Shanker B,Ergin A A,et al.Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm [J].IEEE Transactions on Antennas Propagation,2003,51(3):628-641.
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 任仪,赵延文,聂在平,马文敏.基于高阶叠层矢量基函数的时域电磁场积分方程方法[J].电子学报,2008,36(3):516-519. Ren Yi,Zhao Yanwen,Nie Zaiping,Ma Wenmin.Time-domain integral equations using higher order hierarchical vector basis functions [J].Acta Electronica Sinica,2008,36(3):516-519.(in Chinese)
 Rao S M,Wilton D R.Transient scattering by conducting surfaces of arbitrary shape [J].IEEE Transactions on Antennas and Propagation,1991,39(1):56-61.
 Vechinski D,Rao S M.A stable procedure to calculate the transient scattering by conducting surfaces of arbitrary shape [J].IEEE Transactions on Antennas and Propagation,1992,40(6):661-665.
 Davies P J,Duncan D B.Averaging techniques for time-marching schemes for retarded potential integral equations [J].Applied Numerical Mathematics,1997,23(3):291-310.
 Dodson S J,Walker S P,Bluck M J.Implicit and stability of time domain integral equation scattering analysis [J].Applied Computational Electromagnetics Society Journal,1997,13(1):291-301.
 Manara G,et al.A space-time discretization criterion for a stable time-marching solution of the electric field integral equation [J].IEEE Transactions on Antennas and Propagation,1997,45(3):527-532.
 Weile D S,Pisharody G,Chen N W,Shanker B,Michielssen E.A novel scheme for the solution of the time-domain integral equations of electromagnetics[J].IEEE Transactions on Antennas and Propagation,2004,52(1):283-295.
 Hu J L,Chan C H,Xu Y.A new temporal basis function for the time-domain integral equation method [J].IEEE Microwave Wireless Components Letters,2001,11(1):465-466.
 Wang P,Xia M Y,Jin J M,Zhou L Z.Time-domain integral equation solvers using quadratic B-spline temperoral basis functions [J].Microwave Optical Technology Letter,2007,49(5):1154-1159.
 Pingenot J,chakraborty S,Jandhyala V.Polar integration for exact space-time quadrature in time-domain integral equations [J].IEEE Transactions on Antennas and Propagation,2006,54(10):3037-3042.
 Shanker B,Lu M,Michielssen E.Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields [J].IEEE Transactions on Antennas and Propagation,2009,57(5):1506-1520.
 Shi Y F,Xia M Y,Chen R S,Michielssen E,Lu Mingyu.Stable electric field TDIE solvers via quasi-exact evaluation of MOT matrix elements [J].IEEE Transactions on Antennas and Propagation,2011,59(2):574-585.
 Yücel A C,Ergin A A.Exact evaluation of retarded-time potential integrals for the RWG bases [J].IEEE Transactions on Antennas and Propagation,2006,54(5):1496-1502.
 Ülkü H A,Ergin A A.Analytical evaluation of transient magnetic fields due to RWG current bases [J].IEEE Transactions on Antennas and Propagation,2007,55(12):3565-3575.
 赵庆广,赵延文,毕海燕,聂在平.利用时间步进算法精确稳定求解时域积分方程[J].电子学报,2008,36(6):1135-1139. Zhao Qingguang,Zhao Yanwen,Bi Haiyan,Nie Zaiping.Accurate and stable solution of time-domain integral equation using marching on in time method [J].Acta Electronica Sinica,2008,36(6):1135-1139.(in Chinese)
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In this post, we’re sharing with you amazing class Notes called Abhinay Maths PDF, which is very useful for learning maths and preparing for the competitive exams.
Abhinay Sharma Maths Notes Book PDF is important for almost every competitive and govt. Exams. This book covers previous exam questions and normal essential questions with detailed explanations and examples, which will help you understand the topics quickly.
Play with Advanced Maths PDF
Abhinay Sharma is the best and popular maths teacher among all students; everyone wants to read his book.
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Play with Advanced Maths by Abhinay Sharma PDF
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Abhinay Sharma Maths Book PDF Details
- Book Name: Abhinay Maths PDF Download
- Size: 53 MB
- Pages: 508
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Abhinay Maths PDF Chapters
All the topics of Abhinay Maths Book PDF book are divided into six chapters, such as –
- Percentage (प्रतिशत)
- Compound Interest (चक्रवृद्धि ब्याज)
- Average (औसत)
- Boat & Streams ( नाव और धारा )
- Geometry Triangles (रेखागणित )
- Mixture & Allegation (मिश्रण)
- HCF & LCM
- Number System (संख्या पद्धति)
- Profit & Loss ( लाभ और हानि )
- Ratio & Proportion (अनुपात)
- Simple Interest ( साधारण ब्याज )
- Time & Work ( कार्य और समय )
- Distance Speed & Time ( समय दूरी और चाल )
- Work & Wages ( कार्य और वेतन )
- Partnership ( साझेदारी )
- Pipe & Cistern ( नल एवं टंकी )
- Algebra (बीजगणित)
Abhinay Sharma Maths Book PDF Download
You can easily download this book from the given link below and open it with any pdf viewer or office software through your mobile or computer.
Abhinay Maths PDF Chapter Wise Notes Download
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Frequently Asked Questions (FAQ)
How do you calculate a quadratic equation.
- To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
What is the quadratic formula?
- The quadratic formula gives solutions to the quadratic equation ax^2+bx+c=0 and is written in the form of x = (-b ± √(b^2 - 4ac)) / (2a)
Does any quadratic equation have two solutions?
- There can be 0, 1 or 2 solutions to a quadratic equation. If the discriminant is positive there are two solutions, if negative there is no solution, if equlas 0 there is 1 solution.
What is quadratic equation in math?
- In math, a quadratic equation is a second-order polynomial equation in a single variable. It is written in the form: ax^2 + bx + c = 0 where x is the variable, and a, b, and c are constants, a ≠ 0.
How do you know if a quadratic equation has two solutions?
- A quadratic equation has two solutions if the discriminant b^2 - 4ac is positive.
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Quadratic Formula Calculator
This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax 2 + bx + c = 0 for x, where a ≠ 0, using the quadratic formula .
The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots. Calculator determines whether the discriminant \( (b^2 - 4ac) \) is less than, greater than or equal to 0.
When \( b^2 - 4ac = 0 \) there is one real root.
When \( b^2 - 4ac > 0 \) there are two real roots.
When \( b^2 - 4ac < 0 \) there are two complex roots.
The quadratic formula
is used to solve quadratic equations where a ≠ 0 (polynomials with an order of 2)
Examples using the quadratic formula
Example 1: Find the Solution for \( x^2 + -8x + 5 = 0 \), where a = 1, b = -8 and c = 5, using the Quadratic Formula.
The discriminant \( b^2 - 4ac > 0 \) so, there are two real roots.
Simplify the Radical:
Simplify fractions and/or signs:
Example 2: Find the Solution for \( 5x^2 + 20x + 32 = 0 \), where a = 5, b = 20 and c = 32, using the Quadratic Formula.
The discriminant \( b^2 - 4ac < 0 \) so, there are two complex roots.
calculator updated to include full solution for real and complex roots
Cite this content, page or calculator as:
Furey, Edward " Quadratic Formula Calculator " at https://www.calculatorsoup.com/calculators/algebra/quadratic-formula-calculator.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators
Quadratic Equation Solver
Is it quadratic.
Only if it can be put in the form ax 2 + bx + c = 0 , and a is not zero .
The name comes from "quad" meaning square, So, the biggest clue is that highest power must be a square (in other words x 2 ).
These are all quadratic equations in disguise:
How does this work?
The solution(s) to a quadratic equation can be calculated using the Quadratic Formula :
The "±" means you need to do a plus AND a minus, so there are normally TWO solutions !
The blue part ( b 2 - 4ac ) is called the "discriminant", because it can "discriminate" between the possible types of answer. If it is positive, you will get two normal solutions, if it is zero you get just ONE solution, and if it is negative you get imaginary solutions.
Quadratic Formula Calculator
Enter the equation you want to solve using the quadratic formula.
The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients. For equations with real solutions, you can use the graphing tool to visualize the solutions.
Quadratic Formula : x = − b ± b 2 − 4 a c 2 a
Click the blue arrow to submit. Choose "Solve Using the Quadratic Formula" from the topic selector and click to see the result in our Algebra Calculator !
Solve Using the Quadratic Formula Apply the Quadratic Formula
Solve Using the Quadratic Formula x 2 + 5 x + 6 = 0 Solve Using the Quadratic Formula x 2 - 9 = 0 Solve Using the Quadratic Formula 5 x 2 - 7 x - 3 = 0 Apply the Quadratic Formula x 2 - 14 x + 49 Apply the Quadratic Formula x 2 - 18 x - 4
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Quadratic Formula Calculator
The calculator below solves the quadratic equation of ax 2 + bx + c = 0 .
In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:
ax 2 + bx + c = 0
where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. The numerals a , b , and c are coefficients of the equation, and they represent known numbers. For example, a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). Below is the quadratic formula, as well as its derivation.
Derivation of the Quadratic Formula
From this point, it is possible to complete the square using the relationship that:
x 2 + bx + c = (x - h) 2 + k
Continuing the derivation using this relationship:
Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. The x values found through the quadratic formula are roots of the quadratic equation that represent the x values where any parabola crosses the x-axis. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. This is demonstrated by the graph provided below. Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.
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Quadratic Formula Calculator & Solver
Calculator for solutions to any quadratic equation..
The calculator uses the quadratic formula to find solutions to any quadratic equation .
The quadratic formula calculator below will solve any quadratic equation that you type in. Simply type in a number for 'a', 'b' and 'c' then hit the 'solve' button.
Quadratic Formula Calculator and Solver
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The calculator on this page shows how the quadratic formula operates, but if you have access to a graphing calculator you should be able to solve quadratic equations, even ones with imaginary solutions.
- Step 1) Most graphing calculators like the TI- 83 and others allow you to set the "Mode" to "a + bi" (Just click on 'mode' and select 'a+bi').
- If you can set your calculator's mode to a + bi you should be able to even calculate imaginary solutions.
- The rest of the steps just involve typing in a,b and c. Make sure that you divide the entire numerator by 2a, just use parentheses.
- Quadratic formula worksheets (several free printable pdfs with answer keys on the quadratic formula)
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Quadratic Equation Solver · Step-By-Step Example. Learn step-by-step how to solve quadratic equations! · Example (Click to try) · Choose Your Method. There are
To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). What is the quadratic formula? The quadratic formula gives solutions
This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax2 + bx + c = 0 for x, where a ≠ 0, using the
Quadratic Equation Solver. We can help you solve an equation of the form "ax2 + bx + c = 0" Just enter the values of a, b and c below:.
Quadratic Equation Solver. If you have an equation of the form "ax2 + bx + c = 0", we can solve
Enter the equation you want to solve using the quadratic formula. The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients
A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the
The calculator uses the quadratic formula to find solutions to any quadratic equation. ... The quadratic formula calculator below will solve any quadratic
Inuyasha The Final Act (Subbed) · ALEKS - Drawing the Haworth projection of a ketose from its Fisher projection · Solving Linear Equations Using


