Calculus with compositions–an interesting juxtaposition– the hardest differentiation technique is the chain rule, while the easiest integration technique (other than ooh!  I see a derivative I know!) is simple substitution.

Theorems of Calculus – EVT, IVT, MVT:  mostly a repeat of what we did in October, though we got into a cool discussion. f(2) = 1; f(5) = -2; f(13) = 6, for f twice differentiable.  Thought 1:  Does f have an absolute maximum on [2,13]?  Does f have an absolute maximum on (2,13)?  Does f have an absolute minimum on [2,13]?  How about (2,13)?  I like the thought path that leads you to conclude that based on the closed interval having endpoints higher than a known point on the open interval, there has to be a non-endpoint minimum on the closed, therefore on the open.  Nice.

Second thought here:  1.  y = cuberoot(x).  Can’t apply MVT, but the conclusion of the MVT holds.  Piecewise f(x) = (1) sqrt(1-(x-1)^2), 0≤x≤2; (2) -sqrt(1-(x+1)^2), -2≤x≤0.  Conclusion of MVT holds here too.  Maybe a measure of a graph where you can draw all tangent lines?

Cognitive leaps in mathematics:  Quantities, Place Value, Operations, Symbols, Expressions involving variables, Find the numbers that make equations true; Functions – all expressions that have a relationship; calculus – the rate of change of output with respect to input.  differential equations – equations whose solutions are families of functions.

Jeopardy – working with cool problems in a fast paced format.  Easy to edit file for creating calculus jeopardy games.

Exploring some weird functions:  f(x)=.2x+x^2sin(1/x); f'(0)=.2, however, no interval around 0 is increasing.



FTC Ramblings

Over the last couple of days, I’ve thought a lot more about the FTC than I have in a while.  For those of you not entirely immersed in the wonderful world of calculus, the FTC is the Fundamental Theorem of Calculus.  It beautifully and usefully ties together the worlds of differential and integral calculus, and provides a way for us to handle a huge cross-section of interesting problems.

And it’s not like I don’t think about the FTC–I teach Calculus for Euclid’s sake–I just haven’t given it much more thought than as a tool.  I guess that’s more a comment on life in general–you learn something at a basic level (perhaps as a concrete application), then come back to it again (perhaps in a more abstract or symbolic manner), and finally begin to see how something is put together.  The problem is, when something like the FTC is internalized in a teacher’s mind, after high school calculus, college calculus, graduate school calculus, and workshop calculus, it’s easy to forget that there’s more to discover, and harder to remember how confusing something is when first you encounter it.

Discovery/Rediscovery 1:  Re-thinking point-slope form of a line.  (Reflection # 1)

What if you take something as simple as the FTC:

Well, latex isn’t working in wordpress right now, so I’ll do some fun type-y stuff. int(f'(x)dx,a,b) = f(b)-f(a), and rewrite it as a starting point and a sum: f(b) = f(a) + int(f'(x)dx,a,b). You end up looking at something kind of profound. (1) we’ve arrived at a place where we’re saying that a function value is equal to an initial value, plus the accumulated changes between the two x-values.  This has ramifications in terms of stats, where everything is linearized, and everything is written as a starting point (y-intercept) plus the accumulated change (slope*x); and in algebra, where we can revisit the idea of point-slope form of a line y-y1=m(x-x1) as a starting point y1, added to the accumulated change m(x – x1).  You end up with something I’ve never really had a strong affinity for, though I’ve seen some students write point-slope lines as y = y1 + m(x-x1).  I like that this ties to the FTC.  I like more that this ties beautifully into Taylor Polynomials–a super-hard topic for students to wade into, made easier by the fact that they have been doing linear approximations with Taylor Polynomials since kindergarten (that’s when you learn to write equations in point-slope form, right?).  Anything that can give someone a leg up in life (or calculus), I’ll take.  

Discovery/Rediscovery 2:  The Other Part of the FTC (Journal Part 2)

I figured it might be nice to combine my journal entries, since I found myself writing thematically about the fundamental theorem of calculus again.  This will be shorter–getting students to not only compute, but understand the process of applying the FTC (the one showing derivatives and antiderivatives as inverses) when there’s a chain rule involved.  There were two instances I’d like to share here, in hopes that I remember to discuss both later this year.  

