More Useful Trig Graphics

First off, wow, that unit circle post was by far the most viewed I’ve had.  So maybe people would also be interested in this too.

The thing that’s difficult about graphing trig functions for the first time is that regular old graph paper just isn’t up to the job.  When we have points like \left(\frac{\pi}{4} , \frac{\sqrt2}{2} \right) and \left(\frac{\pi}{3} , \frac{\sqrt3}{2} \right) to plot, you need something that is suited to the task, and the numbers involved.  I have met teachers that say they just have students plot the multiples of \frac{\pi}{2} , and then tell the students to trust them, it’s not really a zig-zag.  And it seems to me that having them spend time plotting points in order to develop a wrong intuition is an amazingly bad use of time.  You could just use the calculator and it would be better. But in order to make the time spent plotting points valuable you have to be able to see the shape of the curve from the points are plotted. So you need y-values of \pm\frac{\sqrt2}{2} and \pm\frac{\sqrt3}{2} and x-values that are multiples of \frac{\pi}{6} and \frac{\pi}{4}.

So behold:

triggraph

Of course, you need a key for this:

triggraph-key

You also need to call the students’ attention to how the x-gridlines aren’t evenly spaced, and that the unlabeled ones are multiples of \frac{\pi}{6} and \frac{\pi}{4}.  But after that…

triggraph-dots

you have something that can allow student to connect their knowledge of the trig functions in the context of the unit circle to the traditional graphing context.

I don’t feel like wrestling with the code tags, so you can check out the latex source here, at sharelatex.  (You don’t need a ShareLaTeX account to access that link, but if you’d like to sign up for one, please use this referral link, which lets me earn points toward referral bonuses. Thanks!)  That document has two types of graphs– one for sine and cosine, and another for tangent and cotangent.  There’s a second set with the functions graphed as well, which I don’t give to students, but it’s useful to have to point at in later classes.

TikZ Unit Circle

After getting several requests from IRL folks for my unit circle, I figured I’d make it a post here. (Seriously, when I get tackled in the hall for a unit circle handout, it means that there’s a real need out there, even though I would have figured that there’s enough unit circles in the world.)

Blank unit circle
blank unit circle

blank unit circle as pdf

latex code on sharelatex.
latex code:

\documentclass[border=4pt]{standalone}
\usepackage{amsmath,mathpazo,gensymb}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture} [>=stealth, scale=2.5,
% Toggle commenting on the next four lines for the completed unit circle:
angle/.style={draw,text=white,fill=white,minimum height=1cm, minimum width=1cm},
point/.style={white},
%angle/.style={fill=white},
%point/.style={},
]
	\draw [white] (-3.6,-3.6) rectangle (3.6,3.6);
	\draw [thick,fill=white] (0,0) circle (3cm);
	\draw [thick, ] (-3.3,0) -- (3.3,0);
	\draw [thick, ] (0,-3.3) -- (0,3.3);
	\draw (0,0) --  node [angle] {$\frac{\pi}{2}$} (90:3)
				node[point, above right] {$ \left(0,1\right) $};
	\draw (0,0) --  node [angle] {\footnotesize{$\pi$}} (180:3)
				node[point, above left] {$ \left(-1,0\right) $};
	\draw (0,0) --  node [angle] {$\frac{3\pi}{2}$} (270:3)
				node[point, below right] {$ \left(0,-1\right) $};
	\draw (0,0) --  node [angle] {\footnotesize{$2\pi$}} (0:3)
				node[point, above right] {$ \left(1,0\right) $};
	\draw (0,0) --  node [angle] {$\frac{\pi}{4}$} (45:3)
				node [point, above right] {$ \left( \frac{\sqrt2}{2} , \frac{\sqrt2}{2}  \right) $};
	\draw (0,0) --  node [angle] {$\frac{3\pi}{4}$} (135:3)
				node [point, above left] {$ \left( -\frac{\sqrt2}{2} , \frac{\sqrt2}{2}  \right) $};
	\draw (0,0) --  node [angle] {$\frac{5\pi}{4}$} (225:3)
				node [point, below left] {$ \left( -\frac{\sqrt2}{2} , -\frac{\sqrt2}{2}  \right) $};
	\draw (0,0) --  node [angle] {$\frac{7\pi}{4}$} (315:3)
				node [point, below right] {$ \left( \frac{\sqrt2}{2} , -\frac{\sqrt2}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{\pi}{6}$} (30:3)
				node [point, above right] {$ \left( \frac{\sqrt3}{2} , \frac{1}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{\pi}{3}$} (60:3)
				node [point, above right] {$ \left( \frac{1}{2} , \frac{\sqrt3}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{2\pi}{3}$} (120:3)
				node [point, above left] {$ \left( -\frac{1}{2} , \frac{\sqrt3}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{5\pi}{6}$} (150:3)
				node [point, above left] {$ \left(- \frac{\sqrt3}{2} , \frac{1}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{7\pi}{6}$} (210:3)
				node [point, below left] {$ \left(- \frac{\sqrt3}{2} , -\frac{1}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{4\pi}{3}$} (240:3)
				node [point, below left] {$ \left( -\frac{1}{2} , -\frac{\sqrt3}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{5\pi}{3}$} (300:3)
				node [point, below right] {$ \left( \frac{1}{2} , -\frac{\sqrt3}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{11\pi}{6}$} (330:3)
				node [point, below right] {$ \left( \frac{\sqrt3}{2} , -\frac{1}{2}  \right) $};