The 18th Annual Abacus International Math Challenge runs from September, 2014 through April 30th, 2015.
Categories & Contacts
In your first e-mail, please indicate your name, your grade/age, the name of your school, and the city where you live. Make sure that you indicate the number of the problem you are responding to. For proper identification, every e-mail you send should include your name.
SHOWING YOUR SOLUTION:
The solution to a problem should include the results and your reasoning. Make sure that you try to find all the possible solutions for a problem. Your reasoning has to be given in English, but do not be discouraged if English is your second language. If we have a question about your answer, we will contact you.
Try to send your answers by the last day of the month for the posted problem, however solutions to any of the problems will be accepted until April 30, 2015. (You are welcome to participate in higher grade groups by solving their problems.)
You get five points for a thorough solution with reasoning; fewer points for a partial solution or solution with no reasoning, and more points if you find additional different solutions or prove more than required.
You may earn extra points if you design your own problems and they get posted in the challenge. So, try to make up some problems and solve them.


In a surprisingly large part, our reality consists of computable problems. Should I buy or rent this? What is my ideal calorie intake? Can I take this loan? Or even how many items do I need to sell at what price in a business venture to reach a point where profits equal costs? Often we are unable to solve these problems because we lack the knowledge, skills, time or will to do the calculations. And then we make bad, uninformed decisions.
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This site seeks to solve all the little math problems that people do on an everyday basis. You can always use the calculator on your device or do a quick search, but for more complex answers there are 1585 calculators on this site to try to help.
Omni Calculator is a web/mobile calculator that solves tangible problems like mortgage payment, gross margin, body mass index, cooking, zombie invasion and everything else you can think of (within reason).
Omni Calculator exists to give you a superpower – to see the world through the lens of science. There are 1000+ custom calculators built here that take just seconds to use and give you exactly the numbers you need.
It’s simple to use, find the calculator group you want, click it, and choose from several specialized calculators in the following categories:
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Omni Calculator works on the web, mobile and as an embedded web widget.


Quite simply, slots use mathematics to function, so to win at slot games, youâll have to use maths too! Donât worry, you donât need to be a mathematical whizz kid to win at slot games, playing free slot games here is really quite simple, but understanding the maths behind the game will help you to get to grips with how they work and ultimately choose the best slot game to play on.
If you havenât heard of a random number generator, this is the algorithm that determines the outcome of every spin of the slot machine. Often abbreviated to RNG, a random number generator, as you might have guessed from the name generates the symbols that appear on the reels at complete random. This means that they cannot be influenced externally by anyone, including players. This is in order to protect the game, players and the industry. After all, if you could predict the outcome of the slot, thereâd be no fun in that, it wouldnât be fair for other players and the industry would collapse, meaning no slots for anyone! The number codes generated at random correspond to symbols on the reels and if you land a certain combination, you win!
A maths-based approach to winning slot games includes working out the probability of you winning a slot game and to work out the probability of a particular slot, you will need to know what the winning combination pays out in addition to what the probability will be of getting that combination.
Your first port of call should be the paytable, which will list how much certain combinations payout. Then you will need to calculate the probability of getting that combination during the game. Itâs easy enough to calculate, all you need to do is calculate the amount of stops there are in the slot. For example, if the slot has three reels with five symbols on each reel, simply multiply the number of symbols on each reel to calculate the possible combination: 5 x 5 x 5 = 125 possible winning combinations! If the jackpot in the same slot then pays out for three lemon symbols, and only one lemon symbol occurs on each reel, to calculate the probability of a jackpot would be: 1/5 x 1/5 x1/5 which means you would have a 0.008% chance of landing the jackpot in that slot game.
So, as long as you know the number of times a certain symbol appears of each reel, itâs easy to work out the probability of getting any combination! It can get tricky however with slot games which have a large number of reels and winning combinations, especially as many modern slot games have up to 10 reels with 200+ stops.
To help you decide which slot to play, check the Return to Player percentages (RTP) which shows the overall percentage of the total bets which a slot game would theoretically payout over a prolonged period of time.


McGill provides Common Core math through its Houghton Mifflin Expressions (TK-3rd) and Go Math (3rd-5th) programs. Math manipulatives and school-to-life connections are saught for the internalization of math concepts.
To support the various mathematical concepts from number sense, to algebraic thinking and problem solving, we offer online math programs for students to practice their daily math skills. Our TK/K and 1st grade teachers use Sumdog.com, our second grade teacher uses Turtlediary.com, and 3rd-5th use Aleks and LeverEd.
Recognizing the dilemma of multiple programs, McGill is in the process of selecting one math curriculum to create a TK-5th grade continuum.
McGill assesses students with Renaissance STAR assessments four times a year to keep track of learning progress and need for Response to Intervention supports.


