Monthly Archives: November 2011

My New Classroom Poster

After I put it up a student asked me if I drew it. Yes, yes I did. Because before I was a math teacher I was a Norwegian expressionist painter who died in 1944.

By the way, I just photoshopped this and split it up onto 9 pages, then used the Math Department color printer to print it off (oops). If you want the files, here they are in a compressed folder (nine 8×10 images and the original full file).

Wannabe Math

There are a lot of times in Calculus where stuff that you want to be true is (like the limit of a sum is the sum of the limits) but then there are a lot of other times where sadly your intuition just doesn’t work (like the derivative of a product is NOT the product of the derivatives). This is a random little thing, but I’ve been doing “Wannabe Math” in thought bubbles on the board to distinguish between rules we know that work and ones that we don’t know are solid:

I want to encourage students to try things (so encourage them to use their intuition to explore) but give them tools to be able to test to see if they are correct. I did this for the product/quotient rules and then used it to derive the chain rule with the class – with the Wannabe Math on one side and the actual math on one side, we could easily see what was “missing” and come up with the whole derivative of the inside part of the rule.

Relating those Rates

As the last big topic to end our first term, we explored Related Rates in our AP Calculus class, one of the first topics that involves really in-depth, complicated problem solving (which scares the bejesus out of a lot of students). I did some new things that worked and had some ideas on how to improve what I did, so I wanted to write them down now to reflect, especially because (for some odd reason I go into this topic in about 2 months in my very differently sequenced non-AP course).

 1. Visualizing the BIG IDEA Behind Related Rates Problems

I think that one of the hardest things for students to do is to visualize related rates problems. And the big idea behind them that makes them interesting problems in the first place – mainly that there is this relationship between different aspects of an object or situation and because of this relationship, when one changes, others do too. However, these things often don’t have linear relationships, so even though they change relative to each other, one changing at a certain rate doesn’t mean that another will change at a constant rate too. Right? Kind of wordy, but this is what I’ve always views as the important idea. Students tend to see them as more static situations and because they don’t have the big (changing) picture, get totally lost in the calculations. This is totally the kind of thing that gets brushed aside in my AP class because “we don’t have time,” but it’s wholly unsatisfying and not good teaching – I find myself cutting things out that I would never cut out normally just to keep up with my curriculum map. Sigh.  

So I was looking for good (and let’s be honest, quick) ways to help students visualize these problems. I introduced Related Rates with the simple GeoGebra applet above (click to go to the dynamic view). As time increases, both balloons blow up, but one blows up with a constantly increasing volume and one blows up with a constantly increasing radius. I gave them the applet and gave them about 10 minutes or so to figure out everything they could about rates with the two balloons and pick which one fits the actual situation of blowing up a balloon (I gave them actual balloons to blow up too). Through this, we talked about the idea of how in one object, many things can be changing with respect to time (so it’s possible to have dV/dt AND dr/dt and many other things), which was helpful to do before throwing notation up on the board. From this, we talked about how those relationships can be related and did some quick calculations to confirm their predictions of how when dV/dt is constant, dr/dt is decreasing etc. It ended being a great way to introduce the idea and gave us good language to talk about future problems. I spent the rest of the next 3 or 4 class periods just solving this monster packet of problems with them and giving them time to solve problems themselves.

This was a great introduction, but I think that I will do more big picture activities like this with my non-AP class (and less calculation/pen on paper problem-solving), but we did do a few other quick things to help visualize problems. I made some GeoGebra applets based on problems we were solving (see below, click for the dynamic applet). In the future, an assignment I want to do is have my students pick a problem to animate it – I think it would be a great way for them to deconstruct a problem and REALLY understand the relationship.

The last visualization exercise that we did was with the classic ladder sliding down a wall problem (I got this from my AP Workshop this summer). With WinPlot, I made an animation of this situation. To help students see the non-intuitive fact that if one end is moving at a constant rate, the other end moves at an increasing rate, I had the whole class clap every time the end of the ladder passed a tick mark. First we did that with the bottom, which was moving at a constant rate, and with the clap you could really see/hear that it was moving at a constant rate. Then, we did it with the top as it was falling, and with the claps speeding up and speeding up, it was easier to visualize that the rate was different. What I want to do with my non-AP class is get a ladder and model this situation by putting it against the wall and pulling it away 1 ft at a time and measuring how far the top goes down the wall.