	\foreach \n in {1,2,3,4}
		\foreach \t in {0,30,45,60}
			\fill (\n*90+\t:3) circle (0.03cm);
	\foreach \t in {30,45,60, 120,135,150, 210,225,240, 300,315,330}
		\node [font=\tiny, fill=white,inner sep=1pt] at (\t:.75) {$ \t\degree $};
\end{tikzpicture}

\end{document}

Completed Unit Circle
completed unit circle

completed unit circle as pdf

latex code on sharelatex
Get the latex code here.

\documentclass[border=4pt]{standalone}
\usepackage{amsmath,mathpazo,gensymb}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture} [>=stealth, scale=2.5,
% Toggle commenting on the next four lines for the completed unit circle:
%angle/.style={draw,text=white,fill=white,minimum height=1cm, minimum width=1cm},
%point/.style={white},
angle/.style={fill=white},
point/.style={},
]
	\draw [white] (-3.6,-3.6) rectangle (3.6,3.6);
	\draw [thick,fill=white] (0,0) circle (3cm);
	\draw [thick, ] (-3.3,0) -- (3.3,0);
	\draw [thick, ] (0,-3.3) -- (0,3.3);
	\draw (0,0) --  node [angle] {$\frac{\pi}{2}$} (90:3)
				node[point, above right] {$ \left(0,1\right) $};
	\draw (0,0) --  node [angle] {\footnotesize{$\pi$}} (180:3)
				node[point, above left] {$ \left(-1,0\right) $};
	\draw (0,0) --  node [angle] {$\frac{3\pi}{2}$} (270:3)
				node[point, below right] {$ \left(0,-1\right) $};
	\draw (0,0) --  node [angle] {\footnotesize{$2\pi$}} (0:3)
				node[point, above right] {$ \left(1,0\right) $};
	\draw (0,0) --  node [angle] {$\frac{\pi}{4}$} (45:3)
				node [point, above right] {$ \left( \frac{\sqrt2}{2} , \frac{\sqrt2}{2}  \right) $};
	\draw (0,0) --  node [angle] {$\frac{3\pi}{4}$} (135:3)
				node [point, above left] {$ \left( -\frac{\sqrt2}{2} , \frac{\sqrt2}{2}  \right) $};
	\draw (0,0) --  node [angle] {$\frac{5\pi}{4}$} (225:3)
				node [point, below left] {$ \left( -\frac{\sqrt2}{2} , -\frac{\sqrt2}{2}  \right) $};
	\draw (0,0) --  node [angle] {$\frac{7\pi}{4}$} (315:3)
				node [point, below right] {$ \left( \frac{\sqrt2}{2} , -\frac{\sqrt2}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{\pi}{6}$} (30:3)
				node [point, above right] {$ \left( \frac{\sqrt3}{2} , \frac{1}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{\pi}{3}$} (60:3)
				node [point, above right] {$ \left( \frac{1}{2} , \frac{\sqrt3}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{2\pi}{3}$} (120:3)
				node [point, above left] {$ \left( -\frac{1}{2} , \frac{\sqrt3}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{5\pi}{6}$} (150:3)
				node [point, above left] {$ \left(- \frac{\sqrt3}{2} , \frac{1}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{7\pi}{6}$} (210:3)
				node [point, below left] {$ \left(- \frac{\sqrt3}{2} , -\frac{1}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{4\pi}{3}$} (240:3)
				node [point, below left] {$ \left( -\frac{1}{2} , -\frac{\sqrt3}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{5\pi}{3}$} (300:3)
				node [point, below right] {$ \left( \frac{1}{2} , -\frac{\sqrt3}{2}  \right) $};
	\draw (0,0) --  node [near end, angle] {$\frac{11\pi}{6}$} (330:3)
				node [point, below right] {$ \left( \frac{\sqrt3}{2} , -\frac{1}{2}  \right) $};

