Tag Archives: AP

Recentering a Normal Curve with CAS

Sometimes, knowing how to ask a question in a different way using appropriate tools can dramatically simplify a solution.  For context, I’ll use an AP Statistics question from the last decade about a fictitious railway.

THE QUESTION:

After two set-up questions, students were asked to compute how long to delay one train’s departure to create a very small chance of delay while waiting for a second train to arrive.  I’ll share an abbreviated version of the suggested solution before giving what I think is a much more elegant approach using the full power of CAS technology.

BACKGROUND:

Initially, students were told that X was the normally distributed time Train B took to travel to city C, and Y was the normally distributed time Train D took to travel to C.  The first question asked for the distribution of Y-X if the mean and standard deviation of X are respectively 170 and 20, and the mean and standard deviation of Y are 200 and 10, respectively.  Knowing how to transform normally distributed variables quickly gives that Y-X is normally distributed with mean 30 and standard deviation $\sqrt{500}$.

Due to damage to a part of the railroad, if Train B arrived at C before Train D, B would have to wait for D to clear the tracks before proceeding.  In part 2, you had to find the probability that B would wait for D.  Translating from English to math, if B arrives before D, then $X \le Y$.  So the probability of Train B waiting on Train D is equivalent to $P(0 \le Y-X)$.  Using the distribution information in part 1 and a statistics command on my Nspire, this probability is

FIX THE DELAY:

Under the given conditions, there’s about a 91.0% chance that Train B will have to wait at C for Train D to clear the tracks.  Clearly, that’s not a good railway management situation, setting up the final question.  Paraphrasing,

How long should Train B be delayed so that its probability of being delayed is only 0.01?

FORMAL PROPOSED SOLUTION:

A delay in Train B says the mean arrival time of Train D, Y, will remain unchanged at 200, while the mean of arrival time of Train B, X, is increased by some unknown amount.  Call that new mean of X, $\hat{X}=170+delay$.  That makes the new mean of the difference in arrival times

$Y - \hat{X} = 200-(170+delay) = 30-delay$

As this is just a translation, the distribution of $Y - \hat{X}$ is congruent to the distribution of $Y-X$, but recentered.  The standard deviation of both curves is $\sqrt{500}$.  You want to find the value of delay so that $P \left(0 \le Y - \hat{X} \right) = 0.01$.  That’s equivalent to knowing the location on the standard normal distribution where the area to the right is 0.01, or equivalently, the area to the left is 0.99.  One way that can be determined is with an inverse normal command.

The proposed solution used z-scores to suggest finding the value of delay by solving

$\displaystyle \frac{0-(30-delay)}{\sqrt{500}} = 2.32635$

A little algebra gives $delay=82.0187$, so the railway should delay Train B just a hair over 82 minutes.

CAS-ENHANCED ALTERNATIVE SOLUTION:

From part 2, the initial conditions suggest Train B has a 91.0% chance of delay, and part 3 asks for the amount of recentering required to change that probability to 0.01.  Rephrasing this as a CAS command (using TI-Nspire syntax), that’s equivalent to solving

Notice that this is precisely the command used in part 2, re-expressed as an equation with a variable adjustment to the mean.  And since I’m using a CAS, I recognize the left side of this equation as a function of delay, making it something that can easily be “solved”.

Notice that I got exactly the same solution without the algebraic manipulation of a rational expression.

My big point here is not that use of a CAS simplifies the algebra (that wasn’t that hard in the first place), but rather that DEEP KNOWLEDGE of the mathematical situation allows one to rephrase a question in a way that enables incredibly efficient use of the technology.  CAS aren’t replacements for basic algebra skills, they are enhancements for mathematical thinking.

I DON”T HAVE CAS IN MY CLASSROOM.  NOW WHAT????

The CAS solve command is certainly nice, but many teachers and students don’t yet have CAS access, even though it is 100% legal for the PSAT, SAT, and all AP math exams.   But that’s OK.  If you recognize the normCdf command as a function, you can essentially use a graph menu to accomplish the same end.

Too often, I think teachers and students think of normCdf and invNorm commands as nothing more than glorified “lookup commands”–essentially nothing more than electronic versions of the probability tables they replaced.  But, when one of the parameters is missing, replacing it with X makes it graphable.  In fact, whenever you have an equation that is difficult (or impossible to solve), graph both sides and find the intersection, just like a solution to a system of equations.  Using this strategy, graphing $y=normCdf(0,\infty,30-X,\sqrt{500})$ and $y=0.01$ and finding the intersection gives the required solution.

CONCLUSION

Whether you can access a CAS or not, think more deeply about what questions ask and find creative alternatives to symbolic manipulations.

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.

Tangent Perspectives

I assigned AP Calculus BC 1975 problem #7 to my class a couple weeks ago.  I got a 100% legitimate answer I didn’t expect from a student, so I thought I’d share.  It’s what can happen when you encourage students to follow their instincts.

Paraphrasing, the students first had to find an equation of a line through the origin tangent to the graph of $y=ln(x)$.  Most had no problems concluding that this was $\displaystyle y=\frac{x}{e}$.