1.  d/dx(int(f(x)dx,a,g(x)

This is the method that I worked out with my students this year–certainly not what I did in school (think follow the pattern), and not what I’ve seen most math students do (follow the pattern).  

Actually perform the symbolic integration, and you end up with something like d/dx(F(g(x))-F(a)); when you take the derivative, the chain rule for the first function is easy to see, f(g(x))g'(x), and the second value (a constant) goes to zero.  This is nice way for students to see (1) where the chain rule part comes from, and (2) why the lower bound doesn’t affect the FTC-based result.

2.  d/dx(int(f(x)dx,a,g(x)

Let A(x) = int (f(x)dx,a,x); A'(x) = f(x); A(g(x)) = int (f(x)dx,a,g(x)), A'(g(x))g'(x) = f(g(x))g'(x).

I like the algebraic reasoning here, and can reach another chunk of students, for whom algebra comes plainly, and some of the calculus is a bit overwhelming. 

MMSI Day 2

A collection of gathered thoughts and activities from MMSI (Massachusetts Math and Science Initiative) APSI (AP Summer Institute) AP Calculus AB Experienced workshop.

We revisited 2009 AB5/BC5, a table problem I worked last fall with Mike, which was enlightening again–I found it cool how much more comfortable in predicting the expected pieces required by AP Graders in running through the open response–difference quotients written out for slope (MVT); products written out for Riemann sums et cetera.  Having presented this packet to a group of AP students last spring, I think refocusing on the problems gave me more insight into ways to approach them next year.

The Infamous Hot Wire Problems – 2005 AB3  Lowest AP score on a free response (to 2005).  One thing new here:  sometimes points are assigned separately for units being correct throughout a problem or in pieces of the problem.

Then we moved on to the all time lowest scoring problem on AB:  2007 AB3!  I liked the focus on jumping between forms of the data in tables.  I didn’t like the inverse function part–which is fair enough, I just don’t ever remember how to do inverse derivatives from tabular data, so I create a inverse composite (g-inverse(g(x))=x) take the derivative of both sides and reason from there.

Nice discussion of FTC composition based functions–derivatives of inverses based on A(x) as the area function


A collection of gathered thoughts and activities from MMSI (Massachusetts Math and Science Initiative) APSI (AP Summer Institute) AP Calculus AB Experienced workshop.

First activity–a variation on graph matching.  Using name tags, students match derivative graphs with function graphs.  I liked the physicality of the activity–I generally enjoy matching activities, and particularly liked the moving around twist.

Common student mistakes – some good discussions on individual error types, followed up by adding some notes that I can give to students.  I picked up some additional thoughts on simplifying problems in free responses for students, which is nice.

Calculus Reasoning from Graphs – this was similar (I think) to something Mike presented last October, but I got more out of the activities this time for use in my classroom.  We started out by looking at 2013 AB 4, which shows a polynomial looking graph over a small interval, and provides the areas between the curve and the x-axis.  We had a nice discussion about an alternate form of the FTC as a sum instead of a difference, which ties nicely to point-slope in function form, and Taylor polynomials.  This was also the only good (judged by my) argument I’ve ever heard for moving point-slope into function form–to make it easier to move into Taylor polynomials with even a little more comfort is rather lovely.  I hope to use this packet, if not in class for a good several day lesson, for a Saturday session working with other students and my own.

Thought – Have students derive definitions for concavity using secant lines and tangent lines.  Will secant lines always be above a concave up interval?  Always be below a concave down interval?

The last thing we did was explore calculator use in calculus, and played with some places that calculators have issues graphing or solving problems due to the algorithms the calculator uses to perform calculus.  Some cool examples to share with students:  graphing y = sin(3x), then y = sin(47x) (a problem with calculator resolution); solving 0 = (x-2)^2 by graphing and finding zeroes (a problem with the way a calculator finds zeroes (looks for where the function changes sign)); evaluating the derivative at 0 and 1 of 1/(x-1)^2 (symmetric difference quotient issue); and the definite integral of (3+cosx)/x^4 dx from 1 to 3, and from 1 to 10^5 (a problem with the summation that the calculator uses to approximate based on intervals from 1 to 10^5).  Cool stuff.  Especially since learning that my students really struggled more on the calculator sections last year, and I need to help them improve this year–it’s always fun to have places that the calculator doesn’t work.  I want my students to be thinking calculator users, not using a calculator to think.


