Definition of Fraction:
In math, a fraction is used to denote the portion or the part of the whole thing. It depicts the equal parts of the whole. A fraction has two parts, which are identified as the numerator and the denominator. The number on the head is understood as the numerator, and the number on the bottom is named the denominator. The numerator maintains the number of equal parts taken, whereas the denominator displays the total number of equal parts in a whole.
For example, 8/10 is a fraction.
What are Improper Fractions?
Improper fraction, besides the name, means the fractions that are not done correctly for any number, object, or part. In terms of Math, an improper fraction has a numerator that is bigger or greater than the denominator, such as 8/2. While, in a proper fraction the denominator is bigger than the numerator, like 6/9.
Simplifying Improper Fractions
We have explained the concept of an improper fraction. Now, let us discuss how to clarify them:
- Step 1: To discover if the given fraction is improper or not
- Step 2: Then understand the denominator and check for how many parts it is sharing the numerator with
- Step 3: Then check the common factors of the presented numerator and denominator
- Step 4: Lastly, eliminate the like terms both from numerator and denominator to find the final answer
Interesting Ways To Learn Improper Fractions?
Don’t Memorize, Learn
A lot of students start learning the concepts by memorizing improper fractions and their facts. While this method can work, it could also create difficulties and confusion in the future. An improper fraction is very simple and can be broken down into a stepwise manner that will make it very straightforward to perform and use in mathematical problems.
Once the students have mastered the basics of factors, they can set goals as to what they want to focus on and excel in while solving and exercising. If you are looking to be good with improper fractions, then set the right goals and work on them. Setting goals is one of the most essential steps of getting a hold of the concepts and mastering the working of fractions.
As with any subject or method, the most authentic way to master it is through daily and persistent practice. Improper fractions will be easier once you get the hang of their concepts. Practice will always help students to outshine and be confident while solving math. Practice is the only thing needed to completely comprehend the basics as well as the complex problems in math.
Learn from Your Mistakes
Improper fractions are one of the topics where it’s extremely necessary to take notes of your mistakes and failures. One cannot afford to repeat the same mistakes while working with fractions. Also, if we look at it from a whole changed perspective, mistakes can help students understand the entire concept of factors that can help them in the long run.
While improper fractions are more about numbers and their representatives than theory, there are also a few comprehensive and complicated concepts to help understand different types of fractions. Most students work on memorizing and don’t pay much thought to the actual concept that is involved in the fractions. Hence when it comes to improper fractions, one should always focus on the core concepts rather than focusing on memorizing them.
Ask Help When Needed
It’s okay to ask for supervision while learning something new. Even if the students think they know the concepts and understand the fractions, they might need extra guidance to lead in an authentic direction and be crisper. Students can consider getting a math tutor, asking a friend, or seeking help from their parents. Remember, it’s okay to ask for help when needed while practicing improper fractions. There are various online learning platforms that provide free worksheets that can help you understand the concept of improper fractions. Cuemath is one such platform that uses interactive worksheets in order to engage students. You can visit website and download the pdf interactive worksheets by Cuemath for free.


Dinheiro forex on-line Passos Wednesday, 22 November 2017. Strategy forex con fibonacci ratio ... The Golden Ratio mysteriously appears frequently in the natural world, architecture, fine art, and biology. Using Fibonacci Extensions. That said, many traders find success using Fibonacci ratios and retracements to place transactions within long-term price trends. Fibonacci retracement levels often indicate reversal points with uncanny accuracy. Nov 26, 2017 - Explore Speedy's board "Trading" on Pinterest. See more ideas about Fibonacci spiral, Fibonacci sequence, Golden ratio. Jun 12, 2020 - Important Facts and Information about Fibonacci Percentages The Fibonacci Percentages are primarily based upon the sequence of Fibonacci numbers, one of which (0.618) is known as "the golden ratio" because it occurs many times in nature. The Fibonacci... Dec 1, 2017 - Explore Passive Cashflow Academy's board "Only Fibonacci" on Pinterest. See more ideas about Fibonacci, Forex, Forex trading basics. Only Fibonacci Collection by Passive Cashflow Academy. 35 Pins. golden mean 16 x 20 rectangle grid - Google Search. Creation Research. Finonacci. Learning this now. --This world is really awesome. Everything You Need to Know About the Golden Ratio ... The Fibonacci Sequence/Golden Ratio is integral in geometrical design. Knowing Joel’s love for intricate geometrical designs and perfect shapes, it would only make sense that he appreciate the Golden Ratio for its precision. The Sequence itself is found in a plethora of real objects and figures, including the human body. It is also represents the idea of theoretical Rithmatic lines, a highly ... For us forex traders, the discovery of the fibonacci sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 and the golden ratio of 1.618 has proven to be very useful in determining high-probability turning points in the market. Why is this, simply because the day to day movement of any one asset, particularly a forex pair, is not that great. To understand let’s touch base quickly on what Fibonacci Retracements are. They are price levels determined by the Golden Ratio as laid out by the famed mathematician Fibonacci. The levels are percentages of a given price ... Search the world's information, including webpages, images, videos and more. Google has many special features to help you find exactly what you're looking for. The Golden Ratio. The significance of this number-sequence extends beyond its apparently self-generating nature. If you divide any Fibonacci number with the one that precedes it in the sequence, you get approximately 1.618 – the Golden Ratio. The first few numbers in the sequence are exempt from this rule.
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Traditionally, win-probability systems
are purely context-dependent. In fact, however, I do not think that this is necessarily the appropriate starting point for measuring player value. Rather, I am interested in beginning with an assessment of players’ performances in the absence of the contexts in which the players actually performed. That is, what would the expected won-lost record be for a player, given his actual performance, assuming that performance had come in a neutral context? To answer this question, I construct a set of context-neutral Player Game Points. Once these are constructed, I can then add back in the contextual information
in a way that clearly identifies how players’ values were affected by the context in which they performed.
Context-Neutral win probabilities are constructed as follows. Player Game Points are divided into three categories for the purpose of calculating context-neutral win probabilities: independent events, base-state dependent events, and purely contextual events.
1. Independent Events
Most events can happen regardless of the base-out situation. One can strike out at any time, regardless of how many baserunners or outs there are. Similarly, a triple could happen at any time regardless of the number of baserunners. All batter results, except for double plays (which are base-state dependent), intentional walks, and bunts, fall into the category of independent events. Intentional walks and bunts are treated as purely contextual events, which are described below.
For independent events, the expected win probability of such an event is calculated for each event within the league-year using the Win Probability Matrix for the ballpark in which the event took place.
For example, the win probability of a home run at Wrigley Field in 2005 is calculated by taking every plate appearance that took place in a National League ballpark in 2005 and calculating, for that plate appearance, what the added win probability would have been had the game been played in Wrigley Field and the batter hit a home run. The context-neutral win probability of a home run at Wrigley Field in 2005 is then equal to the average of all of these probabilities. In this case, the average win probability added by a home run at Wrigley Field in 2005 was 0.141 wins.
In the case of events which may or may not lead to baserunner advancement – e.g., outs, singles, doubles – expected results are calculated based on average baserunner advancement, just as is done with contextual Player Game Points.
2. Base-State Dependent Events
Some events can only happen given certain baserunners or a certain number of outs. For example, one can only ground into a double play with at least one baserunner on and less than two outs. Any Player Game Points accumulated by a baserunner on third base can, of course, only be accumulated in a base-out state that includes a runner on third base.
For baserunner game points (except for stolen bases, which are treated as purely contextual events and discussed below) and double plays, the context-neutral win probability of the event is calculated the same as for independent events, except that the average win probability is only calculated across events with relevant base-out states.
So, for example, the context-neutral Player Game Points associated with a double play are calculated as the average win probability, given the ballpark in which the game takes place, added from hitting into a double play across double-play situations (runner on first base and less than two out). For a ground ball to the shortstop at Wrigley Field in 2005, the average win probability added by a double play is 0.011 losses (from the batter’s perspective) (on top of the 0.046 losses accrued from the initial ground-out).
For baserunner advancements and baserunner outs, context-neutral win probabilities are only averaged given the specific batting event and hit type. That is, the context-neutral Player Game Points for a runner on third base advancing on a fly out are calculated only considering plays in which a runner on third base advances on a fly out. Similarly, the context-neutral Player Game Points for a runner on first base who only advances to second base on a single are calculated only considering plays in which a runner on first base does not advance to third on a single.
3. Purely Contextual Events
While it is possible to remove much, if not all, of the context from most plays, there are certain plays which are, essentially, purely elective plays, and are therefore inextricably tied to the context in which they take place. In my opinion, it would be wrong to attempt to divorce these plays from their context.
Three types of plays fall into this category: intentional walks, stolen base attempts (including stolen bases, caught stealings, pickoffs, and balks), and bunts (regardless of either situation or outcome). In each of these three cases, the context-neutral Player Game Points are simply set equal to context-dependent Player Game Points.
Context-neutral win values for specific events by season are presented and discussed in a separate article
Context-neutral Player Game Points form the basis for context-neutral, teammate-adjusted Player wins and losses, which I call eWins and eLosses. The calculation of eWins and eLosses is described in more detail in a separate article
All articles are written so that they pull data directly from the most recent version of the Player won-lost database. Hence, any numbers cited within these articles should automatically incorporate the most recent update to Player won-lost records. In some cases, however, the accompanying text may have been written based on previous versions of Player won-lost records. I apologize if this results in non-sensical text in any cases.
List of Articles


