The following article was published in volume 13 of Animation Journal edited by Maureen Furniss. Please note that most of the links below are not active anymore.
Animating the 4th Dimension
Since the later part of the 19th century people have been interested in the fourth dimension. Very popular at that time it fascinated and left room for all kinds of speculations. A lot of assumptions have been made. Some people even assumed it must be the place where god resides, heaven. Especially artists took a great interest in the fourth dimension. Writers like H. G. Wells, Alfred Jarry and Fyodor Dostoevsky and avant-garde artists like Marcel Duchamp, Francis Picabia, František Kupka, Kazimir Malevich, Umberto Boccioni and others were influenced by its theoretical implications. Marcel Duchamp even studied higher dimensions very seriously. In his work The Bride Stripped Bare By Her Bachelors, Even (The Large Glass) the bride (see picture below, left) was meant to be a four-dimensional entity in contrast to the more simply constructed three-dimensional bachelors.
Another example would be Unique Forms of Continuity in Space (see above, right) a sculpture by Umberto Boccioni.
It can be said that some of the avant-garde styles like Cubism, Futurism, and Constructivism emerging at that time were highly influenced by the expanded vision of space the fourth dimension offered. This all changed when Einstein made time the fourth dimension in his theory of relativity. The extra dimension that was undefined and mysterious before now served a scientific purpose to better understand our universe. People became more interested in Einstein’s discoveries and what would result from the assumptions he made. But with the rise of computer technology some scientists returned to the fourth dimension for purely geometrical studies. Through animating four-dimensional objects in motion not only fascinating abstract images emerge but also the almost incomprehensible, dry theory becomes a bit more accessible by rather seeing patterns and structures than reading about them (unless you are a mathematician of course).
With this essay I’d like to share my fascination with the fourth dimension in motion that brought me to making “Gestalt” a computer animation featuring four-dimensional fractals called quaternions. Another subject I’d like to touch briefly is fractals. I have long been fascinated with fractals. When my brother showed them to me on his computer I was in awe that these beautiful patterns not only have such interesting features like self similarity but were also created by using only a simple formula. We tried to make small animations of zooming in their structure by taking single frames from the monitor screen with our dad’s super 8 camera while we were changing the parameters of the formula. The results were not very satisfying for both of us due to the technical limitations.
But first please let me introduce you to the fundamentals of understanding four-dimensional geometry. Next I’ll try to give a brief introduction to quaternions, the fractal entities living in 4d space. Freshly equipped with a basic understanding of the subject matter I will then give some examples of 4d related animations artists and scientists have done so far.
In 1966 Michael Nöll and his associates at Bell Laboratories produced an animation of a hypercube using computer graphics. It was the first time a hypercube was visualized in motion. Later in 1978 Thomas Banchoff and Charles Strauss produced the film “The Hypercube: Projections and Slicing”. This award winning film shows a more complete approach to the animated hypercube. A hypercube is a four-dimensional cube. Now what does that mean?
Before I try to explain the concept of four-dimensionality I’d like to give a few words of advice. To understand the fourth dimension, don’t try to understand the fourth dimension. With this cryptic sentence I’d like to point out that you should not try too hard to ‘understand’ the fourth dimension, e.g. how it looks like in the real world. There is no fourth dimension in reality. It’s a concept of an artificial space that bears no direct similarity to real space. Try to think of it as geometrical extravaganza. It is constructed with mathematical means by analyzing three-dimensional space and how the three dimensions relate to each other. The extra dimension is added due to the geometrical laws found in the first three. Think of it like for example the concept of eternity which you don’t find in the real world either or the concept of negative numerical values. In reality you can count the things you see but things are either there (positive value) or not there (zero value) but how can they be negative? The concept of negative values only makes sense if you work with numbers in an abstract way. Of course we are used to them and they don’t appear strange to us anymore but it is an altogether abstract and artificial concept similar to the one of a four-dimensional space.
To get an idea of the fourth dimension maybe it would be a good start to begin with a point. A point has zero dimensions. It only exists in itself and has no possibilities to move anywhere. If we put another point at the right side beside the first one and connect them we get a line. A line has one dimension. It is now possible for the zero-dimensional point to travel within the line in a left and right manner. Now we move the line perpendicular to itself in a plane and we get a square. A square has two dimensions. The point can now travel left and right as well as up and down. Moving the square perpendicular to itself results in a cube. A cube has three dimensions. The point is happy because it can travel even more. It can now move in three directions: left and right (x-direction), up and down (y-direction) and back and forth (z-direction). So now to make the point even happier we move the cube perpendicular to itself in kata direction to get a hypercube. So here it is, the four-dimensional hypercube. The point can now move in four directions: left and right (x-direction), up and down (y-direction), back and forth (z-direction) and ana and kata (w-direction).
As you might have noticed the point can travel now in a direction (ana and kata or w-direction) that we, bound to three-dimensional space, can’t go. This is the new direction that makes the fourth dimension. But what about time being the fourth dimension? Well, time can be the fourth dimension. It would then be called four-dimensional space-time and is used in Einstein’s famous Theory of Relativity. In geometry it is possible to extend space to as many dimensions as you want. The same way we moved the cube perpendicular to itself into hyperspace (four-dimensional space) to get the hypercube we can lift the hypercube perpendicular to itself into five-dimensional space to get a five-dimensional hypercube. This could be done again and again up to n-dimensions. For our zero-dimensional point each additional dimension means one direction more to move in. Since our three-dimensional space can be described in mathematical language it is possible to extend it by mathematical means even though in our physical world we do not have access to it. How to Visualize the Hypercube
Even though we cannot grasp the fourth dimension directly there are several ways to access it. All of the following methods were developed by Charles Howard Hinton. First, in order to visualize the hypercube it would be good to know its geometrical attributes. Compared to a cube which can be described geometrically as consisting of 8 vertices (0 dimensional object), 12 lines (1 dimensional object), 6 faces (2 dimensional object) and the cube itself (3 dimensional object) the hypercube can be described as being made of 16 vertices, 32 lines, 24 faces, 8 cubes and 1 hypercube. One method to get an idea of the hypercube is to unfold it. Below you find a picture of an unfolded hypercube called a tesseract next to an unfolded cube for comparison:
Whereas the cube unfolds into 2d space and is made of six squares the hypercube unfolds into 3d space and is made of 8 cubes. Another way of visualizing a hypercube would be to project it from 4d space into 3d space. The next picture shows a projected hypercube:
Here you can also see the 8 cubes. Clearly recognizable is the big cube with the smaller cube inside. There are also six cubes around the small one which connect it to the big one. Because of the projection they are distorted so they look like six pyramids with their apex cut off. All cubes have originally the same size but are distorted due to the projection. The small cube is farther away in kata direction (w-direction) from the big cube and the six other cubes connect the two cubes in a way four squares connect the nearest square with the farthest square in a projection of a cube onto a 2d surface. A 4d to 3d projection is quite similar to a 3d to 2d projection. We are used to 3d projections because we see them all the time in pictures or in movies. In order to visualize depth in a 3d projection on a 2d surface we use a vanishing point and make the objects farther away smaller. In the 4d projection into 3d space the objects that are farther away in kata direction are also shown smaller. Now in order to animate the hypercube you can rotate it and the projection will look like this:
You can see the dark inner cube change position due to the rotation in 4d space. Also the cube is squashed during the process.
Another way to visualize a hypercube is by continually slicing it from top to bottom. Again let’s start with a cube for comparison. If you take a cube and want to slice it you have several possibilities to do so. You can slice it beginning with a face which would result in several identical squares. Also you can slice it edge first. This would result in a series of rectangles starting and ending with a line. The rectangles will widen while keeping the same length until, halfway through the cube, the transformation will revert itself. Now another method of slicing would be to begin slicing vertices first. First we get a point which becomes a triangle. The triangle increases in size until it transforms into a hexagon halfway through the cube. There, again, the process reverts itself. Please consult the picture for details:
What now if we slice a hypercube? First of all we’ll get 3d slices. If we slice cube first we only get a bunch of identical cubes. By slicing face first we get a square which will get thicker until it becomes an oblong box halfway through the hypercube. It then will diminish in thickness until it will be the square it was in the beginning. So basically the slices are very similar to the slices of the cube beginning with the edge. Just an extra dimension (thickness) was added (cube => hypercube: line => square, rectangle => oblong box). If we now slice the hypercube edge first the slices will be similar to the ones of the cube beginning vertices first. You just have to add that extra dimension of thickness to it. So it would start with a line expanding to a triangular prism with its height exactly the same as the original edge. Halfway through the hypercube the triangular prism changes to a hexagonal prism and than changes back to the triangular prism and finally the line (point => line, triangle => triangular prism, hexagon => hexagonal prism). But the most interesting slices we do get if we slice the hypercube vertices first. First we get a point that transforms into a small triangular pyramid, a tetrahedron, which increases in size until its vertices are cut off . Its size increases until it changes into an octahedron when it is in the middle of the hypercube. It then shrinks back to the tetrahedron (rotated by 45 degrees around the x-axis) with its vertices cut off until it becomes the full tetrahedron again which shrinks further until it’s the point it was at the beginning. You can see the different slices here:
Now you can just put all the slices you made for one frame each and you have an animation of a continuous 3d slide of a hypercube. Quaternions
Quaternions are a group of numbers that consist of one real number and three imaginary numbers. So they are made of actually four numbers altogether. They are fundamental for quaternion fractals a special group of fractals that reside in 4d space. Those quaternion fractals can be projected into 3d space and so be made visible as 3d objects. You may have heard about Mandelbrot sets or Julia sets. These are 2d fractals. A common Mandelbrot fractal looks like this:
The base of 2d fractals is complex numbers which consist of one real number and one imaginary number. Complex numbers were discovered by Leonard Euler trying to solve the equation x² = -1. Since this is not possible with real numbers alone he had to invent imaginary numbers to solve it. Almost a century later William Rowan Hamilton discovered quaternions which even had three imaginary numbers beside the real number. Quaternions play an important role in quantum mechanics. But included in a fractal geometry framework they also provide a powerful visual stimulus. Here a picture of a quaternion fractal:
Instead of further explaining the technical side of quaternion fractals I’d rather like to give some personal account of my own work in the realms of 4d quaternion animation.
While I was investigating 4d geometry, I stumbled upon quaternion fractals. I was fascinated by the universe of bizarre shapes and forms that could be expressed by a single mathematical formula. Wondering how it would look if I changed some parameters over time, I experimented with animating them. I was captivated by the results. The program I used for renderingquaternions I found on the web. It's called Quat and was written by Dirk Meyer. It’s a very powerful quaternion fractal generator that is freeware. Unfortunately it is only possible to render single images with this software. So I had to write my own little program that would feed Quat with data for sequences of slowly transforming shapes to render. I fiddled with every parameter that could be changed and did series of images while gradually changing one or more parameters to test if something promising would develop. This procedure posed a real challenge to my patience. Though I did only small resolution test renderings they sometimes took half an eternity to finish. But at the same time I was really enthusiastic about the results I got. When I had enough material I began rendering the sequences I liked most in a higher resolution. That was the biggest challenge to my patience ever. It took almost a year and would have taken even longer without the help of family and friends who donated computer time. So after the renderings were finished I just had to put the sequences together in a way that made sense to me. I tried to show many facets of the mathematical body and transformations of it. For the soundtrack I referred to FM-Synthesis. I like the sound of it and found it matched the bizarre, highly artificial, somehow alien shapes. I named the film ‘Gestalt’ and it’s available for download on my website.
Now please let me elaborate further on my fascination with fractals, the fourth dimension and mathematics in general. It was Pythagoras who made the belief in numbers central to his worldview. He founded the Pythagorean Order who tried to live science, to directly integrate scientific observation in practical life. Their aim was to understand the divine order of the world. It was believed that this order was a mathematical one. For them numbers weren’t arithmetic alone but entities of their own. It was these entities that were divine because in their worldview they alone had the power to fully explain the universe. In modern physics today there is one branch that dedicates their efforts to finding the one theory to describe the order of the universe. Superstring Theory. This theory is by no means completely understood even by the people working on it. It’s said that scientists stumbled on this theory that was originally meant for the next century to discover. And it’s also not quite clear if this theory actually works. There is a lot of controversy. But the scientists who dedicate their life to it are captivated by its elegance. Superstring Theory uses higher dimensional space. It places unimaginably small vibrating strings as the basis for all matter and comes in different flavours of which the ‘smallest’ is embedded in a 10 dimensional space and the ‘biggest’ makes use of a 26 dimensional space. Superstring Theory is an attempt to unify all four forces (gravity, electromagnetic force, weak nuclear force and strong nuclear force) which according to physicists constitute our universe. These aspects of numbers, mathematics and science are central for my approach to ‘Gestalt’. Actually working with the tools (numbers, mathematics) used to explain the fabrics of space I sometimes felt like a painter who some centuries ago used to paint biblical scenes to let the beholder feel a glimpse of the divine.
If you browse the internet looking for 4d related sites you find very diverse applications of 4d geometry. I’ll try to give a short summary of what I found during my quest for 4d stimulus.
Hyper is a playable demo of a first-person-shooter dealing with an extra fourth dimension. Added to the conventional FPS control keys are two extra keys that move the player in ana and kata direction in 4d space. If you go in this extra direction the space around you changes, the room transforms. Doors may appear that weren’t there before. Unfortunately this demo presents only a rough sketch. The gameplay as such is not very elaborated. But it introduces a totally new element to the FPS genre.
Other 4d extensions of popular games are:
4D Rubik’s Cube
4D Rubik’s Cube Java Applet
4D Maze another version
I’d like to focus on three contemporary artists influenced by the fourth dimension. First the painter Tony Robbin, who also wrote a book on the fourth dimension and art called ‘Fourfield: Computers, Art and the 4 th Dimension’. He also wrote some programs to visualize higher dimensional objects which can be downloaded at his website.For his paintings he transfixes complex patterns derived from his research in 4d geometry on canvas. On some of them wire arrangements are added to provide even more depth. With the wire structures changing in perspective when passing the painting the painted structures remain fixed on the canvas-space. This interplay of perspective and motion directly refers to phenomena that can be observed in hyperspace.
Another contemporary artist working with 4d geometry among other tools is the ‘transarchitect’ Marcos Novak. His liquid architectures / transarchitectures as he names them are highly elaborated spaces that exist in cyberspace. cyberspace here means the space that only can be provided by a computer and far surpasses the notion of space as we know it. Whereas architects use the computer as a tool to construct for the real world he rather constructs for the same space he constructs with, the virtual space. Since the virtual space of the computer is made of mathematics alone it can be extended and broadened by mathematical means. Marcos Novak makes full use of this fact.
Last but not least there is Manfred Mohr. He came from Action Painting and later developed his unique style of cube semiotics that are calculated and plotted by computers. From the 3d cube he moved on to higher dimensional cubes. He says about his work: “In my research, since 1973 I have been concentrating on fracturing the symmetry of a cube (including n-dimensional hypercubes since 1978), using the structure of the cube as a "system" and "alphabet". What interests me, are the two-dimensional signs (êtres- graphiques) and their visual ambiguity resulting from the projection of the lines of the cubes from higher dimensions into two-dimensions. I describe them as unstable signs because they evoke visual unrest.”
A lot of 4d animations can be found on the web. Most of them are very short. They show for example what a 4d object looks like when rotated in 4d space. Also the slicing method is sometimes used to explore certain properties of 4d objects.
4d Star Polytope Animations
Stereoscopic Animated Hypercube
Thomas Banchoff’s Project List Very interesting animation studies by the author of ‘Beyond the Third Dimension’ and producer of ‘The Hypercube: Projections and Slicing’.
Gallery of Mathematics Very nice animations of for example a hyper-torus and an inflated hypercube. Also very thorough information on the mathematics involved.
Hyperspace Star Polytope Slicer A java applet with lots of parameters to fiddle with. Should not be missed.
Lots of quaternion animations [unfortunately hypercomplex.org no longer exists]
Among other things a vrml model of a Calabi-Yau manifold a 6d object that is important to Superstring Theory.
Beyond the Third Dimension This virtual exhibition consists of mathematical objects that are visualized in the computer. It shows relations between Mathematics, Art and Computer Graphics. Its images try to suggest geometrical intuitions evoking the mathematical notion of dimension.
Outside In Turning a Sphere Inside Out
Recommended reading: Fourth-dimension and fractal geometry
Beyond the Third Dimension – Thomas F. Banchoff (New York: Scientific American Library, 1990)
Hyperspace – Michio Kaku (New York: Oxford UP, 1994)
Chaos – James Gleick (New York: Viking, 1987)
Flatland – Edwin Abbott Abbott (1884. Princeton: Princeton UP, 1991) available online here
And He Build a Crooked House – Robert A. Heinlein (New York: Amereon House, 1983)
Tangents – Greg Bear (London: Viczor Gollancz, 1989)
The Fourth Dimension and Non-Euclidean Geometry in Modern Art – Linda Henderson (Princeton: Princeton UP, 1983)
Fourfield – Tony Robbin (Boston: Bulfinch, 1992)