Circling a Solution

One of my old students emailed me about her summer work heading into calculus–

A circle is tangent To the y-axis at y=3 and has one x-intercept at x =1. Determine the other x-intercept.

What I enjoy about the problem, is that the harder parts of it are more deeply related to finding a way to analyze the problem, then the actual math involved here.

First:  If we know the circle is tangent to the y-axis, we know that a radius to this point from the center will be perpendicular to the y-axis (i.e. horizontal), which is really useful:  it gives us a y-coordinate for the center to use in our general equation for a circle.

Let’s take a look at the algebra now:

The general equation for a circle given center (h,k) and radius r, is:


Throwing our value for k in:


We also happen to know a couple points on the circle:  (1,0) and (0,3).

Let’s plug in (0,3), and we arrive at:



So, h=r, which makes sense, because if we are starting from the y-axis (where we’re sitting at the point of tangency), the distance to the center, is going to be equal to the difference in the x-coordinates, or (h-0), or h.  Cool.

Let’s plug in (1,0), and r for h, and we arrive at:



Through some algebraic manipulation:



Once we have a value for the x-coordinate of the center of the circle, we can think about this graphically or algebraically.

Graphically, we have a circle with center (5,3), and an x-intercept at (1,0).  We can reflect the point over the vertical diameter, x=5, and get the point (9,0), which is the other x-intercept.

Algebraically, we can plug in y=0 to the final equation of our circle, and solve for x.





x = 1, and x = 9.

This is a really hard problem for my students to do, but not one that’s impossible—and when a student can make the first leap or second leap, they get excited and build resiliency.

Hope for my children* (

When I have children (*sometime in the not-so-near future), I hope identity comes easy, and gender-roles are anything but traditional–little girls catching snakes, before going to tap class and playing ice-hockey; or boys who learn to sew while building lego forts for their pirates to take back from Raggedy Ann.  When we grew up (maybe growing up isn’t quite the right word), I don’t think many of us realized the indoctrination the world had in for us.  Why should it be possible that I can still hear echoes of what young women (or young men for that matter) can or can’t do, can or can’t become?  

Let’s not even consider the current disgust I hold for politicians who feel they have the right to take away from half the population, the right to control their own bodies.  Let’s focus on the messages we send our students. 

It’s bad enough that I have students entering my classroom with the idea that math is too hard for them, or that they’re just plain bad at it.  It’s worse that I’ve had administrators thank me for making math interesting, because they know how hard it must be to make something so hard and boring anything but… 

When we make anyone represent more than themselves (i.e. one young woman representing her sex), we deny her the chance to make a space for herself as an individual, and run the risk of creating an oversimplification and overgeneralization that is harmful to not only the person, but the society we live in.

As teachers, isn’t it our job and duty to fight these types of mindsets?

First Daze of School

There are some seriously big changes heading my way this year–I’m starting in a new district after teaching in the same school for six years, with new coworkers, new administrators, new students, and new classes.

Eventually, I’ll need to come to terms with the idea that I’m going to be teaching students a great deal more privileged than I’ve become comfortable with (and a great deal whiter than I’m used to), but for now, I can’t help but focus on my class schedule.  

I’m teaching calculus for the first time.  And I can’t wait.  And I’m terrified–not because I’m not ready for the math of it all, because I am, but because it’s something new, and I have had a rather overt affair with the beauty of calculus since I first encountered it.  I’m terrified, because I want my students to see the wonder and the interconnectedness of the universe that calculus brings, and I know they won’t all find it as beautiful as I do.  

But I can dream.

And I do.  

Of refracted rainbows, and right angles and arch angels of varying degrees (stolen from Taylor Mali), of discoveries and the failures that come from near misses.  

I suppose after the first week of school, I’d love for my calculus students (not all of them, but at least a few) be open to the idea that the study of math can actually make the world a more vibrant and beautiful place. 

If I can’t have that, well, I think I’d at least like to live long enough for the weekend to give me a breather and the last few warm days of a late summer in New England.  

I don’t always rest my hopes on the shoulders of giant dreams, but usually, I think, it’s hope enough.