Comparing Areas of Parallelograms in Mathematics Education
What is a Parallelogram?
A parallelogram is a geometric shape that has two pairs of parallel sides. It is similar to a rectangle or a square, but the opposite sides are not necessarily perpendicular. The opposite sides of a parallelogram are equal in length, and the opposite angles are also equal.
Parallelograms are important because they are used in many real-world applications such as engineering, architecture, and construction. They are also commonly used in mathematics to illustrate concepts such as vectors and transformations. Comparing the areas of parallelograms is an important aspect of studying them, as it helps us understand their relationships with other shapes and their geometry.
How to Compare the Areas of Parallelograms
The formula for calculating the area of a parallelogram is base x height, where the base is one of the parallel sides and the height is the perpendicular distance between the two parallel sides. If two parallelograms have the same base length but different heights, then the one with the greater height will have the greater area. Similarly, if two parallelograms have the same height but different base lengths, then the one with the greater base length will have the greater area.
Another way to compare the areas of parallelograms is to use the formula for the area of a triangle, which is 1/2 base x height. If we draw a diagonal line through a parallelogram, we can split it into two equal triangles. The area of each triangle is half the area of the parallelogram. So, we can compare the areas of two parallelograms by comparing the areas of the triangles that they are split into.
Examples of Comparing Areas of Parallelograms
Let’s take two parallelograms with the same height but different base lengths. The first parallelogram has a base length of 6 cm and the second parallelogram has a base length of 8 cm. Since they have the same height, the second parallelogram will have the greater area because it has the greater base length.
Now, let’s take two parallelograms with the same base length but different heights. The first parallelogram has a base length of 5 cm and a height of 4 cm. The second parallelogram has a base length of 5 cm and a height of 6 cm. Since they have the same base length, the second parallelogram will have the greater area because it has the greater height.
Comparing the areas of parallelograms is an important aspect of geometry and is used in many practical applications. To compare the areas of two parallelograms, we can compare their base lengths and heights or use the formula for the area of a triangle. By understanding how the areas of parallelograms compare, we can better understand their properties and geometry.
What is a Parallelogram?
A parallelogram is a two-dimensional shape that has four sides and four vertices. A parallelogram has two pairs of parallel sides, which means that the opposite sides of the parallelogram are both parallel and congruent. This means that if you were to draw a line across the parallel sides of the parallelogram, you would create two identical triangles. The properties of a parallelogram include opposite angles being congruent and opposite sides being congruent, the diagonals bisecting each other, and the sum of the interior angles equaling 360 degrees.
How do the areas of parallelograms compare?
The area of a parallelogram can be found by multiplying the base (one of its sides) by its height (the perpendicular distance between the two parallel sides). The formula for finding the area of a parallelogram is:
Area = base x height
The base and height of the parallelogram can be any pair of opposite sides, as long as the height is perpendicular to the base. Therefore, the area of parallelograms with the same base and height will be equal to each other, regardless of the congruency of its sides and angles.
However, if the parallelograms do not have the same base and height, their areas will differ. This means that if two parallelograms have the same base, but different heights, the one with the longer perpendicular distance will have a larger area. Similarly, if two parallelograms have the same height, but different bases, the one with the longer base will have a larger area.
It is also worth noting that the area of a parallelogram is equal to the area of a rectangle with the same base and height. This is because a rectangle is simply a special type of parallelogram where all angles are right angles, which makes calculating its area easier since it only requires multiplying its length and width.
In conclusion, when comparing the areas of parallelograms, it is essential to consider the length of its base and height. The area of two parallelograms with the same base and height will be equal, but if either of those measures differ between the parallelograms, then so will their area.
Area of a Parallelogram
Parallelograms are geometric shapes that have two pairs of parallel sides. The area of a parallelogram is the amount of space inside the parallelogram. To calculate the area of a parallelogram, you need to know the length of the base (b) and the height (h). By multiplying the base by the height, you can determine the area of the parallelogram.
The formula for the area of a parallelogram is:
Area = base x height
Area = b x h
For example, if the base of a parallelogram is 6 units and the height is 8 units, the formula would be:
Area = 6 x 8 = 48 square units
The area of the parallelogram would be 48 square units.
It is important to note that the unit of measurement used for the base and height must be the same. If the base is measured in feet, then the height should also be measured in feet. Otherwise, the formula will not produce the correct answer.
You can also use trigonometry to calculate the area of a parallelogram if the lengths of two sides and the angle between them are known. The formula for the area using trigonometry is
Area = a*b*sin(θ)
where a and b are the lengths of two sides and θ is the angle between them.
Overall, calculating the area of a parallelogram is a simple process once you have the length of the base and the height. By using the formula, you can determine the amount of space inside the parallelogram and compare areas of different parallelograms.
Comparison of Areas
Parallelograms are geometric shapes that have two pairs of parallel sides. They come in different shapes and sizes and can be classified based on their properties. One of the most important properties of a parallelogram is its area. The area of a parallelogram is the amount of space it takes up, measured in square units.
When comparing the areas of different parallelograms, it’s important to consider the dimensions of each shape. The formula for finding the area of a parallelogram is base x height. Therefore, the area of a parallelogram will increase or decrease depending on the length of its base and height.
Example 1: Let’s consider two parallelograms: parallelogram A and parallelogram B. Parallelogram A has a base of 5 cm and a height of 8 cm. Parallelogram B has a base of 10 cm and a height of 4 cm. To find the area of each parallelogram, we can use the formula base x height:
Area of parallelogram A = 5 cm x 8 cm = 40 cm²
Area of parallelogram B = 10 cm x 4 cm = 40 cm²
In this example, we can see that even though parallelogram A and B have different dimensions, they both have the same area. Therefore, we can say that the areas of parallelogram A and B are equal.
Example 2: Let’s consider two more parallelograms: parallelogram C and parallelogram D. Parallelogram C has a base of 6 cm and a height of 9 cm. Parallelogram D has a base of 4 cm and a height of 12 cm. To find the area of each parallelogram, we can use the formula base x height:
Area of parallelogram C = 6 cm x 9 cm = 54 cm²
Area of parallelogram D = 4 cm x 12 cm = 48 cm²
In this example, we can see that parallelogram C has a greater area than parallelogram D. Therefore, we can say that the area of parallelogram C is greater than the area of parallelogram D.
It’s important to note that the area of a parallelogram can also be expressed in terms of its diagonals. The formula for finding the area of a parallelogram using its diagonals is ½d₁d₂, where d₁ and d₂ are the lengths of the diagonals.
Example 3: Let’s consider two parallelograms with the same base and height, but different diagonals: parallelogram E and parallelogram F. Parallelogram E has diagonals of 10 cm and 6 cm. Parallelogram F has diagonals of 8 cm and 4 cm. To find the area of each parallelogram, we can use the formula ½d₁d₂:
Area of parallelogram E = ½ x 10 cm x 6 cm = 30 cm²
Area of parallelogram F = ½ x 8 cm x 4 cm = 16 cm²
In this example, we can see that even though parallelogram E and F have the same base and height, their areas are different because their diagonals are different. Therefore, we can say that the area of parallelogram E is greater than the area of parallelogram F.
In conclusion, the areas of parallelograms can be compared using their base, height, and diagonals. By understanding and applying the formulas for finding the area of parallelograms, we can determine which shapes have greater or lesser areas.
Why the areas of parallelograms are comparable?
When we talk about geometry, one of the basic shapes that come to mind is the parallelogram. You may have learned that the area of a parallelogram equals the base times the height, and while this is true, have you ever wondered why this formula works? The answer lies in the fact that all parallelograms are similar to each other.
Similarity means that two shapes have the same shape but may differ in size. Two parallelograms are similar if they have the same shape and all angles are congruent, but not necessarily the same size. This is important because when we compare the areas of parallelograms, we can use proportions to easily calculate them.
Relationship between parallelograms and triangles
Another important relationship to note is the one between parallelograms and triangles. Both shapes have similar features, such as having bases and heights that can be used to find their areas. Additionally, any parallelogram can be divided into two congruent triangles, which helps us understand the similarities between these shapes even further.
One way to find the area of a triangle is to use the same formula as a parallelogram and divide the result by 2. For example, if you have a parallelogram with a base of 6 and a height of 4, the area would be 24. If you divide 24 by 2, you get 12, which is the area of the triangle formed by half of the parallelogram.
Parallelograms and rectangles
Another shape that is closely related to parallelograms is a rectangle. While a parallelogram can have any angle measure, a rectangle has four right angles, making it a more specific type of parallelogram. One way to think about a parallelogram is as half of a rectangle, where the height is not perpendicular to the base.
To find the area of a rectangle, we use a simpler formula: length times width. This formula can also be used to find the area of a parallelogram if we know the height and the length of the side that the height intersects. Essentially, we can think of the height as the width of the parallelogram, and the base as the length, which makes it easier to compare the areas between these shapes.
Parallelograms and trapezoids
Lastly, we can see the relationship between parallelograms and trapezoids. A trapezoid is a shape with one pair of parallel sides and one pair of non-parallel sides. While the area formula for a trapezoid is a bit more complicated than a parallelogram, they share the same base concept of using the height and the length of the side that the height intersects to find the area.
A parallelogram with a right angle is a special type of trapezoid called a right trapezoid. In this case, we can use the formula for finding the area of a rectangle to find the area of our right trapezoid since one of the legs is perpendicular to the base.
The relationships between parallelograms and other geometric shapes are important to understand. Parallelograms are similar to each other, allowing us to compare their areas easily using proportions. Additionally, we can see the similarities between parallelograms and triangles, rectangles, and trapezoids, which helps us better understand these shapes as a whole. Overall, geometry is a fascinating subject full of connections and relationships between shapes that can help us better appreciate the world around us.
Real Life Applications
The concept of comparing areas of parallelograms is not just a math problem that students learn in school but has many applications in everyday life. One of the fields where this concept is widely used is architecture. Architects and engineers use parallelogram-shaped structures extensively in their designs, and they should have good knowledge of finding the correct areas of those structures.
When designing a building, architects have to work with measurements related to floor areas, wall areas, and roof areas. A building with a larger area often requires a stronger foundation, more support, and higher costs for construction. This is where the concept of comparing areas of parallelograms comes in handy as it allows architects to efficiently determine the size and shape of a building that would fulfill the requirements within a budget. For example, in a commercial building, an architect may need to calculate the area of a parallelogram-shaped roof for installation of solar panels or HVAC units.
In civil engineering, finding the area of a parallelogram is also essential for calculating the amount of material required for road construction, pavement, and drainage systems. Engineers use the formula to estimate the amount of asphalt required to pave a parallelogram-shaped stretch of road. It is also useful for stormwater management, where the engineers need to calculate the areas of the parallelogram-shaped drainage basin that can hold a specific amount of water to ensure that the water is channeled appropriately.
The concept of comparing areas of parallelograms finds extensive use in product design. Designers visualize and create a product in a 3D design software before manufacturing it. The software helps the designers to estimate the material required for manufacturing the product.
For example, if a designer wants to create a parallelogram-shaped phone case, it is essential to know the area to determine the amount of material required for manufacturing the case. In this case, the length and width of the parallelogram and the perpendicular height are necessary for calculating the area of the shape. Moreover, artists and sculptors use the concept of comparing areas of parallelograms to create various art pieces such as paintings, pottery, and sculptures. They may use this concept to calculate the area required for creating a specific shape or to create different ratios of areas to make their artwork more visually appealing.
Land surveying refers to the measurement and mapping of land and the related natural features. It finds application in the construction industry, environmental management, and agriculture. The concept of comparing areas of parallelograms is essential in land surveying as the surveyor needs to determine the exact size and shape of plots of land.
Land surveyors use specialized tools and equipment to measure the length and width of the land parcel to calculate its area. The land parcels are sometimes in the shape of parallelograms. For example, a farmer can use the measurements to determine the area of a farm and decide how much fertilizer would be required to accommodate the crop. A land surveyor also uses this concept in construction projects to calculate the land area required for the construction of buildings, roads, or bridges.
The concept of comparing areas of parallelograms is essential in everyday life, and it finds extensive application in architecture, engineering, product design, and land surveying. By understanding this concept, people can make more informed decisions before constructing buildings or purchasing land. The importance of this concept cannot be overstated as it plays a significant role in various industries and has practical implications.
Importance of Understanding and Comparing the Areas of Parallelograms
Parallelograms are one of the most common shapes in geometry, and understanding their areas is essential in various real-life applications. The formula for finding the area of a parallelogram is simple; it is the product of base length and height. The straightforwardness of this formula is exactly what makes it such a useful tool for measuring surface areas in various fields and disciplines. Here are some of the reasons why understanding and comparing the areas of Parallelograms is essential:
1. Geometric Calculations
In geometry and trigonometry, the area of a parallelogram plays a significant role in calculations related to shapes and angles. For example, understanding and comparing the areas of parallelograms is essential when calculating the area and perimeter of a polygon. It is also used in determining the angle between vectors in physics and engineering.
2. Construction and Architecture
Parallelograms are very useful shapes in the fields of architecture and construction. Architects and builders use parallelograms to create a variety of structures and designs. For instance, they can use parallelograms to create roofs, gables, and other shapes. By understanding the areas of parallelograms, they can determine the amount of material required to build these structures, which helps them estimate project costs.
3. Transportation and Logistics
The transportation and logistics industry uses parallelograms in a variety of ways, such as calculating the volume of cargo required for shipment. By understanding the areas of parallelograms, transportation and logistics professionals can determine the size of the containers required to ship products efficiently. This knowledge helps them optimize space in storage and shipping containers, resulting in fewer trips and lower costs.
4. Land Surveying
Land surveyors use parallelograms in their work, determining the size of plots and land ownership boundaries. They can also use the area of parallelograms to calculate the location of roads, buildings, and other structures. Without an understanding of parallelograms, accurate land surveying would be impossible.
5. Agriculture and Farming
The cultivation of crops often requires area measurement since it is crucial in determining how many plants or seeds are needed to fill a given field. Farmers and agriculturalists use the area of parallelograms to estimate the potential yield of their crops and create accurate planting plans. This knowledge helps them maximize their harvests, resulting in better crop productivity.
6. Everyday Life
Parallelograms are prevalent in our everyday lives, such as in packaging and home décor. Understanding the areas of parallelograms can help us measure and calculate the amount of material required for packaging products, such as gift wrapping paper. Measuring the areas of floor spaces, wall surfaces, and ceiling spaces in home decor is also possible by using parallelograms.
7. Education and Learning
Understanding and comparing the areas of parallelograms is essential in education and learning, specifically in geometry and trigonometry. A clear understanding of the formula for finding the area of parallelgrams, as well as other common geometric shapes, is required for students to progress through math courses and understand various real-life applications. Additionally, knowledge of the areas of parallelograms is required when studying more advanced mathematical concepts, such as calculating the surface area and volume of three-dimensional shapes.
In conclusion, understanding and comparing the areas of parallelograms are essential in various fields, disciplines, and everyday life applications. As highlighted in this article, different sectors rely on the formula for finding the area of parallelgrams to calculate surface areas, optimize spaces, improve productivity, and estimate project costs, among others. Additionally, knowing the areas of parallelograms is a basic requirement for learning geometry, trigonometry, and more advanced mathematical concepts.