The programs in Mathematics Education at UT are coordinated and taught by a collaborative group of professors, adjunct instructors, post-doctoral fellows and graduate assistants. With a shortage of teachers of mathematics at all levels, our programs are critical to meet the growing needs of society. The Masters programs seek to move a competent teacher toward the role of Master teacher, defined as one who not only teaches well but can communicate to others through mentoring, etc., by providing opportunities to:
- examine and revise their assumptions about the nature of mathematics, how it should be taught, and how students learn;
- observe and analyze a range of approaches to mathematics teaching and learning, focusing on the mathematics, discourse, environment and assessment;
- work with a diverse range of students individually, in small groups and in large class settings with guidance from and in collaboration with mathematics education professionals;
- analyze and evaluate the appropriateness and effectiveness of their teaching;
- develop and communicate a disposition toward teaching mathematics that mirrors its importance.
The doctoral program builds on the previous work to produce individuals with the knowledge, skills and dispositions to prepare beginning teachers, master teachers and professional educators and to produce research that furthers the knowledge base in the teaching and learning of mathematics.
Students are required to develop a mastery of mathematics that will give them credibility with a mathematics department (usually at least a 18 hours of graduate level mathematics), to develop knowledge of the mathematics curriculum and pedagogy pre-K through 16, and to gain a knowledge of the current status of research in the teaching and learning of mathematics.


TERRE HAUTE, Ind. – Honey Creek Middle School finished eighth in the Indiana MATHCOUNTS competition hosted by Rose-Hulman Institute of Technology on Saturday.
Two Honey Creek mathletes, eighth-grader Aadil Parvaz and seventh-grader Jayce Dix, finished in the top 24 in the competition, which featured 200 of the state’s best young mathematicians from 43 middle schools.
“I’m proud of all of the kids,” said Bob Fischer, coach of the Honey Creek squad. “To have 200 talented young mathletes here at Rose-Hulman is a great thing.”
The students with the top-four highest overall scores will represent Indiana at the national MATHCOUNTS competition May 12-15 in Washington, D.C. Those four students will be Jack Liu of Carmel Middle School, Taiga Nishida of Happy Hollow (West Lafayette), Aniket Biswal of Creekside Middle School (Carmel, Indiana) and Jacob Zhang of Clay Middle School (Carmel, Indiana).
Honey Creek’s Parvaz finished 22nd overall and Dix finished 24th. In addition to Honey Creek, Terre Haute was represented by students from Otter Creek and Woodrow Wilson middle schools and also students from the Terre Haute Christian Home Educators Association. Greencastle Middle School also qualified for the state competition.
Clay Middle School (Carmel, Indiana) had the top overall team score with West Lafayette Junior-Senior High School finishing second, Creekside Middle School (Carmel, Indiana) third, and Carmel Middle School fourth.
The competition featured several rounds of math problems in the morning, testing the students’ ability to answer difficult questions and work in teams. That was followed by a high-pressure Countdown Round in the afternoon where the top 24 students competed directly against each other in a game-show format.
Honey Creek’s Parvaz finished second overall in the Countdown Round. Carmel’s Liu finished first.
The Indiana MATHCOUNTS competition is sponsored by the Indiana Society of Professional Engineers to promote mathematics development among youths throughout the state.
MATHCOUNTS was developed to improve math literacy in middle schools, and as a way to make parents, educators and the public aware of career opportunities in mathematics. The program impacts these student “mathletes” at an age when they are formulating attitudes about math and technology, and encourages them to pursue careers in science, engineering and mathematics.


are a huge topic.
They're based on mathematics, and deep explanations of
fractals cannot avoid math for long. They can require
billions of calculations, which means they cannot
realistically be done without computers, so some
consideration must be given to the software used to
produce the images. But they are also art, and
if fractals weren't aesthetically pleasing we wouldn't
pursure them with nearly as much vigor.
Collected here are a number of pages describing various
aspects of fractals. Most of this information is stored
here, but pages on sites elsewhere are marked with a


of Mathematics and Computer Science,
We address understanding in mathematics in relation to visualization. We
will use a description of understanding suggested by Ajdukiewicz (as is also
explored in Sierpinska,1994), where understanding is described in the following
way: ``a person P understands an expression if on hearing it he directs his
thoughts to an object other than the word in question'' (Ajdukiewicz, quote
from Sierpinska 1994, pp. 28-29). We illustrate this description by examples
where we take what is understood to be some expression (e.g., a definition or a
proof) and the object that the thoughts are directed to consists of (possibly
mental) diagrams or pictures. The major example comes from a part of analysis,
more precisely from free probability theory, that was introduced by Voiculescu
in the 1980's. The example is intended to illustrate that mathematicians also
think in terms of pictures and diagrams, but that these often are removed in
the final representation of the results. If we grant that the description of
understanding given above is accurate, then I suggest that understanding for
research mathematicians is not necessarily different from understanding for
students. Finally, I will relate the description of understanding to previous
work on the nature of mathematical objects.
topics of visualization and explanation in mathematics are currently discussed
in the philosophy of mathematics. Although the topic of understanding can be
seen as connected both to visualization and explanation, not much has been
written about it. This paper considers the
topic of understanding in relation to visualization.
will explore a description of understanding formulated by Ajdukiewicz. This
description is also explored in Sierpinska, (1994). Understanding is described
in the following way: ``a person P understands an expression if on hearing it
he directs his thoughts to an object other than the word in question''
(Ajdukiewicz, quote from Sierpinska 1994, pp. 28-29).
there may be many ways to understand this statement, we shall here only be
interested in when the object that the thoughts are directed to is a (possibly
mental) diagram or picture. I will present a number of examples illustrating
how I understand this statement. The major example is taken from actual
research mathematics and is intended to show that mathematicians also think in
terms of pictures. Because of readability, this example is omitted in the
present paper. A full version can be found in (Carter 2008). One consequence is
that understanding for a mathematician can be described in the same way as
understanding for a student.
the last part of the paper, I will connect the description of understanding to
a philosophical description of mathematical objects presented in (Carter,
in terms of using pictures and diagrams in proofs, has played an immense role
in mathematics. When talking about proofs relying on diagrams one can not but
respect to understanding proofs, pictures seem to be very useful. There are
even examples of proofs using only pictures. One example is the proof of the
equality 1+2+…n = n(n+1)/2 by drawing a picture
consisting of dots representing the numbers 1, 2, 3,... . The sum can be
obtained from the area of this figure:
is, of course, connected to understanding, as explanations are given in order
to provide understanding. When discussing explanation in mathematics, it is
often referred to a classic paper by Steiner (1978). Steiner also discusses
proofs and draws a distinction between proofs that demonstrate and proofs that
explain. Steiner's aim is to describe what is meant by an explanatory proof.
One of the descriptions that Steiner discusses is that a proof is explanatory
if it can be visualized. But he finds that this criterion is too subjective.
Instead he settles for the following description: A proof is explanatory if it
makes ``reference to a characterizing property of an entity or structure
mentioned in the theorem, such that from the proof it is evident that the
result depends on this property'' (p. 143).
Note that it may also be possible to visualize in a picture or diagram
how a result depends on properties of certain objects. Consider the diagram of
the dots representing the sum 1+2+…n = n(n+1)/2 above. This diagram shows that the numbers can
be arranged so that they represent part of a square, and this gives the desired
are also other descriptions of explanation in mathematics. Sometimes when
mathematicians talk about explanations, they refer to facts that are more general
than what is to be explained. As an example of this could be the fact that
complex analysis explains the behavior of certain things in real analysis. I
doubt, however, that this description is very different from Steiner's
characterization. By this characterization, a proof of a fact, for example in
real analysis, would have to refer to certain properties of the objects
embedded in complex analysis.
to the question, whether there is a difference between understanding for
mathematicians and students, referring to more general or underlying facts
could hardly make things easier to understand for students. Hersh (1993) claims
that proofs play different roles for mathematicians and for students. He writes
that the role of proofs for a mathematician is convincing, whereas for students
it is understanding. This claim seems to be questioned by other mathematicians:
``There is no doubt about it, many results are, in the first place, proved by
sheer brute force. ... Subsequently other people, impressed with the result,
look at it, try to understand it and finally dress it up in a manner which
makes it look appealing, makes it look elegant. ...If you want other people to
understand the essential ingredient of an argument, it ought in principle to be
simple and elegant'' (Atiyah (1974), p. 234).
wish to insist that understanding can be described in the same way for students
and for mathematicians. We now investigate the description of understanding
referred to above. We will exclusively consider this description in the case
where what is understood is some expression (e.g., a definition or a proof) and
the object that the thoughts are directed to consists of (possibly mental)
diagrams or pictures. In the next section, we shall give a number of examples
illustrating this description.
that several authors have claimed that there are different levels of
understanding. Connes and Changeux (1995) distinguish 3 levels. The first level
consists of mechanical operations, operations that follows a certain procedure
and could also be performed by a computer.
The second level of understanding involves being able to adapt and criticize a
method to a particular problem. This kind of activity results in an
understanding of both the aim and the mechanism of the procedure (Connes and
Changeux, p. 87). Finally the third level of understanding refers to the
creation of new methods or results. I do not find it appropriate to
characterize the first level as understanding. For a person to understand
something, it requires that the person understands the underlying mechanisms.
simple example illustrating the act of understanding an expression by directing
ones thoughts towards a picture consists of an understanding of the fact that 7 x 8 = 56. A picture that could illustrate this is a rectangle, where
7 and 8 corresponds to the (length of the) sides and 56 is the area. It could
also be pictured as 7 rows of 8 dots. Such a picture could also help understand
(visualize) other properties of the product, for example, that 8 x 7 = 7 x 8 . Note that the pictures may provide more information than
what is contained in the expression. In the case of the product, we can
visualize that multiplication is commutative.
I wish to show how the description of understanding presented here connects to
my previous work on characterizing mathematical objects.
`Ontology and Mathematical Practice' (2004), I present a position on the nature
of mathematical objects based on a case study in modern mathematics. According
to this position mathematical objects are introduced or constructed by
mathematicians. Claiming that mathematical objects are human constructions
raises the question whether there is any difference between mathematical
objects and other objects that are invented by human beings, such as fictional
characters. I claim that there is a difference. One difference lies in the fact
that for mathematical objects it is possible to distinguish between two acts of
presenting a given object. Take as an example,
some mathematical objects, it is possible to exhibit the object by a picture.
Consider, for example, a circle. It can be defined as a curve fulfilling the
equation x2 +y2 = 1, but it can
also be drawn. Obviously, it is not always the case that a mathematical object
can be visualized in such a way. However, I argue in the paper that it is still
possible to distinguish between two different acts of presenting an object.
the case study, we have discussed above, there are obvious exhibitions of the
permutations and equivalence classes. Thus, one may state that in the case where
there is a visual exhibition of a mathematical object, then one can understand
the definition of this object by directing ones thoughts to the exhibition. In
this way, it is possible to link the description of what it is to be a
mathematical object to what it takes to understand a definition of the object.
J. (2004): Ontology and Mathematical Practice. Philosophia Mathematica 12,
J. (2008): Visualization and understanding in mathematics. In: Sriraman,
Michelsen, Beckman and Freiman: Proceedings of the 2nd
International Symposium on Mathematics and its Connections to the Arts and
J.-P. and Connes, A. (1995): Conversations
on Mind, Matter, and Mathematics, Princeton:
R. (1993): Proving is convincing and explaining. Educational Studies in Mathematics 24, 389-399.
P. (2005): Mathematical reasoning and visualization (Visualization in logic and
mathematics. In: Mancosu, Jørgensen and
Pedersen (Eds.), Visualization,
explanation and reasoning styles in mathematics (pp. 13-30).
M (2003): Mathematical proofs. Synthese
A. (1994): Understanding in Mathematics.
M. (1978): Mathematical explanation. Philosophical
Studies 34, 135-151.
Tappenden, J. (2005):
Proof Style and understanding in mathematics I: Visualization, unification and
axiom of choice.
In: Mancosu, Jørgensen and Pedersen (Eds.),
Visualization, explanation and reasoning styles in mathematics (pp.
 A few authors have addressed these topics, for example, J. Tappenden (2005) and J. Avigad (Understanding proofs. To appear in The Philosophy of Mathematical Practice forthcoming for Oxford University Press). Avigad explains understanding in terms of certain abilities, i.e. describing what you should be able to do in order to claim that you have understood a proof. Tappenden's approach is more in line with this paper, as he makes a case that visualization plays a role in explaining fruitfulness of certain frameworks.
 Interestingely, Connes also
places much of what goes on in high-school mathematics, ``tracing graphs of
curves and doing kinematic calculations'' in this level.
 This distinction is taken from Panza (2003).


