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Pythagoras Theorem and Irrational Numbers

It is interesting to see how Pythagoras Theorem helps in identifying the location of an irrational number on the Number Line. Consider a number x which is a rational number but not a perfect square. It follows that the square root of x must be irrational, that is, a non-terminating and non-recurring decimal number. Now, our interest is to determine where this lies on the Number Line. To do this, let us consider a right triangle whose base equals (x-1)/2 the hypotenuse equals (x+1)/2. What would be the height of this right triangle, we mean, the other arm of the right triangle? Let us suppose it is y. Pythagoras Theorem tells us that the sum of squares of the arms of a right triangle equals the square of the hypotenuse. So, in the triangle we have just considered, we can write:

[(x – 1)/2]^2 + y^2 = [(x + 1)/2]^2

Or y^2 = [(x + 1)/2]^2 – [(x – 1)/2]^2

= [(x^2 + 2x + 1) / 4] – [(x^2 – 2x + 1) / 4]

= [(x^2 + 2x + 1) – (x^2 – 2x + 1)] / 4

= [x^2 + 2x + 1 – x^2 + 2x – 1)] / 4

= 4x / 4

= x

i.e. y^2 = x ⟹ y = √x

This is exactly what we were looking for, the square root of x which is irrational. Now, the length of the third arm of the triangle we constructed can be marked on the Number Line using a compass.

So, if you are looking for the size of √x, this is how we go about it. Mark a point A. Mark B such that AB = x units. Mark C such that BC = 1 unit. That is, AC = x+1. Bisect AC. If D is the point of bisection of AC, AD = DC = (x+1)/2. Now, what would be the length of DB? Since DC = (x+1)/2 and BC = 1, DB = DC – BC = [(x+1)/2] – 1. That is (x+1-2)/2 or (x-1)/2.

Let’s construct the triangle now. Draw a line perpendicular to AC at B. From D, intersect the vertical line at E such that DE = AD. Now we have a right triangle in which the base is (x-1)/2 and the hypotenuse is (x+1)/2. Can you see what the measure of BE is going to be? Of course, as we have shown above, it’s going to be √x. You may transfer this length of BE to the number line now, using the compass.

Try representing √5, √7, √11, √6.8 and √9.5 on the Number Line. Each of them must hardly take a couple of minutes or less.

Math can be fun. As you explore, it is exciting to see how arithmetic, algebra and geometry converge eventually.

Teaching High School Mathematics in One Hour Time Slots

In the mid-1990s, the administration of the school in which I taught decided to change from using 40 minute teaching periods to 70 minute periods. It allowed the administration to gain extra teaching time from each teacher within the industrial award provisions. In fact, it allowed the administration to have English, Science and Mathematics teachers teach an extra class without having more time in the classroom.

My school became one of the first to do this and became an example for other schools to follow in the following years. As a result of this, I was asked to present a workshop to a nearby high school Mathematics Department explaining how my Mathematics Department had gone about adjusting to this major change.

Below is a synopsis of what I spoke about during this workshop.

For the teachers, personally:

  • It is hard work.
  • The class time must be regarded as “untouchable” and you must fight to prevent it being “borrowed” even by the administration.
  • Detailed planning is essential. It is easy for the teacher to waste/lose time without realising it is happening.
  • They need to develop a strategy to cope with absent students as even one period missed is a great chunk of their learning time.
  • Additionally, teachers need to develop a strategy for any absences they may have. In fact, teachers would be tempted to teach on even when they are not well so as to not lose valuable teaching time.
  • Their lessons must become a series of mini lessons to cover the course and to survive physically.
  • It is possible to teach a whole unit in one period.
  • They need to work smart. They must use every available tool or pedagogue to get the message across to the students.
  • Group planning by teachers will improve the quality of lessons presented to the students.

For the teachers and students:

  • There is a lack of continuity created by less teaching periods spread over the week. (In some schools, there was a two week rotation of periods.)
  • It is difficult to create a work ethic when you see the class less frequently.
  • Learnings skills must be taught more thoroughly because students must become more accountable for their learning, homework and study.
  • Learning to think mathematically must become a priority to help the students accept more accountability for their learning.
  • Mentoring becomes a useful tool to consolidate learning.
  • Learning the basic skills and procedures is paramount to gaining worthwhile success in their learning.
  • There is time to pursue problem solving in unfamiliar contexts provided the teacher’s planning covers the mandated learning.

Many of the ideas raised above had become part and parcel of Mathematics teaching since the late 1980s brought about by the introduction of new syllabuses in Mathematics that opened up the teaching of Mathematics moving away from the traditional “Chalk and Talk” Maths lesson to lessons using a variety of pedagogue.

Personally, I found teaching with 70 minute periods challenged me to use a greater degree of teaching pedagogue. Initially, I found I was rushing to cover the course. I did find that teaching had become more stimulating.

As head of Mathematics in my school, I did not see any significant change in the standard of the work produced by teachers and students. It just goes to show how adaptable teachers and students can be.

Rick Boyce has taught for over forty-five years, the last fifteen years as the Head of Mathematics in a large Australian school. There, he supervised the induction of many young teachers. This article is but a small taste of what can be found in “The Classroom Management Compendium”.

