In Maths, you must have come across many concepts and theories, which are applied in mathematical calculations, both in academic and real-life. One of the useful concepts is the concept of the area of different shapes, defined in geometry. There are a number of geometrical shapes which we learn in our academics, such as circle, square, triangle, cylinder, etc. All these shapes have their own properties, areas and volumes.
The area of a given shape signifies the region occupied by the shape in a given 2-D or 3-D plane. Therefore, calculating the area helps us to measure the space occupied by a given object of a certain figure. To each shape which are introduced in geometry, there are formulas to calculate the area for them. These shapes have dimensions defined for them, based on which the surface area for them is calculated. Let us go through some of the real-life applications, where the area is measured for a given object.
- By measuring the area of a wall, which has equal length and height, we can calculate the amount of paint required to cover the surface area. Here we can use the formula of area of square for the given wall because a square has all its edges equal in length.
- To cover a floor with a carpet, we need to know the area of the floor by measuring its dimensions.
- Measuring the agricultural fields or plots for purchasing or selling purpose.
- Even to build a Pyramid, the architect must have required the area covered beneath the structure with the help of formula of the area of a two-dimensional triangle.
- To buy a table cloth, we need to know the surface area of the table.
- To construct overhead tanks in an apartment or independent houses.
- Wrapping a gift which are in rectangular shapes, will require to measure the surface area of the gift to wrap it completely.
- To use rods of cylindrical shapes during the construction of buildings, we can measure its lateral surface by using the formula of area of cylinder, by knowing the diameter of the rod.
‘Area’ is a major part of mensuration topic in Maths just like the volume. We may see many real-life examples, where the surface area of any object or place is measured.
Download BJYU’S – The Learning App and learn Maths in a creative way with the help of interactive videos.


Mathematics grade 12 exam papers 2011
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Add an equation with MathType
If you have MathType 6.7d or later installed, you can use it to include mathematical expressions and equations in a presentation.
Use MathType to add mathematical expressions and equations
Click where you want the equation to appear.
Choose Insert > MathType Equation (from the Insert menu at the top of your screen).
Type or paste your equation in the field provided.
See MathType Help for instructions on using the MathType tools.
To save your equation, choose File > “Close and Return to Keynote” (from the MathType File menu at the top of your screen), then click Yes in the window that appears.
The equation appears at the insertion point in your presentation. To edit the equation, double-click it to open MathType again.
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00:11 = Overview
01:34 = Top Row Bins
03:25 = Bottom Row Bins
04:18 = Square Wave Example
06:10 = Context Menu
12:49 = Creating A Custom Wavetable
All the way at the top of the Wavetable Editor is the FFT Area. FFT stands for Fast Fourier Transform and it allows us to generate and change the shape of our waveforms by drawing in harmonics and their relative phases to one another.
The draw tools provide an easy hands on way to draw our waveforms, but if we really know what we're doing, the FFT Area will allow us to take complete control of our waveform shape using a little bit of sine wave math.


For this Dover edition, Professor Schwerdtfeger has added four new of complex numbers and of the elements of analytical geometry and. Read Geometry of Complex Numbers by Hans Schwerdtfeger by Hans Schwerdtfeger by Hans Schwerdtfeger for free with a 30 day free trial. Read eBook on the. Geometry of Complex Numbers has 9 ratings and 0 reviews. Illuminating, widely praised book on analytic geometry of circles, the Moebius.
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Geometry of Complex Numbers
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How do you write improper fractions as a mixed number or a whole number.
In this case, the greatest factor present in both numbers is 3. This calculator will also simplify improper fractions into mixed numbers. Only recently been discovered but is used daily with all my classes.
Keep it up and thank you. Brilliant and much appreciated. The range of questioning provided is excellent as are some of the images. They are of immence help and the students enjoy them very much. If you are solving a word problem, the word "of" in math translates to multiplication.
Check that you know what inequality signs mean and how they are used to compare two quantities. Multiply the Numerators Once you have turned your whole number into a fraction, follow the rules for multiplying fractions. Click on any heading to view the worksheet. Could we have some on angles too please.
An improper fraction is any fraction where the numerator is greater than the denominator. The product of your denominators is the denominator of your answer: Match shapes and numbers in this fractions activity.
By Lisa Maloney; Updated May 14, A mixed number is any expression that combines a whole number equal to or greater than 1, plus a fractional reminder. Combine the Whole Number and Fraction Reunite the two parts of your mixed number: Equivalent fractions have the same value.
Parents - if you'd like to help your child learn math we suggest you start with our math tutorial section before returning to use these worksheets. Either way, your math life will be a lot easier if you get into the habit of reducing all of those fractions to lowest terms.
Lovely resource for stimulating learning and getting eveyone off to a good start. Vulgar fractions can be proper, improper or mixed.
Thirty pairs of numbers or calculations. One is when a radical or square root sign shows up in the denominator of the fraction: Well, make them to have one and, subsequently, add the two: List Factors Make a list of the factors for the numerator of the fraction, followed by a list of factors for the denominator: Multiply the denominator by the whole number, add the the numerator to that total and put the new total over the original denominator.
A self marking quiz about the application of the four operations to mixed numbers.
So useful and handy, the children love them. I don't know were I would turn to with out you. Forget finding common denominators when adding dissimilar fractions - this way is much faster and easier.
Proficiency also depends on reasonable numeracy skills particularly the multiplication tables and finding the lowest common multiple of two numbers. Next,you just reduce it. A great resource - thanks a million. How do you write an improper fraction as a mixed number.
Sciencing Video Vault Multiply the Denominators Multiply the numbers on the bottoms of the fractions, called the denominators. That boy who holds three quarters of his heart and all the best of him.
This process is called Prime Factorization. This is where the numerator is larger than the denominator. So if you have to divide the numerator by a number, you must also divide the denominator by the same number. Jan 27, · Write it as a fraction, so the complete division would be 3 1/4.
Simplest form: if the improper fraction was (say 44/12) the mixed number would be 3 (44/12 = 3) with a remainder of eight which is shown as 3 and 8/Status: Resolved.
Now let’s look at Simplifying Improper Fractions. You may remember that an improper fractions is where the numerator has a greater value than that of the denominator. So each time you perform an operation on fractions and your answer ends up as an improper fraction, you will usually need to simplify your answer.
When writing fractions in simplest form, there are two rules to follow: Ask if the numerator and denominator can be divided by the same number, which is called a common factor.
See if at least one number in the fraction is a prime number. Fractions represent part of a whole number. Fractions, decimals and percentages can all have the same values and you may find it useful to know how to convert between them. You can keep an improper fraction in that form, or you can convert it to a mixed number.
Either way, your math life will be a lot easier if you get into the habit of reducing all of those fractions to lowest terms. A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal holidaysanantonio.com spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.How to write an improper fraction in simplest form


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Thursday December 19 at the Argenziano School, Somerville
Girls game – 6:15 p.m., Boys Game – 7:45 p.m.
The Somerville Mathematics Fund recently awarded two out of five scholarships given this year to PHA Class of 2013 graduates Rishab Sharma and Djinnie Timoleon. Rishab will attend Northeastern University in the fall and Djinnie will attend Hamilton College. As recipients of this award, they will receive renewable mathematics scholarships of $1,000 per year.
PHA is delighted to announce that two alumni, Naggai Eloiseau and Leo Costa, have been chosen as the first hires for the initial Prospect Fellows cohort. Click "read more" to learn more about them!