	\foreach \n in {1,2,3,4}
		\foreach \t in {0,30,45,60}
			\fill (\n*90+\t:3) circle (0.03cm);
%	\foreach \t in {30,45,60, 120,135,150, 210,225,240, 300,315,330}
%		\node [font=\tiny, fill=white,inner sep=1pt] at (\t:.75) {$ \t\degree $};
\end{tikzpicture}

\end{document}

As a sidenote, I am really pleased that WordPress’s code tags seem to have improved greatly since the last time I tried to use them.  Never mind.  It’s better than it was, but it’s changing < and > to < and >.  And it took the away entirely.  For the actual code, you can check out the versions on sharelatex, as well as this document, that has them both embedded in it.  (You don’t need a ShareLaTeX account to access these links, but if you’d like to sign up for one, please use this referral link, which lets me earn points toward referral bonuses. Thanks!)

First post of a new semester

So we just finished the third fourth week of school, and I haven’t written anything here yet.

The math department was in an especially desperate situation this semester– one week before the semester started, there was somehow a surge of unexpected new students, and 800 people were on a waiting list wanting to take college algebra.  I really don’t understand how something like that happens, but then I don’t know much about how that sort of administrative thing works.  The university was frantically trying to get us more classroom space, and the department was moving people around in an effort to create more College Algebra sections.

At the same time, for the first time since 2008, I had a class whose initial registration count was under 30.  Way under.  It was originally 14 students in one of my sections of Calculus for Life Science.

I cannot emphasize enough how bizarre that is.  I haven’t had a class for the past two years that started with less than 40.  Usually it’s 48, which is the fire-marshal limit for our classrooms.  And since this happened at the same time that the department was scrambling to get more college algebra, I just kept my head down, and hoped that they wouldn’t notice me and kill my class.

And now I’ve got a class of 17, for the first time since I was a grad student teaching developmental math.  And it’s glorious.  It’s just… amazing.  I’m doing the blog project in that class, and I can keep up with everything that everyone posts.  When we do group activities, I can actually check in on each group.

I know I complain a lot about class sizes, but there’s been this little cynical voice in the back of my head saying that it’s probably not the class sizes, I’m just burning out on teaching.  But no!  It really is the class sizes.  This class comes at the end of a very long day, and I have more energy for it than I do for my 10 AM precal that has 48 students in it.