Fit for Reading – The Cause of English Dominance

Little doubt, English is the single dominant global and scientific language in the world today. Mainstream opinions attribute its rise to political, economic, technological and cultural causes, through the influences of the British empire and the USA.

However, from the standpoint of our previous papers, our argument is straightforward: legibility [1] is the cause. The evolutionary account was addressed in my paper “visual evolution of written language towards Latin alphabet – a hypothesis”. Writing’s effects on the world were discussed in “effects of written language on the non-textual world” [2]. We now explicitly discuss this topic is to correct the misconceptions and affirm our argument which can be used to explain the history and present societies, and to predict the future.

1. “Being fit for reading” is the cause

We should treat writing as a tool, just like other tools, such as a pen. One pen is better than another not because the person who uses it has better writing skills, although he/she would likely to choose a better pen. Likewise, a writing system is better not because its people are more powerful. Writing is useful because it is uniformly arranged. The legibility of the arrangement certainly has an effect on its usefulness. Different writing systems have varied levels of legibility. When other conditions are the same, people prefer reading/writing legible [3] texts to illegible ones. Legibility (fitness for reading) is the rudimentary criterion that people choose a writing system from existing ones or develop new systems.

The letters, words and sentences are refined visually by millions of people. The visual refinement of written English is unmatched by other systems. That enables the vast publications and spread of the English writing system, maximizing the established visual, textual information. Legible systems attract more people to contribute to its expansion. Although Mandarin and Spanish have more native speakers than English, their spreads are limited due to most advanced knowledge, which is the reason for language learning, are written in English. With fitness for reading, English writings are relatively fixed and reasonable. Its speakers are firm about that, influencing the people of illegible systems. When two or more writing systems meet, each absorbs others’ vocabulary, grammar, etc. to grow themselves. The English turned out to be stronger in this aspect.

Of course, legibility is not the only reason for a writing system to spread. Official status, the existing number of users, available publications, etc. all influence the spread’s momentum. But legibility is the ultimate cause, either due to voluntary adoption of legible systems or legible systems growing strong to overpower illegible ones. Illegible systems will evolve in legible direction or be replaced by legible systems, which might be initially smaller.

2. The misconceptions

2.1. Nonlinguistic influences, or non-textual causes

This view attributes the causes to external factors instead of language itself, for instances, political, economic, technological, and cultural causes. These factors empower the people, whose languages will consequently dominate. People from less advanced societies would voluntarily or be forced to learn the languages of more advanced societies. To an extreme, illiterate people will eliminate all texts if they are powerful enough. Generally, this view would consider that advanced societies need legible systems, but not necessarily.

We argue that legibility factor is ultimate and deterministic. We consider texts as the center of the human-created world. Legible texts establish advanced societies, which can be implied from The Paper. That is to say, social progress is built upon a specific writing system, which spread with its societies and people. Political, economic and other situations might fluctuate, for months or even years. An illegible writing system might prevail temporarily. On the long run, legible systems would build advanced societies and overpower illegible systems.

2.2. Spoken form is the primary form of language

This view draws people’s attention away from the written form. Many people concentrate on the spoken forms when comparing two languages, marginalizing writing or not even starting to think about the effects of writing’s visual aspect. They certainly cannot get into the heart of the problem, not to mention finding solutions. Sound representing is a major disguise of the legibility of scripts. The truth is: the spread and dominance of oral English is due to the success of English texts. The course of English globalization is the governance of speech by the written text.

2.3. Simple spelling contributes to English spread

Many people know that English spelling is simple, thus easy to learn and use, contributing to English spread. This opinion starts to look at the written form. However, simple spelling is usually considered a trivial factor. Moreover, their opinion assumes spoken English is primary, which the spelling represents.

In sum, the root causes of the wrong understanding are not realizing written form as the primary form of language, and written language as the central cause of social progress.

3. Latin alphabet-based writing systems, English and other systems

Like English, the historical dominances of Latin, Spanish, German and French were all due to the Latin alphabet, although their spoken forms set them apart as different languages. Latin alphabet’s ultimate contribution is neglected. But these systems are not as refined in basic-unit level as English. The use of diacritics, ligatures and additional letters can make the scripts worse for reading. Complex spelling-sound correspondences and irregularities of spelling also provide English the flexibility to achieve legibility and facilitate vocabulary growth.

It is important to compare Latin alphabet-based systems with other systems, because the distinctions between Latin alphabet-based systems are relatively small, obscuring the significance of the legibility factor. Compared with other scripts, Latin script’s letters, particular lowercase letters, are better shaped with clarity, simplicity and inter-letter differentiation. Latin letters also make up syllables, words in legible manners. The prevalence of romanization indicates the high legibility of Latin alphabet, often under the disguise of spelling to represent sounds of the languages of non-Latin scripts.