Here you can download a copy of the unit circle. It has all of the angles in Radians and Degrees. It also tells you the sign of all of the trig functions in each quadrant. Or if you need, we also offer a unit circle with everything left blank to fill in.
What makes it special is having a radius of 1, which is a unit in our number system. The circle is centered on the origin, x=0, y=0, or coordinates (0,0).
These simple settings give lengths and sin/cos/tan values that are easy to work with compared to neverending decimal numbers, and are very useful, as 30°, 45°, and 60° angles show up a lot in the world!Memorizing these common values makes for a valuable tool for any mathematician. From trig to geometry and calculus, these values will come in handy many times over.
How long did it take you to fill in your blank unit circle? Let us know in the comments, or show off by typing out the sin, cos, and tan values for 30°, 45°, and 60° without looking at the reference sheet!
Remember, those special right triangles we learned back in Geometry: 30-60-90 triangle and the 45-45-90 triangle? Don’t worry. I’ll remind you of them.And after you know your Radian Measures, all we have to do is learn an amazing technique called the Left-Hand Trick that is going to enable you to find every coordinate quickly and easily.
Everything you see in the Unit Circle is created from just three Right Triangles, that we will draw in the first quadrant, and the other 12 angles are found by following a simple pattern! In fact, these three right triangles are going to be determined by counting the fingers on your left hand!
If you place your left hand, palm up, in the first quadrant your fingers mimic the special right triangles that we talked about above: 30-60-90 triangle and the 45-45-90 triangle.Unit circle – trigonometry online worksheet for high school. You can do the exercises online or download the worksheet as pdf. HOMEWORK: Practice completing the entire unit circle with angles in radian measure with their corresponding (x,. Complete the Unit Circle Worksheet. Practice these unit circle worksheets that focus on completing the unit circle diagram, finding the values of trigonometric functions, charts and more! Find the exact value WITHOUT USING THE UNIT CIRCLE AND TABLE! 15. sin 240°. 16. cos(−225)°. 17. tan 315°. 18. csc(−90)°. Find the exact value WITHOUT. The given point P is located on the Unit Circle. State the quadrant and find the angle, also sine, cose and tan. 1) (-. Quad: sin 0: 1 √3. 2 2. 2) P(0,-1). Unit Circle Worksheet. Use the unit circle and the first quadrant chart to find the given values. Θ. 4560 90. Name Jey. 30. ΘΕ. Tice π/4. 1/4 I. UNIT 6 WORKSHEET 5 USING UNIT CIRCLE. Unit Circle Worksheet. 1. Draw a unit circle. 2. Put the radian and degree measure next to each angle. 4. Find the ordered pair that would go with each one. What is the reference angle for 240°? Put your answer in degrees and in radians. 7.) Evaluate sin, cos, and tan -150° without a calculator. 8.) Evaluate csc.2 pages Here you can download a copy of the unit circle. It has all of the angles in Radians and Degrees. It also tells you the sign of all of the trig functions in. Unit Circle Trig. (Degrees) worksheet (see below), #1-12. Teacher: Mr. Whetstone. Class: Algebra 2 Pre-AP. Periods: 4 and 5. Unit Circle Trigonometry. Drawing Angles in Standard Position. UNIT CIRCLE TRIGONOMETRY. The Unit Circle is the circle centered at the origin with radius 1. Unit Circle Worksheets. Blank Unit Circle Worksheet: Practice your skills by identifying the Radian Measure, Degree Measure and Coordinate for. Worksheet by Kuta Software LLC. Algebra 2/Trig. Practice: Angles and Unit Circle Values. Name___________________________________ ID: 1. Find each exact value. Do not use a calculator. Name. Key. 1. sec 45°. 2. cos 120°. 3π. 3. sin. √2. -1/2. العالم. 4. tan. -√√3. Worksheet by Kuta Software LLC. Kuta Software – Infinite Algebra 2. Name___________________________________. Period____. Date________________. Homework 10.1 Unit Circle. Name. Section: 1 2 3 4 5 6 Date: Draw the picture!. Use reference angles and the unit circle to find the exact values of:. 1. Use the unit circle to find the sine, cosine, and tangent ratios of an angle with a measure of 135°. . Use the blank unit circle worksheet to test yourself and keep the filled unit circle handy for a reference. Memorizing these common values makes for a. The Unit Circle is probably one of the most important topics in all of Trigonometry and is foundational to understanding future concepts in Math Analysis, Calculus and beyond.Additionally, as Khan Academy nicely states, the Unit Circle helps us to define sine, cosine and tangent functions for all real numbers, and these ratios (that we have sitting in the palm of our hand) be used even with circles bigger or smaller than a radius of 1.
Unit circle trigonometry worksheet pdf
Unit circle worksheet pdf kuta
Unit circle worksheet questions and answers pdf
Unit circle worksheet answer key pdf
Unit circle worksheet answer key
Unit circle worksheet with answers
Unit circle worksheet pdf
I will show you how to remember each angle, in radian measure, for each of your fingers and also how to find all the other angles quickly by using the phrase:Now, I agree that may sound scary, but the cool thing about what I’m about to show you is that you don’t have to draw triangles anymore or even have to create ratios to find side lengths.The Unit Circle has an easy to follow pattern, and all we have to do is count and look for symmetry. Moreover, everything you need can be found on your Left Hand.Well, these special right triangles help us in connecting everything we’ve learned so far about Reference Angles, Reference Triangles, and Trigonometric Functions, and puts them all together in one nice happy circle and allow us to find angles and lengths quickly. Determine the values of six trigonometric ratios by applying the measure of the angle encompassed by the terminal side on the unit circle. Instantly validate with the answer keys provided. These two-part worksheets for high school offer exercises to determine the exact value of a specific trigonometric ratio given as degrees in Part-A and radians in Part-B using the unit circle.