This paper is concerned with the dynamical stability analysis for almost periodic solution of memristive neural networks with time-varying delays. Under the framework of Filippov solutions, by applying the inequality analysis techniques, the existence and asymptotically almost periodic behavior of solutions are discussed. Based on the differential inclusions theory and Lyapunov functional approach, the stability issues of almost periodic solution are investigated, and a sufficient condition for the existence, uniqueness, and global exponential stability of the almost periodic solution is established. Moreover, as a special case, the condition which ensures the global exponential stability of a unique periodic solution is also presented for the considered memristive neural networks. Two examples are given to illustrate the validity of the theoretical results.
"Almost Periodic Solution for Memristive Neural Networks with Time-Varying Delays." J. Appl. Math. 2013 1 - 12, 2013. https://doi.org/10.1155/2013/716172


Pensacola State College has Math Labs located on the Pensacola, Milton, and Warrington campuses. The labs provide the following services for any currently enrolled PSC student:
- Math Tutoring: Developmental Math, Intermediate Algebra, College Algebra, Math for Liberal Arts, Statistics, Pre-Calculus, Trigonometry, Business Calculus, Calculus I, II, & III, and Differential Equations.
- Textbooks and Solutions Manuals: Current textbooks and solution manuals for most Pensacola State College math courses are available for use in the Math Lab. Photo ID is required to check out these materials.
- Calculators: Scientific and graphing calculators available for in lab use.
- Computer Lab: 32 computers available for homework, online work, and printing.
- Science Tutoring: Physics and/or Chemistry tutoring may be available at the Pensacola Campus Math Lab. Please call 484-2003 to check about current availability.


Maths Spring 1
The year 6 teaching team are committed to ensure that children are given regular opportunities to revisit their learning and embed a range of numerical skills.
This half term, we will be focusing on the follow areas in maths:
* Identifying the equivalence and relationship when converting between Fractions, Decimals and Percentages
* Identifying percentages of shapes and amounts
* Calculating the area and perimeter of a range of shapes
* Calculating the volume of cubes and cuboids
The children will practice their fluency skills before moving onto reasoning questions and problem solving which requires critical thinking.
We have uploaded some information and games that you can use to support your child with these mathematical topics.
Facts and Questions


It's a blog.
This comic was pretty funny, and the age/2 + 7 formula got tossed around a lot by my roommates.
Of course, it gives us the minimum age one can date without being creepy. At 22, it’s [(22/2) + 7], or 18. (I, however, maintain that this discrepancy would, in fact, be creepy.)
But what about the upper age limit? The formula itself is silent on this, but we can easily do some substitution to make it work. If the minimum acceptable age (“M”) is your own age (“A”) divided by two, plus 7, we get:
M = A/2 + 7
We typically solve for M, knowing A. However, the oldest person I could date would have my A as their M, e.g.:
22 = A/2 + 7
With this realization, it’s a simple Algebra 1 question. Subtract 7 from both sides and then multiply by two.
Thus, the maximum age one can date is 2(a-7), where a is your age. For me, it’d be 2(22-7), or 30.
What interests me, though, is that this means I’m allowed to go back four years, but forward eight, within the margin of creepiness.
I built a spreadsheet for people aged 1 to 100 showing this and various other statistics. It’s online here as an HTML document. A few interesting trends emerge that aren’t intuitively obvious working with just the formulas:


ASIANET NEWS NETWORK
Every school goer has studied the Pythagoras' theorem name after the ancient Greek mathematician Pythagoras. The theorem states that "In a right angled triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides". However, way before Pythagoras, an Indian mathematician by the name of Baudhayana had come up with a similar theorem. How many of us knew about this piece of trivia? It is interesting to note that an Indian has contributed to this theorem way before the Greek mathematician came up with it.
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Quantitative Trading firm seeks a Quantitative Analyst that will work on different mathematical, programming and analytical skills in performing data analysis, mathematical research and quantitative model implementation.
The primary purpose for the position is to assist senior quantitative traders in implementing a variety of projects.
Position requirements include a degree in mathematics, physics, and engineering or a related field, strong C++ programming experience is a must have. Finance experience is not necessary though a strong plus, but proven excellence in math and good programming skills are essential. This position requires a highly self-motivated and detail-oriented candidate able to work closely with and communicate effectively with senior quants and developers. This is a multi-platform environment, using both Linux and Windows, so a high comfort level working in both is very desirable, including command line skills, debugging experience using Visual Studio or gdb. Other desirable skills include experience with SQL, python and Excel, and knowledge of options/derivatives and options pricing models.


