One problem with math education is that access is very uneven. In theory, the internet can level things out a bit. This post is the next in Occupy Math’s series of activities that parents and teachers can use for enrichment and enhancement. For free.
Here is an example of a “guess-the-next-number” puzzle — “what is the next number in the sequence 2,5,8,11,14,?” The answer is 17. The student should figure out that the terms of the sequence increase by three every time, and 14+3=17. This post is about constructing this sort of puzzle with some notes on how to make harder and easier puzzles. Puzzles like this are good arenas to practice math skills. They can be structured as contests which is motivational, and with the information in this post, they can be tuned to your student’s needs.
Occupy Math just got back from Schloss Dagstuhl, a castle in the woods in Germany not too far from Luxembourg, where there are week-long research meetings. This meeting was on using artificial intelligence for games. One of the outcomes of this meeting was a simple game used to explore cooperation between humans and artificial intelligence — which Occupy Math thought had potential as an activity to introduce the mind set for programming. This activity is in today’s post. One of the groups was looking at making artificial intelligence to cooperate with human beings. They invented what they thought was the simplest possible coordination game. The game works like this.
- Both players pick a number from 1-100 and write the numbers down secretly.
- The numbers are revealed. If they are the same, the game is done.
- If the number are not the same, the players write numbers again and repeat step 2.
- Other than revealing numbers, the players are not allowed to communicate.
The goal of the game is to coordinate — get to the same number — in the smallest number of steps. A play of the game that takes 8 steps is shown at the top of the post. In the rest of the post, we give a couple of ways to use this game as an activity, and we also explain why the researchers do not think, after having people play it, that this game is simple.
This post is intended for the teachers of students who are just starting out with fractions, or who need to improve their understanding of fractions. The basic idea is simple: sort a list of fractions into ascending order. This can be done a number of ways, from reducing the fractions to decimal numbers and then sorting those, to using the cross-multiplication trick shown near the bottom of the post. There are also several special purpose shortcuts that solve parts of the problem, making this an exercise in problem solving as well, especially if the fraction sorting is done as a race or under time pressure.
In this post we look at several ways traditional textbook publishers are poisoning math instruction — charging insane prices for books, writing lectures for professors (bad ones), and doing a bad job of generating problems for practice, homework, and examinations. We begin with the price issue. A while back, Occupy Math announced he had written a calculus book, Fast Start Calculus for Integrated Physics. The material from the book is being republished as the three books pictured at the top of the post. This is part of Occupy Math’s war on outrageous textbook prices. These books are distributed at low cost to a university and zero cost to students at participating universities.
In this week’s post we look at a game for practicing recognition of multiples of a number. It can be used in class by a teacher or it might be a good game to play with your kid by way of some math practice. The game prepares a student for multiplication by practicing how to recognize which numbers are multiples of five. The numbers 5, 10, 15, and so on generate scores when they come up in the course of the game. There are multiple versions of the game (for other numbers besides five), but the demonstrations are based on recognizing multiples of five. The game is a little like dominos and it’s a little like a crossword puzzle.
This week on Occupy Math, we have a new puzzle to help make arithmetic practice go down more smoothly. Look at the array of numbers above. To play the puzzle you take groups of three numbers, either in a row or in an corner-shaped group of three squares. Examples appear below the fold. The numbers in the example above are chosen so that the groups of three numbers, when added, include all the numbers from zero to twenty seven. The goal? Gotta find them all. Warning: if you write down a 4×4 array of digits at random, then your odds of being able to get all the numbers are bad. Occupy Math used a design trick to create that example.
Whole numbers are generally easier to work with than numbers with a lot of digits after the decimal place. The puzzles posed in this post appear to be purely about whole numbers; they ask students to put squares on a grid with a spacing of one. The answers to the puzzles, however, require the use of radicals. The post thus opens the door to irrational numbers. Occupy Math’s editor said that she thinks her personal journey through these puzzles would be increasing frustration ending in rage and a sense of having been deceived when the notion of rotating the square finally occurred to her or was suggested by a collaborator. This perhaps speaks to the need for the teacher or parent to be ready with a hint if the frustration is too deep. It is also worth stating that the reward for the pain of frustration is the sense of accomplishment that comes from understanding or, even better, discovering a new thing.
This post fits into Occupy Math’s series of activity posts. It is easy to learn that the area of a square is the square of the length of its side. In fact we named the mathematical operation “squaring a number” after the area formula for geometric squares. Here is a puzzle problem: Put the corners of a square on the points of a regular grid with points at a spacing of one so that the square has an area of 5. Initially, since the square root of five is not a whole number, this seems like a very hard problem — the trick appears in the picture above. Each side of the tilted square is the hypotenuse of a right triangle with side lengths 2 and 1 — making the side length √5. It is possible to practice and review quite a bit of geometry with puzzles on regular grids with grid spaces at distance one. These puzzles require the Pythagorean theorem and the area formula for rectangles and triangles.
In today’s post, Occupy Math will show you a family of puzzles that help you sharpen your logic skills. These puzzles are Occupy Math’s expanded version of some wonderful exercises developed by Peter Harrison called Bovine Math. Dr. Harrison’s exercises are a bridge from arithmetic to algebra, trying to ease the mental transition from concrete numbers to the abstraction of having variables. If you are a teacher unfamiliar with bovine math, definitely follow the link. Occupy Math notices that these wonderful exercises could be formalized as graph theory and found a family of puzzles. This is not Occupy Math’s first foray into educational puzzles that use graph theory. The puzzles in this post are for grades 2 and up, unless you learn to add before that. The larger the puzzle, the harder it is.
One of the big problems with teaching arithmetic is that we carefully structure the learning process to be boring. Drill is effective, but only if the students do it, and some will not. In this post we present a challenging series of puzzles (and techniques for creating more puzzles) to wash some of the boredom out of learning arithmetic. The picture below shows the solution to a viral puzzle called four fours. The puzzle asks the student to make the numbers 0 to 20 out of four copies of the digit four. The term digit is used judiciously because it’s not quite the number four. Notice that “44” counts as two fours. The number of other symbols used is unlimited and sum, difference, product, quotient, power, square root, factorial, and parenthesis are all available. The solutions shown are far from unique. Consider 4+4+4+4=16 or 4×(4/4+4)=20, for example. This post extends the problem and provides keys for five fives and six sixes as well as proposing additional activities.
This post is an activity post with activities for grades four and above. It uses a kind of math that Occupy Math has talked about before in A Wonder of Mathematics, but the post is structured to help a teacher or parent present this material as a discovery activity. The core of the exercise is a simple three-part rule:
- If a number is a multiple of three, divide it by three.
- If a number is one more than a multiple of three, multiply it by five.
- If a number is two more than a multiple of three, add one to it.
So, for example, 4 equals 3+1 and so gets multiplied by five, yielding 4×5=20. It is a little startling how many odd properties this three-part rule has.