Here’s a free 1-hr. webinar, Thursday, December 13, 12:00 PM EST, covering robot specification and how fabricators can ensure they select the right robots for the tasks at hand. Come a understanding robot reach, work envelope, payload, inertia, moment of arm and cycle time. Presented by the Robotic Industries Association with support from its members ABB, Kawasaki and Omron STI, the webinar also addresses the concepts of mathematical modeling and simulation. Register here.


Webserver to compute b> Expected 5-3 distance in RNAb>
The C program averageDistance5to3.c and its Python prototype distance5to3.py are efficient and exact computations of the expected distance between 5' and 3' ends of all secondary structures for a given RNA sequence. The expectation or average is taken with respect to the Boltzmann probability exp(-E(S)/RT)/Z, where E(S) is the Turner energy for secondary structure S, R the universal gas constant, T absolute temperature, and Z the partition function.
If you use our code or this server for your work, please consider citing the paper:
Expected distance between terminal nucleotides of RNA secondary structures. P. Clote, Y. Ponty, J.-M. Steyaert. J Math Biol. 2012 Sep;65(3):581-99. Epub 2011 Oct 9.Supplementary information is here.


After high school, the majority of the population throws out their advanced calculators and rejoices in the thought of never having to look at another equation again. That is, until you have to help kids with their homework and your phone’s calculator just doesn't cut it.
Photomath claims to solve all of your maths problems (literally and figuratively) and it's being dubbed a “smart camera calculator”.
It's as simple as pointing your camera at a math problem and the app will produce the answer as well as a step-by-step guide on how to solve the problem. While the app can literally do kids’ homework for them, it can also give them an understanding of the logic behind the answer in a few steps.
The app was created by a company called MicroBlink, which uses text recognition technology that interprets the math problem once it has been scanned – providing instant feedback. Photomath supports basic arithmetics, fractions, decimal numbers, linear equations and several functions like logarithms. The app doesn't recognise handwritten text, for now, meaning that users can only scan textbook equations by placing the equation within the red frame.
There are a few kinks that need to be worked out along the way, aside from only recognising printed text, such as the text recognition technology mistaking an "X" variable for a multiplication symbol, and the red frame limiting the length of the equation that can be scanned.
However, maths equations are just the beginning for MicroBlink as they plan to make advances in using the technology to revolutionise online banking.
Photomath is available on iOS and Android operating systems at no cost!


ENSTA ParisTech - Université Paris 1 Panthéon-Sorbonne
Master MMMEF - Track "Optimization, control and operations research"
Academic year 2014/2015
"Stochastic Optimisation: Numerical Methods"
The aim of this course is to provide a framework for extending the optimization methodology already studied in the convex deterministic case to the stochastic case, both from the theoretical and the numerical points of view.
The course consists of two parts:
During the first part of the course, we put the focus on large-scale optimization problems and decomposition/coordination methods.
- we are first interested in stochastic open-loop optimization problems, and we thoroughly study the stochastic gradient method and its variants,,
- we then move to closed-loop optimization problems, and study on the one hand the significance of the information structure for such problems, and on the other hand the difficulties related to the discretization of these problems.
The course takes place on Tuesday afternoon from 14:00 to 17.30 at ENSTA
(Getting to ENSTA),
and is given in English.
Lesson 1 (January 13, room 2.4.30)
Introduction to Stochastic Optimization: motivations and objectives
of the course, reminders from the deterministic case and transition
to the stochastic case. Overview of the stochastic gradient method.
Lesson 2 (January 20, room 2.4.30)
Generalized stochastic gradient method: introduction to the Auxiliary
Problem Principle, convergence in the cases without and with constraints.
Lesson 3 (February 3, room 2.4.12)
Exercises on the stochastic gradient method and presentation of extensions
of the method to the case of constraint in expectation and in probability.
Lesson 4 (February 10, room 2.4.12)
Dual effect in stochastic optimization: information structure, dual
effect issues and application in stochastic optimal control.
Lesson 5 (February 17, room 2.4.12)
Discretization: issues of discretization in stochastic optimization,
counterexample, convergence theorem.
Lesson 6 (March 3, room 2.4.30).
Course notes about stochastic optimization (in English)
Course notes about the stochastic gradient method (in French)
Toolbox ''Stochastic Gradient''(in French)
Le but de cette boîte à outils (écrite en langage Scilab) est d'illustrer sur un exemple simple le comportement du gradient stochastique ainsi que sa vitesse de convergence, et de montrer ce qu'apporte la technique de moyennisation.
Page managed by P. Carpentier
(last update: February 18, 2015)


As I sat in the elementary art class, on assignment as a substitute at the school, I listened to the teacher’s lesson to the children on concentric circles. The exercise would begin with a dot being drawn. A small circle would be drawn around the dot, then a larger one around that, and then a larger one, and so on, and so on. Each circle shared the same center. In this particular exercise, several series of concentric circles would be drawn on a large sheet of paper, overlapping each other. The different color patterns made quite a cool effect.