By: Shane Embury
When students are given two electives each semester, those periods can be filled with a variety of classes. Many students opt for fine arts, foreign languages or athletics, while others choose to double up on their core classes. There is no wrong choice for the chosen electives, but in some cases, the choices appear uneducated.
Many students gravitate toward doubling up on their math courses. If those students want a career centered around math, or they thoroughly enjoy the subject, this decision would make sense. However, I know of many students who are already struggling in their math class for the fall semester, yet they still plan to take another math class in the spring. At this point, doubling up on math is pointless.
There are two main objectives people seek by taking extra math courses: 1) they want more credits, or 2) they genuinely want to pursue a career involving math. By taking more math than they can handle, these students will not be able to accomplish either of those goals. The credits will not be valuable if the students cannot make it out of the class with a high grade, and if the students want a math-centric career, it would be more beneficial for them to get tutoring in their current math class than to continue on in higher levels of math without first having an understanding of the lower levels.
Another core class that many students want to double up in is science, but this subject is more understandable than math. There is a wide variety of science classes offered at the school. If a student is struggling in chemistry, they may turn out to excel at biology. However, if a student is doubling up on chemistry after already struggling with it in the fall, then that would be an unwise decision. As a general rule, it is not fair to judge someone for doubling up on a science course because they could be suffering in one aspect of science in the fall but then take a course on a completely different aspect of science in the spring. The situation is similar for history-related courses because they all cover different areas of social studies. Psychology, Bible History and AP European History are all drastically different courses, so it can hardly even be considered doubling up since none of them are related.
The overarching difference between doubling up on math or science and history is that math courses follow a linear path whereas science and history courses branch out into several different areas.
Math courses tend to build off of previous levels of math. Skills learned in one class will follow you to all of your next math classes, so it is important to get help as soon as the struggling begins; do not add more stress by attempting to double up. In science and history classes; however, the options are so varied that the student could be taking two completely different courses for each semester of the school year.
If a student is considering using one of their elective periods to double up on a core subject, s/he needs to make sure s/he is prepared to take the class. Do not make the uninformed decision of taking an additional math course in the spring after barely staying afloat in the fall semester’s math course. Be careful of science and social studies choices. Most students want to excel in high school, but understanding your own academic abilities and catering to those needs is the best way forward.


PhD MathEd students celebrate International Day of Mathematics
BICOL UNIVERSITY, Legazpi City – Graduate students, Doctor of Philosophy in Mathematics Education had celebrated the International Day of Mathematics, with the theme “Mathematics is everywhere”, on March 14, 2020. The celebration has commenced with the opening program – Mr. Engelbert T. Baloloy, PhDMathEd President delivered his opening remarks highlighting the importance of Mathematics in various facets of life; followed by Ms. Aiko B. Adonis, PhDMathEd student, explaining the purpose of the celebration.
Professor Dioleta B. Borais, in her message, underscored the importance of teachers in inspiring the learners to love mathematics. She added that it is a challenge to teachers to make mathematics fun, engaging and meaningful to learners; teachers must be creative, innovative and resourceful to cater the best experience among learners and optimize their learning capabilities.
The activity proper of the celebration is composed of group presentations. The first group demonstrated and presented MATH and ARTS: Creative Origami with Mr. Michael Domanais as presenter. This activity is good for eye and hand coordination, sequencing skills, reasoning, spatial skills, memory, patience and attention skills which are essential in learning mathematics. The Hidden Art of Tangrams was presented by Mr. John Christian D. Santos, representative of the second group. Geometrical Craft, Math-Arte, and Excursions in Sudoku and Dymestifying Mathematical Puzzles were presented by the three other groups, respectively.
Awarding of certificates to the participants followed after the group presentations – presenters of MATH and ARTS: Creative Origami declared as BEST GROUP PRESENTER. The celebration is ended with a simple food party. The masters of ceremony were Ms. Olive N. Mancera and Mr. Rowel S. Ramos, both PhDMathEd students.
© 2020 Ryan Bernido Network


In grades K-2, students build a strong foundation of basic skills in a faith-filled and loving learning environment. Students engage in lessons in the areas of religion, reading and language arts, math, social studies, and science. Other areas of focus include communication, problem-solving, higher level/critical thinking, and relating learning to real life experiences. The faculty work together to ensure a chronologically consistent and complete curriculum for all students.
Assumption students learn about and interact with their faith throughout their day. In addition to daily prayer, weekly Mass, seasonal prayer services and activities, students participate in age-appropriate lessons and discussions that help them to learn about the Catholic faith and grow in their relationship with God. The four pillars of the Catechism of the Catholic Church are incorporated into each of the grade level themes. The children learn through scripture, statements of belief, worship, and morality and prayer in addition to inspiring stories of the Saints.
The students in kindergarten and first grade are immersed in a phonics based learning environment. The students in second grade review these skills as they make the transition from learning to read, to reading to learn. We utilize a research-based curriculum grounded in systematic, explicit instruction of phonemic awareness, phonics and word knowledge, comprehension skills and strategies, inquiry skills and strategies, and writing and language arts skills and strategies.
In math, students begin learning basic concepts. The math curriculum helps to form a strong base of learning in the younger grades so students may build upon what they have learned. There is a focus on the use of manipulatives for hands-on lessons, the practical application of concepts, as well as the learning of basic math facts.


Math games are a very important aspect of recreational activity that students must divulge. The subject of mathematics can be confusing and difficult at times. Engaging in these activities can help you in the enhancement of skills like analyzing, thinking, interpreting, problem-solving, and so on. These skills are of utmost importance for you to excel in the subject of math. There are different varieties of math games like sudoku, 2048, tic-tac-toe, and so on. These games may have simple as well as complex rules and procedures to solve them. In this article, we will go through the various types of math games and also discuss the advantages of playing them.
Learn from the Leading Live Online Platform and Excel in Math
Cuemath is the leading live online math classes platform designed to help students learn math as a life skill. Teachers on the platform of Cuemath are very friendly, fun-loving, and dedicated to the development of the students. They go through many interviews before they start to teach you. This helps in ensuring that you are getting the right kind of education for the teachers who have expertise in the subject of math. Many recreational activities like math games and math puzzles are conducted periodically which helps the students to relate the concepts of math with real-life situations. Visit Cuemath and book a free session to learn and understand math, the Cuemath way.
Various Types of Math Games
There are hundreds of math games ranging from simple games like 2048 and tic-tac-toe to complex math games like sudoku. The various types of math games are as follows:
- Tic-tac-toe: Who has not heard about tic-tac-toe? It is one of the most popular games played across the entire globe by the students of lower classes as well as higher classes. Tic-tac-toe is a very simple and easy math game that can be played between two players. One player chooses ‘X’, while the other player chooses ‘O’. The players in this game aim to make a straight cross either horizontally, vertically, or diagonally. This game helps us develop observational and tactical skills.
- 2048: It is a math game that is available on an Android or an IOS device. This math game has been recently developed and is quite addictive. It is a single-player game where players aim to make the number 2048 by sliding tiles of the numbers starting from the number 2. Although 2048 is the most common number that a player has to reach, various variants of the game include higher numbers too.
- Sudoku: We can classify sudoku as a math puzzle too. Most of the students find this game very tough and complex. However, once you understand the proper techniques involved to solve it, this math game becomes very interesting. You can find sudoku games in newspapers as well as on mobile devices. Math games like sudoku apply the concept of permutations and combinations primarily. We can find various variants of this puzzle starting from simple and easy level to complex and tough level.
- Prodigy: This math game can be customized by the teachers. In this math game, students need to answer a set of questions to pass on to further levels. If questions are customized nicely by the teachers, this game becomes very interesting. For students of the lower classes, easy questions can be set while for the higher classes, tough and complex questions can be set.
There are various other math games like 101 and out, math bingo, math baseball, 24, angels and devils, hopscotch math and so on that you can try out. If you want to learn math in a fun and interesting manner along with playing these interesting math games, visit Cuemath.


A team of students from CPCS competed against nine schools in the 13th Annual SCCS Trinity Math Triathlon last Friday. Each student competed in three areas of math skills. The first competition was an individual grade level test on paper. The second competition was a group event. The final competition was a relay race — teams answered math questions within a six minute time frame. Schools were ranked based on the sum of the event scores and rankings. Congratulations team on a successful day!
1st Place Individual Medals:
6th–Adam Gibson, Bradley Reece
Math Team Members:
6th–Adam Gibson, Oliver Postma, Bradley Reece, Rachel VanderZee
5th–Brynne Noordermeer, David Price, Ryan Sederstrom, Jonathan Slager
4th–Heidi Gorter, Avery Martin, Simon Postma, Evan Reece
3rd–Isaac Gorter, Kaimbria Noordermeer, Ben Robbins, Raelyn Sederstrom