In most instances, the information will be from sources that have not been peer reviewed by scholarly or research communities. Please report cases in which the information is inaccurate through the Contact Us link below. Subjects A limited number of items are shown. Click to view More Topology. Content Types A limited number of items are shown. Click to view More Electronic books. Series K-monographs in mathematics ; v. Notes Includes bibliographical references and index. Contents Algebraic constructions -- Homotopy theoretical localization -- Completions in homotopy theory -- Spherical fibrations -- Algebraic geometry -- The Galois group in geometric topology.
Geometry 19 , Ergodic Theory and Dynamical Systems 4 Advances in Mathematics, 51, no. Hermes, A. Equations 55 no. Solution of the Fatou-Julia problem on wandering domains. Annals of Mathematics, , Ergodic Theory and Dynamical Systems 5 , Ghys, L. Ergodic Theory Dynam.
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Biswas, S. Geometric Topology Seminar Notes. The Hauptvermutung book, , K-Monogr. Cui, Y. Systems 17 , The Teichmuller space of a holomorphic dynamical system with C. Body, M. Mumura, H. Topics in lowdimensional topology, In Honor of Steve Armentrout. Banyaga, H. Publishing, River Edge, NJ, Subj-class: Geometric Topology; Quantum Algebra.
To appear in Annals of Mathematics In "Michael R. Herman" Gazette des Mathematiciens No.
Societe Mathematique de France, Paris, , Duke Math. Tillman, Cambridge University Press, , Proceedings of Symposia in Pure Mathematics, Vol. K-Monographs in Mathematics, Vol. Submitted to The Journal of Topology, Cohen, J. Ranade, C. Basu, J. McGibbon, M.
AMS :: Bulletin of the American Mathematical Society
Differential Geom. Volume 97, Number 1 , Gorokhovsky, Z. Journal of Topology and Analysis Triangulating Homotopy Equivalences.
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Very often we have a long sequence of data with a very high variance due to noises. The typical approach to this situation is to create a circular vector of size N, put the new incoming value in the vector in a sequential way replacing the older one and finally calculating the mean of the values.
The “smart” approach let us take memory of the sum of the values adding the new value and removing the older before dividing for the size of the vector, not too much complicated, but prone to error.
Math comes to us…
Lets start by this formula:
“m” is the mean at time “t“, “v” is the new incoming value at time “t“, “α” is the weight. The only value that we must take care to is “m” that propagates through time.
For example: we are at time “t=100” and we have reached this status:
going on replacing each with the value calculate at (t-1) we can see how the weight of older values contributes less to the total sum in an exponential way according to the value of “α”.
But why can we use this formula to approximate a “moving windows average algorithm”?
The reply is simple: we can approximate the “width” of the windows with the following formula:
and we obtain the following window sizes for typical “α” values
If we want to calculate the value of “α” that must be used to obtain a searched window size we can use the formula
So many line of codes can be replaced with a single line
double mean = 0;
double alpha = 0.9; // mean of about last 10 values
while( acquiring )
[... get "newValue" in some way ...]
mean = alpha*mean + (1-alpha)*newValue;
[... do something with the mean ...]
The only problem with this approach is that for the first values the mean is really “slow” and very near to zero. We can overcome this problem using some kind of Bias Correction:
A complete C++ class:
That’s all for this tutorial. I have used this approach more than one time since I discovered it following the “Deep Learning” specialization on Coursera by Andrew Ng.
If you want to learn more follow the links to the three videos about this argument:
- Exponential Weighted Average
- Understanding Exponential Weighted Average
- Bias correction in Exponential Weighted Average


HOW TO SOLVE A CODED PUZZLE: In a coded puzzle, all of the letters have been replaced by numbers. For each letter in the puzzle, there is a number. If a letter appears more than once, the number for that letter remains the same every where in the puzzle.
This puzzle is made up of words that all start with the letter "A". Notice that all of the words in the puzzle have the number 21 in the first square of that word. So, we would guess that the number 21 represents the letter "A". Click on any square where you see the number 21 and press the letter "A". The computer will put a letter "A" in every square numbered 21.
Click on any square with the number 14. Press the letter "D". The computer fills-in all square numbered 14 with the letter "D" and reveal the first word of the puzzle which is the word ADD.
In this same manner continue substituting the remaining letters for numbers until all of the words in the puzzle reveal themselved.
(c) copyright 2004-15 by Lance Micklus - All rights reserved


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I don’t often rave about apps but this one is definitely ‘rave-able’! Craig posted yesterday about using this app in his classroom so I thought I’d download it and show it to my class today. Well – talk about excited! They got together with buddies and scripted some quick maths videos.
I found a ‘relatively’ quiet place for them to do the recording and left them to it. Half the class has had a go already and the rest will have a go tomorrow. They really enjoyed working like this and reckon they won’t forget their demos when it comes time to use them during maths lessons.
Here are four of the videos they created.


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BOL -> SFOL semantics: translation for R* insufficient?
BOL → SFOL semantics: translation for R insufficient?*
The BOL → SFOL semantics translates occurrences of R* (for possibly complex R) non-compositionally as described below:
But what prevents a (set-theoretic) model of the generated SFOL theory from setting C_R to the trivial full relation?
Concretely, if M is the model, [m]M[[ι]][/m] its universe for the sort ι, then [m]M[[C_R]] = M[[ι]] x M[[ι]][/m] is perfectly acceptable and does not contradict the generated axioms.
Indeed, Wikipedia says at https://en.wikipedia.org/w/index.php?title=Transitive_closure&oldid=945383137#In_logic_and_computational_complexity: [quote]one cannot write a formula using predicate symbols R and T that will be satisfied in any model if and only if T is the transitive closure of R[/quote]
Does this mean the translation BOL → SFOL is actually “wrong” (i.e., not as intended)?
Reformulating your criticism, why does the axiomatization only require that C_R is some transitive relation bigger than R?
To be the transitive closure, shouldn’t there be axioms that make sure it is the smallest such relation?
In some situations, it would indeed be helpful to do that.
It would in particular ensure that C_R is the transitive closure in every model.
But we would need a stronger logic for that, namely one in which we can quantify over all relations P - then we can add axiom like
forall P. (R subset P and transitive(P)) => C_R subset P
But we do not need that because the semantics of an SFOL theory is not determined by what can be true in some model.
It is determined by what is true in every model (or equivalently, by what is derivable in the calculus).
Because the model in which C_R is the transitive closure of R is among all the models in which C_R is a transitive extension of R, the semantics can at most prove properties that hold about the transitive closure.


Thank you to everyone who made the 11th Annual Joe Holbrook Memorial Math Competition a success! Tests, solutions, and results are now posted in our Archive. Please note that participants will need their Participant ID number to interpret results.
Hosted at the Bergen County Academies
The BCA Math Team proudly presents the 11th Annual Joe Holbrook Memorial Math Competition, formerly known as the BCAMC, on October 16, 2022. This competition brings the best and brightest elementary and middle school students from the greater Bergen County area to meet and compete with fellow students.
To see more information about the Bergen County Academies Math Team, visit its official website at https://bcamathteam.org.


With Halloween earlier this the week, it’s a fitting time for us to reflect on some of the more “terrifying” topics of our times. Starting with some scary STEM stats that might make you think twice about the state of science, technology, engineering, and math in the U.S.
But don’t worry — while they may be daunting, they are all within our power to fix!
U.S. students are lagging in STEM proficiency. STEM feeds innovation, yet students in the United States are consistently underperforming in science, technology, engineering, and math — the U.S. is ranked 22nd and 31st for Science and Math, respectively. This issue gained attention when the Federal Government expressed concern over the dropping performance of STEM students in the U.S. and launched the Educate to Innovate campaign.
Most teachers do not have a degree (or formal training) in STEM. Seventy-eight percent of U.S. teachers say they have not received the training necessary to teach STEM topics. This causes a lack of confidence in incorporating STEM into the classroom, or teachers who might not know where to begin when doing so.
Minorities have become a minority in STEM education. Less than half of African-American (42 percent) and Latino (49 percent) students complete the STEM courses they start, compared to 70 percent of Caucasian students. This statistic assumes a greater significance when you consider that minority groups pursue STEM courses much less frequently than non-minority groups in the first place — often as a result of inadequate academic preparation.
There are fewer women in STEM. Women are graduating college at greater numbers than ever before. Since 2014, more women than men earn college degrees each year; and while they are increasingly finding a place in tech companies, they are still not fulfilling their share of technical roles.
The gender gap in STEM develops earlier than we think. By middle school, only 11 percent of girls expect to take up STEM careers versus about 36 percent of boys. Gender stereotyping begins from a very young age with girls being discouraged — both directly and indirectly — from taking up math and science. As a result, many young women end up feeling unprepared and diffident for STEM careers when pitted against their male counterparts.
While there is no magic spell to completely address all of the problems with STEM education in the U.S., experts suggest that the earlier kids are exposed to STEM education, the better. So find any opportunity that you can to get your kids interested in discovering the joys of STEM! Maybe the future won’t be so scary after all.


One fine fall day, Fred, the famous football player, decided to do a little math activity with his favorite sport. So he put his ball down on one of the goal lines of the 100-yard field. He then moved it forward half the distance — that is, 50 yards — toward the other goal line. Next he moved it half the remaining distance, 25 more yards.
He planned to continue in this manner as long as he could, always advancing the ball half the remaining distance toward the other end. Of course, it should be obvious that sooner or later his moves will be quite small, so small in fact that we could say he was truly “inching along”.
The question now is: on which move of the ball did the distance for the first time become less than one inch?
Bonus: what was the total distance that Fred had moved the ball after that move? Give your answer rounded to the nearest 100th of an inch.
When you get this problem’s two answers, please e-mail me. Be sure to explain “how” you arrived at your conclusions.


