Handouts: Calculator worksheet

I put together a worksheet on the basics of calculators, looking at the maximum number of digits and seeing what happens when the number is a little bit bigger. In addition to the problems it creates for displaying the number correctly, it also leads to overflow and underflow errors.

The worksheet makes use of Google's calculator, and the results you get from using that calculator change over time. Several years ago Google was giving about 10 digits. Now it appears to be 12. The errors exposed in the worksheet might not be errors in another 2 years; in the future you should confirm the worksheet for the numbers given and modify the tex file accordingly.

You can find the calculator worksheet on the Handouts page.


Here's a game I use to demonstrate deductive and inductive thinking. I'm calling it "Coins" but it's one heap Nim. Calling it an unusual name will prevent your students from finding the answer on the internet. One heap Nim has many names and versions: here it's called "The 21 game" and I've seen other versions using "matches" (as in matchsticks) in the name. Here are some rules for the game.

  1. Start with a pile of coins.
  2. Two players take turns choosing either 1, 2, or 3 coins each turn.
  3. The player who takes the last coin loses the game.

Obviously the rules can be modified: the number coins in the pile, the number of coins that can be chosen each turn, and criteria for winning. Play against your class several times to make sure they understand how the game is played; I have the class take a vote on their collective move. Make sure to vary the number of coins in the pile and let them choose who goes first. By mixing it up you'll prevent them from seeing a pattern (or keep them from copying the moves you made in an earlier game).

After they understand how the game is played, they're ready to learn how to determine the winning and losing positions (deductive thinking) and see a pattern in the answers (inductive thinking). The fact that the pattern is created by the rules of the game means that the pattern they find is, in fact, correct. This makes it easy to figure out who should win (with best play) in a game with 1,345 coins. Modifying the rules of the game so that the  players choose 1, 2, 3, or 4 coins at a time can easily be solved by the same method.

You can find all the mathematical details on the Handouts page (tex file and PDF). One heap Nim is a simple game that illustrates the power of deductive thinking.

Inductive versus deductive reasoning

If you teach geometry, you cover two types of reasoning. Inductive reasoning is based on patterns or probabilities and its conclusions aren't certain. We saw an example of that in this post. It looked obvious that the next number in the sequence 2, 4, 6, 8, 10, . . . had to be 12 because it was "obvious" what the pattern was: 2n for n a positive integer. Obvious and wrong; we saw any real number can be justified as the next number.

Inductive reasoning is the reasoning of science. Use experimentation to find a relationship/fact and then test it. If it works over and over and over then eventually it becomes a scientific law, like the Law of Conservation of Mass. If further experimentation shows that the Law of Conservation isn't always correct then the law will be scrapped or modified to account for the exceptional cases.

Deductive reasoning separates math from other sciences because deductive reasoning is infallible. Yes, people can make mistakes in reasoning but if a statement is true because it follows logically from other facts then the conclusions must be true. The most amazing example I know is the De Broglie Wavelength: there is a wave-particle duality to matter because the equations say so.

I've put together a simple example to explain the difference between them. By noticing

1+2+3 = 2(3)         2+3+4 = 3(3)          3+4+5 = 4(3)         4+5+6 = 5(3)           5+6+7 = 6(3)

we can use inductive logic to form a conjecture. But to prove that pattern is always true takes deduction. Now a computer or calculator can help you verify the formula is correct for a finite number of cases but there are an infinite number of equations that need to be checked. With deduction, we can prove the conjecture will be true for an infinite number of equations in a finite amount of time. That makes deductive reasoning and math very special. I've written up the details for this example and it's posted on the Handouts page.


Chess Endings: King, Bishop, Knight versus King

It's the most challenging "basic" endgame out there: King, bishop, knight versus king. Many strong players have trouble with this ending. Some feel it's so unlikely to occur that it isn't worth the time to learn and every so often someone strong shows just how weak they are. That's muddled thinking at best: an important reason for studying the ending is to learn how to coordinate your pieces effectively to follow the basic checkmate plan: force the lone king into a quarter of the board which is the color of the bishop. Force the king back further until it has 2 squares that it can move between: the corner and an adjacent square. Time the movement of your pieces so that a knight check forces the king into the corner, whereupon the bishop will administer checkmate. Just remember, you've only got 50 moves to get it done!

I've posted the tex file and PDF on the Chess page.


In an earlier post, I mentioned the Ted-Ed as a promising site to watch if you were looking for videos  to use in the classroom. The Mathigon website is also in the early stages of development but you can see the potential. It promises videos, slide shows, problems, and interactive games to help explain the beauty of mathematics. Many sections of the website say "Coming Soon" but from sections which are done, see the index (notice the scroll bar at the bottom!) you'll see the professional quality that's gone into make the website look so slick. This page has a nice demonstration of the Sieve of Eratosthenes.