 2. Finding Ways to Go About Complex PROBLEM SOLVING

 The other new thing that I did with Related Rates was that I found a great way to get students to both organize information and get started in the problem solving. For every problem, we simply made a table to organize all the variables and their rates (I know, not revolutionary), like this:

I found that this helped for a couple of reasons. First, the most obvious is that it keeps them organized with all their information, which can totally get jumbled during a Related Rate problems. Second, it helped them figure out what the relationship was that they needed to differentiate – by filling in all the variables in the table, they usually had to use some equation that related all of them and thus stumbled upon the relationship that they needed to use in the problem solving. Third, it helped them decide what variables were actually constants and what variables could be replaced by relationships with other variables (like the height and radius in a cone) – by forcing them to write down the rates for each variable, they are forced to think “Is this thing changing or not?” Then, it was easy to tell them to substitute anything that was constant (or could be related to another variable) into the expression BEFORE differentiating, as it makes the derivative so much easier. I just graded finals and it was really nice to see how many of the students took hold to this idea and made good use of the table (though overall the AP Free Response Question did NOT go well – I guess it’s still November, so that’s okay!).

I’d love to hear any good ideas for Related Rates – I find this to be a really interesting, but REALLY TRICKY topic to teach. I love Sam’s Related Rates Logger Pro Investigation, and might try something like that in the future with my class. I feel like I could spend a whole semester talking about these problems… if only there was more time!

“The Hardest Part of Calculus is Algebra”

One of the mantras in my Calculus (non-AP) classroom is that the hardest part of Calculus is Algebra. Seriously. I find it really depressing because often when I am correcting tests and quizzes, they will have the Calculus PERFECTLY correct and then just royally mess up the algebra. This year, the students in this class have incredibly weak backgrounds and I’m not sure why (well, I do have an idea that I will keep to myself…), but I do know is that many are really smart kids who just have serious holes in their math education. I’m really struggling to find a balance between teaching Calculus and remediation of basic math, which forcing me to go really slow and snip out wonderful projecty-type applications that really make the class what it is. It’s a balance between “Teach them where they are” and “Teach what I am supposed to be teaching”

I’m posting about this as a call for help to try to fix the crazy problems that I am having. I think the biggest problem is that the notation is so abstracted that they don’t have a basic feel for what you can and can’t do when manipulating expressions. I’d say the problems fall into two main categories…



In all six of the examples I picked above, the Calculus is EXACTLY correct, but it just gets completely and unfortunately ruined. I think that I will show them this when we go back to school on Sunday:

My AP class literally laughed out loud when I put that up on the board and I hope they have the same reaction and maybe help them realize that I have the same reaction when I see them manipulate some of the algebraic expressions the way that they do.

Any ideas on how I can do an Algebra boot camp of sorts to remediate some of these misconceptions? Are these normal Algebra mistakes for a senior in high school? Do other people have similar problems?





Experiments with Math Whiteboarding

Getting five mini-whiteboards was a real game changer for my classroom. It has completely changed the way that I go about skills-based instruction (which is what a large chunk of the first term of Calculus ends up being) and has added so many new tools to my instruction toolbox. I think that too often in the past I relied on variations of the learn-practice-apply model, and the “practice” part not only always seemed to make class drag, but never really felt effective. Well, doing practice on the whiteboards with some sort of extra little component to make it interesting has proven to be not only interesting (I’ve never seen students more engaged while practicing skill-based math), but far more effective at teaching skills. The only downside to the whiteboards is that they don’t have anything in their notebooks to study at home, but I’m brainstorming ways to fix that (mostly posting pictures of the whiteboards on our course website or not caring because they can always find examples of solved problems in our textbook – it’s the practicing part that matters).

Here have been my favorite things to do so far with the whiteboards and a few ideas for experiments that I want to try in the future. I would love any and all comments about different non-topic specific modes of instruction that you use with the whiteboards to expand my repertoire.


One of the great parts about whiteboards is that you can get get students to use different colors like you do on the board up front to get them to focus on different things. For example, above is a whiteboarding exercise I did with the Chain Rule. Students were in groups of threes – for each problem, one person had to rewrite the problem in different colors to indicate which was the outside and which was the inside function, the next person had to differentiate it still using the colors to point out where each part of the new expression came from, and then the last person had to rewrite the expression in a simplified form. This was perfect because the hardest parts of the chain rule are recognizing when you need, seeing inside vs. outside and then seeing where the parts of the new expression come from.