And the silly saga comes full circle as the elite puppet Novak Djokovic was deported from Australia after the government’s lawyers argued he’d become an ‘icon’ for anti-vaxxers. And that is what this is partly about, a false idol for the gullible diet-woke people.
I’ve already covered this obvious staged ritual twice, showing you the numbers and gematria behind it.
And now, Mr. 53, has been deported on the 16th.
Djokovic Deported = 256
The Square Root of 256 = 16 as in January 16.
And the 16th Prime Number is… 53!
Novak Djokovic = 53
Djokernole = 53 (what he calls himself on Instagram)
Covid = 53
Deceive = 53
The ruling came from three judges, and the first mentioned in the articles was…
Chief Justice James Allsop = 241
241 is the 53rd Prime Number.
January 16 leaves 349 days remaining in the year. 349 is the 70th Prime Number.
Djokovic Deported = 70
January 16 can be written as 1/16 = 116
Coronavirus Disease = 116
Australian = 116 (as in Australian Open)
The ruling came from ‘three’ judges.
Three = 56 (the pandemic code)
Unvaxxed = 56 (what they claim he is)
Deportation = 56
Coronavirus = 56
Covid Vaccine = 56
Virus Outbreak = 56
Wuhan Corona = 56
Society of Jesus = 56
Novak was born on May 22, 1987. This deportation happened 127 days before his upcoming birthday.
Djokovic = 127 (a perfect date for a ritual about him)
This deportation comes 239 days after his birthday. 239 is the 52nd Prime Number.
Government = 52 (the Australian government played along in this ritual)
Authority = 52 (to show the people its authority)
Wuhan Corona = 52 (keyword for many covid/vaccine stories)
Vaccine Mandate = 52 (what these staged stories are used for)
Novak was first detained on January 6, as in 1/16 = 16.
He was then deported ten days later on January 16, the 16th day of the month.
Ten = 39
January 16, 2022 = 16/1/22 = 16 + 1 + 22 = 39
Visa Appeal = 39
Deported = 39
Australia = 39
And of course…
Ten Days = 56 (here we go again)
Djokovic’s lawyer was Nicholas Wood Barrister SC
Nicholas Wood SC = 1318 (in the important Jewish Cipher)
This is referencing Revelation 13:18 where the number of the beast is revealed.
Number of the Beast is often used in vaccine stories, because…
Vaccination = 666
And remember, he was first detained on January 6, the day that came exactly 666 days after the declaration of the fake Pandemic on March 11, 2020.
Nicholas Wood = 66
Anti-Vax Sentiment = 66
Number of the Beast = 66
Revelation is the 66th book of the Bible. Also notice the 57 and 75, reflections of each other.
Yet another ritual by the Jesuits to divide people and to manufacture false icons and idols for the gullible anti-vaxxers while mocking them.


At Ouston Primary School, we view mathematics as essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, and a sense of enjoyment and curiosity about the subject.
Our intent is to provide children with a mathematics curriculum that will allow them to become confident individuals through developing their mathematical skills to their full potential. We also aim to present maths as a challenging, exciting, creative and relevant subject in order to promote a positive and confident attitude.
Click on any thumbnail below to view a larger image in our gallery slideshow
UKS2 Formal Written Methods Demo
OPS Calculation Policy
Progression Map Fractions and Reasoning
Progression Map Algebra and Reasoning
Progression Map Addition and Subtraction with Reasoning
Progression Map Multiplication and Division with Reasoning
Progression Map Measurement with Reasoning
Progression Map Geometry properties of shapes with Reasoning
Progression Map Geometry position and direction with Reasoning
Progression Map Statistics and Reasoning
Progression Map Ratio and Proportion with Reasoning
Progression Map Place Value and Reasoning
POLICY FOR MATHEMATICS


Commercial arithmetic is a branch of mathematics that deals with money matters. Commercial arithmetic is used in many areas of our daily activities. Under commercial arithmetic’s we have the simple interest, compound interest, appreciation and depreciation, hire purchase and taxes.
Interest is the price for the use of borrowed money for a specific period of time. If a sum of money is deposited in or borrowed from a financial institution for a period of time, it earns interest. The sum of money deposited or borrowed is called principle (P). The ratio of interest earned in a given period of time to the principle is called rate (R) of interest. The rate of interest is usually expressed as a percentage per annum of the principal. Per annum (p.a.) means yearly.
When interest is calculated using only the initial principal at a given rate and time it is called simple interest (I)
What is the simple interest earned on sh. P at a rate of R% per annum for T years?
Simple interest (I)=Principal (P) *Rate (R) * Time (T) /100
Abbreviated as I=PRT/100 this is the simple interest formula
What is the simple interest earned on $ 2,000 at 12% per annum for (a) 1-year (b) 2years (c) 3 years?
Interest earned is:
(b) $ 12/100*2000*2=$480
(c) $ 12/100*2000*3=$720
If money is borrowed from or deposited in a financial institution It earns interest after a specified period of time at the stated rate of interest. Instead of interest being paid out to the owner, it may be added to (compounded with) the principle and thereafter also earn interest.
Find the amount after 2 years if $1,000 is invested at 15% p.a. compound interest
1st year interest=$.15/100*1000=$sh150
Amount after 1st year. =$. (1000+150) = $1150
2nd year interest =$ 15/100*1150 = $. 172.50
Amount after 2nd yr. =$. (1150+172.50)
Explanations such interest which is added to the principal is called compound interest. The period after which it is compounded to the principal is referred to as the interest period. The compound interest may be calculated annually, semi-annually, quarterly, monthly, etc.
After every interest period the principal increase and therefore the interest earned also increases. The sum of the principal and the compound interest is called the accumulated amount.
Appreciation and depreciation
Machines and building experience wear and tear with time. Because of this they lose their value. The loss in value of an asset with time is called depreciation if however, an asset gains in value with time it is said to have appreciated. For example, the value of land in urban centers keeps on increasing due to high demand.
Example an iron box costs $.500 and depreciates each year by 10% of its value at the beginning of the year. What will be its value after 4 years?
Value after the first year =$. (500-500*10/100)
Value after the second year= $. (450-450*10/100)
= $. 405
Value after the third year = $. (405-405*10/100)
= $. 364.50
Value after the fourth year = $. 364.50-364.50 *10/100)
= $. 328.05
Show that the value of the iron box after 4 years is
Frequently, many individuals find themselves compelled to make huge cash outlays for worthwhile purchase or for emergencies. Most often, they do not have the necessary cash and have to either borrow the money or buy on credit. To buy on credit means to make a down payment and then pay the balance by instalment (usually regular).
In this method interest has to be paid on borrowed money. The rate of interest charged depends on the time, the amount and the lending body.
Peris wants to buy a sewing machine on hire purchase. It is marked $. 7,500 cash. She can pay the price or make a down payment of $2,250 and 15 monthly instalments of $ 550 each
How much interest does she pay under the instalment plan?
Total amount of instalments = $. 550*15 = $.8,250
Down payment =$.2,250
Total = $.10,500
Amount of interest = $. (10,500-7,500) =$. 3,000
Note Peris will be able to take the sewing machine as soon as she pays the down payment.
The interest charged for buying goods/services on credit is called a carrying charge which is calculated as a compound interest. Such a method of buying items by instalment is called HIRE PURCHASE
To provide services such as health, education, roads, etc. the government has to raise funds. One of the main sources of revenue for the government is through taxation. Tax is charge levied on individuals and companies for the purpose of providing government services. There are two main types of taxes: direct and indirect taxes
Direct taxes include income tax (levied on the income of individuals) corporate tax (levied on profits of companies), capital gain tax (levied on properties), VAT and estate duties (levied when a person who has property dies)
Indirect taxes include sales tax (levied on some consumer goods and services) excise duties (levied on goods manufactured locally), Custom duties (levied on goods imported from outside the country) and export duties (levied on goods exported from the country
The government also earns revenue by charging various fees such as business and trading licenses, road licenses, airport tax, etc.
If a 110% duty is charged on a bottle of beer, which costs $ 4.50 to produce, calculate the selling price
Duty =$. 110/100*$. 4.94
Selling price =$. (4.95+ 4.50) = $. 9.45


Title: Harmonic B-splines, Poisson Solverand Poisson Vector Graphics
Date: 10:00am – 11:30am. December 22, 2016
Venue: 310 room, Office building (行政楼310会议室)
Host: Prof. Zhang Caiming
Abstract: Solving Poisson equations playsan important role in various computer graphics tasks, such as imagecomposition, surface reconstruction, vector field design, just name a few.Traditional solvers, such as the finite element method (FEM), provide numericalsolutions only. In this project, we are interested in solving Poisson equationswith piecewise constant Laplacians. We show that this special type of Poissonequation has a closed-form solution, which are represented by harmonicB-splines. We demonstrate our closed-form solver on 2D/3D Poisson vectorgraphics, a new type of vector graphics that overcomes the limitations of theexisting first- and second-order diffusion curve images.
Bio: Ying He is currently anassociate professor at School of Computer Engineering, Nanyang TechnologicalUniversity, Singapore. He received the BS and MS degrees in electricalengineering from Tsinghua University, China, and the PhD degree in computerscience from Stony Brook University, USA. His research interests fall into thegeneral areas of visual computing and he is particularly interested in theproblems which require geometric analysis and computation.