Unfortunately, the details aren't always correct. The page on Graph Theory talks about Euler living in Konigsberg and thinking about what later becomes known as the Konigsberg bridge problem. Except that's not true. I've got 2 references that say otherwise. Reference 1: "The Truth about Konigsberg" by Brian Hopkins and Robin J. Wilson. Yes, the Robin Wilson.

Reference 2: "Early Writings on Graph Theory: Euler Circuits and the Konigsberg Bridge Problem"; the PDF can be found here on the internet.

Page 4 of the reference 1 says that 3 letters from the Archive Collection of the Academy of Sciences in St Petersburg give us some information on the origins of the Konigsberg bridge problem. Euler learned about the problem through correspondence with Carl Leonhard Gottlieb Ehler. A scan of Euler's letter is in Fig 6 of the PDF. These references also make it clear that the modern depiction of graphs weren't used by Euler; you can even see the diagrams he used in the original paper. As reference 1 indicates (page 199) those pictures of graph that we use today first appeared in the "...second half of the nineteenth century".

It would be best, of course, if all the information was accurate but for now the Mathigon website looks like it will be a promising way to supplement the your class teaching. Bookmark the page now and check back in a couple of months.

Problems: de Moivre's Formula

Officially, mathematical induction is taught at the high school level but unofficially it doesn't seem to be taught much at all. It's easy to see why. Start with an ambitious pacing guide that has every class covering something. Now remove time to get classes started (books handed out, time to explain the rules and procedures, students adding and dropping classes) and you've lost at least a week of class. Now take out time that administration uses for special assemblies, school trips, and projects and you've lost at least 2 weeks more. Remove yet more time for giving and taking quizzes and tests and reviewing the results. None of that gets counted on the pacing guide so a lot of class time is lost and needs to be made up. Where can the time be found? Since the teacher performance is based on how their students perform on multiple choice tests from the county/state (and induction doesn't get covered on the exams) and many teachers lack the math background to teach it, induction tends to get axed. The result? By the time they get out of high school, students have seen lots of math that is essentially worthless (orthocenter and circumcenter) but have no exposure to induction. That's tragic, because induction is one of the the most useful and amazing "tools" that mathematicians use frequently.

For those of you who teach induction, de Moivre's Formula is a natural exercise for induction because seeing de Moivre's formula again (they learned it for complex numbers) helps to reinforce the result.

Use induction to show, for all positive integers n,

$latex (\cos(\theta)+i \sin(\theta))^n = \cos(n\theta)+i\sin(n\theta)$

I've posted the problem on the Problems page.


If you have Linux or Windows operating system and you teach at the high school level, then you should take a look at Xournal. This note taking software program reminds me of an interactive smarboard. The standard background is white college ruled notebook paper, but you can change the color of the paper as well the style (blank paper or graph paper); you can even take notes on pictures and PDFs. In addition to the standard tools of pens (with varying thickness), eraser, choice of pen color, and the option to use a stylus/pen for input you also get a highlighter, the ability to type text (which includes a font that looks handwritten), a tool to let you create straight lines, and a shape recognizer that will take your drawing of a triangle, quadrilateral, or circle and make it perfect. Here's a screenshot of Xournal in action (click on the image to make it larger):

If you haven't tried it before, you should. Without a pen/stylus, I found drawing with the mouse is a little clunky but with a stylus this would be a really useful program. Xournal gives you the option to export to a PDF file which means you can work out the solutions for homework in Xournal and then distribute the results as a PDF file.

Euclid: The primes are infinite

One of the most beautiful and timeless theorems of mathematics is Euclid's proof that the number of primes is infinite. The proof can fit into the indirect proof section of the geometry curriculum: Euclid, after all, is considered to be the father of geometry.  I haven't seen the proof in a high school geometry book but it's such an important result. Every math student must see it. But you should develop Euclid's idea before going over the proof. Have your students add 1 to the product of several primes and observe that the resulting number is prime bigger than the others or is a composite number with a factor which is a prime different from what you used to create the number. Is that always true? Once they understand the process of creating a number by adding 1 to the product of several primes show them the following process: start with any prime and add 1 to it. You either have a new prime or a composite with some new prime factor (maybe more than one new prime). Multiply the primes together and add 1 to create a new number which will lead to another prime that can be added to your list. As the number of primes increases, use technology to help you form 1 plus the product of primes as well as to factor this new number. This algorithm creates a list of primes that grows after each iteration, so no matter how many primes you have we can find another. That is, the number of primes will be infinite. This seems, to me, to be the best way to explain the theorem before you prove.

Some background is required for the proof of Euclid's theorem, but it's rather straightforward. You can find that background, and Euclid's proof, on the Handouts page.