I am a Ph.D student supervised by Prof. Carlo A Furia at the Software Institute, USI Lugano. Previously, i was a Doctoral Research Assistant at the Institute of Information Security, ETH Zurich from 2015-2018. I have an Integrated Bachelors and Masters degree in Mathematics and Computing from IIT Kanpur, where my Masters Thesis was advised by Prof. Mohua Banerjee .
My areas of interest are in the intersection of formal methods and software engineering with focus on automated program repair. I have also worked on projects aiming to solve scalability issues for runtime monitoring on large and high velocity data traces, and explanation of counterexamples generated by model checkers.
PhD (Contd.) in Computer Science, 2018-present
Software Institute, USI Lugano
PhD in Computer Science, 2015-18
Institute of Information Security, ETH Zurich
BSc-MSc in Mathematics and Computing, 2010-15
Indian Institute of Technology Kanpur
Teaching Duties at ETH Zurich:
Teaching Duties at USI Lugano:


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What do you need to master in GRE Math section?
The basic misconception about GRE math section is that GRE maker wants you to calculate and solve the sum step by step. If it sounds pretty enticing and convincing to you, then you are in an unending trap.
You may get some easy questions done with your magical approach. But you must be jolting in medium difficulty questions. By this time, the tick of clock is no more favoring your brave but unsmart approach.
Finally, you are dumped without answering last five questions. I still feel the memento of that stupidity. I cannot help breathing a sigh.
Let’s flip the coin and think for a moment!!! Rudimentarily, Mathematics is a language. It has thoughts that are expressed through signs.
Are you crazy? I never thought it in this way.
Your reaction is not quite a new. I did have the same before I was told by my mentor. I think most of us have got the same.
Let’s dissect the issue. We all speak in our mother tongue. If we just ramble willy-nilly, it does not make sense. Every language possesses specific letters like computer code. Mathematics must have some codes. We gotta explore them. By the way, mathematics is not about calculation, it’s about logic.
Once we fathom the thread of logic in mathematics, we can beat and tame it like our mother tongue. We were taught mathematics in clumsy and fearful way. We are now brave enough to dismantle any disorder of this language. WOW!!!
Just follow these 5 steps to crack GRE Math-
- Create a mindset to understand and realize the language of mathematics. Try to grasp test makers mindset.
- Think every math question as a dual dialogue between you and math.
- The less you need to calculate, the better is your understanding. If you think that you gotta do lots of calculation, you are far away from GRE math touchstone.
- Each question must have a trap. If you can’t unveil it, you are doomed.
- Practice makes you perfect. WRONG!!! Perfect practice makes you perfect.
Let’s face the music………
If you like this post, subscribe for free to receive newsletter. When you subscribe, you will instantly get a GRE vocabulary book pdf.


During your time at Skyline College you will almost certainly need to take at least one Mathematics Course. In order to make your experience in Mathematics as successful as possible we have prepared this web site to help you make informed decisions about which classes you should take. It is in your best interest to make sure that you enroll in the appropriate class and that you do so as early in your program as possible. Mathematics courses are prerequisite for many business and science courses. There are specific Mathematics requirements for an Associates Degree, and for transfer to most four year institutions. The sooner you complete your requirements the more flexibility you will have with your program choices.
How To Choose The Right Course The First Time?
The answers to these 3 questions will help determine which Math course you should start with at Skyline College.
- What Math courses have you had before coming to Skyline?
- How long has it been since your last successful Math experience?
- What is your Math placement results?
Before taking any Math class at Skyline College you should take the placement test. You cannot pass or fail a placement test. It simply provides information to you and your counselor about your current skills. If you have materials or notes from your last math class available, it would be helpful to review them before you take your placement test.
After taking a placement test, make an appointment with a counselor to discuss the results of your placement test and your prior experience in Mathematics. Be sure to bring a copy of your transcripts (if they are available) to the meeting with the counselor. The counselor will help you decide which Math course is most appropriate for you. Be sure to take into consideration all of the prerequisites for any course.
If you feel that the placement test did not adequately reflect your skills, it is possible to retake the test, but this is done on a limited basis. You may wish to talk to one of the Mathematics instructors before determining an appropriate class.
Another important consideration in taking a Math class is your proficiency in English. All mathematics classes are taught in English and the instructors will expect that you have a reasonable command of English.
Here are some guidelines and SAMPLE QUESTIONS/ PREPARATION- TUTORIALS on your math assessment:
Students who intend to measure proficiency in Basic Arithmetic and pre-algebra computations. Students may register in Math 811, Fundamentals of Mathematics without taking the math placement test. If you have never had an Algebra class or if you have had one year of high school algebra, but it has been more than two years since your last Math class.
Students who expect to enroll in Elementary Algebra (Math 110 or Math 111) or if you have one year of high school Algebra or if you have had two years of Algebra but it has been more than two years since your last Math class.
Intermediate College Algebra
Students who expect to enroll in Geometry (Math 115) or Intermediate Algebra (Math 120 or Math 122), or if you have two years of high school Algebra but it has been more than two years since your last Math class.
Students who intend to enroll in Trigonometry (Math 130), Statistics (Math 200), Quantitative Reasoning (Math 201), Precalculus (Math 222), or Applied Calculus (Math 241) to measure the proficiency.
This test is designed for students who intend to enroll in Calculus with Analytic Geometry (Math 251) or if you have had four or more years of high school Math.


"Search for Breakthrough by Mathematical / Mathematical Sciences Researches toward the Resolution of Issues with High Social Needs (Focusing on Collaboration with Wide Research Fields in Science and Technology)" (Set in FY2007)
1. Title of the Strategic Sector
Search for Breakthrough by Mathematical / Mathematical Sciences Researches toward the Resolution of Issues with High Social Needs (Focusing on Collaboration with Wide Research Fields in Science and Technology)
2. Relationship with the relevant strategic priority for science and technology
In relation to this Strategic Sector, it is referred to in the information and communication field of the promotional strategy by field as follows: "It is clear that any information and communication technology utilizes mathematical achievements. Strengthening the fostering of mathematical researchers is an essential policy for the progress of information and communication technology as well as science and technology in other fields for the next 30 years." In addition, in other fields including life sciences as well, the needs for mathematics are shown in the form of simulation or system researches.
Furthermore, the 3rd Basic Plan for Science and Technology says that "it is necessary to create an environment to encourage intellectual inspiration and integration among different fields by, for example, promoting R & D in which the wisdom of researchers required for problem solving can freely be brought together beyond the existing field classification in order to create new wisdom."
3. Reasons, urgency, and needs from specialists and the industries for the priority over other strategic priorities for science and technology, etc
Considering the present situation of mathematical researches in Japan, the following characteristics can be found:
(1) Though Japan maintains a certain level of mathematical researches, there is no quantitative increase as seen from the number of papers. It is equivalent to the number in Europe, but far less compared with that in the United States (4-5 times than that in Japan).
(2) Promotion of mathematics hardly gets into the political limelight, and investment in mathematics in Japan is quite small compared with that in major countries in Europe and in the U.S. (The U.S.: approx. 40 billion yen, France: approx. 19 billion yen, Japan: several billion yen)
(3) More emphasis is placed on pure mathematical researches, and fewer researches on application are carried out in Japan.
(4) There are barriers among different cultures, including the difference in technical terms among fields, leading to difficulty in having contact points with researches in different fields.
Meanwhile, mathematics is a study that serves as a basis for various sciences and provides a breakthrough in R & D of many areas through collaborative researches with other fields. In the 3rd Basic Plan for Science and Technology as well, it is mentioned as follows: "When developing eight promotional strategies by field, they should be proceeded while paying careful attention to flexibly responding to these emerging / multidisciplinary areas and appropriately linking to innovations."
In fact, according to the questionnaire survey of industry-academic-government researchers of eight priority fields in Japan at National Institute of Science and Technology Policy, the answer that there are issues for which mathematics' contribution is expected accounted for 81%, showing high needs for mathematics.
Therefore, it is necessary not only to promote mathematics itself but also to immediately launch activities that can contribute to further development in different fields by utilizing the achievements in mathematics.
4. Image of innovative achievements expected to be achieved under this goal (expected innovative outcome with specific examples), background, and social and economical imperatives.
With regard to the issues raised at present, truly innovative solutions (innovations) should be obtained through the processes of logical discussion, organizing issues and selecting approaches which lead to concrete solutions. This "logical discussion" had conventionally been made only with the knowledge within the field, and sufficient in-depth discussions were limited. By introducing mathematical approaches here, a breakthrough in organizing issues and in getting concrete solutions can be expected.
To this end, as seen from various technological innovation cases utilizing mathematical theories in the past (application of the fuzzy set theory into home electric appliances, etc. announced in 1965, application of the wavelet analysis into data compression, etc. in image processing announced in the early 1980s, etc.), the breakthrough should be realized, the innovations be created and improvement in economic and social values be planned by applying mathematical approaches into issues with high social needs including the development of new products.
5. Research and development goal to be achieved during this strategic project (creation of knowledge as the source of innovation, technology seeds, technology concepts to be demonstrated etc.)
In order to build deeper cooperation between mathematics and different fields, it is desirable to flexibly combine activities to promote individual researches with theme setting based on the mathematical researchers' intentions first under certain conditions and those to cultivate collaborative researches seeking the possibility of cooperation with other fields, developing them into collaborative researches with other fields. In addition, in order to judge if the intended Research Subjects can effectively be solved by utilizing mathematics, it is necessary to make the best use of personnel also with a broad view of other fields on the side of mathematical researchers. To this end, the system as shown below should be established for collaborative researches between mathematics and other fields:
(1) To appoint mathematical researchers also with a view of different fields (Research Area) as research supervisors.
(2) To collect mathematical subjects in which development to other fields can be expected from mathematical researchers and advance the researches after selection by research supervisors.
(3) To hold workshops, etc. as places of exchange between researchers of mathematics and other fields, developing an atmosphere conductive to collaborative researches between mathematics and other fields.
(4) To make arrangements so that the research forms (individual researches / team researches) can flexibly be settled according to collaborative phases with other fields at the time of application by the research proposers.
6. Scientific justification for research and development goal to be achieved during this strategic project (progress status of related research, potential for development of this field in the future and prospect that many outstanding research plans are proposed)
At the Department of Mathematics in Hokkaido University, they are working on the project "Mathematics Center for Advanced Researches" to invite mathematical questions from researchers who are carrying out advanced researches in various fields in the university as part of "the 21st century COE Program." This is an experimental project to try to search for clues for problem solving by brainstorming with mathematical researchers after obtaining information about the present situation of researches relating to questions from questioners. Fourteen questions have been provided so far. In the process of this project, not only direct solutions are found, but also the following results are obtained:
· Logical structures of the questions become clear following discussions with mathematicians
· Discovery of new mathematical issues and young human resources development with it (development into research themes for students)
· Development into collaborative researches
· Better understanding of the characteristics of each other's fields leading to gaining in breadth as researchers
As mentioned above, the advanced efforts in a university are achieving results, and the accumulation of successful cases for the development into collaborative researches of mathematical researchers with other fields can be expected with experiences of such projects as a beginning.
7. Specific issues to be overcome in achieving this research and development goal, points to be cared , and any overlapping with existing policies, projects, etc.
With respect to collaborative researches between mathematics and different fields proceeded under this Strategic Sector, consideration should also be given to cooperation with the existing policies and projects by handing them over to R & D projects with the existing competitive research funds , etc. depending on the development phases.