The Department of Basic Engineering was established in 1977 as one of the departments of Faculty of Agricultural Engineering. The department deals with various supporting courses like Applied Mathematics, Applied Physics, Computer Science and Engineering mechanics with special application in Agricultural Engineering. The supporting course of Functional Mathematics is offered to the faculties of Crop Production, Crop Protection, Agriculture Social Sciences and Animal Husbandry and Veterinary Sciences. The department also offers two courses of Mathematics to the Information Technology Centre and one course to Institute of Food Sciences Technology.
The mission of the department is to provide leadership and excellence in teaching, research and outreach in Engineering Mechanics and Computer Science with Applied Mathematics at the graduate and postgraduate levels. Engineering Mechanics and Applied Physics laboratory and Computer Lab are available for teaching to graduate and post graduate students.
|Mr. Muhammad Aslam Samo||MSc (University of Sindh, Jamshoro)||Assistant Professor and Incharge Chairman||[email protected]|
|Mr. Ghulam Shabbir Bughio||MSc (University of Sindh, Jamshoro)||Assistant Professor||—|
|Mr. Saleem Ahmed Naich||M.Phil (Mehran University of Engineering and Technology)||Lecturer||[email protected]|
|Mr. Amanullah Tunio||MPhil (MUET)||Lecturer||[email protected]|
|Mr. Veero Mal Bheel||BS (University of Sindh, Jamshoro)||Lecturer||[email protected]|
|CODE||TITLE OF COURSE||CR. HRS|
|BE-403||Advanced Mathematics for Engineers||3(3+0)|
|BE-404||Computer Programming and Applications||3(1+2)|


Roulette is a famous game whether you join brick and mortar casino or online casino. This is because of its fairly basic concept – you just need to predict which number the small ball will rest on after the croupier has spun the wheel. If your prediction is right, without a doubt you win.
While it is simple, there are many things to know. If you are a newcomer, you will be overwhelmed by all the options you see on the board and the wheel. You have to overcome the basic things so you can play your best. If you are wondering about the numbers on Roulette Wheel, here are the things that you need to know about it.
The roulette variant
The wheel will depend on the roulette variant. On American roulette, there are 38 numbers – 1 to 36 then two more pockets for 0 and 00. The American roulette has a house edge of 5.26% because of the additional 0 pocket. On European roulette, there are 37 numbers – 1 to 37 with an additional 0 pocket. Many beginners consider this variant because it presents a lower house edge, which is 2.70%.
Regardless of the variant, the numbers are either red or black. As for the 0 or 00 slots, they are green in color. When you take a look at it, the black numbers are 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, and 35. The red numbers are comprised of 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, and 36.
Numbers in total
The result of adding up all the black numbers is 334. When you add all the red numbers, the result is 332. Adding up all the numbers will give you 666. One interesting story about the number – Francois Blanc, a magician in Homburg brought roulette into the spotlight. Rumors circulated that he made a deal with the devil to learn roulette secrets thus the total number.
When you look at the even black numbers, there are 10 while there are 8 even red numbers. There are also 5 black prime numbers – 11, 13, 17, 29, 31, and 5. The red prime numbers are 1, 3, 5, 7, and 23.
The number of combinations is strange in a glance because they are not balanced. Aside from not being balanced, the numbers do not give the impression of being random. If you want to increase your chances of winning, you have to focus on the betting style and their payouts. Since you are a beginner, you should just stick to outside bets.


Predicting stock exchange developments?
Since the mid-1990s a team of ULg researchers has applied a mathematical model to certain financial phenomena. This model has the special feature of integrating notions of chaos and unpredictability into a determinist system. Professors Jacques Bair and Gentiane Haesbroeck have jointly written a publication offering an overall synopsis of this research (1).
Between the linearity of models and the chaos of life
In the middle of the 1990s scientists from different countries were looking to develop determinist models which could be applied to the chaotic world of finance. At the University of Liège one research group was built around Professor Jacques Bair: it took a very close interest in the subject and was one of the first to transmit this new mode of thinking into francophone Belgium. ‘With Professor Crama we created the GEMME (Groupe d’Etude des Mathématiques, du Management et de l’Economie). We were supported by many fledgling researchers, at the time, and were very active on the question of the integration of mathematics in the world of the economy, and we published a whole series of articles, notably with Professor Haesbroeck.’
(1) Bair J, Haesbroeck G., Modèles chaotiques en économie, Tangent sup., Prévoir pour décider, POLE, 2012.


Differential Forms in Algebraic Topology
The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology. Accord ingly, we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology. For applications to homotopy theory we also discuss by way of analogy cohomology with arbitrary coefficients. Although we have in mind an audience with prior exposure to algebraic or differential topology, for the most part a good knowledge of linear algebra, advanced calculus, and point-set topology should suffice. Some acquaintance with manifolds, simplicial complexes, singular homology and cohomology, and homotopy groups is helpful, but not really necessary. Within the text itself we have stated with care the more advanced results that are needed, so that a mathematically mature reader who accepts these background materials on faith should be able to read the entire book with the minimal prerequisites. There are more materials here than can be reasonably covered in a one-semester course. Certain sections may be omitted at first reading with out loss of continuity. We have indicated these in the schematic diagram that follows. This book is not intended to be foundational; rather, it is only meant to open some of the doors to the formidable edifice of modern algebraic topology. We offer it in the hope that such an informal account of the subject at a semi-introductory level fills a gap in the literature.
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Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that".
Order theory focus on transitive binary relations, we start our definitions from a basic one --
A preorder is a thin category --
hom_set(A, B).size <= 1.
PreOrder, we can add antisymmetric to get
Note that, there will be no cycle in a partial order, and we can implement topological sort for a partial order.
TODO We can implement sort algorithms for
PreOrder again, we can get equivalence relation by adding symmetric.


