One of Occupy Math’s research areas is figuring out how to get computers to produce game content more-or-less automatically. This post announces a book that summarizes many of Occupy Math’s findings. You can buy a copy from my publisher Morgan and Claypool, but if you are part of a university or other institution that subscribes to the Morgan and Claypool synthesis series then an e-book with unlimited use for students and faculty will show up in your library presently. Profits go to Occupy Math’s consulting company, which funds students and research!
Occupy Math’s image of the week, this week, is another generalized Sierpinski fractal. This one is a bit spiky.
Occupy Math looks at many different situations where mathematics (or the lack of mathematics) are important to people. One of our many threads is activities and information for teachers and parents. This post provides an index to these activity posts and then an index to some of the informational posts that might give helpful background. One thing to keep in mind — if there is a type of activity you might like to see, let Occupy Math know (e-mail email@example.com).
A Julia set with an added rotation is this weeks Image of the Week on Occupy Math.
This week’s Occupy Math looks at a type of puzzle where you want to fill a rectangle with a shape. We will be using the L-shaped 3-square polyomino, used to fill a 5×9 rectangle below, as our example shape. The goal is to figure out every possible size of rectangle that can be filled with this shape. If you are constructing puzzles for other people — e.g., your students — knowing which problems can be solved gives you an edge. The post will not only solve the problem for our example shape, but also give you tools for doing this for other shapes. The answers, and the tools, are at the bottom if you don’t feel like working through the reasoning.
The week’s image on Occupy Math is a generalized Sierpinski fractal.
This post is the next in Occupy Math’s series of activities for teachers, students, and parents. There is a good deal of basic algebra and geometry that can be practiced with puzzles on grids. These puzzles are intended for grades 6-10, depending on how much the students know about polygons and finding areas. You can use standard graph paper to supply the grids for these puzzles. We start with the following puzzle. If we assume the grids have spacing one, draw a square whose corners are on grids and that has an area of exactly five.
A student will have a natural desire to draw a square with vertical and horizontal edges — but the squares that can be drawn that way on this grid have areas of 1, 4, 9, and 16 — that is all. How do we get five? Read on to find out.
A two-parameter cubic Julia set! Quadrant convergence and lurid colors.
We all know what the average of a bunch of numbers is: you add up the numbers and divide the total by the number of numbers you added up and you get something in the middle. This sort of average assumes that all the numbers you are adding up are equally important. There are many situations where the numbers are not equally important. This is why we have weighted averages — a weight is another number that says how important a number is in the group of values you are averaging.
The simplest example is when the weights are positive and add to one. This is something that often happens with grades. The teacher might say that “The final grade is 50% of your homework grade plus 30% of your quiz grade plus 20% of your final exam grade”. In this case the weights are 0.5, 0.3, and 0.2. If a student had 86% on homework, 72% on quizzes, and a 91% on their final then their grade would be 0.5×86%+0.3× 72%+0.2× 91%=82.8% for their final grade.
This is another generalized Sierpinski fractal, but with a Markov chain controlling the transitions. It is like a super-snowflake.