to be sure I get right in the Algebra 1 classes to get the kids off to a good start. So a few of my blogs in the coming year may be on Algebra 1 topics with the view of having students do things right from the start and not having to relearn things when they get to calculus.

So here is the first thing I want to be sure to work on: the m-dash also known as the minus sign.

According to Wikipedia:

The minus sign () has three main uses in mathematics:

  1. The subtraction operator: A binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.
  2. Directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5.
  3. unary operator that acts as an instruction to replace the operand by its opposite. For example, if x is 3, then −x is −3, but if x is −3, then −x is 3. Similarly, −(−2) is equal to 2.

Using the same symbol understandably can confuse beginning math students. I am not going to invent new symbols so I will just have to be careful with what I say and let the kids say. And I have to say it right , if I expect them to.

When used between two numbers or two expressions with variables the symbol means subtraction. That’s pretty easy to spot and understand in context.  But when used alone in front of something the minus sign means different things.

The m-dash may always be read “opposite.” So “–a” is read “the opposite of a” and not “negative a.”  Likewise, –5 is read “the opposite of five.”

There is only one instance where the m-dash may be read “negative.” When it is used in front of a number it indicates a negative number so      “–5” is also correctly read “negative five.” This is the only time the m-dash should be read “negative.”  Things like “–a” should always be read “the opposite of a” and never read “negative a.”

There was a time when the folks who write math books tried to make the distinction by using a slightly raised dash to indicate negative number so negative 3 was written  “3.” This has carried over into calculators where the key marked “(–)” is used for “negative” and “opposite.” and is printed on the screen as a shorter and slightly raised dash. The subtraction key is only used for subtraction.

Oh, if it were only that simple. What do you do with –(–5)? Not really a problem the “opposite of the opposite of 5” and the “opposite of negative 5” are both 5.

I’ll know I’ve succeeded when everyone can get 100% on this little True-False quiz:

  1. The opposite of a number is a negative.
  2. x < 0
  3. x > 0
  4. | x | = x
  5. |– x |= x

Answers are in my next post.

Revised 10-27-2018

A Note on Notation

For quite a while I’ve been writing sin(x), ln(x) and the like with parentheses instead of the usual sin or ln x .

The main reason is that I want to emphasize that sin(x), ln(x), etc. are the same level and type of notation as f(x). The only difference is that sin(x) and ln(x) always represent the same function, while things like f(x) represent different functions from problem to problem. I hope this makes things just a little clearer to the students.

I also favor using (sin(x))² instead of sin²(x), again to make clearer just what is getting squared. Notation can be inconsistent: I don’t think I’ve ever seen ln²(x) or even ²(x).  So this helps in that regard as well.

Of course, when entering functions in calculators or computers you almost always must use the “extra” parentheses in both cases. (Except for the new Casio PRIZM which will understand sin x and ln x, but not sin²(x).)

Now we can use that spot in the notation exclusively for inverse functions, as in {{\sin }^{-1}}\left( x \right) and {{f}^{-1}}\left( x \right). Maybe that will lessen the confusion there.

Another possible inconsistency is trying to write sin′(x)  for the derivative as you do with {f}'\left( x \right)Although, if I saw it I would understand it. (LaTex won’t even parse  sin′(x).)