If the total number of students in this class is 20, how many students scored below 60? Find the maximum value. How long would it take the same boat to go from A to B in still water? An airplane flies against the wind from A to B in 8 hours. The same airplane returns from B to A, in the same direction as the wind, in 7 hours.
Could the same problem be represented with thinking blocks or bars? Here is one possible set-up: Again, after setting up the problem, we need to work backwards, solving 64 19 to find the first unknown 45 and then subtracting 45 from 87 to find the number of nickels for the cookie. After the model is set up, it functions as a bridge to algebra.
A blue block can be labeled b, and from there, we can write equations to express what is shown in the model and solve for a. Students and teachers need to appreciate that there can be a number of ways to visualize a problem, and a number of ways to solve a problem non-visually.
Some students might benefit from visualization more than others. Sometimes students resist using visual models when a solution is readily apparent to them. It might be helpful to designate certain class periods as times for learning to use a new tool; a time when students practice using the tool on very simple problems even though they don't really need it.
In that way, they become comfortable with how the tool works, and are ready to apply it to more challenging problems. Other class periods can be designated as "choose your own approach" problem-solving sessions, during which students might opt to represent problems with manipulatives, other pictorial representations, algebraic notation, or mental math.
Fraction problems often are represented with pies or rectangular slices; area and perimeter problems are often represented with geometric diagrams. If Bob has 5 more toys than Tom, how many toys does each one have? John can eat a quarter of a pizza in one minute. How long does it take John to eat one pizza and a half? John can eat a sixth of a pizza in two minutes.
It takes 3 minutes for Billy to eat one quarter of the same pizza. If John and Billy start eating one pizza each, who will finish first?
John read the quarter of the time that Tom read. Tom read only two-fifth of the time that Sasha read. Sasha read twice as long as Mike. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
.In the model area, the bars are provided at the outset, and the student must drag them into position. If Bob has 5 more toys than Tom, how many toys does each one have? Mary wants to make a box. After the student calculates that one block represents 18 children, it will be clear that the total number of children is 5 x The average of students who scored below 60 is
The inequalities section lets you solve an inequality or a system of inequalities for a single variable. Bar modeling helps make abstract problems more concrete and facilitates algebraic reasoning in students of all ages. It is feasible, although many using the full program advise against it. The average of students who scored below 60 is
John can eat a sixth of a pizza in two minutes. A hint is available below the blocks in some problems; a "math tutor" responds to the student's latest action; and the student is prompted for the next step. Students and teachers need to appreciate that there can be a number of ways to visualize a problem, and a number of ways to solve a problem non-visually. Also Solutions and explanations are included. How many chocolate bars are in the large box?
The student must determine which operations to use and in what sequence. How many chocolate bars are in the large box? Maths Problems with Answers for Grade 5 Grade 5 maths problems with answers are presented.
How many degrees does it rotate in one second? It is feasible, although many using the full program advise against it. A "taste" of bar modeling is probably not very effective; part of the power of bar modeling in the Singapore Math program is the structure and sequencing of the lessons. In that way, they become comfortable with how the tool works, and are ready to apply it to more challenging problems.
What is the total cost of 1 shirt, 1 pair of trousers and 1 hat?


Santacesaria, E.; Tesser, R.; Di Serio, M.
Fluid Phase Equilibria,84 (1993) 1ll-122
A simple and predictive for describing the high P-T vapour-liquid equilibria (VLE) of binary mixtures is described. It is based on the use of group contribution method such as UNIFAC combined with the Redlich-Kwong-Soave (RKS) equation of state (EOS) and quadratic mixing rules. VLE data are generated by UNIFAC at a prefixed temperature. These data are used for determining by regression analysis the binary interactin parameters of the RKS/EOS. Then the EOS with the same parameters can be used for describing VLE at higher temperatures.


We are active in the fields of computational biology, bioinformatics, biostatistics, and systems biology. Our activities include the development of mathematical and statistical models, their efficient implementation in computer programs, and application to biomedical problems. We are interested in disease-associated biological networks and in pathogen evolution associated with disease progression. Our ultimate goal is to predict the effect and to support the design of medical interventions in complex, rapidly evolving, biological systems, such as virus populations, bacterial colonies, and tumors. We make use of high-throughput molecular profiling data and apply high-dimensional statistical modeling in order to study intracellular as well as intercellular biological networks and to predict the effect of genetic alterations. We model the evolutionary dynamics of pathogen populations to investigate virulence, disease progression, immune escape, and drug resistance development. Our models aim at understanding evolutionary escape and improving diagnostics, prognostics, and rational treatment design.


Kimchuk enters her third year as the Faculty Athletic Representative for the University of the Sciences Athletics Department.
A Senior Lecturer of Mathematics at the University, she has been teaching mathematics for 24 years, 20 of which have been at USciences.
While having taught many courses, Kimchuk's main teaching responsibilities have been Precalculus and Calculus. She serves as an academic advisor for first and second year students, and has been actively involved in STEM outreach. She also works closely with the freshmen in her courses to ease their transition from high school to college.
Kimchuk has received the William F. Homiller Award for Excellence in Teaching, the Lindback Award for Distinguished Teaching, the Outstanding Academic Advisor of the Year Award, and the Patricia Leahy Innovation in Teaching “OWL” Award.
She holds degrees from Gwynedd Mercy College and Villanova University.


Factorising can be defined as a revised process of expanding brackets. This article is about how to factorising and its example in the math problem.
What is Factorising?
Factorising is the reverse process of expanding brackets. A factorised answer will always contain a set of brackets. To factorise an expression fully, take out the highest common factor (HCF) of all the terms.
How To Factorising
Factorising – Expanding Brackets
This section shows you how to factorise and includes examples, sample questions, and videos.
Brackets should be expanded in the following ways:
For an expression of the form a(b + c), the expanded version is ab + ac, i.e., multiply the term outside the bracket by everything inside the bracket (e.g. 2x(x + 3) = 2x² + 6x [remember x × x is x²]).
For an expression of the form (a + b)(c + d), the expanded version is ac + ad + bc + bd, in other words, everything in the first bracket should be multiplied by everything in the second.
Expand (3x + 1)(x – 1)
(3x + 1)(x – 1)
= 3x² – 3x + x – 1
= 2x² – 2x – 1
See also : Introducing Algebra
Factorising is the reverse of expanding brackets, so it is, for example, putting 2x² -2x – 1 into the form (3x + 1)(x – 1). This is an important way of solving quadratic equations.
The first step of factorizing an expression is to ‘take out’ any common factors which the terms have. So if you were asked to factorize x² + 2x, since x goes into both terms, you would write x(x + 2).
There is no simple method of factorizing a quadratic expression, but with a little practice, it becomes easier. One systematic method, however, is as follows:
Factorise 12y² – 20y + 3
= 12y² – 18y – 2y + 3 [here the 20y has been split up into two numbers whose multiple is 36. 36 was chosen because this is the product of 12 and 3, the other two numbers].
The first two terms, 12y² and -18y both divide by 6y, so ‘take out’ this factor of 6y.
6y(2y – 3) – 2y + 3 [we can do this because 6y(2y – 3) is the same as 12y² – 18y]
Now, make the last two expressions look like the expression in the bracket:
6y(2y – 3) -1(2y – 3)
The answer is (2y – 3)(6y – 1)
See also : Cara memeproleh Akar Persamaan Kuadrat
Factorise x² + 2x – 8
We need to split the 2x into two numbers which multiply to give -8. This has to be 4 and -2.
x² + 4x – 2x – 8
x(x + 4) – 2x – 8
x(x + 4)- 2(x + 4)
(x + 4)(x – 2)
Once you work out what is going on, this method makes factorizing any expression easy. It is worth studying these examples further if you do not understand what is happening. Unfortunately, the only other method of factorizing is trial and error.
The Difference of Two Squares
If you are asked to factorize an expression which is one square number minus another, you can factorize it immediately. This is because a² – b² = (a + b)(a – b) .
Factorise 36 – x²
= (6 + x)(6 – x) [imagine that a = 6 and b = x]
Factorise the following :
- 49 – x²
- x² – 81
Grouping terms may allow using other methods for getting a factorization.
For example, to factor
one may remark that the first two terms have a common factor x, and the last two terms have the common factor of y. Thus
Then a simple inspection shows the common factor x + 5, leading to the factorization
Lihat Juga : Garis Singgung Persekutuan Dalam dan Luar