Mega Millions Number Generator
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Pick six or more numbers and at least one Mega Ball from the grid below and then hit the ‘View’ button. You will be shown all the possible combinations of those numbers – all that’s left then is to enter them in an upcoming draw.


The Centrefor Education in Mathematics and Computing (CEMC) is Canada's largest and mostrecognized outreach organization for promoting and creating activities andmaterials in mathematics and computer science housed within the Faculty ofMathematics at the University of Waterloo. It was founded in 1995 with originsdating back to the 1960s.
We recently received the good news that our students Joe, Sherry, Larry, Eva, Jason, Chris and Kenny had gotten excellent result from the Canadian Math Competition held by University of Waterloo. What’s more, Martin from G12 won the Hypatia medal, while Jasmine won the Galois medal. Congratulations to them.
They won’t have done these well without thehelp and instruction of their math teacher Joanne. Thank her very much. We areproud of them all.


What Does Arithmetic Shift Mean?
An arithmetic shift in computer science involves a bitwise operation shifting binary numbers to an adjacent position, or some other position, either right or left. Arithmetic shifts can help perform multiplication or division on binary numbers.
Techopedia Explains Arithmetic Shift
As computers entered the “personal computer age” and evolved, arithmetic shifts became a way to work with binary, and indirectly, with machine language. Shifting the bits changes the values of the binary numbers, which can relate to memory addresses or other aspects of programming and input.
Arithmetic shifts can be used with signed or unsigned numbers. Experts call some types of arithmetic shifts logical shifts right or left. More can be found in a wealth of specific coding examples online showing, for example, how a function to perform an arithmetic shift changes the value of a binary number for some specific programming purpose, and what these shifts look like in the syntax of particular programming languages.


We have a wide array of events and courses to suit every level. We believe in customising each event or programme to meet individual school's specific needs. So feel free to contact Ms Pearl Tan (email@example.com) to have a detailed discussion.
Delightful & Enriching Courses
These courses promote an interest in Maths and develop a wonderful sense of discovery with interesting Mathematical investigations, intrigues and games.
P1 Maths Is Fun & P2 Discovery Maths Course
Pupils explore mathematical concepts through fun, hands-on activities in stations, investigations, projects, Maths tricks, games, and visualization exercises
Tough Maths Made Easy Course (P1-4)
Pupils master thinking skills and problem solving heuristics and apply them to seemingly difficult real-life situations. Hands-on activities promote the learning of these skills and concepts and create interest in Maths.
Spatial Maths Course (P1-6)
Pupils develop spatial visualization and visual problem solving skills through the use of manipulatives. Activities are very engaging and pupils will benefit from working in pairs or peer groups.
Investigative Maths Course (P3-5)
Pupils develop metacognitive skills as they are incited to ask questions and plan how they would approach non-routine challenging problems with a repertoire of problem solving strategies.
Maths Learning Journey (P5)
Pupils embark on an interesting Maths learning journey with activities and projects to encourage appreciation of Maths, investigation and application.
Calculator Maths Course (P4/5)
Pupils learn how to use the various functions on the calculators and engage in interesting calculator activities to improve their number skills and hence develop a greater interest in Mathematics.
Captivating & Challenging Courses
These courses expose pupils to interesting Olympiad-type questions and impart the necessary process and visual thinking skills required in solving such problems.
Problem Solving Heuristics Course (P4-6)
Pupils master a repertoire of problem solving skills and techniques that are required to solve challenging word
Maths Olympiad Training Course (P3-6)
Pupils engage in intensive training to solve challenging problems, interesting puzzles, and associated rich mathematical topics relevant to Mathematical Competitions.
Nurturing & Coaching Courses
These courses motivate pupils by creating opportunities for success, reinforce basic concepts and number skills, and train pupils in exam skills so that they can attain a higher level of performance.
Upgrading U-Graders Course (P3-6)
Pupils learn to manage undesirable learning behaviours and develop confidence as they are guided to tasks they can perform. Pupils are motivated to learn through activities using manipulatives, object lessons and classroom games
Maths Noteworthy Questions Course (P5-6)
Pupils encounter noteworthy questions which require application of concepts or interesting strategies to solve. They will also be introduced to higher-order challenging problems that will stimulate their thinking and problem solving skills
Equipping for PSLE Courses
These courses prepare and equip pupils of different achievement bands to excel in PSLE through our differentiated, effective, no-nonsense coursework.
P5 Year End Head-start Course
Pupils will re-encounter concepts and topics covered in P5, but extended to the next level of complexity to prepare them for PSLE. Differentiated coursework are available for 3 ability groups: Higher Ability, Middle Ability, Lower Ability.
High Achievers Course (P5/6)
To help higher achieving pupils (A or A* graders) prepare for the PSLE by equipping them with problem solving strategies to handle challenging non-routine problems in Section C. Problems will involve higher order application of mathematical concepts, spatial visualisation and perspective, and understanding and applying strategies
Professional Training Programme for Teachers
Empowerment Programmes for Parents
Equipping Primary 5 cohorts annually with calculator skills using Casio calculators
Annual Mathlympics For All Singapore Primary Schools in collaboration with ACS (Independent) and Casio (S) Pte Ltd
Maths & Thinking Day
P1 Orientation Programme
Happy Cube Challenge
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Using Your Calculator Effectively
Through this book, we help pupils discover how to navigate around the different functions of their calculators, understand mathematical concepts in various types of questions and how to solve elegant Mathlympics
questions taken from previous Annual Mathlympics competitions.
Annual Mathlympics 10-Year Compilation (2008 - 2017)
This book is a compilation of the Annual Mathlympics for All Singapore Primary Schools held for the past 10 years.
It contains both the questions and solutions for the preliminary rounds and both the sprint and vault sections of the final rounds, which can be used for revision and preparation of future Mathlympics.