Lambda Dice Version 1.01
Lambda Dice is a small and compact program for generating random numbers and simulating die rolls. It is intended to be used with tabletop or pen-and-paper gaming, but it can be used for any purpose that requires random numbers.
The program lets you generate any number of rolls using dice with any number of sides, including strange or unrealistic numbers, such as a 7-sided or 1000-sided die.
It requires version 4.5 or newer of the .NET Framework. If you need that, you can get it from Microsoft
Lambda Dice is free software and may be distributed freely.
Lambda Dice has had more than 13000 downloads.
Copyright © 2006-2021 Jason Champion


People of all ages love to play games that are fun and motivating. Therefore one of the best ways to make learning fun is ensuring that you organise exciting and engaging educational games for children during your interactions with them. Educational games for children offer opportunities to deepen an understanding of concepts learnt in school and teaming them up with HP Print Learn Center’s worksheets will ensure that your child’s learning journey is filled with smiles, laughter and conceptual clarity.
Educational games and worksheets give kids opportunities to explore fundamental learning concepts across subjects, for example counting sequence, one-to-one correspondence, computation strategies in math and language conventions, comprehension, fluency, phoneme awareness in literacy. At HP PLC you can avail of age level worksheets from ages 3-12 years. These worksheets include activities and fun educational games as well. All these educational worksheets have been designed by experts. The worksheets target age-appropriate learning goals and reinforce concepts taught in school. These are a list of some educational games for kids you can play with your child.
I Spy is a game that is always a favourite with kids. It can always have an interesting educational twist as well. For example, you can play the game at home or when you’re having a walk in the park or even when you’re thinking about what ingredients to use when you’re making a sandwich.
Here are many variations on this game:
- Something beginning with sound…
- Something that rhymes with the word…
- Something that is shaped like a…
- Something the colour of…
You will find similar educational games online that can also be paired with HP PLC worksheet. For example in this worksheet here the children have to play the ‘I Spy’ game and find the numeral 6
Magnet wand Game
Activities do not always have to be online educational games for kids to learn something, in fact you can make up educational games at home. For this activity attach a magnet to a ruler to make it into a wand. Take a tray with magnetic letters and have your child pick up the letters with the help of the magnetic wand. Team this up with HP PLC worksheet of Alphabet Sequencing.
Deck of Cards Game
A deck of number cards can be used as an excellent educational game for children and can be used in many fun ways to practice math concepts. For example, you can practice the concept of greater, lesser than and equal to using the cards. To play the game, have the child pick up a card and place it in a manner that shows the number on top. Next, let the child pick up another card and place it next to the previous card. Let him/ her compare the 2 numbers after identifying them. Which number card is greater/lesser or equal and place the correct sign in the middle. You may also play the game with 2-digit numbers to increase the complexity of the game. Team this up with the HP PLC worksheet of Comparison of Numbers.
Math with Dice
Manipulatives such as dice are a great learning resource which can be used to play a variety of educational games. For instance, a great way for children to practice addition can be through a quick and fun dice game. Have your child pick up a number chit and read aloud the written number. Next, ask the child to use 2 dice to form the given number. For example, the child may use 2 dice that show numbers 5 and 6 to make number 11. Similarly encourage the child to come up with different number combinations that add up to the given number. Team this up with the HP PLC worksheets of Addition.
Bowling for Sight Words
Movement while learning helps in breaking the monotony and keep children engaged. Write the sight word written on a paper cup and place them like bowling bottles. Let the child bowl using a ball. Next, ask him/her to read aloud the sight word written on the fallen cup. The same activity can be used with any new words that the child has learnt related to different subjects, be it Science/Math/English etc wherein the child can explain the meaning of the new terms learnt using the same activity. This same concept can be explored through the HP PLC worksheet on sight words.
Racing Ramps Game
Science is all about experimenting. Children love playing with toys such as vehicles in particular. Build a ramp to test the speed of different objects. Race the cars and balls on the ramp and discuss why some objects move faster than others. Consider the weight and shape of each object. You may also explore racing the objects on different textured surfaces. For example, place a towel on the ramp and observe the difference in the speed as well as explore the concept of friction. Pair this activity with a HP PLC worksheet that will help children record their observations.
Mathematical concepts such as measurements need not be only practiced through word problems. A more practical approach to understand this concept is by observing and measuring things in your surroundings. The HP PLC printable worksheets on measurement can be used to move around the house and measure the different areas of the home with one’s feet. You may also have the child explore areas outside the home such as in a park or a playground and use a particular unit of measurement later as a variation to the same activity.
Online educational games for kids offer exciting opportunities for children to practice the concepts learnt. Some educational games online that children can play are –
Twelve a Dozen
This is one of the educational games online that introduces the initial thought process involved in solving algebra and exploring longer, more complex algebraic equations. The child has to help Twelve save her family from being destroyed – by solving puzzles based on core math concepts. Team this game up with HP PLC math worksheets that can be downloaded from the platform
Think rolls Play and Code
This is a puzzle online educational game for kids that introduces kids to concepts in physics and develops precoding skills. Kids need to help the character navigate the maze by solving puzzles. They learn about gravity, force, spatial reasoning, elasticity, and more. And because they also learn coding, kids can create their own puzzles in the game as well! Strengthen more such Physics concepts by downloading the science worksheets from the HP PLC platform.
Learning is fun when there is an auditory and visual effect involved in it. This online educational game for kids will allow the animated monsters to teach your kids everything from basic ABCs to complex sentences. It has puzzle games with talking letters and well-illustrated definitions to help your child learn to read. Pair this game with Language and Literacy worksheets available on the HP PLC platform.
Educational games for kids encourages strategic thinking as children find different strategies for solving problems and deepen their understanding of various concepts. When played repeatedly, these educational games support students’ development of computational fluency. Educational games online, also present opportunities for practice, often without the need for teachers and parents to provide the problems. Educational games for children support a school-to-home connection that can be reinforced with Hp PLC printable worksheets. Parents can learn about their children’s level of comprehension by playing these with them at home.


I introduced a new game with my 8th graders yesterday. It is always hard to know whether a game will be a hit or a flop, so I was surprised by the enthusiasm with which they received this very simple activity.
The instructions are very simple: the students are to count a range of numbers, say from 1 -20, 100 and up or whatever you choose (I am also planning to use letters of the alphabet, simple German verb conjugations etc.). Instead of having an order of which student goes when, they are to say the number out when it feels like it's their turn. The only rule is that if two students speak at the same time, they have to start over.
As I was watching the 8th grade, I saw just how deeply this game meets the children. For one, it is a rhythmical activity which helps them relax into class. They also have to get very still, listening to each other. This prepares them extremely well for the rest of class. Finally, the children love a challenge, and this game sets the tone for trying to do better all the time, an element that I have found can be challenging for some of the students.
So this one was a winner which we will be playing regularly.