This ratio is calculated by taking the total cash ﬂow available to service debt during a speciﬁed period and dividing it by the total amount of debt repayment plus interest payment for the same period. The period chosen is normally that from one debt repayment date to the next (typically, six months).
The amount in US dollars paid for a ship sold for demolition.
Interest Cover Ratio is calculated by dividing the total cash ﬂow available to service debt during a speciﬁc period and dividing it by the interest payment for the same period. Where interest is not being paid, the lending bank will have to account for the loan as being non-performing and make necessary loss provisions in its accounts. The bank may, … Read the rest
London Interbank Offered Rate (LIBOR)
This is a ratio that shows the extent to which the balance of the outstanding loan is covered (secured) by the market value of the ship. It is normally found as a covenant in loan terms sheets and loan agreements. Traditionally, it used to be expressed as the ratio of the market value of the ship divided by the outstanding … Read the rest
Monte Carlo simulation is a computerized mathematical technique used to analyze risk. Monte Carlo simulation performs risk analysis by building computer models of possible results by substituting a range of values—a probability distribution—for any factor that has uncertainty. It then calculates results over and over, each time using a diﬀerent set of random values from the speciﬁed probability function for … Read the rest
Daily operating costs include the cost of Crew, Spare Parts, Insurance. The expenses of Drydocking and Special Survey are also operating costs. Always remember to check if daily operating expense quoted is including provision for DD/SS.
For financial analysis and modelling of ship investments, the cost of off-hire should also take into account.
For ships on time-charter, the charterer pays … Read the rest
This ﬁgure shows the minimum market value the ship can have before a loan is in breach of its minimum value covenant. It is calculated by taking the outstanding loan balance and multiplying it by the LTV ratio. For example, if the loan balance is USD10 million and the LTV covenant is 130%, the required minimum value will be USD13 … Read the rest


What is Math & Academics Club?
The Math & Academic Club is an school-sponsored organization for students with an interest in mathematics and academics. We meet weekly to discuss problems and puzzles, play games, and learn about interesting topics beyond the scope of what is taught in the classroom.
When/Where are the meetings?
Math Club meets on Thursdays from 4:00-5:00 in Mrs. Rost’s classroom
Academic Club meetings on Tuesdays from 4:00-5:00 in Mrs. Rost’s classroom


5 edition of Brain-Boosting Math Activities: Grade 3 found in the catalog.
January 1, 1999 by Scholastic .
Written in English
|The Physical Object|
|Number of Pages||64|
Students first hop on one number, then an operation, another number, the equal sign, and finally the answer. The U. All rights reserved. Number Practice - When completing each of the ten worksheets in this packet students will practice saying and writing the numbers and number words one-ten. Comprehension My Name is Albert - Read a story and answer the questions on this reading comprehension worksheet.
Two players play at a time using a ball and everyone's hands instead of paddles. Synonyms - Students will identify synonyms and write their own. Dog Numbers 9 and 10 - Students will draw the correct number of dog bones and practice printing the numbers 9 and How Many Worms? Alphabet Antics - Students will test their knowledge of letter order in the alphabet, alphabetize words, and solve a secret code. If your kids are older, mix things up with ping-pong paddles or larger squares.
Invite students to analyze how the density of the fruit or vegetable affects its weight. As students transition into the new school year, help them sharpen their number skills with some of these fun and effective games. Unscramble a group of words to make a question and a statement. One-Meter Dash Hand groups of students a meter stick, a pencil, and a sheet of paper each. Writing Large Numbers in Standard Form - Students will read large numbers in word form and write them in standard form. Vowels and Consonants Vowel or Consonant Mail - Sort and alphabetize double vowel and double consonant words.
Postmodernism in literature and culture of Central and Eastern Europe
The Construction Of Intercultural Discourse. (Utrecht Studies in Language and Communication, 2)
Modern foundry operations and equipment.
practical treatise on the law of nuisances in their various forms
nonsense verse of Edward Lear
Guide to Western architecture
Cultural continuity and population change on the Isle of Skye
shaving of Shagpat
The Sabbath in Puritan New England.
Not only are playground games for kids Brain-Boosting Math Activities: Grade 3 book fun way to teach, they're also a great opportunity to keep your mind sharp. Which group came closest to one meter? Snowflake Match Worksheets - Count the snowflakes, and draw a line to the number that tells how many on worksheets 1 and 3.
To play, you'll need nine plastic lids and three different colored markers. Number Match - Draw a line to match objects to numbers. You can even use the crazy bounce ball to challenge even the fiercest athletes.
Earthquakes and Volcanoes - Identify sentences and sentence fragments on this science worksheet. In order to avoid being tagged, a runner has to make it to base and tap out this color pattern using feet or hands. All rights reserved. Holidays at PrimaryGames PrimaryGames has a large collection of holiday games, crafts, coloring pages, postcards and stationery for the following holidays: ChristmasHalloweenEasterValentine's DaySt.
This game taps into both sides of the brain by writing with both dominant and non-dominant hands. Count and Color 1 through 10 - Each of the ten worksheets in this set focus on a different number from and provide students with practice counting as well as enhancement of fine motor skills when they trace and print each number and number word.
Then give them five minutes to measure the items and record their lengths and add them together. The Number 5 Worksheet A - Count and color the clothes in each set and circle the set that has 5.
Numbers - This packet includes five worksheets in which students count items in a group and circle the groups that have the given number.
PrimaryGames is the fun place to learn and play! Synonym Worksheet 2 - Draw a line to match the word with the same or almost the same meaning. If your kids are older, mix things up with ping-pong paddles or larger squares.
Seven Fruits - Students will name each fruit and draw additional pieces of fruit in each row to make a total of 7. Your budding brainiac will need to keep her eye on the ball to scoop up the jacks and make the catch.
Syllable Sleuth Worksheet 3 - Students will identify the correct syllabication of various words, identify two and three syllable words, and divide words into syllables.
Playing in pairs, each student lays two cards face up, then subtracts the lower number from the higher. Ask students to stand in groups according to their predictions: those Brain-Boosting Math Activities: Grade 3 book think their arm span is greater than, less than, or equal to their height.
With older kids, you can include the square root symbol and negative integer sign. My First Number Word Search - Find and circle the number words one through ten in this math word search puzzle. Worksheet 2 - Count the worms, and circle the number that tells how many. Watching your child as she tries to learn to hop on one foot or use a non-dominant hand is actually firing up her brain cells.Dec 13, · Parenting» Worksheets» Worksheets and activities for winter break Worksheets and activities for winter break Keep 'em sharp over winter break with these 18 fun, brain-boosting worksheets and activities for grades K-1,and Math Activities for Kids Show kids how creative and applicable math can be with these imaginative math activities designed for children of all ages.
Entire Library Printable worksheets Online games Guided Lessons Lesson plans Hands-on activities Online exercises Interactive stories Song videos Printable workbooks Science projects. Origami Math"Unfold" math learning with hands-on fun and activities.
Quilt Square Math K-2, Make math connections colorful, fun, and inviting to students with a unique unit on the geometry of quilting. Online Activities Teacher Guides Max's Math Adventures Teacher's Guide The teacher's guide to the online activity that focuses on.Math Skills Worksheets.
This pdf contains only a partial index of the math skills worksheets on S.T.W. Full Website Index. View the complete index of all Math, ELA, Spelling, Phonics, Grammar, Science, and Social Studies worksheets found on this website.Get this from a library!
Brain-boosting math activities: more than 50 great activities that reinforce problem solving and essential math skills. [Carolyn Brunetto].Used, new & out-of-print books ebook Offering millions of titles from thousands of sellers worldwide.
Joyful Learning: Brain-Boosting Math Activities: Grade 6. Brain-Boosting Math Activities: More Than 50 Great Activities That Reinforce Problem Solving and Essential Math Skills. by Professional Books. Starting at $Price: $


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|Number of Puzzle Pieces||1|
|Material Care Instructions||Easy to Clean Soft Cloth|
|Remote Control Included?||No|
|Product Dimensions||15 x 10 x 10 cm; 270 Grams|
|Item part number||W4A-Math-03|
|Manufacturer recommended age||24 months - 2 years|
|Country of Origin||India|
|Item Weight||270 g|
Pine MDF Wooden Math Counting & Plus, Minus, divide & Multiply Educational puzzle Board Game
- This wooden Count puzzle board is brightly colored , help to learn the letters and colors.
- Material: Wood + non-toxic paint. Safe and no harm to babies.
- Promotes hand-eye coordination, fine motor skills, and problem-solving skills.
- Early Pre-school learning for kids.
- An ideal gift for kids learning Calcutaion Sum Math Count.
Write Your Own Review


A sixth form community where talented mathematicians can excel. Based close to Cardinal Newman College in Preston, we will provide an intellectually stimulating and challenging curriculum along with high-quality resources for 16-19 year olds who love maths.
As part of the Rigby Education Trust, with Lancaster University and Cardinal Newman College, you will receive exceptional teaching, first-class facilities, and a high-quality enrichment programme that encourages the practical application of maths and brings in industry experts to the classroom.
More than maths
All students study A-levels in Mathematics and Further Mathematics. You will then choose your third and fourth A-level from either the subjects taught at the LUSoM campus, such as Physics, Chemistry or Computer Science, or select from any other Level 3 subject taught at Cardinal Newman College.
Our exceptional teaching is provided by specialist Maths School staff. We will push the boundaries of maths education, inspire and challenge you every single day, and give you the freedom and encouragement to be openly passionate and excited about maths.
The LUSoM guarantee
Students who attend and successfully complete their studies at Lancaster University School of Mathematics will be guaranteed an offer for nominated degree programmes at Lancaster University, provided they meet the entry requirements.
Rigby Education Trust Registered in England as a company limited by guarantee number 12098063. Registered offices for the Rigby Eduation Trust: 37 Fowley Common Lane, Glazebury, Warrington, England, WA3 5JJ