I’ve come to understand that most of my problems stem from thinking, at the most inopportune times, that I’m a lot smarter than I actually am.
I took the first of my hell week exams on Wednesday for my statistics class. This class is doing nothing to progress my degrees—I registered for it when I had a different track in mind, one that didn’t accept credit by exam—and I’ve pretty much taken this exact same class before, so every time I make my way into the classroom it’s essentially my walk of shame.
I came into the exam expecting it to be a blow-off and already planning out when to get my flu shot, which I assume is what all college students spend their waking moments thinking about.
Armed with my free financial calculator, courtesy of my program—I really thought I’d gamed the system by not having to buy a $7 scientific calculator—I received the test packet. Opened it. 25 problems total: ten multiple choice, five short answers, one bonus. Nothing unmanageable, not that I could do anything otherwise.
For the first hour, at least, all was well. I knew how to do every problem and was reasonably sure my answers were correct. I wasn’t getting any weird numbers; they all rounded out nicely to two decimal places, as per the instructions.
Then, with fifteen minutes left, I started to check my work. The multiple-choice section was fine. I got to a question that didn’t quite get me the same result when I worked backward, but I got the same answer calculating it forward and chalked it up to rounding variations in the z-tables. Then I got to the bonus, which involved systems of equations (working with two equations to solve for two variables, one by one.) An algebraic concept, systems of equations are easy until you keep getting answers that don’t match and you realize you’re intrinsically worthless.
Through equation 1, x = 1001.23
Through equation 2, x = 980.19
Erase. Repeat. Ten minutes left.
x = 879.83
x = 1157.37
Erase. Repeat. Five minutes left.
X = 968.72
X = The Pope
Frantically reviewing my work, I realized how odd it was that all my answers had been rounding so neatly to two decimal places.
And then I realized. Stomach sinking, I typed in .333333 into my calculator and pressed enter. The screen flashed and showed .33.
My financial calculator had been rounding all my numbers to two decimal places this whole time. All the numbers. As in, if I did 1 divided by 3 and then multiplied by 3, I’d get .99 instead of 1.
The professor called time. I scribbled one of my recent answers into the box and handed my test in. “I realized toward the end that my calculator had been rounding everything to two decimal places,” I said. I don’t know what I was hoping for, telling her this. Maybe humanity.
“Oh, because you used a financial calculator. Next time, get the decimal places to float.”
“Okay,” I said. And walked out.
Later, when a friend asked how my exam went, I recounted this entire tragedy and he asked, indignantly as a good friend should, if that didn’t defeat the purpose of a having a calculator for accurate calculations. Like, in what scenario would you want this feature? (Not rhetorical. If there’s a purpose for it I might feel slightly vindicated, but only slightly.)
I still have no idea how I did on that test, and I can’t even really talk about it because saying my calculator ate my decimal places is tantamount to the “dog ate my homework” excuse. Although come to think of it, if I’m afraid of people thinking my reluctance to talk about an easy exam means I’m dumb, it might be a bit late for that.
Please consider following this blog via email and/or liking its Facebook page, where I post occasional life updates and quality excuses for the lack of said life updates. Make me feel better about this being my last week as a teenager and having to spend it working.
Last post: Just Biden My Time
If you’re still reading, check out this list of pointless things that exist. It made me unreasonably amused.


Accompanied by statistical tutor for mendenhall beaver and beavers a brief course in business . An edition of a brief course in business statistics 1994 a brief course in business statistics 2nd ed william mendenhall robert j beaver barbara m beaver. This course will give you a start in understanding statistics in general and business application of statistics in particular we will discuss the terminology of statistics basic measurements and the probability concept you will have a taste of the fundamentals of statistics and how it is applied in business settings for simple problems. A brief course in business statistics william mendenhall 1995 this text covers basic descriptive and inferential statistics linear and multiple regression analysis of variance chi square and quality control all problems in the text are based on real data from published sources each chapter
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The Koç School Mathematics Department aims to prepare and develop the mathematical background of the students which will be required in their academic studies in the university.
High School Mathematics Department encourages students to:
English mathematical terminology is taught in prep class. 9th and 10th grade levels cover the MEB/national curriculum and students choose MEB – High Level Mathematics or IB Math programme (Standart or High Level) in grade 11.
The topics covered in the national curriculum are:
The topics taught are enriched with technology and in class activities. Multiple choice questions (in Turkish) are also integrated in the lessons to prepare students for the university entrance exams in the national curriculum.
Starting from grade 9, TI-84 plus and TI-NSpire CX graphing display calculators are powerful tools that students use to understand the new concepts and to check their assumptions.
For detailed information on the IB curriculum (standard and higher level), please follow this link.
Mathematics department offices are located in 210A and 310A in the high school building and can be reached at extensions 6498 & 6499.


This course prepares students for future study of mathematics at university level by developing an understanding and application of algebra, geometry and calculus. Students will practise effective techniques for studying mathematics as well as learning to read and write mathematics clearly and intelligibly. Students will also develop an appreciation of the patterns which arise in mathematics and how they are applied in a number of different subject areas. Before considering a University Preparation Course please review important information to determine if you are eligible for the Mature Age Entry Scheme and how to go about qualifying and applying.
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How can someone learn math through a fantasy comic book story? Learn more about Ryan's Math Adventure and find out.
Ryan, Aris, and Sir Walter are searching for the prophesied Book of Light. Without math, the nearby town has been thrown into chaos.
Suddenly, monstrous larvae creatures appear before Ryan and Aris! Can Ryan, Aris, and Sir Walter find the Book of Light in time to stop the evil Pesia?
1. A Change in Town
①Counting, Writing, and Reading Numbers from 1 to 5
②Ordering from 1 to 5
③Understanding the Idea of 1 More and 1 Less (1)
④Understanding, Writing, and Reading 0
⑤Counting, Writing, and Reading Numbers from 6 to 9
2.Workers at the Robot Factory
⑥Understanding the Order of Numbers up to 9
⑧Understanding the Idea of 1 More and 1 Less (2)
⑨Comparing Numbers up to 9
3. Stone of Life
⑩Different Ways of Grouping the Numbers 2, 3, 4, 5
⑪Different Ways of Grouping the Numbers 6, 7, 8, 9
⑫Understanding, Reading, and Writing 10
⑬Grouping and Gathering the Number 10
⑭Counting, Writing, and Reading Numbers from 11 to 19
4.Birth of Namute
⑮Grouping and Gathering Numbers from 11 to 19
⑯Understanding Numbers up to 19
⑰Comparing Numbers up to 19
⑱Understanding Even and Odd Numbers
The Ryan's Math Adventure series provides a complimentary alternative to traditional math textbooks.
Instead of just telling a child who dislikes math, "study harder," try out Ryan's Math Adventure.
Through this fantasy comic book story, kids can study math in a more positive and fun way.
Ryan's Math Adventure 1: Numbers Smaller than 20 was a very cute math adventure book that teaches simple math to elementary aged children through the lens of an adventure story. The pictures were well done and the dialogue bubbles kept the story form being too wordy. I received this item for free or at a discount in exchange for only my honest and unbiased review, all opinions, thoughts, and views are strictly my own and not influenced by this fact.


If you’re hoping to increase your chances of success with online roulette, it’s a good idea to try out a few popular betting systems and implement some basic strategy.
While betting systems cannot guarantee that you will win, if done correctly, they can greatly increase your chances of success.
However, before getting started with any of the betting systems explained below, we suggest that you get to grips with the basics of roulette and make sure that you understand the ins and out of this popular casino table game.
Martingale Roulette System
The Martingale roulette system is the most commonly used system of all and is simple enough to understand – increase your bets after very consecutive loss and when you win you go back to the original betting amount and start the progression all over again.
Most players use the Martingale on even chance bets and double their bets after every loss. The Martingale will increase your chances of winning in the short term, so use this method sparingly.
D’Alembert Roulette System
The D’Alembert is one of safest roulette systems you can implement because even though you are increasing your bets after each loss, the increases are incrementally much smaller than the Martingale. You will essentially pick a starting bet and increase the bet by 1 after a loss, and decrease by 1 after a win. As this is an even chance betting system it can be used on red/black, odd/even, and 1-18/19-36.
The mathematical progression behind the D’Alembert roulette system makes it unsuitable for use on dozens/columns and inside bets.
Many players find the mathematical equations behind roulette betting systems complicated, and it’s for this reason that some players choose to stick to simpler games such as real money blackjack!
Fibonacci Roulette System
First things first: the Fibonacci didn’t start out as a roulette system. It’s actually a simple mathematical theory where you start with one and add the two previous numbers together to give you the next number in the sequence. For example: 1-1-2-3-5-8-13-21-34 etc.
As a betting strategy, you don’t have to start with 1, but it is safer as the betting progression will be slower, and you will work your way through the strategy until you get a win. Once you’ve hit a win, move back 2 numbers in the sequence and bet that amount.
Labouchere Roulette System
The Labouchere is a slightly more complex system and it may require some practise in order to get it right. This system differs from the others as you dictate the rules before starting play, and you decide how much you will bet and how long you will play for.
This is an even chance betting system so it can be used on red/black, odd/even, and 1-18/19-36 and it cannot be used anywhere else on the table as the maths wouldn’t add up.
The first you do with the Labouchere is to create a ‘betting line’, for example 1-1-1-1-1-1, with the length of the sequence dictating how long you will play for. Once you have your line, take the 1st number on the left, added to the first number on the right, add them together and you’ll have your betting amount.
Using the line above, your bet will be 2. If you win, you cross both the left and right hand numbers off the line, making it shorter. Repeat this process until there are no numbers left in the line, at which point you can walk away or construct a new line.


The formal mathematics curriculum begins with activities to teach sequence, recognition and quantity of numbers 1 through 10. Two parallel lessons formats build on this knowledge: operations with numbers (addition, subtraction, multiplication and division) and memorization of mathematical facts. Eventually used together, these two disciplines form the foundation for working complex math problems. Other lessons introduce math concepts and skills, such as: counting-by-multiples, the squares and cubes of numbers 1 through 10, and fractions. All of these activities are available to young children in a Montessori early childhood program.
Sandpaper numbers are the sandpaper cutouts of the numbers zero through 9 glued to individual boards. They teach the numerical symbol of the quantities the child has already learned from the number rods. This is the beginning preparation for number writing. Sandpaper makes the process of tracing out the number both more interesting to the child and more effective in absorbing the mental image of the number.
Montessori is focused on teaching for understanding. In a primary classroom, three and four-year-olds receive the benefit of two years of sensorial preparation for academic skills by working with the concrete Montessori learning materials. This concrete sensorial experience gradually allows the child to form a mental picture of concepts like how big is a thousand, how many hundreds make up a thousand, and what is really going on when we borrow or carry numbers in mathematical operations.
Teen Board & Beads
With the Teen Boards, the child explores the numbers 11-19. The child puts together ten bars and colored bead bars to create teen quantities, learns the symbols for 11-19 and associates the quantity and symbol.
Memorization 100 board
Remember that academic progress is not our ultimate goal. Our real hope is that the children will have an incredible sense of self-confidence, enthusiasm for learning, and will feel closely bonded to their teachers and classmates. We want much more than competency in the basic skills; we want to them to honestly enjoy school and feel good about themselves as students.


