A Generic Approach to Arclength in Calculus

Earlier this week, a teacher posted in the College Board’s AP Calculus Community a request for an explanation of computing the arclength of a curve without relying on formulas.

The following video is my proposed answer to that question.  In it, I derive the fundamental arclength relationship before computing the length of y=x^2 from x=0 to x=3 four different ways:

  • As a function of x,
  • As a function of y,
  • Parametrically, and
  • As a polar function.

In summary, the length of any differentiable curve can be thought of as


where a and b are the bounds of the curve, the square root is just the local linearity application of the Pythagorean Theorem, and the integral sums the infinitesimal roots over the length of the curve.

To determine the length of any differentiable curve, factor out the form of the differential that matches the independent variable of the curve’s definition.

Tell a Friend

I’ve been in several conversations over these first couple weeks of school with colleagues in our lower and middle schools about what students need to do to convince others they understand an idea.

On our first pre-assessments, some teachers noted that many students showed good computation skills, but struggled when they had to explain relationships.  Frankly, I’m never surprised by revelations that students find explanations more difficult than formulas and computations.  That’s tough for learners of all ages.  But, in my opinion, it’s also the most important part about developing a way to communicate mathematically.

In the other direction, I frequently hear students complain that they just don’t know what to write and that teachers seem to arbitrarily ask for “more explanation”, but they just can’t figure out what that means.


Just like writing in humanities classes, a math learner needs to seriously consider his “audience”.  Who’s going to read your solution?  I think too many write for a classroom teacher, expecting him or her to fill in any potential logical gaps.

Instead, I tell my students that I expect all of their explanations to be understandable by every classmate. In short,

Don’t write your answer to me; write it to a friend who’s been absent for a couple days.

If a random classmate who’s been out a couple days can get it just based on your written work, they you’re good.

Marilyn vos Savant Conditional Probability Follow Up

In the Marilyn vos Savant problem I posted yesterday, I focused on the subtle shift from simple to conditional probability the writer of the question appeared to miss.  Two of my students took a different approach.

The majority of my students, typical of AP Statistics students’ tendencies very early in the course, tried to use a “wall of words” to explain away the discrepancy rather than providing quantitative evidence.  But two fully embraced the probabilities and developed the following probability tree to incorporate all of the given probabilities.  Each branch shows the probability of a short or long straw given the present state of the system.  Notice that it includes both of the apparently confounding 1/3 and 1/2 probabilities.


The uncontested probability of the first person is 1/4.

The probability of the second person is then (3/4)(1/3) = 1/4, exactly as expected.  The probabilities of the 3rd and 4th people can be similarly computed to arrive at the same 1/4 final result.

My students argued essentially that the writer was correct in saying the probability of the second person having the short straw was 1/3 in the instant after it was revealed that the first person didn’t have the straw, but that they had forgotten to incorporate the probability of arriving in that state.  When you use all of the information, the probability of each person receiving the short straw remains at 1/4, just as expected.

Marilyn vos Savant and Conditional Probability

The following question appeared in the “Ask Marilyn” column in the August 16, 2015 issue of Parade magazine.  The writer seems stuck between two probabilities.


(Click here for a cleaned-up online version if you don’t like the newspaper look.)

I just pitched this question to my statistics class (we start the year with a probability unit).  I thought some of you might like it for your classes, too.

I asked them to do two things.  1) Answer the writer’s question, AND 2) Use precise probability terminology to identify the source of the writer’s conundrum.  Can you answer both before reading further?


Very briefly, the writer is correct in both situations.  If each of the four people draws a random straw, there is absolutely a 1 in 4 chance of each drawing the straw.  Think about shuffling the straws and “dealing” one to each person much like shuffling a deck of cards and dealing out all of the cards.  Any given straw or card is equally likely to land in any player’s hand.

Now let the first person look at his or her straw.  It is either short or not.  The author is then correct at claiming the probability of others holding the straw is now 0 (if the first person found the short straw) or 1/3 (if the first person did not).  And this is precisely the source of the writer’s conundrum.  She’s actually asking two different questions but thinks she’s asking only one.

The 1/4 result is from a pure, simple probability scenario.  There are four possible equally-likely locations for the short straw.

The 0 and 1/3 results happen only after the first (or any other) person looks at his or her straw.  At that point, the problem shifts from simple probability to conditional probability.  After observing a straw, the question shifts to determining the probability that one of the remaining people has the short straw GIVEN that you know the result of one person’s draw.