You can use these operators here just like in math, but I'll explain them for the sake of clarity.
When using with numbers, it will add them together:
After executing the code above, the value of num will be 8.06. When used with strings, it concatenates them:
As a result, rdy will be appletree. You can sum up vectors too, this way it will add together the X, Y, and Z coordinates one-by-one:
The value of vector will be (85, -1, 3477.7).
It works the same way as addiction, except the fact that you can't use it with strings. You can subtract two numbers or two vectors from each other in the way described above.
You can use these only with numbers. However, even if you are working with integers, division will always return a floating point number.
You can make these statements shorter in a special way, so instead of this:
You can write this:
This way, your code can be shorter, and cleaner. Obviously, you can use it with the rest of the operators too (ie *=).
Moreover, there is a shorter version if you want to increment or decrement your variable only by one. Instead of these:
You can use these:
It is recommended that you always use the shortest possible forms of these operators to maintain the quality and readability of your code. And believe me; you do want that.
As a result of using these with variables, you will always get true, or false. With the help of these operators, you can check if a number is bigger than another one, if two strings are the same, or if a boolean variable is true or not.
You can check the equality of two objects of the same type by using the == operator. You can move the result of the equation into a variable the following way:
The value of logical will be true if num1 is 4, otherwise false.
Please note that == true can be omitted in case you would like to know if a boolean variable holds a value of true. Basically, this statement:
equals to this:
You will need it soon when checking conditions in a branch, because it will make your code much easier to read. In addition, all numbers can be handled as if they were booleans. Zero means false, while every other non-zero value will be interpreted as true.
You can use these only with numbers:
The logical variable will be true if num1 is greater than num2, otherwise false.
You can mix them the same way you would in math, for example:
The operator of 'and' is &&. By using it, you can list a series of expressions of which all of them has to be true in order for the whole to be evaluated to true.
The operator of 'or' is ||. By using it, you can list a series of expressions of which at least one of them has to be true in order for the whole to be evaluated to true.
If you would like to use these in one expression, you have to use brackets, since the game must be able to decide the order it should interpret the expressions in. Example:
The operator of negation in programming is (usually) the exclamation mark (!).
You can negate the value of a variable, but obviously it can only be used with logical values. If you write this:
Then if logical2 is true, then logical1 will be false, and if logical2 is false, then logical1 will be true.
You can negate an equation too (it can be used with every variable type which you can use the equation with). If you don't want to check if str is "bush", but instead if it is NOT "bush", then you can simply write this:
neg will be true, if str is not "bush". Please do not forget when negating an equation that you must use only one equal sign!
We can also negate a complex logical expression by putting it in brackets, and placing the exclamation mark right before it.
These are used very rarely, but if we use them when really needed, we can reduce the memory usage of the mod, and it can evaluate some operations much faster. Bitwise variables (basically integer variables, but we are handling them using bitwise operators) can be used (for example) if we want to store which items are owned by the player, and which are not.
This is especially useful because we can tell exactly which of these values are true and which aren't without actually storing all of them one-by-one in different variables or a huge array.
Call of Duty 4 uses these special values to tell which intel has been found by the player during the Single Player campaign. In my Hide 'n Seek mod, I use them to tell which map a specific player has already played on, and which are yet unplayed by them.
We must work with these in binary system. The ID of the first bit is always 1, and the ID of every other bit is two times bigger, than the previous one's. So if you want to store 4 values, then the ID of the first one is 1, the second one is 2, the third is 4, and the fourth is 8. If the first and the third bits are true, then the value of your variable will sum up to 5 (1 + 4). More about the binary system can be found here: En.wikipedia.org. It looks scary, but it really isn't a big deal once you get the idea.
You can check easily, if a specific (e.g. 4th) bit is true using the & character:
Since these are simple numbers, a + sign can also add them together, but there is a special sign for adding bitwise values: |. It doesn't matter which one you use, it really is just to keep the code clean and readable:
You will most likely never use these operators, unless you want to do something really extreme. The operator of left shift is <<, while the same for right shift is >>. This shift is done in binary system, so the meaning of 5 << 2 is that we are shifting 5 left by 2 places:
5 in binary system is 101;
we shift it to the left by appending two zeros to it: 10100;
by converting it back to a regular number, we get a result of 20 (4 + 16).
We can shift the same way to the right, but in that case, we subtract zeros from the end instead of adding them.


EMI stands for Equated Monthly Installment which is a fixed payment a borrower must make on a specified date on a monthly basis. EMI consists of your principal loan amount and interest amount, payable every month.
Based on the repayment schedule, we will divide the principal and interest over the repayment time period. The initial payments have a higher interest component, but later payments have a higher principal component.
* If you would like to see how loan payments are calculated, please click here to see a loan calculator.


Math encourages Pre-Kindergarten
children to solve problems, develop their natural curiosity and enhance their self-confidence. Through active exploration and the use of concrete objects, children begin forming important concepts about numbers and using mathematical ideas. Children are also encouraged to use reasoning and math skills in everyday life.
Math provides Kindergartners
the opportunity to develop their analytical thinking through hands-on activities. Children also continue to build strategies and skills for problem solving. Math is an integral part of the curriculum, connecting ideas and applications to the real world in which children live. Math concepts are introduced, revisited and built upon throughout the school year. Cooperative learning is fostered through small groups, games and computer activities.First Grade
Math at the First Grade
level encourages children to become investigators, discovers, predictors, explorers and risk-takers. In these ways children make the connections between the concepts and skills important to understanding and applying mathematical principles. First Graders study number and operations, identify, describe and apply number patterns and properties, work with geometric shapes and measure and explore data and chance.Second Grade
Math in the Second Grade
expands children’s number sense as their understanding of operations, data, geometry and spatial orientation deepens. Students move toward more symbolic representations of mathematics and they record their observations and computations in journals. Second Graders learn from each other and cooperative projects help students appreciate the various strategies that may be used to arrive at a solution.Third Grade
Math in the Third Grade
expands on children’s abilities to make connections between the classroom and the world outside of school. Students increasingly explore more complex mathematical concepts and relationships. They also practice communicating, both orally and in writing, an understanding of numbers, variables, functions, data, geometry, measurement and probability.


Four-time International Mathematical Olympiad contestant, Kyle Patrick Dulay, is no stranger to problem-solving. And Kyle sees mathematics as the solution, which he believes can unlock secrets to many of the universe’s mysteries. That’s where he draws his love for maths from.
“I love mathematics because it feels like it is the language of the universe in a way. So that if I ask a question of, “how is this the way it is right now?” I can usually translate it to a mathematical problem, which I can then try to solve. Other people can appreciate math more if they can look at it not as a school subject with a set number of lessons to learn but as a solution to many of the universe’s questions.”