Best Probability of Winning in Casino Gambling
A casino game is any of a variety of casino games that make use of random chance as a way of determining the outcome of the game. The outcome of a single game is not influenced by the choices of any more than one person. Casino action is often referred to as “payback” or “cash out” compared to conventional casino games where there is absolutely no payout; these games are rather characterized by the “pot”, which is often won or lost. Recently, there has been an increase in using casino game machines as a means of recreation, gambling, and even as income generators for operators of the modern casino hotels.
There are three basic categories of casino game games: table games, gaming machines, and random number machines. Gaming devices, including pachinko and slot machines, are generally played by one person at a time and don’t require the active participation of casino personnel to play. Random number machines alternatively, like roulette, craps, baccarat, and keno, require the lively participation of the players in order to make the odds move in the casino’s favor. The random number machines found in many training video poker casinos are referred to as blackjack, roulette, and poker machines.
All three forms of casino game equipment are governed by the same mathematical laws of probability. Blackjack, roulette, and poker devices all utilize random quantity generators or random variety processors to determine the odds of success. These same regulations of probability govern the results of all other games including slot machines. In most cases, it is possible to alter the odds slightly through the use of a few of the many slot gaming devices on the market, though it should be noted that even this small change will have little effect on the ultimate outcome of the overall game.
The most popular of the casino game devices may be the slots. Slots are carefully identified with gambling thrillers, since they require luck and are made to keep people playing for long periods of time. Of all the games available on the Las Vegas strip, slots are the most popular attraction, drawing thousands of visitors each night. While the prospect of winning major may entice some people, the long wait necessary for a single gain poses another barrier to access, making slots probably the most appealing casino game options for the traveler.
Blackjack, like slots, is really a game of chance. Fortune is involved in a great deal of casino game outcomes, although expertise also plays a part. A well-planned strategy can help one improve the odds of winning, but there is no inherent strategy that guarantees good results. For this reason, no matter which casino game is performed, skill, patience, and a bit of knowledge can help one improve the odds of hitting the jackpot. One valuable strategy would be to analyze casino game outcomes using the binomial distribution.
Binomial distribution probabilities can be used to examine casino games at any degree of play, including the novice person. The binomial distribution capabilities as a robust tool for identifying the 모나코 카지노 best odds of hitting in certain slot odds and table games. In roulette and other table games with typical deviation probabilities, the binomial distribution can be used to identify the best odds of hitting in a set number of trials. Common deviation probabilities take random dimensions of casino results and will be used to examine the probability of casino results from the random event. Once again, a basic understanding of statistics is necessary to understand how these probabilities may be used to identify the best odds in casino video games.
It ought to be noted, even so, that the binomial model will not apply to all casino games also to all casino gaming equipment. It applies most efficiently to games with standardized random outcomes, such as roulette, blackjack, and the slot machines featured in many casinos. Because the outcome of any casino game is totally random, the probability of hitting a jackpot with one machine is the same whether you are playing at an online internet casino or at a offline casino. Hence, in cases like this, the binomial model wouldn’t normally use.
Slot machines are by far the most popular games of all casino floors and so are the focus of much of the entertainment at casinos around the world. Blackjack, craps, baccarat, and other slots are the most popular games in Las Vegas, Atlantic City, and Macao, and there are literally hundreds of slots per floor at most NEVADA hotels. The large variety of slot machines on any given casino ground raises the odds of hitting a jackpot and makes it more likely that players will be successful in their bets. Slots are the most reliable way to increase your likelihood of winning for the most part casinos.


||flat slice histogram
||Software Studies Blog
This is a "flat slice histogram", a slice histogram where each bin is forced to a fixed height. Normally, histogram bins are fixed "width", meaning they cover the same range of the distribution variable. Here, the bins are fixed height, and thus the variable ranges they cover are unknown (without additional markings on the plot, which is of course a possible modification here). Note that the flat histogram would be perfectly useless for traditional statistical plotting, but since we are making image histograms, they can reveal a lot. The left-to-right visual patterns do still map roughly to the distribution variable. Here, hue is the distribution variable, and we sort vertically by brightness (a couple times). We can see quite clearly that in this dataset, blues and reds tend to be darker than the greens and yellows.


Statistical Consulting Services
Our team can help you unleash the power of statistical analysis. The use of an advanced mathematical approach to solve difficult problems can greatly increase the productivity of your business.
Give us a call at 1-818-850-7850 for a FREE initial phone consultation, or send us an e-mail, or send us a FAX at 1-651-691-2616 to tell us about the project you need help with
No matter the type of problem, we can take care of your math and statistical needs. Our commitment is to offer customized help that is designed to fit your specific and unique situation.
We provide thesis help (we work with all major statistical software packages), data mining and analysis, and can help you with business intelligence tools.
• Areas of Specialization
We specialize on a wide spectrum of topics, which include:
- SPSS®, Minitab®, Excel®, STATA® , SAS®, EVIEWS®, JMP®, R projects
- Multiple Regression Analysis, Logistic Analysis
- Analysis of Variance and Hypothesis Testing, Factorial ANOVA, Repeated Measures and Mixed Designs
- Power Analysis and Sample Determination, Meta-Analysis
- Non-Parametric Statistics
- Survey Design, Sampling Methodologies
- Reliability Analysis, Analysis of Scales and Dimensions
- Factor Analysis and Principal Component
- Time Series Analysis
- Linear and Nonlinear Optimization, Models
- Qualitative Analysis, Survey
- Paper Analysis and Review, APA editing
- Longitudinal Analysis
- Business Intelligence (backend, data processing and intelligent reporting)
• Who could benefit from using our services?
The use of applied statistics is arguably one of the most powerful analytical tools today, and it can be
applied to a variety of practical uses.
- Researchers and Students: Many areas of study necessitate the use of statistical analysis, but researchers may not
be familiar with all the statistical tools at their disposal. We can help with conducting research, graduate student theses, experiment design and clinical trials.
- Law Offices: We can analyze data and make statistical conclusions based on sample evidence.
- Business Owners and companie in general: We can use a variety of analytical tools to improve your productivity. We can develop customized business intelligence tools for you.
- Hotel Managers: We can use Time Series to analyze historical trends and make statistical predictions about the
occupancy rate of a given season, week or day of the year.
- Marketing Companies: Statistical Analysis can be extremely valuable to assess the results
of marketing campaigns and to identify patterns and trends.
• What should I do to contact you?
Call us for a FREE initial phone consultation at 1-818-850-7850. You can also
e-mail us or send us a FAX at 1-651-691-2616 describing your problem and we will send you a proposal and an estimate of the time and cost.
Our rates depend on the complexity of the project, and can be on an hourly basis, or on a completed project basis, depending your needs. Our rates are very competitive


One of the most famous anecdotes from Antiquity deals with the philosopher-mathematician Pythagoras (c.570-c.495), who discovered the theorem that is named after him, and sacrificed an ox – or even one hundred oxen – to celebrate this. The joke that ever since the oxen are afraid of scientific progress has been used a bit too often by scientists dismissing critical reviews.
For several reasons, this anecdote is problematic. In the first place, because it is probably one of those unhistorical tales attributed to Pythagoras. Another example is his legendary visit to the ancient Near East, which is referred to for the first time in the second century CE, when Apuleius says that the Samian sage was “believed by some to have been a pupil of Zoroaster” (Apology, 31). In his Refutation of All Heresies (1.2.12), Hippolytus of Rome (early third century CE) implies that he had read this story in a book by Aristoxenus of Tarentum, a contemporary of Alexander the Great. Yet, even if Hippolytus’ is right (which is doubtful), this means that Pythagoras’ eastern trip is unmentioned by earlier authors describing Pythagoras’ life and opinions, even though Herodotus, Plato, and Aristotle had many opportunities to discuss it. The story is almost certainly invented, just like Pythagoras’ visit to India.
The same applies to the theorem that in right-angled triangles the square on the hypothenuse is equal to the sum of the squares on the sides containing the right angle. Pythagoras and his pupils were interested in mathematical proof, certainly, but the first to attribute the theorem to the Samian sage is Proclus (412-485), who lived almost one thousand years after Pythagoras (On Euclid I, 426.6-14 [Friedlein]).
A second problem is that the principle was already well-known prior to Pythagoras. Several cuneiform texts from the twenty-first and twentieth century BCE prove the that the ancient Babylonians not only knew that a²+b²=c², but also knew that this principle was generally applicable. There is a difference in the way Babylonians and Greeks proved this rule, but it is possible to overstate Pythagoras’ importance.
J. Høyrup, ‘The Pythagorean “Rule” and “Theorem” – Mirror of the Relation between Babylonian and Greek Mathematics’ in: J. Renger (red.): Babylon. Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne (1999).


