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.
This 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.
This 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.
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?
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.
This is another in Occupy Math’s series of activity posts. A bouquet puzzle presents some bouquets of flowers, together with their price, and then presents another bouquet with an unknown price. These puzzles are intended for 4th through 8th grade, though they can be fun at any point after a student masters addition. This post starts with an example of a bouquet puzzle. These puzzles also prepare students for abstract reasoning and for solving simultaneous linear equations, an important skill that appears from business to calculus. In this post we will give several examples of bouquet puzzles and also give a technique for creating your own puzzles that yields puzzles that can be solved.
Some calculators actually work on fractions directly. This week’s post is not about those. This week we look at how to add fractions with a regular calculator and get fractions back at the end. Occupy Math looked up “adding fractions with a calculator” on the web and found some remarkably unhelpful pages. They give instructions on how to add fractions with a pencil and paper, only you could use a calculator to do the arithmetic. This post is not deep, but it may be helpful, and it shows you how to simplify adding fractions with a calculator.
This week Occupy Math has another activity post, this one is not only a pretty good puzzle, it uses the kind of thinking you use to learn programming. It can be used in grades 2-12, though what is emphasized will be different at different grade levels. Here’s how the game works. You work with a simple calculator with a display window and a memory, like the one shown above (but on paper, at least until an app it built). There are only three keys that do the following:
- “Load one” puts a one in the display.
- “Save” writes the display to the memory.
- “Add” adds the contents of the memory to the display.
You start with a number on the screen (and zero in memory). There is a target number N. The puzzle is to generate N, from the starting number, using the calculator. For example, “Starting with the number 2, generate the number 31” can be done this way:
This example shows how to work these puzzles on paper.
A closed curve is a shape you can draw, without lifting your pencil, that begins and ends at the same place. An example of a closed curve is the boundary of the shape shown above. In this post Occupy Math is going to use a type of shape he discovered by accident that makes a large number of interesting closed curves. This is one of Occupy Math’s activity posts. For the accompanying booklet with questions for students and fifty shapes click here. Here are the shapes as clip art PNG files in a zip archive. This activity is intended for K-4 students. The rest of the post is about the math used to build the activity.
This week on Occupy Math we are looking at the Ultimatum Game. This is a two-player mathematical game with two roles. The game has $100 at stake. The first player, the proposer, suggests a division of the money. The second player, the responder can accept or reject the proposal. If he accepts, the money is paid out according to the proposed split; otherwise, no money is paid out. This game is used by economists to understand economic behavior. In this post we discuss some of the issues with using this game and propose an activity based on the ultimatum game.
Look carefully at the M. C. Escher picture to the left. If an ant keeps moving forward, he will end up on both sides of the strip he is walking on. This suggests that the strip has only one side (neglecting the edges of the strip) which is pretty cool. The funny walkway is a Möbius strip which does have only one side. Making one-sided objects is only a small part of what the math that makes Möbius strips can do. The same trick can also show you that 1980s video games were being played on the surface of a doughnut, give you a way to understand four-dimensional objects like Klein bottles, and even let us describe the shape of the observable universe.