So, the writer was correct in all of her claims; she just didn’t realize she was asking two fundamentally different questions.  That’s a pretty excusable lapse, in my opinion.  Slips into conditional probability are often missed.

Perhaps the most famous of these misses is the solution to the Monty Hall scenario that vos Savant famously posited years ago.  What I particularly love about this is the number of very-well-educated mathematicians who missed the conditional and wrote flaming retorts to vos Savant brandishing their PhDs and ultimately found themselves publicly supporting errant conclusions.  You can read the original question, errant responses, and vos Savant’s very clear explanation here.


Probability is subtle and catches all of us at some point.  Even so, the careful thinking required to dissect and answer subtle probability questions is arguably one of the best exercises of logical reasoning around.


As a completely different connection, I think this is very much like Heisenberg’s Uncertainty Principle.  Until the first straw is observed, the short straw really could (does?) exist in all hands simultaneously.  Observing the system (looking at one person’s straw) permanently changes the state of the system, bifurcating forever the system into one of two potential future states:  the short straw is found in the first hand or is it not.

CORRECTION (3 hours after posting):

I knew I was likely to overstate or misname something in my final connection.  Thanks to Mike Lawler (@mikeandallie) for a quick correction via Twitter.  I should have called this quantum superposition and not the uncertainty principle.  Thanks so much, Mike.

SBG and AP Statistics Update

I’ve continued to work on my Standards for AP Statistics and after a few conversations with colleagues and finding this pdf of AP Statistics Standards, I’ve winnowed down and revised my Standards to the point I’m comfortable using them this year.

Following is the much shorter document I’m using in my classes this year.  They address the AP Statistics core content as well as the additional ideas, connections, etc. I hope my students learn this year.  As always, I welcome all feedback, and I hope someone else finds these guides helpful.

SBG and Statistics

I’ve been following Standards-Based Grading (SBG) for several years now after first being introduced to the concept by colleague John Burk (@occam98).  Thanks, John!

I finally made the dive into SBG with my Summer School Algebra 2 class this past June & July, and I’ve fully committed to an SBG pilot for my AP Statistics classes this year.

I found writing standards for Algebra 2 this summer relatively straightforward.  I’ve taught that content for decades now and know precisely what I want my students to understand.  I needed some practice writing standards and got better as the summer class progressed.  Over time, I’ve read several teachers’ versions of standards for various courses.  But writing standards for my statistics class prove MUCH more challenging.  In the end, I found myself guided by three major philosophies.

  1. The elegance and challenge of well designed Enduring Understandings from the Understanding by Design (UbD) work of Jay McTighe the late Grant Wiggins helped me craft many of my standards as targets for student learning that didn’t necessarily reveal everything all at once.
  2. The power of writing student-centered “I can …” statements that I learned through my colleague Jill Gough (@jgough) has become very important in my classroom design.  I’ve become much more focused on what I want my students (“learners” in Jill’s parlance) to be able to accomplish and less about what I’m trying to deliver.  This recentering of my teaching awareness has been good for my continuing professional development and was a prime motivator in writing these Standards.
  3. I struggled throughout the creation of my first AP Statistics standards document to find a balance between too few very broad high-level conceptual claims and a far-too-granular long list of skill minutiae.  I wanted more than a narrow checklist of tiny skills and less than overloaded individual standards that are difficult for students to satisfy.  I want a challenging, but reachable bar.

So, following is my first attempt at Standards for my AP Statistics class, and I’ll be using them this year.  In sharing this, I have two hopes:

  • Maybe some teacher out there might find some use in my Standards.
  • More importantly, I’d LOVE some feedback from anyone on this work.  It feels much too long to me, but I wonder if it is really too much or too little.  Have I left something out?

At some point, all work needs a public airing to improve.  That time for me is now.  Thank you in advance on behalf of my students for any feedback.

Chemistry, CAS, and Balancing Equations

Here’ s a cool application of linear equations I first encountered about 20 years ago working with chemistry colleague Penney Sconzo at my former school in Atlanta, GA.  Many students struggle early in their first chemistry classes with balancing equations.  Thinking about these as generalized systems of linear equations gives a universal approach to balancing chemical equations, including ionic equations.

This idea makes a brilliant connection if you teach algebra 2 students concurrently enrolled in chemistry, or vice versa.


Consider burning ethanol.  The chemical combination of ethanol and oxygen, creating carbon dioxide and water:

C_2H_6O+3O_2 \longrightarrow 2CO_2+3H_2O     (1)

But what if you didn’t know that 1 molecule of ethanol combined with 3 molecules of oxygen gas to create 2 molecules of carbon dioxide and 3 molecules of water?  This specific set coefficients (or multiples of the set) exist for this reaction because of the Law of Conservation of Matter.  While elements may rearrange in a chemical reaction, they do not become something else.  So how do you determine the unknown coefficients of a generic chemical reaction?