Math and reading scores for 9-year-olds in the US fell between 2020 and 2022 by a level not seen in decades, a foreboding sign of the state of American education two years after the Covid-19 pandemic began.
The results were part of the National Assessment of Educational Progress long-term trend reading and math exams, often called the “Nation’s Report Card,” conducted by the National Center for Education Statistics. The exams were administered to age-9 students in early 2020 before the pandemic and then again in early 2022, the group said.
The average scores in 2022 declined 5 points in reading and 7 points in math compared to 2020 – the most significant decline in reading since 1990 and the first ever decline in math, the organization said.
US Secretary of Education Miguel Cardona said the drop was due to remote learning versus in the classroom. “In-person learning is where we need to focus. We need to double down our efforts. I’m very concerned about those scores, and I know that we have the resources now, and we need to maintain the same level of urgency we had two years ago to get our students back into making sure that our students get support.”
It was also reported that states with more stringent lockdowns had the most severe underperformance in testing. With places that pushed ‘at home learning’ the longest seeing the most significant declines.
For more on this story, please consider these sources:
- Student test scores plummeted in math and reading after the pandemic, new assessment finds CNN
- The Pandemic Erased Two Decades of Progress in Math and Reading The New York Times
- Students’ math, reading scores during COVID-19 pandemic saw steepest decline in decades: Education Department Fox News
- Reading and Math Scores Plummeted During Pandemic, New Data Show The Wall Street Journal
- Reading, math scores fell sharply during pandemic, data show CTPost


APR is an acronym for Annual Percentage Rate. The purpose of the mortgage APR is to inform the borrower during the home loan process that there are costs associated with a home loan beyond the interest paid with each payment.
APR is a combination of the interest rate, mortgage closing costs, and if applicable mortgage insurance, all converted into a one-year yield. The APR is not the mortgage interest rate. The amount of interest paid on a mortgage is a result of the actual interest rate, not the APR.
The APR’s true purpose is to help consumers shop and compare mortgage offers. In a perfect world borrowers would be able compare two APRs and determine which loan program has the overall lower costs. Unfortunately, the world is not perfect and this isn’t always the case.
APR vs. Interest Rate
The APR is supposed to be higher than the interest rate because it accounts for the fees. The only instance when the APR will equal the interest rate is when there are zero fees and it’s a “free” loan.
APR Calculation Explained
Here’s how to calculate the APR:
- First, determine the monthly principal and interest payment using the actual interest rate. You can use our mortgage payment calculators, a scientific calculator, or an Excel spreadsheet to determine the P&I payment.
- Next, subtract all APR fees from the loan amount. Let’s call this the “APR loan amount.” Note: you will not know the APR fees on your own. A mortgage lender would need to disclose the APR fees.
- Finally, using a scientific calculator or excel spreadsheet and solve for a new interest rate (the APR) using the P&I payment as PMT and the “APR loan amount” as the loan amount.
For example, a $300,000 loan with a 6% interest rate that is fixed for 30 years has a P&I monthly payment of $1,798.65. Let’s assume the loan has $3,000 of APR fees.
To calculate the APR we solve for an interest rate using the following: a monthly payment of $1,798.65, a loan amount of $297,000, a term of 360 months (12 x 30 years), and 0 (zero) for a future value. The APR is calculated to be 6.094%.
APR Can Be Misleading
There three major flaws that exists when comparing APR’s. First, APR calculations don’t consider credits toward closing costs. Second, the APR fees can vary from lender to lender. Finally, the calculation doesn’t consider variable interest rates.
Different loan programs can have different methods to pay for closing cost. This makes comparing APRs moot. For example, one loan program may have a slightly higher APR but offers a large lender credit. Even though the lender credit is absorbing a large amount of the costs, the APR calculations don’t take who’s paying those costs in to consideration.
Another flaw when comparing APRs from different lenders is that APR fees vary from lender to lender. As a result two identical Closing Disclosures (CD) from two lenders could have different APRs. This makes it difficult to compare apples to apples between lenders if one lender is including more fees in the calculation.
The final flaw with the APR, as if the previous two aren’t big enough, is that the APR isn’t truly accurate for Adjustable Rate Mortgages (ARMs). Because the monthly payment changes throughout the term of the loan the initial APR isn’t accurate for the life of the loan.
Ultimately the APR is meant to help you be an information consumer. Just realize that it’s an imperfect method to compare loan options. As always, you’re welcome to contact the Mortgage Mark Team with any questions.
Loan Officer, NMLS # 729612


We see Year 7 as an introductory year, laying the foundation for secondary school. Subjects offered at this level include English, Maths, Art, Science, Humanities, French, Dance, Music and Philosophy. Students have several free periods a week, to enable them to explore their own interests.
In Years 8, 9 and 10, the only mandatory subjects are English, Maths, and Music. All others are electives, involving choices from VCE and other subjects which include Beginners French, Intermediate French, Advanced French, Art, Drama, Music, PE, Chemistry, Biology, Physics, Forensic Science, Outdoor Education, Dance, “Nerd Club” (ICT), Chinese, Philosophy and Anthropology, Motion Media, Guitar Club, and History. Classes in Years 8, 9 and 10 are frequently grouped according to subject choice, allowing for greater diversity of interaction, and a greater range of electives.
We encourage keen and able students in Years 9 and 10 to tackle at least one VCE subject during this time. In exceptional circumstances, Year 8 students may also take on a VCE subject.
During free time, many optional activities are offered, including sport, martial arts, chess and writing workshops.


The best score for each applicant is calculated and the results are forwarded to the CAO in July of each year.
You can calculate your score using the free online points calculator at www.careersportal.ie/qqi/, which is based on the following scoring system:
Each level 5 and level 6 component is scored:
- 3.25 for a Distinction
- 2.16 for a Merit
- 1.08 for a Pass
This number is then multiplied by the individual component credit value to a maximum of 120 credits (a total of 390 points).
It may be easiest to multiply the individual component credit value by 3 for Distinction, 2 for Merit, and 1 for Pass, multiplying by 13 and dividing by 12.


Year 4 will be taking a times table check this year. Click here for a good online resource to support their learning. There are games as well as tests and activities to check their speed recall of the facts. Please encourage your child to practise them whenever they have free time.
All the children have log in details for the website Sumdog. This is a safe learning and games environment that will encourage your child to practise their maths knowledge. Click the image below to take you to the website. If you need your childs details please contact Mrs Rolfe who will be pleased to re-issue them.


I want to design a space in which the user is the architect of the space and constructs it using the help of others stepping in within the defined boundaries. Users act as anchor points and restructure the space dynamically as they move around. My explorations do not intend to be a game, or an interactive installation but merely a reinterpretation of how space and architecture is created/defined. In these diagrams, users are anchor points that form Delaunay triangulations.
In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulationDT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Connecting the centers of the circumcircles produces the Voronoi diagram.
In these diagrams, users act as points that form Delauney Triangulaltions. After a certain number of points, then a Voronoi is created and the users (with the help of each other) will then see the fruits of their labour: the construction of a 3D geometric shape.


You can find instructional videos by clicking on: topics in the table of contents below, the categories to the right, the tags to the right and down, or by searching in the search bar to the right and up.
- Essential Algebra
- 1 – Substitution, like terms, expanding, binomials, fractions, absolute values, index laws
- 2 – Factorising, substitution, surds
- 3 – Changing the subject of formulas, solving linear equations, solving quadratic equations
- 4 – Linear equations and straight lines, gradient/slope, intercepts, gradient intercept form of a line, one point gradient formula, two point equation of a line, general form of a line, midpoint, distance
- 5 – parallel and perpendicular lines, simultaneous equations, surds
- 6 – Relations and functions, domain, range, independent/dependent variables, vertical line test
- 7 – Examples of functions, piecewise, absolute values, polynomials
- 8 – Exponential functions, logarithmic functions, log laws
- 9 – Solving exponentials, inverse functions
- 10 – Trigonometric ratios, identities, cosine and sine rules
- 11 – Unit circle for trig functions, reference/special triangles, radians, arc length, area of a sector
- 12 – Graphs of trig functions, shifts, modelling, solving equations
Learning and Practicing Mathematics via the Internet
The Khan Academy, available at http://www.khanacademy.org/, has over 2600 videos, many of which are mathematics. You can watch videos without logging in, but signing up only requires a Google account or a Facebook account and means your activity is recorded so that points and badges are earned by watching videos.
When each problem in the video is presented you should pause the video and attempt to solve the problem yourself. Once you have had an attempt or two, continue watching the video to see the approach taken by the presenter. At the end of the video you can often click the green ‘Practice this concept’ button above and to the right of the video.
Some of the videos start very basic and progress, some are just basic as an introduction for the next video, but if you can’t understand something in the video, there is help available. You could leave a comment on the post that sent you to the video, leave a comment on the video in the Khan Academy, search YouTube (Bullcleo1, AlRichards314 and PatrickJMT are some recommended channels) or other sites for another explanation.
GAIM Foundational – Getting Ahead in Mathematics Foundational – MATH1002 University of Newcastle, Australia


Prior to the early 1980s, all running events were run in yards. Since running event records exist in both meters and yards it is necessary to convert these earlier performances.
The basic procedure is that of interpolation, taking the following steps (I've also given an example below):
Suppose a person runs the 100 yard dash in 12.1 seconds. Since the time is only in seconds, we can skip step one above. The yard conversion factor from the table below is 91.44. Dividing 12.1 by 91.44, we obtain an answer of 0.1323272. Multiplying this answer by the race distance of 100 meters, and then rounding to the nearest tenth of a second, the final time becomes 13.2. Running a 100 yard dash in 12.1 is equivalent to running a 100 meter dash in 13.2.
Calculating the yard conversion factor
The yard conversion factor is simply the result of converting the distance in yards into its equivalent distance in meters. For example, take the distance of 100 yards. 100 yards is the same as 300 feet, which is the same as 3600 inches. Since 1 inch equals .0254 meters, 3600 inches equals 91.44 meters. That is, 100 yards is equal to 91.44 meters (making 91.44 the conversion factor for 100 yards).
The table below lists the yard conversion factors for each race.
(equivalent meter distance)
|100 yards||91.44 meters|
|110 yards||100.584 meters|
|220 yards||201.168 meters|
|330 yards||301.752 meters|
|440 yards||402.336 meters|
|880 yards||804.672 meters|
|1760 yards (1 mile)||1609.344 meters|
|3520 yards (2 mile)||3218.688 meters|