EXPERIMENT I WANT TO TRY: Mistake Marker. I want to have the students solve problems and then whenever they make a mistake, instead of just erasing part and fixing it, they write over their mistake with the mistake marker color (or make a note with the mistake marker if it’s not possible to write something in that color) and then continue the problem in a new line below with their original color. Then, at the end we can collect as a class the most common mistakes that are made when doing a complicated problem like, say, the Quotient Rule (not complicated you say? Then you have never taught students Calculus who have a terribly weak Algebra background).


I know I have mentioned this like twelve times already, but I absolutely love it. This is stolen from Kelly (read her description here), but the basic idea is that groups present solutions to semi-complicated/involved problems on whiteboards, but while presenting their solution, purposely make a mistake (and not an silly arithmetic mistake like 3+4=8, real hardcore-misconception-style mistake). Then, they present their work to the other students in the class, trying to sell their mistake as having been made for real. Then other students ask thoughtful questions about the presenting group’s solution to try to help everyone find the mistake. This is always great with a quick class followup at the end collecting the most common mistakes.


When I have wanted to show how a topic from my AP class could be applied to many different situations, I have done some sort of rotation so that students could be exposed to a wide variety of problems (without taking the time in class to do ALL of them). The first exercise with this is a simple gallery walk – each group of 2-3 students solves a problem individually. Then, when everyone is done, the groups rotate around to each station, taking a few minutes at each one. This is far more successful if you give them specific tasks, like “First, check if their answer makes sense, then see how they set up the limit definition of the derivative” or something concrete like that. The second exercise is when each group starts doing a problem, and then the groups rotate after about 5 minutes and they pick up where the last group left off (someone on Twitter gave me this idea, sorry, I forget who!). Then, they rotate every 4 to 5 minutes until all of the problems are completed. I tried to emphasize that while doing this, you must show your work neatly and clearly (an important skill for all math, but especially an AP test) so that the next group can quickly see what has been done to solve the problem and what still needs to be done. The thing that I liked about this was the meta cognitive mapping out of problem solving, though I don’t think I left enough time for students to really think about each one.

EXPERIMENT I WANT TO TRY: Rotate Marker. I didn’t love the rotating problems mid-problem solving, mostly because it stopped students in the midst of great problem solving, but I think one thing I am going to try is having the students rotate within their group who is writing. So a group would be solving a problem and every 2 minutes, the next person in the group would become the writer. This would ensure that all the students in a group are engaged in the problem solving process and that they are all talking math with each other.


One of my many goals this year is to step back and focus more getting students to figure out how to go about problem solving. One thing I did that I really liked was I photocopied a few pages from the book and cut out like 50 functions for each group that all required a variety of differentiation rules. Then the students made categories on their whiteboards and sorted the functions based on which differentiation rule they needed. It was a really interesting process, especially because many of the functions needed more than one rule. I really enjoyed seeing how students solved this – most just made a bunch of different categories (like Quotient & Chain, Product & Chain, Product & Quotient & Chain), but one group made a crazy complicated Venn Diagram and another made a table kind of like one of those that shows the distances between cities (so like Quotient, Product and Chain both across the top and down the side) then placing the functions at the intersections of the rules they needed. It only took 15 minutes, but after learning so many differentiation rules, I think it was great to give them a chance to step back and figure out what types of rules they needed to use and where. The next day on the quiz, I saw tons of students circle parts of functions and write “Product” and “Chain,” which is something I have never seen them do before. To me, this is a wonderful problem solving strategy that was explicitly identified and strengthened by a quick activity.


Overall, I just think the added presence of the whiteboards has given my classroom a much more dynamic feel. When I asked for feedback from my students about how class was going these were my two favorites:

I like how we change up the routine. We do not sit and do the same thing over and over again, its changes up and keeps me interested.


One positive thing is the different types of work you give us because it is not all the same thing it is diverse so it keeps things interesting.

I guess it was a good indicator to me that switching up the routine for switching-up-the-routine’s sake is not a bad thing. Knowing some basics about how the human brain works, keeping the kids from sinking into a comfortably numb routine will certainly make everything a little bit stickier.

And in case you forgot in all of my blathering, I’ll repeat my plea from above…  I would love any and all comments about different non-topic specific modes of instruction that you use with the whiteboards to expand my repertoire!