The next part asked if the tangent was above or below $y=ln(x)$.  In class, we had discussed why the position of tangent lines was dependent on the underlying function’s concavity, so I fully expected successful solutions to end up at $\displaystyle y''=\frac{-1}{x^2}$ which is negative for all $x\neq 0$ therefore making the original curve concave down and the tangent line above.  Most successful solutions did this, but one was different.

Paraphrasing M’s work, he concluded that if the tangent line was entirely on one side, then $\displaystyle g(x)=\frac{x}{e}-ln(x)$ must have an extremum.  From there, $\displaystyle g'(x)=\frac{1}{e}-\frac{1}{x}=0$ confirms the tangency point at $x=e$ from earlier, but this time as a critical point on g.  From here, he concluded that $\displaystyle g''(x)=\frac{1}{x^2}>0$ for all $x\neq 0$ making his critical point a global minimum.  From the construction of g, the tangent line then had to be above $y=ln(x)$.

Admittedly, M’s algebra work took a bit longer, but what impressed me was his completely different visualization of the problem.  I’m betting he didn’t remember the down-concavity-means-tangent-above factoid from class, so he had to invent his own approach.  And he did this by turning a concavity problem into an optimization problem.  Nice.

The Problem with the Test

Following is a snip from a 12.19.2011 post by James K. to the AP Calculus EDG .  It was a response to a teacher’s frustration with her district increasing the number of students in her AP Calculus course and then evaluating her performance in part on the scores her students receive AP exam.  Sorry for the length of the block quote, but it’s worth absorbing.

I have some separate thoughts about tests and teacher evaluation, which I’m hoping to carve out time for sometime soon, but I’m eager to develop the conversation around the following comments:

[citing another post] … I do get students with weaker and weaker math backgrounds.  How weak? I have student who can’t evaluate sin(pi/3), and a student [who] can’t understand it’s the same to multiply a number by 1/6 and to divide it by 6. How did they pass precalculus? …. It feels nice to be able to say “I’m always willing to teach those willing to learn”, but how do you teach a class with huge gap in students’ capability? Teach fast, and you get lots of complaints. Teach slower, and the more capable students suffer. They (the more capable) are the ones that pay the price when we say “I’m always willing to …” [end internal citation]

What’s at the heart of this discussion is differentiating instruction: that is, developing and implementing activities (which can include lecture, as just one type of activity) which provide rich, effective learning for a wide variety of students.  This requires acknowledging not only that students come to the class with diverse background knowledge and conceptual frames, but also that they are likely to leave the class with different knowledges and conceptual frames– in short, serving [intellectually] diverse students … means valuing significant growth at least as much as valuing specific pre-determined objectives.

This is, I believe, the greatest and most tragic failure of mathematics instruction over the past 4000 years: the teaching of mathematics almost invariably has been designed and implemented around an assumption that the cognitive development of mathematical ability is linear in nature, with the overwhelming result that math courses are substantially about distinguishing successful students (those who are able to reproduce a specific set of mathematical procedures and algorithms in novel contexts) from unsuccessful students (who are unable to do so) … as opposed to being about the development of better mathematical thinking in all of the students in the course.

That’s gotten a bit better, just in the past 100 years or so, but the culture is still overwhelmingly focused on determining who can “do the math” (and, implicitly, who cannot).  I recognize that, particularly within the context of an AP course (where the desired outcomes of the course are richly and rigorously defined by an external examination), it’s hard to embrace the notion that it’s okay for students to come away from a lesson, a unit, and the entire course as a whole with different degrees of knowledge and understanding, and even with different ways of *approaching* their own understanding.  That said, I think it’s the work that we ought to be doing:  Teaching is about fostering growth.  It’s certainly easier to do with a roughly uniform group– but I don’t let my students get away with ignoring the hard stuff, and I don’t let myself get away with it, either.  That means we ought to be teaching at both (really, all) levels, simultaneously: give the weaker students what THEY need, while also giving the more capable students what THEY’RE ready for, all at once.  Not easy, as I say, and especially hard to do through lecture (which is why that’s just one of the types of activity we should be using…)

I certainly haven’t achieved all I want to do with respect to differentiating instruction in my classes.  I’ve also been part of numerous discussions like the one James advances above.  His argument for reaching all students where thy are with what they need is, in my opinion, the great calling of all teachers.

What struck me as unique about James’ post was his focus on interpreting the outcome of an AP Course.  I’ve never clearly thought about it in this way, but the standardization of a single international test for all students with only five possible outcomes (scores ranging 1 to 5) does drive one to believe that all students (if they would just try) will exit the course with the same curricular and comprehension results.  I’ve worked for years (with varying success) to meet my students in their areas of need and have tried to promote their unique perspectives.  I hope all of my students (no matter what their mathematical background) say my classes push them to think in new ways and encourage them to apply what they have learned in ways unique to their own perspectives.  What I hadn’t fully considered was the pervasive way external tests can leverage teachers away from that ideal.

So, I recommit myself to

the notion that it’s okay for students to come away from a lesson, a unit, and the entire course as a whole with different degrees of knowledge and understanding, and even with different ways of *approaching* their own understanding,

especially when teaching courses with external final assessments.