2017 Math Olympiad
Eye Level Math Olympiad is an annual global math contest that began in 2004. Eye Level members and non-Eye Level members around the world can participate.
The Eye Level Math Olympiad is a test designed to challenge students’ math skills in a variety of areas including problem-solving, reasoning, communication, and critical thinking.
-Event Date: Nov 4th
-Location: Ohlone College - Building 5 (SF Area)
-Registration Deadline: 10/15/2017
-Online registration at http://olympiadusa.myeyelevel.com
-Registration Fee: $20 members; $30 non-members


Tools are designed to help their users do their tasks more efficiently and crunching numbers is no exception. I’ve been humming along with Excel and my trusty scientific calculator just fine, but as I’m getting more involved with calculations such as discounting I’ve decided it may be worth the time to pick up a financial calculator that has many of these formulas built-in.
I ended up buying the HP 10BII, which is among the most popular financial calculators out there. It’s very reasonably priced, has training modules on the HP web site, contains advanced functions such as IRR and NPV, and has a user-friendly keypad. For $30-40, this is a must have for anyone in finance or real estate.


|Number 20: 24/05/2003
A scientific publication by SGF and NEODyS
A rather popular word in NEO jargon is "keyhole". It was introduced by Paul Chodas to denote the following: suppose that a certain small body (a comet, an asteroid or, far more often, a small meteoroid) encounters the Earth in a certain year. It can happen that the perturbation imparted by the Earth to the small body puts the latter in a resonant orbit such that, on the same day of some later year, when the Earth is again at the same place, also the small body comes back there, and a collision takes place.
The "coming back" of the small body after a certain number of years is called a "resonant return", and the collision takes place only if the small body passes through a certain small region of the "target plane" in the vicinity of the Earth; this small region is the keyhole associated to the given impact.
Examples of keyholes can be seen on the JPL website (http://neo.jpl.nasa.gov/news/news018.html); they were computed by Paul Chodas in May 1999, when the impact possibilities of 1999 AN10 in the years following the 2027 encounter with the Earth were being analyzed.
To obtain such results one needs a state-of-the-art orbit computation program, able to take into account all the perturbations one can possibly think of, so as to compute with the highest accuracy the precise positions of the collision solutions. One wonders whether it is possible to have an idea of the location, size and shape of impact keyholes, leaving aside the achievement of great precision, in exchange of some geometric understanding. In fact, with the help of Opik's theory of close encounters, we can say a lot about location, size and shape of impact keyholes.
Let us start by discussing the location. As we said before, for a resonant return to be possible, the orbital period of the small body after the first encounter must be in a certain resonant relationship with the orbital period of the Earth: its value in years must be equal to a fraction like p/q, with p and q integer and not too large (say, up to something like 50 or so).
It turns out that, if we compute, using Opik's theory, the target plane points where the small body has to pass in order to be put by the gravitational pull of the Earth in an orbit of prescribed period, we find that these points lie on a circle, whose radius and center are simple functions of the initial orbit of the small body, and of the given period of the post-encounter orbit.
On the other hand, we know that, on the target plane, the region of uncertainty is essentially very narrow, so as to resemble a line segment. Its intersections with the "resonant" circle thus give us the locations of the keyholes; in fact, there are generally two keyholes associated to the same resonant return, and one of them is much smaller than the other, for reasons that will be clear in a moment.
We now discuss the size of keyholes, and especially why they are in most cases very small. Let us see what happens to two imaginary particles that cross the target plane of the first encounter, and let us further suppose that they are very close to each other at the beginning. As a consequence of their different positions on the target plane, between their orbital parameters after the encounter there will be slight differences; the most important of these is the one in semimajor axis, i.e., in orbital period, since it is the one that forces the two particles to become more and more spatially separated over time.
|The dotted line shows the circle, in the b-plane of the 7 August 2027 encounter of 1999 AN10 with Earth, leading to a resonant return in 2044. Units are Earth radii and the Earth is shown as a circle in the center. The continuous lines on the circle enclose the keyholes resulting in impacts on the Earth in 2044. The vertical line at about 5.8 Earth radii from the centre corresponds to the region of uncertainty.|
Thus, the recipe is the following: first, take the disk of the Earth of the second target plane and put it on the first target plane, at one of the intersections between the appropriate resonant circle and the region of uncertainty; second, squeeze the disk by the appropriate compression factor, in the direction normal to the resonant circle (i.e., radially). Voila', it's done: the keyhole resembles an extremely thin lunar crescent, lying on the resonant circle.


The mathematics of poker by bill chen
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Chen holds a Ph.D. in mathematics (1999) from the University of California, Berkeley. He was an undergraduate at Washington University in St. Louis triple-majoring in Physics, Math, and Computer Science, and was also a research intern in Washington University's Computer Science SURA Program where he co-wrote a technical report inventing an Argument Game. Top 100 Poker Books for Texas Holdem: Places 11 to 20 12. The Mathematics of Poker. by Bill Chen, Jerrod Ankenman. Poker has been dominated by players who have learned the game by playing it, people who cultivated an intuition for the game and are adept at reading other players’ hands from betting patterns and physical tells. The Mathematics Of Poker by Bill Chen - Booktopia "For those who think poker math is only about probability, pot odds, and straightforward, rote play, think again. Chen and Ankenman do a terrific job explaining how math can, among other things, show you exactly how to mix up your play in such a way that even champion players cannot get the best of you. Chen Formula - The Poker Bank
Book Review: The Mathematics of Poker – Thinking Poker
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Course: Elementary Math – Grade 5
Grade Level: Advanced 4 – 5 (struggling 6th graders may apply)
Day/Time: Monday 9:00 – 10:15am
Instructor: Cathy Youngblood
Cost: $48 per month + $20 application fee
Books: Saxon Math Homeschool 6/5, Student textbook (3rd Ed.), ISBN# 1591413184. Saxon Math Homeschool 6/5, Tests and Worksheets (3rd Ed.), ISBN# 1591413222; Saxon Math Homeschool 6/5, Solutions Manual (3rd Ed.), ISBN# 1591413265. You can purchase all three as a kit at CBD.
Description: We will cover the concepts students need to master as they prepare for Middle School Math. Various concepts include: order of operations, geometry and measurement, integers, divisibility concepts, ratios, statistics and probability, prime and composite numbers, patterns and sequences, decimals and percentages, powers and roots. Students will specifically learn about making a multiplication table, adding/subtracting fractions with a common denominator, multiplying by multiples of 10 and 100, perimeter, simple probability, decimal parts of a meter, reciprocals, volume, square roots, graphing points on a coordinate plane, and more. In addition to classroom teaching, other resources will be used to enhance the class and help students to better understand the course material when they are working on their own. Homework will be assigned for each day of the week, and tests will be given periodically to solidify the student’s understanding of the material. If your child is advanced in math as a 5th grader, and will be heading into 6th grade math, please look into Middle School Math #1 class.
Course: Middle School Math # 1
Grade Level: Advanced 5 – 6 (struggling 7th graders may register)
Day/Time: Wednesday 9:00 – 10:30am
Instructor: Ellie Ingle
Cost: $58 per month + $30 application fee
Books: Prentice Hall Mathematics: Course 1 (2003/2004 ed.), ISBN# 0130631361; Prentice Hall Math Course 1 Study Guide and Practice Workbook (2004 – paperback), ISBN# 0131254553. Order used online or call 1-800-848-9500.
Description: This is an introduction to what students will face as they enter middle school math. We create a path for students to develop an in-depth understanding of key mathematical concepts needed to succeed in higher level math classes. This foundational class introduces the concepts needed for Middle School Math #2 (7th grade math) and some early Pre-Algebra. We will cover whole numbers, rational numbers, decimals, number theory, fractions, ratios, proportions, percentages, data/graphs, geometry concepts, and integers in preparation for one-and two-step equations and inequalities. In addition to classroom teaching, some on-line teaching resources (videos & examples) will be used to enhance the class and help students when they are working on their own to better understand the course material. Homework will be assigned to enhance classroom lessons and given for each day of the week, while tests will be given at the end of each chapter. If your child is advanced in math as a 6th grader, please look into Middle School Math #2 class.