Combinatorics teaches us to count without counting

Requirements: One die and a piece of cardboard (or paper). Put the die in the center of the cardboard and trace the square outline of one of the faces.

Background: The students need to have learned the Fundamental Principle of Counting; If Task 1 can be done in m ways and Task 2 can be done in n ways then there are mn ways to do Task 1 followed by Task 2.

Take out the die and ask, "What's this?". Make sure the class know that die is singular and dice is plural. Now take out the cardboard and show them how the die fits by putting the face numbered 1 (or any face) onto the square. Holding the cardboard parallel to the ground, rotate the die to each of the 4 positions that keeps the die in the square and tell them that these are all different positions. Your final question is, "How many ways are there to place the die onto the square?". If someone has the answer then get them to explain it. In either case, summarize the problem by writing on the board:

Task 1: Choose a face of the die to rest on the square. There are 6 ways this can be done.

Task 2: Choose a face of the die to face the class. There are 4 ways this can be done.

By the Fundamental Principle of Counting there are (6)(4)=24 ways to place the die in the square.

Conclude by noting that a seemingly difficult problem has been made simple and say, "That's what we mean by 'Combinatorics teaches us to count without counting.' ".

You can download this information on the Teaching Points web page. If you plan to use the tex file you will need to copy the picture above as well for it to compile.

Integration has lots of applications (1)

Today's post is the first teaching point example. So if you have an outline for calculus you can easily slip in the teaching point "Integration has lots of applications". Now the goal (over time) is to have several examples of applications so each year you can decide which examples you'll use. That helps keep teaching fresh. I've also found students pass on whatever homework, notes, and tests they have from a teacher to their friends; changing the content forces them to keep up with the class.

This example assumes the student have studied unrestrained exponential growth in precalculus and have learned how to integrate $latex \frac{1}{x}$. Start by eliciting the formula for (unrestrained) exponential growth and what each variable stands for. The formula should look like $latex N(t)=Pe^{kt}$ where P is the initial population, t is the time, k is some (growth) constant and N(t) is the population at time t. Remind the class that the formula is a mathematical model which helps scientists predict how the population would grow in the future if there was no constraints (food, space, etc.) and the model can even be used to predict the population back in the past. Ask them "How can someone create a formula which can help us predict the growth of a population?". The answer: Scientific observation combined with the deductive power of math! It starts with scientific experiments concluding: The change in population over time is proportional to the population. The power of calculus comes from its ability to take a fact like this and produce a formula that can be used by a scientist to model their problem and make predictions about the future. In this example there are 2 quantities: time, t, and population. Since the population depends on time, let N(t) represent the size of the population at time t. Translating the information "The change in population over time is proportional to the population" into math gives us $latex N'(t)=kN(t)$ where k is an unknown constant. If $latex N(t) \neq 0$ then $latex \frac{N'(t)}{N(t)}=k$. Integrating both sides will result in the formula $latex N(t)=Pe^{kt}$ where P is the initial population. The mathematical details are given on the Teaching Points page along with the tex file for you to add into your notes, if you want. The concluding remark to your students is "If the scientists' observation is correct then calculus tells us this formula must be true!".

Basic Shapes, part 2

The basic shapes that every high school student should know are complete with today's addition of trig functions, logarithms, and exponentials. They can be found on the Graphics page. As mentioned in Basic Shapes 1, graphics created with PS Tricks can't be compiled using pdfLaTeX.

Good Math Habits

A strong mathematical foundation depends, in part, on good math habits. As problems get more complicated, students who don't have good math habits begin to struggle as careless mistakes and errors in understanding increase. Good math habits are designed to help students communicate math effectively as well as catch careless mistakes and errors in understanding sooner, rather than later. Here are some good math habits that you should consider making your class responsible for.