Warning: This post may cause post traumatic stress disorder. Plus, prolific profanity to follow because, math.
Boy-Child#1 has his first junior high pre-algebra benchmarks today.
They're akin to mid-terms, if it were the middle of the term. Which it isn't. So: benchmarks. I had a difficult time getting the point across that this test would cover everything from all of the chapters they've learned so far when I told him, ad nauseum, that this test would cover everything from all of the chapters they've learned so far. Exactly. I tried another tactic.
You know all of that homework you've done since school started? It's going to be on this test.
And... you see this bloody bruise on my forehead? It's from all of the beating against that brick wall.
I figured I'd help him study, being his first
midterm benchmark and all. And this way I could instill some rather bitchen study habits. Plus? I don't like to grab my own boobs toot my own horn but I did get an "A" in algebra eventually in high school and also in community college [which Mr. Farklepants condescends by referring to it as my "little college" and I'm all, at least I went. Oh SNAP!] so I was all, no need to hire a professional! I can do this! Tutor schmooter.
I try to reiterate whenever possible that textbooks are marvelous tools for learning. If you know how to use them. The study plan I designed for Boy-Child#1 included various exercises selected from the chapter review section at the end of each chapter, odd numbered problems only. Because the answers are in the back of the book. Which I could use to correct the problems and see which areas he needed additional instruction.
He took a gander at the volume of math problems I assigned and after fighting back the tears and probably the urge to put a voodoo hex on me; he set about adding and subtracting integers, evaluating expressions, multiplying and dividing integers, [now is probably a good time to climb inside a bottle of Xanax if you feel
a tingly sensation the onset of a panic attack] familiarizing himself with distributive properties, remembering lowest common denominators, multiplying fractions, converting fractions into decimals and vice versa. Etcetera. I'm super math talky.
Then we came across a problem where a mixed number had to be subtracted from another mixed number but the fraction part was smaller than the fraction it was to be subtracted from... And who just left? Bueller? Anyone? ...And my help was needed. Except I couldn't remember how to do it. So I consulted the oh-so-handy textbook of wisdom to find this:
When you subtract mixed numbers , compare their fraction parts. If you are subtracting a larger fraction from a smaller fraction, rename the mixed number before you subtract.
Huh. This book fancies me omniscient. I appreciate that the book informs us that we NEED to do that. I don't want to split hairs but it might be helpful to tell us HOW to do that. And show us with a detailed example. Otherwise I'm just going to rename the mixed number Ricky.
Then I threw the textbook across the room with some fuck you on my lips. And looked my son in the eye and told him to pray that question does not appear on the test. Because when I tutor my children, I'm thorough like that.
*photo by Dorothy Z.


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Advanced Calculations on Curves Toolkit is designed to help with measurements and analysis of waveforms, as well as all types of curves. The toolkit includes 4 VIs that are developed to perform advanced calculations on curves (XY graph data), where the X axis data can be non-monotonic (non-functional curves, closed curves).
The VI library includes:
- Calculate Crossings.vi: Calculates crossing points of input curves (XY graphs) with the given accuracy and returns these points in terms of XY graph data.
- Curve to Functional Sections.vi: Splits the given curve (in terms of XY graph) into functional sections.
- Levels.vi: Returns the points of the given curve that correspond to equidistant levels (Y axis), specified by Delta (difference of 2 subsequent levels).
- Surface.vi: Calculates the surface of the given curve, open or closed, with the sign of the surface depending on the direction of the curve (X axis direction).


Primary schools are yet to receive Sh9.6 billion for more than 1.26 million students in Grade 7.
Education CS Ezekiel Machogu said primary schools are still calculating the data of enrolled students.
Machogu made the announcement on Monday when he attended the St. George’s Girls, Nairobi Form 1 admission day.
“We will send the money after we get the necessary information about who can enroll in the high school,” said Machogu.
Primary schools should receive Sh15,042 as capitation for each grade 7 student.
“Once we get the information we need, we will send the money to all the secondary schools.”
He reiterated that more than 30,000 teachers will report to various schools this week.
The board said 9,000 of these posts are for permanent high school teachers, 1,000 for primary teachers and 21,550 for intern teachers for secondary school.
Students who enter JSS will study 14 subjects of which 12 are compulsory.
English, Kiswahili, Mathematics, Integrated Science, Health Education, Technical and Pre-Career, Social Studies, Religious Education, Business, Agriculture, Life Skills, Physical Education/Sports.
Foreign languages (German, French and Mandarin) will be offered as additional subjects along with local languages.
Other optional subjects include performing/visual arts and computer science.
English, mathematics and social studies are allocated the most hours and there are five lessons per week


Many of you know that I returned to school a few months ago. Most of you also know that MATH gives me fits. But, finally, a math assignment I could sink my teeth into---because it involved WRITING! The assignment: to put together a creative, fun piece on a person who made a major contribution to the field of mathematics. We could go back as far as we wanted. I went back all the way to 287 B.C..... and here is the result. With tongue in cheek and hat tipped to a mathematical genius...
Archimedes was more interested in things of the earth. Forget Jupiter and Mars.
He used mathematics to invent and to solve. His weapons of war brought him fame.
Those weapons helped hold off the enemies who attacked his homeland with war games.
But, Archimedes was killed in 212 B.C as Romans attacked Syracuse.
The exact manner of death is a mystery, but the irony you just can’t refuse.
His weapons of war may have won many battles. In the end, they did not save his life.
But his contributions to mathematics live on, and indeed, they continue to thrive!
When it came to integral calculus, Archimedes was ahead of the curve.
2000 years before Newton and Liebniz, the rate of change Archimedes observed.
Volumes, areas and solutions were all running around in his head.
He worked out his formulas, solutions and theorems which eventually became so widespread.
His most famous theorem, Archimedes’ principle, gives the weight of a body submerged.
His contributions to the area of geometry also gave mathematics a surge.
Angles and planes and shapes and degrees-- understanding them is not always easy.
Archimedes got it, though, but was his knowledge a gift? Did he really find math to be breezy?
Plutarch said his genius might be a gift---Or perhaps the result of hard work
If that that is the case, it gives us all hope that we’ll get it without going berserk!
Today, we use his work in all areas of life, from construction, to manufacturing… to counting!
And, though I’m a writer, and not into big cipherin’, my knowledge of math is mounting.
"Eureka, I have found it!"- Archimedes once said, as he searched for a ‘royal’ solution.
It’s a cry we can all utter when we study and try and eventually gain mathematical ‘absolution.’
So, thank you, Archimedes for all of your work and for helping increase my brain power.
While I still prefer words over numbers and shapes, from mathematics I no longer do cower.


What is a Principal Calculator?
If you have an active loan and are wondering how much principal amount you have already paid off or how much of the principal amount remains to be paid, you can use the Principal Calculator. This is a free tool available online for your convenience. By entering the required information the calculator will give you the breakdown of the Principal amount balance that remains due.
When you take out a mortgage loan, it is subject to interest rates. The interest rates will be a percentage of the loan amount that is included in your monthly payments. The longer your loan term the higher your interest rate charges will be. When you start repayment of your loan amount, the payment is mostly used to pay off the interest rates first. As you continue to keep making payments and the interest goes down, the majority of the payment starts going towards your balance amount. When it comes to loan payments your balance principal amount gets covered at a later period. Using a principal calculator you can assess how many payments are due until your loan is completed.
What information is required for Principal Calculators?
To use the calculator you need to input the following information:
Loan Amount: Enter the total amount of the loan.
Interest Rate: Enter the interest rate that applies to your loan.
Payments made: Enter the number of payments you have made. You can also think of this as the number of months that you have consecutively made repayments towards your loan.
Loan Length: This is the number of years on your loan place for repayment of the loan.