As I mentioned in my post about Ford Spheres, the system can also be defined by partitioning the unit interval into three rational intervals. On looking into this formulation, I’ve found a way to simplify it. Rather than choosing three rational numbers that add to 1, we just need to choose three coprime positive integers a, b and c, say. We can then define the partition by taking a/(a+b+c), b/(a+b+c) and c/(a+b+c) as our three rational numbers that add to 1. These can be envisaged as the coordinates of a point in an equilateral unit triangle. The three coordinates represent the distance of the point from each of the three sides of the triangle (measuring each coordinate along a line parallel to one of the other two sides). And if the time dimension is represented by a+b+c, the sum of our three integers, then we would be deriving space as a partition of time. The most exciting thing about this definition is that it naturally extends to higher dimensions of space. So watch this space (even if right now it is still only an area).
OK, so here’s the latest update. I’ve been working on extending Fareyland into higher dimensions. I’ve now found a two-dimensional version of Fareyland, based on a unit triangle (i.e. an equilateral triangle of side 1). At each rational point on this triangle it is possible to place a sphere that touches the triangle at that point, with a radius of 1/2d, where d is the denominator of the square of the point’s distance from any corner of the triangle. (If different corners give different values of d for a particular point, then take their lowest common multiple.) Amazingly, none of these spheres overlap but they just touch their neighbors nicely. (This system is known as the Ford Spheres.)
This arrangement can also be generated by placing three spheres of unit diameter, one on each corner of the triangle, so that they touch each other, and then filling in the gaps between the spheres by placing smaller spheres between every three touching spheres. Keep doing this over and over and you eventually generate every sphere described above.
I haven’t worked out the details yet, but I’m fairly confident that this system can also be defined by partitioning the unit interval into three sections of rational length, or equivalently, by choosing three rational numbers that add to 1, which in turn is equivalent to choosing two rational numbers that add to not more than 1 (since they will define the third). So here we have a relationship between integers (OK, rational numbers, which are just pairs of integers) and a two-dimensional space. I’m still working out the details of the time dimension but it looks as though each sphere represents a “bubble” of spacetime. If this works, it will give me a system with two dimensions of space and one dimension of time, and I can start looking for the physical laws that govern this system. One interesting facet of this approach is that the space it defines does not expand, but instead becomes more detailed over time. (Of course, if you were inside the system, these phenomena would both look pretty much the same.) And of course, as we had in Fareyland, the edges wrap around so that each corner of the triangle is actually the same point. This gives a symmetrical space like the one I was speculating about here. (This particular space has a 6-way symmetry.)
A couple of notes here: I haven’t proven this mathematically yet, but I have verified it by computer for spheres down to a radius of 1/800 (I found 19,072 of them). And to clarify what I mean by “rational point”, I should add that this is a point which has rational coordinates in a system where the x- and y-axes are at 60° to each other, instead of the more usual 90°. (I may find it convenient to change the way that the coordinates are measured when I start to consider partitions.)
The most encouraging thing is that as I investigate this system, I expect it will become fairly clear how to extend it into three spatial dimensions. Stay posted for more details as I find them.
Here’s a thought that I’m examining. If we were living in a symmetric universe, would we know it?
My idea is that perhaps the Universe is not only deterministic, but symmetric like the line-land I described in my post about Fareyland. Now Fareyland had only one dimension and a two-way symmetry, but when we extend this idea from one up to three dimensions, it gives us a universe with 24-way symmetry. In other words, what if our Universe is composed of 24 identical sectors (implying that there are 23 other copies of ourselves “out there”)? Would we be able to detect this?
Well, unless we were near to the middle of our sector, there would probably be vastly different distances between ourselves and each of the other 23 sectors, so that when we looked out into space we would observe them at greatly varying stages in their development (as well as from different angles). Would we notice the symmetry?
Have you ever eaten in a restaurant which has a wall of mirrors? At first glance (especially if you are not near the mirrors) the room looks to be twice as big as it really is. Or have you ever been in one of those mirror-mazes that seem to be much bigger than they really are? You may not notice the mirrors until you see yourself reflected in one. And if the mirrors were to reflect, not what is happening now, but what happened at some time in the past (as they would if they were zillions of light years away) would we notice them at all?
Food for thought…
Here’s what I’ve been working on lately. I’ve been trying to show how a universe can be defined by numbers. I started by attempting to define a universe with one dimension of space and one of time. I did this using the so-called “Farey Fractions”, which are simply the rational numbers between 0 and 1.
We can generate these numbers in stages as follows:
The first generation is just the numbers 0/1 and 1/1, our endpoints. There is nothing between them yet.
The second generation adds all the fractions with 2 on the bottom, so we now include ½.
Next, the third generation includes the fractions with 3 on the bottom, which gives us 1/3 and 2/3.
The fourth generation also includes 1/4 and 3/4. (We don’t list 2/4 because that’s the same as ½ which we already got in the 2nd generation.)
We keep on doing this and by the 6th generation we have
0/1, 1/6, 1/5, 1/4, 1/3, 2/5, ½, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1.
Note that this series displays a two-way symmetry with the number ½ in the center. Everything that happens on the left of the center is reflected on the right. Note also that each fraction first appears in the generation that corresponds to its denominator (the bottom number in the fraction). More interestingly, it is always “generated” by the fractions on either side, in the sense that if at some generation a/b and c/d are adjacent fractions in the series, then there will appear between them (at generation b+d) the fraction (a+c)/(b+d).
Does this system of generation have a physical interpretation? Well, you could say that each fraction a/b represents a “particle” (of matter or energy) that comes into existence at “time” (i.e., generation) b. We consider the two endpoints to represent a single particle, so making a circular universe that “wraps around” from 1/1 to 0/1. (Let’s call this particle “G,” after the “GO” square on a Monopoly board.)
We define the mass (or energy content) of a particle at any given time as equal to half the difference between the fractions that define its left and right hand neighbors. This definition makes the total mass/energy of the system equal to 1 at all times, so mass/energy is conserved. Thus, for example, at time 6 the particle 1/4 will have mass/energy of 1/15th (=1/2*(1/3-1/5)), while at time 7 its mass/energy would be 3/70ths (=1/2*(2/7-1/5)). (The particle G has mass 1/t at time t.)
If at some stage two particles a/b and c/d are neighbors, then these particles will interact at time b+d to create a new particle between them. The mass/energy content of this new particle will be ½|c/d-a/b| at the time it is created. Of course its “parent” particles will between them have lost an equivalent amount of mass/energy.
We can also define a distance function by setting the distance between two particles at time t as the number of steps from one to the other (going via G if this is the shorter way round). This gives us an expanding universe where the distance between G and ½ represents the diameter of the universe. Of course this is only a one-dimensional universe so nothing can get past anything else, and as a result it’s not very interesting.
But here’s something that I think is really cool: if you imagine these fractions strung out along the rational line between 0 and 1, you can draw circles touching the line at every rational point. At each point (represented by the fraction p/q) draw a circle of radius 1/2q². The amazing thing is that these “Ford Circles” never overlap, but each point’s circle just touches its neighbors’ circles. For example, the circle of radius 1/50 that belongs to the point 2/5 will touch the circle of radius 1/18 that belongs to the point 1/3, as well as the circle of radius 1/8 that belongs to the point ½, the circle of radius 1/98 belonging to 3/7, the circle of radius 1/128 belonging to 3/8, the circle of radius 1/288 belonging to 5/12 and so on–see the diagram below.
Right now I’m working on extending this idea into two dimensions, then (hopefully) I’ll be able to take it to the third dimension and see how similar a universe this gives to our own. There are some exciting things emerging–in two dimensions I have already noticed the emergence of structures that look and behave like black holes. Watch this space for more details.
In the meantime I will leave you to ponder the philosophical problem associated with Fareyland: did it exist before I defined it?


What is angle wordle? If you haven’t gotten hooked on the Wordle geometric angle game yet, it won’t take you 2 minutes to want to play this innovative online math game. We are going to explain to you what the angle game wordle consists of, how to download it and why play it. You have this free online game available if you love new challenges, we think you will like it!
▶ Play ANGLE WORDLE here 👇
This game can only be played once a day inspired by the Wordle game. You can leave your questions in the comments.
What is the word angle? The game is very simple. It consists of guessing a mathematical operation in 4 attempts. If you succeed, angle shows you the time you have left for the next challenge and in what attempt you have solved it. Then it gives you the option to share it on your social networks. It can only be played once a day and you can surely show it to your friends.
▶How to play Angle Wordle
Angel game is a new cloned word that has gone viral thanks to its simplicity, since you only have 4 daily attempts to solve its challenge (the same for everyone). If the user solves the mystery notes, they will not have the option to play the game again for 24 hours. Quite a success from our point of view, since it is presented as an addictive daily challenge.
Guess the angle in 4 tries or less!
Every time you guess, it will tell you how close you are and which way to go.
Angle wordle examples
The game allows you to share the results on social networks, illustrating the sequence of attempts with the colors, but without revealing the solution, in such a way that it increases the intrigue and curiosity to try to reveal the solution of the day.
To play angle you don’t need to download the game angle wordle you can play online you don’t need to download the game to see your stats on the icon above where you can see the result if you don’t hit it.
Today’s Angle Suggestion
Here are different clues for you to find. today’s angle for yourself:
Play in incognito mode so you can play as many unlimited games as you want.
You can view the source code to guess the answer of today’s angle.
if you are looking for angle answer you can solve angle today.
🔥 Alternatives to angle wordle game
We are seeing a lot of web and mobile applications that mimic the way Wordle works and seek to take some of the popularity away from this new platform. Some developers take advantage of this success to launch applications and provide us with alternatives to spend our free time simply using our browser.
We leave you some alternatives to angle game:
You might be interested…
We at Wordgames360.com are interested in games like wordle, we are fans of mind games and we like to report viral games online. Leave us your comments! We want to find all word clone games for you.


(*) Pro/ENGINEER ®, WildFire ® and Creo ®
are trademarks of Parametric Technology Corporation (www.ptc.com)
DeepCoder is a program to encrypt, decrypt
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DeepCoder is available as freeware. Info and download
NUMS is an arbitrary precision floating point and
integer math library in the form of a dll.
can be used by programmers to implement arbitrary precision floating point and
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Mathematics is a creative and highly inter connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy.
Crucially, a sound knowledge of mathematics is vital for young people seeking employment, and securing a qualification in mathematics is a fundamental requirement for the majority of employers.
A high quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.
At Steeple Bumpstead Primary School, children study mathematics daily following the White Rose Maths Scheme of Learning. WRM is a blocked scheme, which allows for depth and breadth of learning within each strand of mathematics.
In line with the National Curriculum Objectives for Mathematics, our intent is that all pupils:
become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
can solve problems by applying their mathematics to a variety of routine and non routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions
Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects.
Please click on the attached Subject Handbook to view the Intent, Implementation and Impact of our Curriculum.