The lift will undoubtedly fit in your garage. The post height is only 84" on the 8000 and 92" on the 8000XLT.
The real question is whether your cars will fit on top of and underneath the lift and that is dependent upon the height of your vehicles and the height of your garage ceiling. Simple math will help here. Add the height of both vehicles together and add another 8" to that. The result must be less than your ceiling height in order for the lift to work in your garage.
(A) Height of vehicle 1- + (B) Height of vehicle 2-
Plus + (C) Track Thickness & Clearance- 8"
Equals = (D) Total height:
(E) Your Ceiling height:
(E) measurement must be equal to or greater than (D) measurement.
Next, the lift will have a maximum clearance underneath of 70" on the 8000 model and 77" on the 8000XLT model when resting on the highest safety lock position. The vehicle you park underneath must be lower than 70" to use the 8000 model or 77" for the XLT model. The vehicle you park on top must be lower than your ceiling height less 74" for the 8000 or 81" for the 8000XLT. (Also, consider the location of your garage door when open and garage door opener if you have one.)
Safety locking positions are spaced at 4 ½ " intervals so if you need to stop the lift at a lower height you may do so.


It was a seemingly contradictory day today: we spent much of the morning doing practical Maths in various guises (but not Maths Whizz) - and I realised why I still feel the need to have the boys use an online Maths curriculum. Friends who advocate 'unschooling' or child-led learning often say that Maths is everywhere, and there is no need for a curriculum. They do have a really good point, but regular readers will know this is the area where I am least confident. This morning's fun at least demonstrated what they are talking about...
First of all, prompted by Middle's struggles with MathsWhizz addition/ subtraction using carrying (or "regrouping" for the up-to-date among you), I had bought a set of Cuisenaire Rods (a friend was going to loan some to us but then I realised how stressed I would get desperately trying not to lose any small bits, so we found a cheapish set to buy for ourselves!). He is such a visual learner, I think it will help him get to grips with the grouping/ carrying concept. Anyway, he found the set of rods first thing this morning and immediately wanted a go - the bright colours appealing to his creative side and the stacking rods appealing to his love of order (this was before we even had a chance to have breakfast). We went through the set finding out what colour all the different number-rods were, then we matched them up to see how many different ways we could make 'tens' - the teacher in my head was mentally ticking off 'number-bonds', while he was just playing and having fun! He then progressed to using them to play a game that basically revolved around times tables - not that he knew that. His enthusiastic squeals soon called his brothers, and next thing I knew they were all playing with "Mummy's New Maths Game" - amongst other things making one huge rod two-hundred-and-fourteen squares long (according to Eldest).
The rods had to go away for breakfast then, but were such a success, I'm confident Middle will be happy to play some more with them when we have another look at the Maths he got stuck on :)
Our next Maths encounter was a result of the car-boot sale I did at the weekend - I tipped out the coins onto the table, and the two older boys helped me sort them into piles, count the totals and bag them, ready to be banked :)
Following that, there were various maths-based games and puzzles, including Electronic Battleships (Eldest v the computer); a jigsaw puzzle aimed at 5-year-olds that Youngest did without batting an eyelid, and the Bus Stop game from Orchard Toys. Today was also Eldest's turn to bake, and he made some scrummy Orangle Drizzle cakes (after an emergency dash to the smallholding up the road where we buy our eggs) - there's plenty of Maths to be found in measuring out ingredients, dividing cake mixture etc - and we got to eat them too... edible Maths is my favourite!
Of course, it wasn't all Maths today - Eldest (who is recovering from some nasty virus picked up this weekend) and I had some lovely snuggle time on the sofa, watching our garden wildlife - including a very shy mouse and some beautful long-tailed tits - and we all played a game or two of Animal Soundtracks (great for encouraging listening skills), and Middle and Youngest spent a fair time on Reading Eggs too! It's just that it did help me to see how Maths really does fit naturally into our everyday lives, without them going near MathsWhizz.
So that said, why do I still feel the need for online (or even workbook-driven) curriculum, such as MathsWhizz? Well, bearing in mind that this is still all fairly experimental (we're still in novice-territory, so may well change our mind next month/ week/ tomorrow), it was partly a conversation that I had with Hubby that crystalised my thoughts. I can't remember how it came up, but he basically mentioned that he never learned his times tables at school (his family moved around quite a bit when he was young, so presumably it just fell by the way in one of the gaps between schools). This was supposed to be a positive point, showing that even though he didn't complete all the basic requirements conventionally expected by a formal education, he still went on to do very well for himself (got Maths A-level, has good job etc). On the other hand, I can vividly remember learning my times tables (mostly I remember a little green paperback book purchased for me by my Mum who wanted to help consolidate what I was presumably learning at school). Hubby said that not having learned his times tables hadn't hindered him at all - he could still work out whatever mental maths he needed to do; it just takes him a little longer than those who had learned them (eg me). So if I was in a situation where I needed to know - say - 6x7, I would instantly know the answer was 42. Hubby would still get the same answer (he's a clever chap), but he would have to work it out. Not much of a draw-back, really. It just got me to thinking then about when I missed two weeks of secondary school due to illness, and consequently never learned how to do long division. "No big deal" you might think, but you'd be surprised how many times it came up throughtout my adult life... not on a daily basis, but often enough to frustrate me that I couldn't do it. It wasn't until I needed to teach my students how to do it that I finally got round to teaching myself how to do it - and now it's another skill under my belt :) Of course, some may argue that my teaching myself is proof that my boys could teach themselves if they ever find themselves wanting to know how to do it - and they'd have a point - but all these little things come together to where I am at the moment, thinking that these maths skills traditionally learned at school actually do have value. If we didn't have MathsWhizz, would Middle ever learn to do "regrouping/ carrying"? Maybe, maybe not. How about times tables or long division? If he didn't, it certainly wouldn't be the end of the world - neither Hubby nor I were held back by our learning gaps - but how many so-called 'learning gaps' am I comfortable to risk him having? And basically, as we do have the opportunity at our fingertips for him (and his brothers) to learn those skills, as long as he is (they all are) enjoying it, we're going to make full use of the online curriculum available - and also carry on having fun seeing how Maths fits into our every day lives too... for now, anyway! It's not a set philosophy - I certainly wouldn't try to tell anyone else they should or shouldn't do it this way... it's just a record in my diary of where we're at right now ;)
And finally: I've said this before, but was reminded again today of one of the benefits of HE. Having had a nasty virus this weekend with accompanying alarmingly high temperature, Eldest seemed to be nicely on his way back to full health today. Had we been in school, I might have been tempted to send him in so he didn't fall behind academically (and so I could fulfil my teaching responsibilities). I'm so grateful we didn't even have to think about it. Sadly we did have to cancel a visit to some lovely friends who we were looking forward to seeing - but we're hoping to reschedule soon, and I was just so grateful that Eldest was free to take the day at his own pace, learning naturally wherever he felt up to it, and resting when he needed... and of course, I was on hand to keep an eye on him, for my own peace of mind. Thank God for Home Ed :)
PS For those who like to share such things, here are our favourite garden visitors from today...


A number is represented as an unevaluated sum of several floating point numbers.
Floating point. Floating point is a peaceful free game about using a grappling hook to swing yourself gracefully through randomly generated spaces. Floating point is used to represent fractional values or when a wider range is needed than is provided by. It represents a wide dynamic range of numeric values by using a floating radix point. The only objective is to collect points and the only thing that increases your points is swinging swiftly smoothly and elegantly.
Examples of floating point numbers are 123 87425 and 90394542. An example is double double arithmetic sometimes used for the c type long double. The ieee standard for floating point arithmetic ieee 754 is a technical standard for floating point arithmetic established in 1985 by the institute of electrical and electronics engineers ieee. A real number that is a number that can contain a fractional part.
In programming a floating point or float is a variable type that is used to store floating point number values. The last example is a computer shorthand for scientific notation. Health partners international canada hpic is a not for profit organization that improves the health and well being of more than one million people around the world every year. The following are floating point numbers.
1 normalizedfloating point. A floating point number is one where the position of the decimal point can float rather than being in a fixed position within a number. Floating pointfp floating point number. Double precision floating point format sometimes called fp64 or float64 is a computer number format usually occupying 64 bits in computer memory.
The standard addressed many problems found in the diverse floating point implementations that made them difficult to use reliably and portably.