A bit more formal on Linear Regression
Let’s formalize a bit our Kitchen Style analogy from the previous post, into a more suitable for coding notation, still being a bit too verbose, as it is typical for Kitchen talks. We’ll start with the same 3D case, and generalize it to bigger dimensions later. Let’s denote scalars as the lower case italic letters (s.a. a), vectors as the lower case bold letters (s.a. x), spaces and sets as the upper case non-bold letters (s.a. A), non-vector members of spaces or sets as lower case letter (s.a. f), and transformations (or in this case liner ones, or matrices) as the upper case bold letters (s.a. M).
Let’s take a 3D vector space X on which we define basis i,j,k. Let’s take a 3D set of our Data and represent it as a set A in the space X. An arbitrary member a of the set A <b.t.> X could be represented as a linear combination x1*i+x2*j+x3*k. Let’s choose a dimension ( represented by, say, the base vector k) through the projection of which we are going to regress our 3D space X into a 2D space Y, or rather one, not yet determined, member of the quotient space X/Y. Torturing a bit notation, let’s call it (x/y)l (or Yl), where l <b.t.> I (-inf…0…+inf). Let’s define a basis in Yl as u and v.
Let’s denote T as a transformation of an arbitrary a<b.t.>A to b<b.t.>B, where B is a equivalence class of all projections of A to the element of the quotient space Yl (or the plane we are seeking in this regression), as T(a)=b=x1*i+x2*j+h*k (in respect to the basis ijk, where h=x3-dx3).
The same transformation may be achieved through the chain of the following transformations of a to f: Pij(a)=c (projection of k to the plain ij), M(c)=d(projection to Y0, with the origins of uv basis in the ijk origin), N(d)=e (transformation to ijk basis back), Q(e)=b (linear shift h from Y0 to Yl along the k (or h) – the quotient space thing). Let’s define f as a representation of the same element in respect to the basis uv: f=y1*u+y2*v, where f<b.t.>Yl, b<b.t.>X. d=f because where b=h+N(f)=N(d)+h, hence f=d (obviously, because of the commutativity) in the vector space (not in general case, though).
We may be are too talkative, defining too many transformations, but let’s see, we may need them later to better understand behaviour of our data in the reduced-dimensionality spaces themselves, and not only in the original space after the reverse transformation, as we usually do it working with regressions.
Looking at the schema 2 we can see that T(a)=QNMP(a)=b=a–dx3*k=a+h–x3*k,
or T(a)-h=a–x3*k=b, or:
(b11 b12 b13) (x1) i – (0) i (x1) i
(b21 b22 b23) (x2) j – (0) j = (x2) j
(b31 b32 b33) (x3) k – (h) k (x3-dx3) k
And if Pij(a)=c, or:
(1 0 0) (x1) i (x1) i
(0 1 0) (x2) j = (x2) j
(0 0 0) (x3) k (0) k
(a11 a12 0) (x1) i (y1) u
(a21 a22 0) (x2) j = (y2) v
(0 0 0) (0) k
of course we could have done direct projection Puv=MPij, without the intermediate step of Pij, though that matrix could be less intuitive to get. However if it is not, just forget the intermediate steps:
(a11 a12 0) (x1) i (y1) u
(a21 a22 0) (x2) j = (y2) v
(0 0 0) (x3) k
(c11 c12 0) (y1) u (x1) i
(c21 c22 0) (y2) v = (x2) j
(c31 c32 0) (e3) k
(c11 c12 0) (a11 a12 0) (c11*a11+c12*a21 c11*a12+c12*a22 0) (t11 t12 t13)
(c21 c22 0) (a21 a22 0) = (c21*a11+c22*a21 c21*a12+c22*a22 0) = (t21 t22 t23)
(c31 c32 0) (0 0 0) (c31*a11+c32*a21 c31*a12+c32*a22 0) (t31 t32 t33)
(1 0 0) (x1) i (0) i (x1) i
(0 1 0) (x2) j – (0) j = (x2) j
(t31 t32 0) (x3) k (h) k (x3 – dx3) k
Now, how do we choose which indexed element l of the quotient space X/Y (or what value of h (or intercept)) is the best on for our purposes? Actually, there could be many of the reasonable criteria we can use, but the usual, default one is the minimisation of the sum of the squares of deltas (or residuals), i.e. RSS (residual square sum), which looks reasonable and is the nice looking one in the matrix form, and also gives nice analytical equations for the first and second derivatives needed for the minimum calculation.
Let’s forget for a moment bases ijk and uv, and let’s won’t depict them in our notations, and denote i <b.t.> I (1…n), where n is a size of our data set A. Then i-th element’s delta x (dx=b–a) can be written as:
(x1i – x1i) = (0)
(x2i – x2i) = (0)
(t31*x1i +t32*x2i – h – x3i) = (dx3i)
Leaving only non-trivial dimension k we do the regression on, we can write system of equations for all 1..i..n data elements in a matrix form:
(x11 x21 1) (t31) (x31) (dx31)
(… ) (t32) (… ) (… )
(x1i x2i 1) ( –h ) – (x3i) = (dx3i)
(… ) (… ) (… )
(x1n x2n 1) (x3n) (dx3n)
or, compactly: Xt–x3=dx, or, as usually it denoted in literature, Xb–y=e. Let’s partially borrow that notation for easiness of the mental mapping of this rubric to books, leaving, though, t in place, because we use b for other purposes. We’ll also drop index 3 in t and rename –h to t0.
Having expressed RSS= e1*e1 + … + ei*ei + … + en*en = eT*e (where eT is a transposed vector e), we can do the same with the left part of the equation as well: (Xt–y)T*(Xt–y)=eT*e; and we want to minimize eT*e (or (Xt–y)T*(Xt–y)). But take a note, – here, from one problem of mapping our data set from a space with the same dimensions as the original data elements have (T(a), ijk) into a subspace with the reduced dimensions (b vu), we moved to another problem, – using our data as a transformation matrix, we map coefficients of our transformation matrix (or we can say transformations T themselves (yes, transformation also may a space element)) of the original problem (X(T), ijk) (with the original number of dimensions (say, T-space)), into a space of deltas (or errors (say, E-space)) with number of dimensions equal to the size of our original data set (e, 1..n). And we want to find such an element (for the one return linear regression it will be a vector t (while for the multi-return regression we look at later, that will be a matrix T)) from the T-space, which our data would transform into the smallest element of E-space.
As usually we do that in Calculus, to find a minimum point (actually, a critical point, which includes maximum and saddle points) of a graph (curve, surface, or generic multi-dimensional data set, or product of 1-dimensional ones), we take Gradient Grad y = @y/@x1*i + @y/@x2*j + … + @y/@xn*n in that point expecting it to be equal 0. Or, for one-return parameter functions (Rn->R), it may be intuitively easier to look for a null Differential dy = @y/@x1*dx1 + @y/@x2*dx2 + … + @y/@xn*dxn (which is, anyway, related to Gradient (Grad y)T*dx=dy), and in either case, we end up looking for null Partial Derivatives of RSS in respect to the basis of t, and generalizing it from the initial 3D case to 1..j..m size, we get:
@Sum[i=1..n]((t1*x1i + … + tj*xji + … + tm*xmi + t0 – yi)*(t1*x1i + … + tj*xji + … + tm*xmi + t0 – yi))/@t1 = @RSS/@t1 = 0
@Sum[i=1..n]((Sum[j=1..m](tj*xji) + t0 – yi)*(Sum[i=1..m](tj*xji) + t0 – yi))/@tj = @RSS/@tj = 0
@Sum[i=1..n]((Sum[j=1..m](tj*xji) + t0 – yi)^2)/@tm = @RSS/@tm = 0
2*Sum[i=1..n]((x1i )*(t1*x1i + … + tj*xji + … + tm*xmi + t0 – yi)) = 0
2*Sum[i=1..n]((xji )*(t1*x1i + … + tj*xji + … + tm*xmi + t0 – yi)) = 0
2*Sum[i=1..n]((xmi )*(t1*x1i + … + tj*xji + … + tm*xmi + t0 – yi)) = 0
and converting equations above into matrix form we’ll get (actually, multiplication signs are not strictly necessary, they are just a visual convenience to navigate between the transpose and inversion notations, and parentheses):
2*XT*(Xt–y) = @RSS/@t = 0
In this transformation we map t of 1..j..m (in our original 3D case ijk) size basis to the same size basis of the partial derivative space, looking for the Null Transform of t, or Kernel t. Which will be:
XT*Xt = XTy
and after multiplication of both sides by the inverse (XT*X)-1:
where (XT*X)-1*XT*X = I – identity matrix, i.e. diagonal with coefficients 1, hence It=t, then:
t = (XT*X)-1*XTy
However, we yet to find out if our critical point is in fact a minimum, and not a maximum or a saddle point.
As usual, and obvious for the 1D mapping (function of one variable) y=f(x), we are looking that second derivative were positive. If the first derivative, usually expressed as slope line to the point where we take it, may be more intuitively envisioned as a “speed” of moving along the graph y in Y space, while we are moving along the x graph in X space, the second derivative is an “acceleration” of that movement. For example, a plane diving (for example for a zero-gravity simulation), and then climbing back up, changes its vertical speed from negative value to positive, transitioning through 0 at the critical point of minimum, its acceleration remains positive (which defines that the critical point indeed was minimum – i.e. whatever large negative speed the plain has, sooner or later (if it doesn’t hit the ground) it will be attenuated by the positive acceleration to 0 speed, and then the plane will start climbing up). If we get rid of the time variable, first derivatives along the function graph dy/dx (“speed” of changing y with changing x) to the left and right from the minimum will have the same sign as dx, while second derivatives d^2y/dx^2 (“acceleration” of changing y with changing x) will be positive. Actually the latter ensures the former, because positive “acceleration” makes dy positive (the further away from the minimum, the greater y is), and sign of the first derivative comes from the dx direction.
The similar approach works in a multi-dimensional case too, but in that case the measure of “acceleration” of the f:Rn->R is called Hessian matrix H:
(@^2f/@x1^2 … @^2f/@x1@xi … @^2f/@x1@xn)
(@^2f/@xi@x1 … @^2f/@xi^2 … @^2f/@xi@xn)
(@^2f/@xn@x1 … @^2f/@xn@xi … @^2f/@xn^2)
We still want make sure that whatever our (positive, or same sign) movement dx is in our X domain, it will be mapped in a positive movement dy in our range space Y , after applying the H transform (which is actually what brute force gradient descent methods do). In the literature such a transform (matrix) is called the positive definite one, but for better intuitive understanding we rather start from the eigenvectors and eigenvalues idea (anyway these things are closely related). Eigenvalue is such a value lambda (of a transform M) that gives to a particular vector(s) v the same transform as a regular Mv mapping. There could be multiple eigenvalues and vectors for particular transform. The nice thing about eigenvectors is that if we find enough (number of X space dimensions, so we can span it) linearly independent eigenvectors (so it will be basis of the space) we may express any vector x=x1*v1+x2*v2+…+xn*vn in the new eigenbasis. If it happens that all eigenvalues lambdai (coefficients in the diagonal matrix L) for that eigenvector basis are positive, M transform will be positive for any positive vector of our domain space X. In case of the H transform that would mean that the point with null Gradient/Differential is indeed minimum.
H*x = x1*H*v1+x2*H*v2+…+xn*H*vn = x1*lambada1*v1+x2*lambada2*v2+…+xn*lambadan*vn =L*x
or variation on the positive definite matrix definition:
xT*H*x = (x1, x2, …, xn) (x1*lambada1, x2*lambada2, …, xn*lambadan) T = x1*x1*lambada1*+x2*x2*lambada2+…+xn*xn*lambadan = lambda>0
In a general case, we may want to find eigenvectors and eigenvalues (anyway they are very handy for the transform analysis (and, in our case, because our data set is used as a transform, for the data analysis, too), however, in special cases, for figuring out only what type of the critical point we are in, we can estimate whether our H matrix is really positive definite, or it can have negative lambdas.
Let’s calculate Hessian for our RSS case, which could be expressed in a compact matrix for as:
@^2RSS/@t@tT = 2*XT*X
Which is quadratic form and it guaranties that whatever our xij data are, xij*xij is positive, and for any dt of our domain space, “acceleration” value of the range space will be positive.
The only thing we have to make sure that we have enough eigenvectors to form eigenbasis, i.e. there are no linear dependency in H, or in other words it’s not degenerate, or is a full rank matrix, otherwise we can’t “control” behaviour of the “acceleration” in some dimensions.
Of course we did not discover any Americas, and all these derivations may be found in many Statistics and Data Science books, and the Linear Regression functions are implemented in many libraries, but we want to experiment a bit further with the regression algorithms on real data, so let’s have our R or/and Python library for it, to play with them…