it is also a game of mathematics, poker value bet poker strategy is a game of skill and using the ability to read situations and opponents to give you the advantage in each hand you play.
Value bet poker strategy
it also does not mean that you value bet poker strategy have made the wrong decision if you lose the hand. Just because you work out the odds in a particular hand and make the correct call it does not mean you are going to win the hand.
betfair trading sure bets Bet calculator value bet calculator system calculator sure bet calculator Betfair 888Sport Betway. Dtd" Value 2018 college football bowl confidence predictions bet calculator - value bet poker strategy calculate lucrative matches - odds calculator. "http www. Betting strategy Betting books Betting theory Bankroll building. W3.org/TR/xhtml1/DTD/xhtml1-strict.
Street Total Big Bets from Both Players Flop 3 Turn 2 River 2 Total Bets: 7 As you can see, three-betting the flop only earned you the same amount of money as a simple raise on the turn would have. You now have a much.
Watch m episode 5 of 10 of the "Advanced SNG Concepts" series. In this free episode, Againstime discusses range unpredictability. The American Poker Awards have announced nominees for the 4th annual event on February 22nd with Doug Polk and Poker Central earning the most nominations.
For example you found a game with odds 2.0 and the estimation is about 60, than the value bet is 1.2. This means your winning amount is 20 of your betting amount on the long run. If the value bet is greater than 1.0 it.
USA: Value bet poker strategy!
does the use of poker mathematics make you a winning player? It is a common question; it is possible to beat the game without incorporating poker mathematics into value bet poker strategy the plays you make?
in Limit Hold'em, it's crucial to understand the nfl predictions sports chat place importance of a single bet, value bet poker strategy and how the bet affects you and your long-term results.
Example of using mathematics in poker. First of all you should estimate that probability that your opponent is bluffing and holds a worse hand than you. Lets say that: Our opponent is a little tricky and bluffs 1 time for every 3 times he has.
whilst a player that has no knowledge value bet poker strategy of mathematics will be unsure. In this situation a player that has good knowledge of poker mathematics will always know whether or not to call,
using mathematics for drawing hands. Where to value bet poker strategy use mathematics in poker. Mathematics can be used in many different situations; however, so where does mathematics come in to play in online poker strategy?after a while things will start to click, it's not an easy read, not even on ThePokerBank!). But you won't find more information on poker mathematics anywhere else (no,) and you understanding of pot odds and expected value will become a lot easier. Look no further than The Mathematics of Poker by Bill Chen. Go back to the awesome Texas Hold'em Strategy. If you want a very thorough book on poker mathematics, more information on poker mathematics.thus if value bet poker strategy we call and have the best hand we will win 14 once, as a result if we called every time, but if we call and have the worst hand we will lose 4 three times.
so does three-betting. Just as waiting until value bet poker strategy the turn to raise makes you money, three-Betting If your goal in every hand of Limit Hold'em is to maximize your big bets, it's important to take a look at three-betting.in the value bet poker strategy above example we had to call 4 to win a 14 pot, which is 3.5-to-1. A simpler way to interpret what we just said is that you should have better odds of winning than the pot is giving you.for simplicity's sake, a single bet on the turn is equal to two bets on the flop; this obvious information becomes valuable when figuring how to play your hand. Let's say you're first in position and flop the absolute, value bet poker strategy un-counterfeitable nuts.
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in this free episode, in this free episode, watch m episode 7 of 10 of the "Advanced SNG Concepts" value bet poker strategy series. Watch m episode 6 of 10 of the "Advanced SNG Concepts" series. Againstime continues to go over shove charts in SNG Wiz.
value bet calculator powered value bet poker strategy by betting-smart.a poorly played hand in No-Limit can cost you your entire stack, while a well-played hand can double up your stack. That means value bet poker strategy every error you commit makes it twice as difficult to make money at the game.you need to make every play for a reason, and focus on conserving your value bet poker strategy losses. Every dollar saved is a dollar earned.
total Big Bets from Both Players. Flop 1 Turn 4 River 2 Total Bets: 7 By waiting until the turn to make your single raise, value bet poker strategy as you can see in the following chart, i'm a celebrity betting tips you've gained yourself one full big bet.
and the likelihood of completing your draw to inform you about whether or value bet poker strategy not you should call or fold. Pot odds take into account the amount your opponent has bet in relation to the pot, expected value.to make the most money, you want to hold out until the turn to make your raise. You are as sure as you can be that value bet poker strategy after you raise your opponent will shut down into check-call mode.while reducing the odds of your opponents' drawing against you. It's the little differences like this that allow good poker players to be consistent winners in Limit games. The difficulty with Limit poker is finding the line in maximizing value bet poker strategy your big-bet winnings,


Small group workshops are a fantastic option for students with special interests or a desire to boost academic skills. Our workshops are research-based, taught by qualified teachers, aligned to the Common Core State Standards, and built from common needs demonstrated by local students. Some workshops even feature two teachers! Our workshops are aligned to curriculum students are learning in their classrooms.
register now & save up to $70 per camp! ends march 27th
summer s.t.r.e.a.m. camps: registration now open
Kids Under Construction is proud to present our Summer S.T.R.E.A.M. Camps this year! We took STEM (science, technology, engineering, and math) to a whole new level by adding READING and ART to each week's curriculum. STEM challenges offer students the opportunity to think outside the box to come up with creative and innovative solutions to everyday problems, and the problem-solving system that students use to complete these challenges will help foster a growth mindset. Our Summer STREAM Camps are packed full of activities that use every part of students' brains, and because we know that all students learn in different ways, we've added in activities to suit visual, audio, and kinesthetic learners.
Each Summer STREAM Camps meet Monday-Thursday from 9a - 2p in our office and feature the following activities:
STEM challenges inspired by some of our favorite children's picture books
binary, directional, loop, and if/the/else coding center rotations
engineering competitions and growth mindset activities based on
well-loved fairy tale characters
common core math activities to prepare for next school year
sensory box and open-ended STEM material exploration time
science and literature crafts
walking field trip every Wednesday to the library to conduct research
light snacks and bottled water
Each Summer STREAM Camp is offered for two consecutive weeks: first to students entering grades 3-5, and then to students entering grades 1-3. Your $295 tuition includes all camp activities, light snacks each day, and all materials. There are only 8 spots available for each camp, so register early to reserve your spot.
Please register for Summer STREAM Camps according to your child's grade level next fall (2020-2021). For students entering third grade in the fall, you have the option to choose either camp depending on your child's needs, maturity, and amount of support needed. Our camps for students entering grades 1-3 will offer modified activities, increased teacher guidance, and literacy support. If your soon-to-be third grader is reading at a mid third grade level, can write a paragraph independently, and is proficient in their math facts, they will be fine in the camp for students entering grades 3-5 in the fall.
Enroll on or before March 27 to save $45 on each camp. Want to save even more? Save an additional $25 per seat when you fill more than one seat in our camps. There are 3 ways to earn this discount:
1) enroll one child in more than one camp
2) enroll siblings in the same or different camps
3) enroll with a buddy in the same or different camps
A $25 per seat refund will be issued to your card after enrollment payments are processed.
check out our summer stream camps below:
JunE 1-4: Grades 3-5
June 8-11: Grades 1-3
INSPIRED BY: My Mouth Is A Volcano
Our Big Volcano Explosion & Candy Shooter Summer STREAM Camp features a STEM volcanic eruption challenge, coding center rotations, candy shooter construction, a Hansel and Gretel growth mindset activity, cause and effect writing, common core math activities, and a BONUS volcano escape room!
June 15-18: grades 3-5
June 22-25: grades 1-3
INSPIRED BY: If I Built A House
Our Dream Home Challenge & Pig Showdown Summer STREAM Camp features a STEM dream home construction challenge, coding center rotations, a pig vs. wolf construction showdown, a Three Little Pigs growth mindset activity, descriptive writing, and common core math activities.
July 6-9: Grades 3-5
July 13-16: grades 1-3
INSPIRED BY: Counting on Katherine
Our Capsule Code Challenge & Plane Design Summer STREAM Camp features a STEM space capsule design and flight experiment, coding center rotations, airplane design challenge, a Tortoise and the Hare growth mindset activity, biographical summary writing, and common core math activities.
July 20-23: Grades 3-5
July 27-30: Grades 1-3
INSPIRED BY: Secret Engineer
Our Best Bridge Challenge & 3 Goat Races Summer STREAM Camp features a STEM Brooklyn Bridge construction challenge, coding center rotations, billy goat bridge races, a Three Billy Goats Gruff growth mindset activity, a supportive details writing lesson, and common core math activities.
August 3-6: Grades 3-5
August 10-13: Grades 1-3
INSPIRED BY: The Water Princess
Our Filtration Challenge & Tower Escape Summer STREAM Camp features a STEM water filtration challenge, coding center rotations, tower escape race, a Rapunzel growth mindset activity, a letter writing opportunity, and common core math activities.
August 17-20: Grades 3-5
August 24-27: Grades 1-3
INSPIRED BY: Everything You Need For a House
Our Treehouse Challenge & Marble Maze Summer STREAM Camp features a STEM treehouse building challenge, coding center rotations, marble maze competition, a Country Mouse and City Mouse growth mindset activity, fictional narrative writing, and common core math activities.


Hi all, how should I enter the step size for staircase in builder if the size is not a fixed value, but a linear equation (of midpoint)? Is the builder able to enter equation or I need to change in the coder?
Here is my step size in detail
I am trying to make a decision-making game between two gambles by using staircase method. The max and min value are (5,30) Basically, subjects will be shown with 2 gambles: 50/50 chance of winning $5/$30 vs. certainty of winning $15. If subjects choose the 50/50 gamble, the amount of certainty money will be increased by the equation (15+max)/2= $22.5 at the next trail; if subjects choose the certainty $30, on the next trail the certainty money will be decreased by the equation (15+min)/2=$10. The same rule continues until the 6th trail.
Thank you so much!!!