1. Write neatly.

2. Write to an audience of your classmates.

3. Use correct grammar and spelling.

4. Use symbols correctly.

5. Use diagrams and examples to help explain complicated examples.

6. Show all of your work (don't skip steps).

7. Always find the exact answer. Do not estimate unless you are told to.

8. Always simplify your answer as much as possible.

9. Always give the units in your final answer.

10. Check your solution for accuracy and clarity.

11. When you cancel something out that might be 0, write a note.

Students should be writing their solution to a problem to show you they understand how to solve the problem correctly. That will help you correct any errors they're making sooner, rather than later. It also means each assignment/test/quiz the student has completed is now helpful study material for the next quiz or test. Points 1 and 2 address the fact that you need to be able to read their work and the solution should have enough detail that others can understand the reasoning process. Points 3 and 4 are setting a "professional" level to their work. Correct spelling and grammar also helps to reinforce English. Using symbols correctly is meant to stop students from writing things like: "Take the $latex \sqrt{}$ of both sides..." instead of "Take the square root of both sides...".

Describing a problem in words can be difficult enough for the student but it can be a nightmare for the teacher trying to read. A diagram of the situation to go along with the words takes some of the confusion away and allows the students to get away with a little less writing.

Show your work is arguably the most important habit on the list. While it may be "obvious" to a student that $latex -2^{3}=8$ some of those students have gotten the answer by thinking $latex -2^{3}=(-2)(-2)(-2)=4(-2)=-8$. This is, of course, wrong because the minus sign isn't part of the base. A student who isn't immediately corrected then reinforces the mistake by solving other problems the same way.

Always find the exact answer is clear enough. Mathematics is an exact science. Getting an expression for the exact answer and then simplifying also forces students to work on the process of getting the answer before going to a calculator. For simplifying your answer, don't give an answer of $latex \frac{56}{14}$ when the answer 4 is much easier to understand. Requiring units is important, too, because math will be used as a tool in their math and science courses.

Checking the solution for accuracy and clarity is another very important habit. So many times I've seen students (especially in word problems) students attempt to solve a problem but because the problem has several steps to it, they end up finding the answer to a sub problem (the length of each piece) and stop. If they had taken time to reread the problem they'd find they were supposed to solve for the total length.

The final good math habit becomes important in later math classes. Consider:

$latex \frac{(x^2-1)}{(x-1)}=\frac{(x-1)(x+1)}{(x-1)}$

Since the student needs to simplify their answer (point eight on our list) and they want to cancel $latex (x-1)$, which might be equal to zero, they should have for their final answer:

$latex (x+1)$ if $latex x \neq 1$.

That's because the original fraction had a domain for all the real numbers except for 1. Writing the note reminds us the answer can't be trusted for $latex x=1$.

As always, you should make sure your students know what good math habits you expect and be ready to answer any questions they might have. I've posted the "Good Math Habits" list on the Procedural page along with its tex file.

Grading Rubrics

Depending on your math class you might find a grading rubric necessary. For low level math classes, such as algebra, the specific write up for a problem isn't that important. I'm happy to see an organized approach to solving a problem and I'm not concerned about the specific approach as much. In this case, a grading rubric is a good way to start getting students to show the steps in their work. It also makes grading easy: A quick look can tell you whether the work is "good" or "bad" and with a little more time you can figure out whether the "good" answer is Excellent or Good. Likewise, a "bad" answer can be easily classified into one of the three bad answers categories.

Excellent (4 points): The answer is correct and all the necessary steps are shown.

Good (3 points): There are minor errors, lack of student understanding, or insufficient work. The answer may be right or wrong.

OK (2 points): There is a lack of evidence that the student understands or there is evidence the student doesn't understand. The answer may be right or wrong.

Poor (1 point): There is a lack of work and/or clear signs the student doesn't understand. The answer may be right or wrong.

No attempt (0 points): No meaningful attempt is made to solve the problem. The problem is blank, the student has rewritten the original problem, or has drawn pictures unrelated to the problem.

For higher level classes, such as calculus, the problems can be much more complicated and the students need to demonstrate more maturity in their answer. I prefer holistic scoring on a 10 point scale because the general explanation of a rubric isn't helpful in describing the specific details that will allow you to classify the problems easily. Likewise, there are too many ways to classify a problem that determining the score isn't so easy. I've put the rubric above, for low level classes, together with a rubric for higher level classes on the Procedural page along with the tex file.