Modern History of the World in under 500 words
Most creditors globally employ a nominal approximation formula to disclose and declare interest rates to borrowers. In technical terms, they disclose and declare a logarithmic-derivative “of” the interest rate, instead of the interest rate itself.
Under the nominal approximation formula, a real annual rate of 1.01% is converted to 1.00%, (and vice versa) with the deviation ever so slight (about 1%, per se).
Under the same formula a real rate of 6.167% per annum is converted to 6.000%. Basically, by increasing the stated rate from 1% to 6%, the deviation increases by a factor of 6 x 6 = 36 times.
By the time a micro-loan business or payday-loaner discloses and declares 300% per annum (as most of them more or less do), the real interest rate is about 30,000% per annum (about 100-times greater than declared).
In 1974, however, the government in the U.K. banned the nominal approximation formula as criminal fraud on the grounds that it is “false and seriously misleading”.
So today in the U.K. the micro-loaners and payday-loaners tell the truth?
No, of course not. That would be silly. In 1974 the U.K. system merely adapted by massively switching (at the higher-end especially) to employing equally illegal pretended loan-fees to disguise and disclose the rate of interest to the borrower, as for example:
Payday-loaner: You get £335 today for an obligation to pay £400 in ten-days time. Legally I am required to disclose and declare to you that the rate of interest is 62,644% per annum.
Borrower: 64,000 per cent! I can’t possibly afford that!
Payday-loaner: OK – I tell you what – We’ll call the £65 difference an application fee plus a cheque-cashing fee. That way you get £335 today for an obligation to pay £400 in ten days. That way the interest rate will be zero percent.
Borrower: Oh that’s much better. I can afford zero percent.
The truly sad state of humanity is that this transparently stupid nonsense actually works on people.
So how much money are we talking about here?
Answer: All of it.
By total amount, all debt and money in the world today is a direct function of the objective math-error in the approximation formula – just since 1974 when it was technically banned as criminal fraud in the U.K.
The entrenched-money-power merely decided at that time that you have no pressing need to know about such things. And so know you don’t (until now).
More generally, the existential problem with lethal danger that grows exponentially is that if you wait until you actually see it coming, then you’re already dead.
Applied to the nominal Covid vaccines, it means that if the coefficient-of-toxicity is a real 64,000, then the pharmaceutical company people will disclose it as about 300.
And they will believe it.


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Integrated Math 1
Integrated Math 3
How we use Khan Academy
How we use Khan Academy at Buckingham, and why we use it.
What is Khan Academy?
Khan Academy is a free website that provides tutorial videos, practice questions, and mastery assessments for all of the California math standards from Kindergarten to AP Calculus.
How we use it?
As the math department of Buckingham Charter, we all use Khan Academy as a formative and summative assessment tool, not as an instructional tool. This means that we, math teachers, are able to look at each student’s individual progress through the skills they are required to master and intervene where we see students who need help, and push students forward who have already mastered current skills. There is also a parent account feature, which lets you keep track of your student’s progress as well.
One of the biggest problems in math education is that students get passed on from one math class to the next, without actually mastering the skills they need in order to be successful in the next year’s course. Khan gives us data on where students have holes, and tells if they are actually ready to move on, not just from year to year, but chapter to chapter and lesson to lesson. Additionally, with 30 or so students in a typical math class, there will be a large range of skill levels, so what is routine for one student may seem impossible for another. Letting each student work on the skill that they need to practice allows each student to be making progress forward.
How does it work?
Students are expected to consistently be making progress through the skills teachers assign. These means they are moving from “NeedsPractice” to “Practiced” when initially assigned, and moving to“Mastered” by the time of the chapter or unit test that corresponds to those skills. They are expected to maintain mastery through the semester final. So in other words, a student’s progress for each assigned skill is measured three different times throughout a semester, first as daily practice, second for mastery by the unit or chapter test, and lastly as part of the student’s final grade. Since it is important that students are always making progress, students can receive up to 70% on any late Khan Academy assignments throughout a semester.


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National Pi Day March 14th Celebrates 3.14 and More
3/12/2018 2:34:12 p.m. - Hobbs, NM.
New Mexico Junior College will host its annual Pi Day Celebration, Wednesday, March 14th, at the Lea County Events Center, in honor of the only holiday to celebrate a number, 3.14. This free event will take place from 4:00 p.m. till 7:00 p.m.
Over 40 unique booths will features games to teach children and adults how math and science are used in our everyday lives, all in a fun, carnival-like way. Since it’s also Einstein’s birthday, there will be free “pie”, pizza and more food and beverage.
Several door prizes will be given away throughout the evening, and each booth has a prize for the kids as they learn math and science.
The U.S. House of Representatives recognized March 14 as National Pi Day in 2009. Pi is one of the oldest and most recognizable mathematical constants. It is never-ending, ad now has over 10 trillion known digits.
The event is sponsored by New Mexico Junior College Foundation, Mu Alpha Theta, French Brothers Homes, and URENCO.
For more information contact Susan Fine, Director of Communications:firstname.lastname@example.org 575-492-2781


Most Popular Articles
Financial Calculator Bond
Michele Said:Where can I Buy a Financial Calculator for class?
We Answered:Walmart sells them for about $20-$25. I bought one for my corporate finance class.
Target, Circuit CIty, Best Buy, and Fry's also sell them for around the same price.
Clifford Said:Can I calculate YTM without Excel or Calculator?
We Answered:Yes. Here are two ways.
1. You can take the present value of a) present value of the sum of the cash flows for each year, b) and then take the present value of the maturity amount in year twelve. Once you have the equaltion set up you would solve for interest rate. This would be a PAIN!!!! So the best way is to use number 2.
Here is a good example of 1. http://www.moneychimp.com/articles/finwo…
2. You would take the present value annuity equation and set the coupon payments as the payments, and fill the rest of the equation with what you have. Then to that equation you would add the present value of the face value of the bond. Again you would solve for the interest rate. You can use the present value annuity because your coupon payments are constant over the 12 years.
The 2 methods may give you slightly different results, but they are both correct ways.
Hope it was not too complicated.... i tried to explain the best I could :)
Lewis Said:How do you get the PV of an ordinary annuity using a financial calculator?
We Answered:The exact procedure depends on which financial calculator you have. I am willing to BET the calculator will TELL you if you can find a help button or one labeled PV (present value). I am 100% positive it you visit the manufacturer's website, you can download a complete manual.


