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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Working together: the harmonic mean

topSuppose that you have a friend helping you rake the yard. If you take three hours to finish the job by yourself, and your friend takes two hours by himself, then, if you work together efficiently, how long does it take for you to finish the job working together? The answer is one hour and twelve minutes. This post is about a tool called the harmonic mean that lets you do calculations like that. The harmonic mean also shows up in some of the laws of physics, which is interesting. You may remember that “mean” is the highfalutin’ mathematical term for average. We will also talk about the sense in which the harmonic mean is an average, even though it does not act like the averaging you are used to.

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Doing the Space Warp!

topHave you ever tried to enlarge a picture and lost detail? Maybe individual pixels became big single-color squares? The skull to the left is made of circles, ellipses, and rectangles, glued together as a series of tests. For a point in the plane, Occupy Math’s computer checks which of these geometric objects a point is inside of or outside of and, from that, deduces a shade: black or white. This may seem to be a lot of trouble to go to for a sorta-okay picture of a skull. The reason for going to that much trouble is that membership in the black parts of the skull, since we are asking about numerical points (x,y), has a resolution of trillions of pixels by trillions of pixels. This means that if we warp space, we will not get pixelization errors or other distortions. In other words, this skull image is far more malleable than usual images stored as pixels. In this post we are going to apply space-warps to the basic skull image. Having explained about the skulls, it is worth noting that Occupy Math created all this stuff while playing around but the math used to make many different skulls turned out to have applications in statistics. Following the fun parts of your work not only relieves stress, it can have big payoffs.

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Parental Sabotage of Education

TestOccupy Math had a difficult year this time around. He taught first, second, and forth semester calculus and four instances of a research capstone course, which is a one-on-one research project to let seniors figure out if they like doing research. These student research projects will show up next week on Occupy Math. This week’s news is this: Occupy Math took of more points from his calculus students for not being able to do arithmetic and basic algebra than he did for not learning the calculus. Overall it was a good year, but there are some disturbing trends that Occupy Math would like to talk about including collapsing instructional resources from high school through university and absolute sabotage of the educational system by deeply confused parents.

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Students of the world! You have nothing to lose but your illusion of control!

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The Illusion of Control is the natural tendency of people to overestimate how much control they have over a situation. When you are in control, it increases your confidence and decreases your fear, so people like to feel in control. In this post Occupy Math will discuss an instance of the illusion of control that decreases students’ grades on a math test. This post raises and discusses a point about how to teach effectively and Occupy Math’s position on the issue is debatable and disliked by many of his students. Interested? Read on!

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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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Adding up infinitely many numbers

topThis week’s post looks at circumstances when you can add up an infinite number of numbers and get a single number as the answer. The classical example is adding one-half plus one-fourth plus one-eighth and so on, forever. The picture at the left shows that the total is (i) bigger than any number smaller than one and (ii) no more than one. This means that the total is, in fact, one. This follows from the fact that the set of numbers bigger than any number smaller than one but not bigger than one is… …one. When Occupy Math teaches this in calculus class, he uses a technical trick called a limit, but the reasoning that says the total is one is solid as presented.

The trick for the picture at the beginning of the post is that, every time we add one of the numbers to the picture, the part that’s left gets half as big. If you divide something in half forever, it becomes smaller than any positive number — which means it becomes zero. This means that the total of one-half plus one-fourth and so on is within zero of one; in other words, it is one.

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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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Grading with rubrics, why and why not.

topThe Eberly Center at Carnegie Mellon University says:

A rubric is a scoring tool that explicitly represents the performance expectations for an assignment or piece of work. A rubric divides the assigned work into component parts and provides clear descriptions of the characteristics of the work associated with each component, at varying levels of mastery.

Occupy Math is sometimes one of the three instructors for a course called Topics in Bioinformatics. This is a course where new students in the bioinformatics program read, discuss, and evaluate papers that introduce the research areas in bioinformatics. The original grading rubric for this bioinformatics course, if followed carefully, would have flunked every student that chose a mathematical or computational tool paper to evaluate. This problem arose from cultural disjunctions between different disciplines in an interdisciplinary field — but it shows that rubrics are potentially dangerous. This post critically examines rubrics which can be useful or deadly.

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