Teaching Points and Themes

Every math teacher seems to go through essentially the same process: they create a set of notes, along with examples, that covers the required course material. Some may develop those notes by focusing on goals and objectives while others are content to summarize the pages of the book they are using with a couple of examples they have thrown in. Regardless of how the notes are created, they can then be written in outline form where superfluous language is minimal and the underlying logic of the notes becomes clearer. Here, for example, is part of a sample outline that could be used for teaching vectors: TeachPointEx

The outline that gets created is made up of what I call mathematical teaching points. Of course, the outline of teaching points will vary from teacher to teacher depending on such factors as the rigor of the course, the level of the student, and so on. Some points, such as a vector has magnitude and direction, should be found in the notes of every person teaching about vectors.

We're not interested in the outline since it can change so much from teacher to teacher, but it's worth noting that organizing your teaching points into an outline forces you to think about the way the lesson is organized and helps to ensure that important material is stated clearly and concisely through well chosen words. Very important teaching points can be given to your class during lessons, on handouts, a class website, etc., to make sure students have seen vital information more than once. Some teaching points and their variants can occur over and over: "A calculator is an inherently flawed tool.". I'll often refer to these as themes.

In addition, the teaching points form the essential notes that you can bring into class and embellish as the material is covered. At appropriate points, of course, we'd expect to give pictures and examples. For example, if I was writing, "A vector can be thought of as a directed line segment." I'd draw the directed line segment and, when the relevant teaching point was reached, would label the arrow of the line segment with the word tip.

Routine examples can be found everywhere, so this website is interested in more unusual examples that make math more interesting, relevant, fun, and/or can be explored with a calculator or computer. By listing teaching points and accumulating multiple examples to illustrate particular teaching points, a teacher can pick and choose those that complement their notes, spice up a lesson, and leave them more time for other duties.

If you have any unusual or interesting mathematics that can liven up a lesson and be freely shared, please contact me!

An introduction to Sage

Sage is a free, open source CAS (computer algebra system) similar to Mathematica or Maple. The Sage website lists close to 100 software packages which are combined under a Python interface to create a very powerful program. Unfortunately, Sage is too complicated for your average high school student (and even some teachers) but the ability to prepare a Sage notebook to bring into class can help a math teacher:

  1. plot 2D/3D functions
  2. illustrate 3D solids
  3. create animated GIFs
  4. build interactive applets
  5. test conjectures by running a Python program
  6. change the values of variables in a calculation and see the results immediately

The documentation for Sage is extensive and intimidating so I've put together a Sage cheat sheet that has information only on those Sage commands that might be useful in a high school math class. Although I wouldn't make the students use Sage, exposing the class to to the power of Sage and handing out a cheat sheet might spark the interest of the strongest students. The cheat sheet is posted on the Python/Sage page along with the lightly annotated tex file.

As this cheat sheet hasn't been tested in the classroom yet, please let me know if you find any errors.

Creating Chess Diagrams

Creating chess diagrams with $latex \LaTeX$ is easy, and every diagram is essentially the same .tex code. Rather than post multiple copies of essentially the same file, I'm posting the code for a complicated example. By making small changes in the code you can, without much time, create any diagram you want. The chessboard package documentation can be found here on CTAN. It can answer questions I haven't addressed.

\usepackage{chessboard} %allows us to work with chessboards
\pagestyle{empty} %no page numbers will be given
linewidth=1pt,padding=2pt,pgfborder,setpieces={Nb4, Nb8, na6, ne6, Re1, Rh5,Bc3,Ka1,kc8,qg7,pa7,Pg2,Ph7,
pb7,pc7,pd7},pgfstyle=cross,color= blue, markfields={a3,a4,c4,c6,h6,h5},
pgfstyle=circle,color= red, markfields={b3,a7,e8},color=black,pgfstyle=straightmove,
markmoves={a1-a2, c3-e5, g1-f3},pgfstyle=knightmove, backmoves={a6-c5, e6-f8},
color=orange,pgfstyle={[left,rotate=340]text}, text=this is text, markregions=\board]

The output of the code, after it is cropped, looks like this: Diagrams

The work is all done following the \chessboard command. The default position of the board has starting position so the clearboard option removes all the pieces. The showmover option will put a colored box (white or black) to indicate the side which will move next. It can be turned on by replacing false with true. The next few options


govern the outline of the board and the second outline of the board. The setpieces option is a listing of the pieces an placement on the board. For White, use P for pawn, R for rook, N for knight, B for bishop, Q for queen, and K for king. For Black, use p for pawn, r for rook, n for knight, b for bishop, q for queen, and k for king. Then give the square it is on. So na6 is a black knight on the a6 square and Pg2 is a white pawn on g2. Easy!