Do you remember math class being so boring or scary when you were in school, that it gave you math anxiety? As a homeschooling mom, I want to make sure my children don’t have the same negative outlook on math as I did when I was in school. Today I’ll be sharing 8 Fun Ways To Teach Young Children Math at Home in ways that your children will enjoy!
My childhood math struggle…
I remember being in public school and wondering why in the world would I ever need (insert really ridiculous math problems that made absolutely NO sense to a normal human brain) in real life.
Am I right?!
I knew I needed to have the basic math skills mastered of course.
You know, like— addition, subtraction, multiplication, fractions, and division, but I never fully understood much else as a school student because it wasn’t “real” to me.
And by “real”, I mean, it didn’t make sense why any normal person would ever need that stuff in “real” life unless you’re a mathematician.
I just didn’t see the need, as a young, naive child.
Take fractions for example
Seeing the circle on the piece of paper cut into equal pieces didn’t bring mathematics to life for me because I ONLY saw it as a circle, with numbers, and couldn’t understand why I would need this math practice.
Not only that, because it was just a circle with numbers, and I didn’t understand it in a real-life sense and how I could use these concepts in real-life circumstances.
And then, insert adding and subtracting fractions into the mix, that also didn’t make any sense to me, because they were just numbers without any real-life application taught along with them.
I promise you, I wasn’t a super rebellious child that just didn’t want to do my math work.
But the math just didn’t make sense to me because of the approach and the way it was taught
Now, a disclaimer— I’m not saying it was any teacher’s fault, because they were only teaching math the way they were told to teach math.
And they did that part right!
But for some children who need creative and fun ways to be taught math to help them to connect math to the real world, as I needed, there are unique approaches you can take.
Sometimes children just don’t catch on to the standard brick and mortar way math is taught.
And because of this, I really struggled with math all of my academic years.
Changing the way math is taught in our home
Math is all around us, we should embrace it and make math more fun!
I didn’t see this until I got older and started homeschooling our kids.
And since I’m a homeschooling mom and choose to give my children a really rich home education, my goal is to apply real-life examples with my children’s studies, in all subjects, when I can!
I want things to make sense to them, so they don’t struggle to try to make sense out of what and why they’re being taught (insert subject).
Their math lessons, being one of them, really needs to be engaging and have a connection to the real world.
With that being said, I have some really fun, engaging, and exciting ways to help teach your child math in a REAL-LIFE way.
Ways that will make sense and show them that math is truly all around us and that we use math every day!
8 Fun and Unique Ways To Teach Young Children Math at Home
These 8 fun examples will help you to teach math skills to your students in a fun, and engaging way that’s geared towards younger children (preschool and elementary school-age).
You can definitely take these math skill activities, and adapt them for older kids as well!
If you enjoy sewing, or would love to learn how to sew, teaching your child right along with you is a great way to help them connect math— with real life.
Here are a few ways math can be applied to sewing:
- Calculations- calculating the amount of fabric for the pattern and other items needed for the sewing project
- Learning or practice with measurements- such as yards, feet, inches, and more
- Addition, subtraction, multiplication, division- when trying to calculate what you need for a pattern, especially if you’re making more than one, or need to alter a pattern
- Measurement with a ruler or measuring stick- correctly measuring out the correct amounts. Teaching your student how to correctly use a measuring stick/ruler.
Here’s a link to an easy sewing tutorial for kids
Who would have ever thought that playing with LEGOS or blocks could be considered math?!
If your child loves building things, you can incorporate math into their building to make math fun!
Next time your child wants to play with legos or blocks, it’s a great time to use some of these tips to “build” math skills at home.
Here are a few ways that math topics can be applied to building:
- Patterns, sizes, and shapes- teaching patterns with building blocks, talking about the different sizes and shapes
- Mathematical problem solving- figuring out what goes where
Cooking with your child is an amazing way to show your child that math is used all around us and an awesome way to use math at home!
In your homeschooling this year, why not add in some cooking skills?
Cooking with your child is one creative way to teach math at home. Pick a fun recipe to cook with your child and show them examples of math in everyday life!
Here are a few ways that math can be applied to cooking:
- Multiplication with recipes- doubling, tripling, etc.
- Measurements– measuring cups and spoons when measuring out ingredients. You can also teach them equivalent fractions here.
- Number Sense- Counting when adding ingredients to the recipe.
- Concept of Time– using timers for cooking, telling time on the clock
- Counting– when adding scoops
- Fractions– when cutting things like pizza, cakes, and using measuring cups/spoons
4. Scavenger Hunts
A scavenger hunt is a fun and interactive way to teach different mathematical concepts to your child. You can do the scavenger hunt in the home, or outside in nature.
After your child collects everything from the scavenger hunt, have your child bring the items inside and work on more math skills by separating the items into groups by color, size, shape, counting the items, measuring items using a ruler, weighing them on a scale, etc.
Here are a few fun ways math can be applied to Scavenger hunts:
- Counting- counting the items, having a certain amount of items to find on the scavenger hunt
- Grouping- group by like sizes, weights, lengths
- Patterns- turn their math scavenger hunt findings into patterns
- Measurements- Have them practice using a ruler to measure items from the scavenger hunt and write them down.
- Weight- using a scale to weigh some items, teaching pounds and ounces, teaching lighter VS heavier
- Sizes- big and small
Gardening is a wonderful way to teach your child some hands-on math skills.
When planning out your garden, let your child help you this year and let them see math in real-life examples with the beauty of gardening!
Here are a few ways math can be applied to gardening:
- Mapping/Graphing- planning out your garden beds
- Addition, Subtraction, Multiplication- figuring out how many seeds should go where
- Fine Motor Skills- placing the tiny seeds into the holes
- Measurement- spacing out rows or garden beds using a ruler, teaching square feet if you choose to do a square foot garden bed
- Counting- when placing seeds in holes or rows
6. Let math become natural in your vocabulary
Using math in your everyday language will help instill math learning skills into your child naturally.
Why is math vocabulary important to include at home? Because it will get numeracy skills ingrained in your child in a natural way.
I like to throw random math word problems at my kiddos at times. Especially when cooking or when I see a situation fit for a nice ol’ word problem. The kids have fun (sometimes) trying to figure out the answer.
Here are a few ways you can apply math concepts in a natural way to your vocabulary:
“Can you please go grab me 3 books from the shelf”- (for a preschooler or younger ages)
“Place 6 plates on the first shelf, and 8 bowls on the middle shelf”
“Can you please grab me 3 large red apples, 4 oranges?”
“Let’s weigh this flour, and measure out 2 pounds of flour”.
“Please place the cans on the shelf in order from smallest to largest”
“Grab one dozen eggs from the chicken coop”
Can playing games be considered math?
Oh yes, it surely can!
And, YES, playing games can be added to your homeschooling math lesson!
Remember this one basic tip about homeschooling- If they’re learning, you’re homeschooling! (But that’s a different topic for a different post ?)
Here are a few fun “math” incorporated games are:
- Yahtzee (this a fun counting game that has a ton of counting involved)
- Sudoku (this great math game and math puzzles are for problem-solving)
- Monopoly (this will teach money management skills)
- Pop for Addition and Subtraction game – (this is a fun addition and subtraction game, which can turn into a fun counting activity)
- Multiplicate and Division Bingo – (the is a fun multiplication and division game, all while playing BINGO)
8. Giving Allowance
If you give your child allowance, turn each time they are given money into a creative way to teach math lessons at home.
I cannot stress the importance of teaching your child about finances, saving/spending, staying OUT OF DEBT!!!!!!! and budgeting! Start from a young age! (That’s again, a topic for another post sometime.?)
Review counting the coins, their names, and their value, and counting the dollar bills.
Help your kids make a budget, or a spending goal to teach them more about money and math concepts.
Here are a few ways that giving allowance can be turned into a math activity and math lesson:
- Money recognition
- Adding money
- Money value
- Spending VS Saving
There are also a ton of real-looking money math manipulatives for practicing money skills. Like these ones found here.
What about you? Do you have any fun and unique ways you like to incorporate math into your homeschooling?
Send me a message, by clicking HERE!