My new, explicit, Mathematical Collaboration Expectations

So I was flipping through my feeds as a way of procrastinating finishing my calculus syllabus when I came across this post from cheesemonkeysf, where she shares a rubric for group work for her middle school math students.

For years I’ve been telling myself that I needed to make my expectations for groupwork more explicit. I’ve told them what not to do: groupwork is not four people working silently side by side, it’s not dividing up the problems and sharing answers, it’s not one person doing everything and not taking input from their teammates. But that’s not the same as saying what it actually is.

But that sort of thing takes a lot of time to develop well, and being pulled in a hundred different directions, I just never took the time to sit down and do it. But when Cheesemonkey gave me such a great starting point, all it took was a little editing to get this:

Mathematical Collaboration Expectations

Active Inclusion

You helped the group to develop its shared mathematical thinking by:

  • allowing others adequate time to express their own thinking
  • demonstrating patience when other group members have difficulty putting their ideas into words
  • making sure that everyone understands why or how a piece of shared thinking or reasoning is so

Individual Participation

You made your own personal contributions to developing the group’s shared mathematical thinking by:

  • developing your own unique insights
  • sharing your thinking and ideas respectfully with the group
  • encouraging and supporting others as they speak their ideas,
    confusion, or questions
  • managing your desire to do more than your fair share of the talking

Deep Listening

You developed your openness as a collaborator by:

  • listening attentively to your teammates
  • asking clarifying questions
  • building on others’ ideas

Exploratory Talk

You developed your voice as a math learner and as a member of a learning group by:

  • noticing and wondering about a problem
  • extracting information and forming questions
  • trying a variety of approaches

Reflective Talk

You developed a sense of self-awareness as a math learner by:

  • noticing out loud other learners’ insights, strategies, or contributions that helped to move the group’s learning forward
  • noticing what you personally did well as a member of the group
  • noticing what you personally need to keep working on to become a more effective member of a mathematical learning group.

I didn’t want a rubric, so I stripped off all the points, and just made it a set of expectations.  If you’re interested in the latex version, you can find it on my sharing handouts page (it’s at the bottom).

And this is my absolute last blog post until I get that stupid syllabus done.

My new favorite latex package

As I was reworking materials for the beginning of the semester (specifically, my Review of Prerequisite Material) and I thought to myself that there had to be a better way to arrange these problems on the page the way I wanted. So I asked on tex.stackexchange (which is full of very pleasant, very helpful people), and got an answer, which I was able to improve on a little to get the new template that I will be using any time I need to create a handout that consists of a bunch of questions/exercises.

\documentclass[12pt]{article}
\usepackage[margin=.75in]{geometry}
\usepackage{exsheets,mathpazo,titlesec}
\usepackage{amsmath}
\setlength\parindent{0in}
\titleformat{\section}[block]{\large\bfseries\filcenter}{}{1em}{}
\setcounter{secnumdepth}{0}
\newlength{\spacing}
\setlength{\spacing}{1in plus 1fil}
\settasks{
counter-format = tsk. ,
item-indent = 2em ,
label-width = 2 em,
resume = true,
after-item-skip = \spacing,
after-skip = \spacing
}
\newcommand{\resetspacing}[1]{
\setlength{\spacing}{#1 plus 1fil}
\settasks{after-item-skip = \spacing,after-skip=\spacing}
}
\begin{document}
\section{PracticeHandoutTitle}
\begin{tasks}(2)
\task
\end{tasks}
\end{document}

This is going to make it a lot easier for me to get things to look like I wanted them to.  Tables never really lined up right if the cells had text of different heights.  And plus, using tables is just a pain.  If you’re interested, here’s how my Pre-Cal Review of Prerequisite Knowledge looks now.  (ShareLaTeX doesn’t have the exsheets package yet, so if you really want to see it, you’ll have to download it and compile it yourself.  I put in a request, though!)