Science, technology, engineering and mathematics are the hot disciplines in higher education but Hispanics, who are a growing segment of the population, are poorly represented in STEM programs.
Mercy College in Dobbs Ferry and Westchester Community College based in Valhalla are partnering to do something about that disparity. The U.S. Department of Education has awarded Mercy College $4 million for a five-year program to increase graduation rates for Hispanic and low-income students in STEM programs.
“When the congressman’s office called, I was so excited,” said program director Nagaraj Rao, a Mercy mathematics professor, referring to U.S. Rep. Nita Lowey. “If we implement this well, so many students will be helped and will have great careers.”
Team STEM, as the program is called, aims to recruit more than 700 Hispanic and low-income students and graduate at least 27 percent with STEM degrees within six years.
Many of the students will start out at Westchester Community College, and the hope is to double the number who transfer to Mercy’s STEM programs.
Team STEM will use several methods to recruit and retain students. A summer program will prepare students to progress from high school to college.
Collaboration will be an important element. Students and professors, for example, will work together on research and community outreach projects that will culminate in conference presentations.
Mentoring could be crucial. Educational research shows that such programs help students stay on course, Rao said. But rather than rely on one type of program, as is usually done, Team STEM will use three levels of mentoring.
Staff members will be assigned to each student to support them through their education, helping with issues such as financial aid and career counseling.
Professors will guide students in their specialties.
And students who have already shown success with difficult courses will be trained to lead collaborative learning teams.
“It’s not their brain power that is limiting them,” Rao said of the Hispanic students he has taught. “It’s the background and environment that is limiting them. Given proper help, they will be very strong.”
Rao began teaching in India, where he earned a master’s in mathematics. He felt drawn to obtaining a Ph.D. in America and he got his wish, earning a doctorate in applied mathematics at the University of Rhode Island. He also got a master’s degree in computer science at City College.
He joined the Mercy faculty in 1983.
Mercy offers majors in six STEM programs: biology, computer information science, computer science, cybersecurity, mathematics and psychology.
Team STEM will begin recruiting students in the spring. Many will be recruited from within the college, among eligible students in other programs.
Westchester County also has a large pool of potential students. Nearly one-fourth of the population, about 250,000 people, is under the age of 18. Inner-city communities have a 10 percent poverty rate, according to census data cited in the grant application. Twenty-four percent of the county population is Hispanic or Latino.
“Their participation in STEM is extremely low,” Rao said, “and that demographic is growing. The need is huge.”
Each time a worthy student graduates, he said, “the whole family is lifted up.”
“That’s what motivates me and the faculty.”


Exciting lesson ideas, classroom strategies, teaching tips, book lists, videos, and reproducibles in a daily blog by teachers from the classrooms of extraordinary. Notes (Solutions) of Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSCII), Punjab Text Book Board Lahore. Follow these simple steps to find online resources for your book. is a free gamebased learning platform that makes it fun to learn any subject, in any language, on any device, for all ages. USATestprep was founded by two teachers who believe that webbased technology could improve student test performance and curriculum mastery. P 'P' stands for primitif, the name assigned long ago by German crystallographers to crystal structures that have the symmetry of a packing of congruent cubes. FLVS (Florida Virtual School) is an accredited, public, elearning school serving students in grades K12 online in Florida and all over the world. Teaching and Math Methodology: Instruction. Introduction to Teaching Challenges A student is not a container you have to fill, but a torch you have to light up. Pearson Prentice Hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. To the modern reader, Shakespeares plays often look like just a bunch of unrelated words mashed together. This is because 1) language is a fluid, constantly. Turnitin is revolutionizing the experience of writing to learn. Turnitins formative feedback and originality checking services promote critical thinking, ensure. Part of a series of articles on the: mathematical constant; Uses; Area of a circle; Circumference; Use in other formulae; Properties; Irrationality; Transcendence Angles 1, 4, 5, and 8 are equal to one another. Angle 1 is vertical to angle 4, and angle 5 is vertical to angle 8. CALIFORNIA STANDARDS TEST GRADE Released T est Questions Math 2 The following nine California content standards are included in the Multiplication, Division, and. Which test are you preparing for? Click for comprehensive study guides and strategies for performing your best on test dayall for free! SAT 1 a: a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; broadly: the study of. Options Select the maximum number of writing prompts to list. You can still pick from among those listed. You are here: Home Articles Help for high school geometry How to help students understand high school geometry? If you read the first part of this article. Print out The Times Tables and stick them in your exercise book. Test Your Tables with an interactive quiz. We would like to show you a description here but the site wont allow us.


Top Math Facts
1. Contrary to popular belief, Einstein never failed math. In fact, by fifteen he had already mastered differential and integral calculus.
2. A new study disproves that math and science homework help your grades.
3. You multiply centimetres by .39 to convert them to inches.
4. In the aftermath of 9/11, rescue dogs found so few living people it caused them great stress because they believed they had failed.
5. No one has ever seen a giraffe swim, but it is mathematically possible.
6. Ancient Babylonians did math in base 60 and not base 10, giving us 60 seconds in a minute, 360 degrees in a circle, etc.
7. Attempting to solve math equations triggers the same region of the brain responsible for reacting to extremely fearful situations.
8. Any number divided by zero is . . . nothing, not even zero. The equation is mathematically impossible.
9. Mathematically speaking, it is actually abnormal to be normal.
10. You can electrically shock a person's brain to greatly improve their math skills for up to 6 months.
11. Our Basketball Coach is a math teacher who thinks he has come up with the equation to never miss a basket. Please send help. -JGH
12. Math anxiety is a psychological disorder which causes stress and anxiety when doing math problems.
13. In 1990, all the world's mathematical knowledge could be written in about 80 books: today it would fill more than 100,000.
14. Starting with the number one, you would have to count to one thousand to use the letter A.
15. For hundreds of years, 'Computer' was a job title for a person who did math problems all day.
16. Only one out of the seven Math Millennium Prize Problems has been solved. Solving each is worth 1 million dollars!
17. Wolves are better at math than dogs.
18. The median score for college-bound seniors on the math section of the SAT in 2011 was about 510 out of 800.
19. Canadians are required to solve a math problem before claiming lottery winnings.
20. A new study suggests that a small electrical current applied to the brain can boost your math performance for up to 6 months.
21. You can get most answers to math assignments online by typing in the name of the textbook and then "answers".
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In 1986, Michael Foot’s appointment as chair of a disarmament committee prompted The Times headline: ‘Foot Heads Arms Body.’ The body of the sea otter has a pouch across the front where it keeps rocks to break open shellfish.
23 Nobel Prizes for Medicine have been won as a result of research on guinea pigs.
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‘The Just Missed It Club’ was for people who almost sailed on the Titanic. Two weeks after it sank, it had 118,337 members.
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Calculate Your Mulch Needs
Mulch is a wonderfull product for dressing up the area around you our shrubs, trees and flower beds, to both bueitify the area and reduce the maintenance required.
Dirt Guy knows that buying mulch or topsoil can be a challenge when you don't know how much you need. We understand that many people do not know how to or want to take the time to do the math to calculate how much mulch is needed. We have an automated mulch calculator to help with this.
There is a fairy easy formula to figure how much mulch or dirt or topsoil is needed for an area. First, it is critical to know there are 27 cubic feet in a cubic yard. One cubic yard will cover a 324 square foot area with one inch of soil or mulch.
Figure out the square footage of your bed, that's width times the length for square- or rectangular-shaped beds. In order to figure out the area of a triangle. It is the base times height, divided by two.
Circular beds are popular in some areas. To find out the square feet, go to the middle of the circle and measure to the outside. This is your radius. Multiply this number by itself.
For example, if the distance was six, multiply by six to get 36. Then multiply 36 by 3.14 (which is pi, remember pi?). This will determine your area in square feet. Won't you impress friends when you tell them you used pi to figure out how much mulch to buy?
Taking the time to calculate the amount can be worth the effort. Although a little extra mulch around the house is kind of like having money in the bank, too much can be overwhelming. With this case then a neighbor may be the beneficiary of over buying.
Multiply your square footage times your depth of inches and divide by 324 square feet, which is one cubic yard, one inch deep. This will tell you how many cubic yards you will need. If you have 100 square feet and want to add three inches to the depth, multiply 100 by three, and then divide by 324 to convert to cubic yards. This equals .92 cubic yards that you need, so you can buy nine of the three-cubic-feet bags. Since 27 cubic feet equal one cubic yard, you will have just a little bit left over.
Mulch prevents moisture loss, and come July and August, those newly planted trees and shrubs will need all available moisture. It keeps soil from being compacted. Compacted soil prevents moisture from reaching those roots and stifles the available oxygen. Mulch keeps the soil cool in the summer and actually stabilizes it in the winter too.
One much overlooked benefit is that mulch helps deter weeds. Weeds that practically need a stick of dynamite to be removed in tight soil can usually be easily plucked in a well mulched bed. Pay attention to what landscapes catch your eye and which ones look most professional. Landscapes with plantings on raised beds with an application of fresh mulch usually will be the ones that receive those appreciative stares.


A new national study shows that while Chicago public school 4th- and 8th-graders have improved in reading and math proficiency over the last decade, they still trail their peers in other large urban districts.
Results released Wednesday from the 2011 National Assessment of Educational Progress, or NAEP, indicate CPS students have improved in step with national averages in core subjects.
Thirty percent of CPS 4th-graders scored at “basic” proficiency in reading in 2011, a mark just off the national average for large districts and seven percentage points higher than in 2002. Still, 52 percent of 4th-graders scored “below basic” in reading proficiency in 2011, 14 points better than nine years earlier.
The performance was better among 8th-graders. Forty-three percent scored at basic proficiency in 2011, a drop from 2002 but on par with both the large district and national averages. Nineteen percent scored proficient in reading, 2 percentage points off the large district mark.
In math, 44 percent of CPS 4th-graders scored basic proficiency, two points higher than the national average and four points better than in 2002. Forty percent of 8th-graders reached that mark, better than both the large district (37) and national (39) averages.
However, the percentage of CPS 4th- and 8th-graders who scored below basic proficiency in math also exceeded large district and national averages.