Using the ethanol example, assume you started with

wC_2H_6O+xO_2 \longrightarrow yCO_2+zH_2O     (2)

for some unknown values of w, x, y, and z.  Conservation of Matter guarantees that the amount of carbon, hydrogen, and oxygen are the same before and after the reaction.  Tallying the amount of each element on each side of the equation gives three linear equations:

Carbon:  2w=y
Hydrogen:  6w=2z
Oxygen:  w+2x=2y+z

where the coefficients come from the subscripts within the compound notations.  As one example, the carbon subscript in ethanol ( C_2H_6O ) is 2, indicating two carbon atoms in each ethanol molecule.  There must have been 2w carbon atoms in the w ethanol molecules.

This system of 3 equations in 4 variables won’t have a unique solution, but let’s see what my Nspire CAS says.  (NOTE:  On the TI-Nspire, you can solve for any one of the four variables.  Because the presence of more variables than equations makes the solution non-unique, some results may appear cleaner than others.  For me, w was more complicated than z, so I chose to use the z solution.)


All three equations have y in the numerator and denominators of 2.  The presence of the y indicates the expected non-unique solution.  But it also gives me the freedom to select any convenient value of y I want to use.  I’ll pick y=2 to simplify the fractions.  Plugging in gives me values for the other coefficients.


Substituting these into (2) above gives the original equation (1).


Traditionally, chemists write these equations with the lowest possible natural number coefficients, but thinking of them as systems of linear equations makes another reality obvious.  If 1 molecule of ethanol combines with 3 molecules of hydrogen gas to make 2 molecules of carbon dioxide and 3 molecules of water, surely 10 molecule of ethanol combines with 30 molecules of hydrogen gas to make 20 molecules of carbon dioxide and 30 molecules of water (the result of substituting y=20 instead of the y=2 used above).

You could even let y=1 to get z=\frac{3}{2}, w=\frac{1}{2}, and x=\frac{3}{2}.  Shifting units, this could mean a half-mole of ethanol and 1.5 moles of hydrogen make a mole of carbon dioxide and 1.5 moles of water.  The point is, the ratios are constant.  A good lesson.


Now let’s try a harder one to balance:  Reacting carbon monoxide and hydrogen gas to create octane and water.

wCO + xH_2 \longrightarrow y C_8 H_{18} + z H_2 O

Setting up equations for each element gives

Carbon:  w=8y
Oxygen:  w=z
Hydrogen:  2x=18y+2z

I could simplify the hydrogen equation, but that’s not required.  Solving this system of equations gives


Nice.  No fractions this time.  Using y=1 gives w=8, x=17, and z=8, or

8CO + 17H_2 \longrightarrow C_8 H_{18} + 8H_2 O



Now let’s balance an ionic equation with unknown coefficients a, b, c, d, e, and f:

a Ba^{2+} + b OH^- + c H^- + d PO_4^{3-} \longrightarrow eH_2O + fBa_3(PO_4)_2

In addition to writing equations for barium, oxygen, hydrogen, and phosphorus, Conservation of Charge allows me to write one more equation to reflect the balancing of charge in the reaction.

Barium:  a = 3f
Oxygen:  b +4d = e+8f
Hydrogen:  b+c=2e
Phosphorus:  d=2f
CHARGE (+/-):  2a-b-c-3d=0

Solving the system gives


Now that’s a curious result.  I’ll deal with the zeros in a moment.  Letting d=2 gives f=1 and a=3, indicating that 3 molecules of ionic barium combine with 2 molecules of ionic phosphate to create a single uncharged molecule of barium phosphate precipitate.

The zeros here indicate the presence of “spectator ions”.  Basically, the hydroxide and hydrogen ions on the left are in equal measure to the liquid water molecule on the right.  Since they are in equal measure, one solution is

3Ba^{2+}+6OH^- +6H^-+2PO_4^{3-} \longrightarrow 6H_2O + Ba_3(PO_4)_2


You still need to understand chemistry and algebra to interpret the results, but combining algebra (and especially a CAS) makes it much easier to balance chemical equations and ionic chemical equations, particularly those with non-trivial solutions not easily found by inspection.

The minor connection between science (chemistry) and math (algebra) is nice.

As many others have noted, CAS enables you to keep your mind on the problem while avoiding getting lost in the algebra.