Profit Mad Publishers Poisoning Math Instruction!

topIn 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.

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The Multiple Game

topIn 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.

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Number Triple Puzzles

topThis 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.

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The Geometry of Frustration and Discovery

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.

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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.

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Graph puzzles to sharpen your skills

topIn 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.

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Arithmetic, hold the boredom: digit puzzles

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.

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An arithmetic activity — with unsolved puzzles!

Count Calculator How To Calculate OfficeThis 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:

  1. If a number is a multiple of three, divide it by three.
  2. If a number is one more than a multiple of three, multiply it by five.
  3. 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.

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The Golden Ratio: Fibonacci Magic

topThis post contains an activity you can do with a calculator, a bit of a magic trick. The post is also about a very special number called the golden ratio. The golden ratio is not as famous as pi or e, but it keeps showing up again and again in multiple contexts. The spiral above is made by choosing quarter-circles that cover two earlier quarter circles, along their sides. After going through the activity, Occupy Math will reveal how this spiral involved the golden ratio.

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Hidden patterns in rational numbers

topThis post discusses the decimal form of rational numbers. A rational number is a fraction whose numerator and denominator are both whole numbers, like 2/5 or 1/3. Whole numbers are rational as well, because you can put them over one – 3/1, for example, is a witness that 3 is a rational number. Every rational number can be written in many ways: 1/2 equals 2/4 equals 3/6 and so on. We usually choose the version of a fraction that has no common factors between the numerator (top) and denominator (bottom), which gives us a unique way to write a fraction. This is all just background. What we want to look into is that every rational number, when written in decimal form, falls into a repeating cycle of digits. For example 1/3=0.333333333 … going on forever.

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Can you add it up? An activity.

This post is the next of Occupy Math’s series on problem factories. Problem factories are a body of mathematical knowledge that, once you understand it, lets you generate many problems and, hopefully, multiple types of problems. The class of puzzles in this post is exemplified by the question “Can you write 25 as the sum of consecutive numbers?” A really clever student might realize that the number 25, by itself, is one consecutive number but if we forbid this answer by fiat, then there is a picture that answers the question. How far can this sort of picture-proof be pushed?

p25

The picture gives us 3+4+5+6+7=25, but there is also 12+13 and 25 by itself. There are also some answers that use negative numbers like -2-1+0+1+2+3+4+5+6+7=25. The fact there are often many answers makes this problem more interesting.

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