To mark up the board, I've used pgfstyle=cross and pgfstyle=circle. Other options are given in the documentation. After choosing how the square will be marked, the color is chosen and the squares to be marked is listed. Arrow styles are chosen with pgfstyle=straightmove for a straight arrow and pgfstyle=knigtmove for the curved arrows by the knights. The list

markmoves={a1-a2, c3-e5, g1-f3}

tells us that one straight arrow starts at a1 and ends at a2, another straight arrow starts at c3 and ends at e5, and a third straight arrow starts at g1 and ends at f3. Finally, I've even printed some text on the board. The default position is the center of the board. See the documentation for details on this, turning the board around, and other issues.

The .tex file for the code above and the cropped PDF output files  is posted on the "Chess" page.

Basic Shapes, part 1

Every student who graduates high school should be familiar with graphs which occur over and over: basic shapes that will help them to understand more complicated functions.

I've posted 8 basic shapes so far on the "Graphics" page which you can get to by clicking on the link given in the sidebar. Most are done using the PS Tricks package, so you can't compile them with PDFLaTeX. I used XeLaTeX although plain $latex \LaTeX$ should be fine. The PS Tricks package for graphics was designed before PDFs existed and .dvi files were the normal output. The PS Tricks website, which is a link on the sidebar, says:

4. You cannot run your files with pdftex or pdflatex, use xelatex instead or the sequence latex->dvips->ps2pdf, or see topic pdfoutput

This will create the PDF file which will need to be cropped. See the previous post for information on that.

More Basic Shapes will be posted later.

The Basics

Although you don't have to learn $latex \LaTeX$, it is an important skill that will help you get more out of this site. If you aren't happy with, say, the color of a particular graphic you can easily change it by downloading the .tex file, changing the code in several places, and running  $latex \LaTeX$ again. I've added some comments to help you find your way around but you'll need to have basic knowledge of $latex \LaTeX$ and be willing to learn a little more.

I assume that you have $latex \LaTeX$ installed on your computer and know how to use it. You don't have to be an expert; modifying code is a lot easier than creating it. The links "TeX: FAQ" and "LaTeX wikibook" located on the right sidebar can help get you started in learning $latex \LaTeX$ and the link "Tex Stack Exchange" is the best place to get answers to just about any $latex \LaTeX$ question you can think of. The "CTAN" links you to where $latex \LaTeX$ package documentation can be found.

All you need to process .tex files are a text editor and the command line. However, there are many IDEs to help make everything easier. I'll mention 3 here, but you can consult the LaTeX page for more options and opinions.

  1. Kile: this well known IDE has a panel that includes the command line. When you open a .tex file with Kile you are automatically in the correct directory. Having a command line is important when you need to run $latex \LaTeX$ with -shell-escape enabled.
  2. Gummi: this isn't well known but it should be, especially once there is a Windows version. Gummi is great because you can see changes soon after typing them. That makes Gummi really useful when you are creating graphics. Gummi's weakness is on large files because it is constantly compiling.
  3. TeXworks: This IDE will be useful because it gives you easy access to compiling your code with PDFLaTeX, XeLaTeX, LuaTeX, ConTeXt and many more. That will be useful for for the .tex files on this site. In general, the files were originally compiled with PDFLatex. However, .tex files that use PS Tricks can't be compiled  that way. I've used XeLaTeX on those. TeXworks also has a hand magnifying glass for inspecting your output.
Note that running PDFLatex on a .tex file which is used to produce a picture results in an uncropped PDF file. To crop the PDF, the command line tool pdfcrop is quick and easy to use. It will remove almost all of the whitespace. If you want to have more control over what gets cropped, then BRISS is a good alternative.