4. The future

It is likely that English will continue to grow and evolve based on current alphabet, vocabulary and grammar. It will speed up its spread during globalization and the revolution of the media. It will be enriched by regional lexicon, creole and pidgin, dialects and varieties. More writing systems and languages might be dying out, regretfully. But the destiny of written language is to regulate, organize, and maximize the capability of the mind, to support larger societies, and to encode the world. Diversity is not the goal of written language. Many writing systems will be preserved or even grow, at least in the near future. With diversity preserved, a dominant writing system will be the foundation of mankind’s common societies and common minds, governing other systems.

[1] Herein, “legibility” and “fitness for reading” have the same meaning.

[2] Referred to as The Paper hereafter.

[3] Herein, “legible” and “illegible” are relative terms.

The Benefits of Learning Indonesian

With the vast territory and tremendous size of its population, Indonesia will dominate the economy in Asia for the next few decades. More and more new businesses will enter Indonesia. Bahasa Indonesia is the official language spoken in Indonesia. Although most people in big cities can speak English, there are major benefits of learning Indonesian:

1. People in Indonesia tend to better relate with you if you speak in their own language. This is particularly important if you do business there. Understanding the culture and the language is the key to your business success.

2. Indonesian language provides an insight into the Indonesian cultures. Understanding how the language works will help you understand the people you are dealing with. This is especially important in a business negotiation.

3. There are some phrases and sentences that do not have equivalent English translations. Or if they do, the meaning will slightly differ. You don’t want this to happen especially when you deal with legal documents or important reports.

4. Not all Indonesians can speak English. Most of the time, in small cities and rural areas, Indonesians speak Bahasa Indonesia or speak in their own dialect. Even if they speak Indonesian, their accents might not be easily understood. You won’t want to buy or say something you don’t intend to convey.

5. Bahasa Indonesia is a fairly easy language to learn. There are no tenses and no gender nouns. However, it has different types of prefixes and suffixes. Learning how to speak Bahasa might be easier than writing it. Based on our experience, within 30 hours of intensive, guided-learning, students are able to construct sentences on their own and speak basic Indonesian.

6. Indonesia has beautiful works of arts such as pantun, sajak, puisi, gurindam, and seloka. All these are different types of poetry. Besides these, there are a lot of interesting fables and traditional folklores associated with different provinces in Indonesia.

7. When you master Bahasa Indonesia, you can also understand Malay. Due to its root, there are similarities between the two languages. Of course the vocabulary is not exactly the same, but chances are if you know how to speak Bahasa Indonesia, you will be able to understand Bahasa Melayu as well. Killing two birds with one stone.

8. Learning language is always an investment. Bahasa Indonesia is no different. You can travel throughout beautiful Indonesia without having to worry about communicating with the natives.

Maria Putera has been teaching Indonesian in Singapore for more than 10 years. Besides face-to-face training, she also teaches Indonesian language online. She is currently the curriculum developer for LIA. Visit our website at http://www.lia.com.sg or contactus@lia.com.sg to start learning Indonesian today.

The Importance of Language in Modern Life

There is no denying the fact that languages are the integral part of the cultural richness of our society and the world in which we try to lead our lives controlled and cleanly. It is significant that learning languages contributes to mutual understanding, a sense of global citizenship and personal commitment. Students learn to appreciate different countries cultures, communities and people. By making comparisons, they gain insight into their own culture and society. The ability to understand and communicate in another language is a lifelong skill for education, employment and technology in this country and throughout the world. Learning languages fascinate opportunities to develop their listening, speaking, reading and writing skills and to express themselves with increasing confidence, independence and creativity. They explore the similarities and differences between other languages and English and learn how language can be manipulated and applied in different ways. The developments of communication skills together with understanding of the structure of language laid the foundations for future study of other languages and support the development of literacy skills in a pupil’s own language.

In using a theme based on educational and cultural affairs, there exist three themes in Germany and France which are specified as environment, media and advertising. In addition, pupils in England studying health in English, French-speaking countries and Impressionism in French, and the geography and history of Berlin in German. The themes and objectives relating to grammar, cross-curricular learning and cultural understanding are defined for each theme and reference grammar sheets and lists of topic-specific vocabulary created. St Marylebone School in London places a strong emphasis on the appreciation of cultural diversity and the languages departments consider the introduction of intercultural understanding as a key concept in the revised programme of study to be the perfect opportunity for a year 9 pupils to investigate the culture of their target language country France or Germany. This would take place through culturally specific topics, as year 9 is an ‘enrichment year’ where learning is thematic following completion of key stage 3 in two years In the unit on Impressionism, pupils were introduced to the movement and shown Impressionist paintings. They were asked to suggest possible titles in English and to match the actual French titles with the paintings, along with more descriptive French phrases for each of the paintings. Pupils then chose an Impressionist artist and were asked to prepare a presentation in French on this artist for their final assessments, using presentational software or other ICT. They spent one art lesson reproducing a picture by their chosen artist and were also given the opportunity to visit the Courtauld Institute to see the original paintings.

The word ‘nationalism’ comes down from the heritage, culture and tradition of a particular country which indicates uniformity in respect of one faith that is the language conventionally uttered from a child which is his actual identity. Nationalism is such which vividly gives an acquaintance in the sense in what language he expresses his mode of his explicit desire as to what he wants or what he would like to do. So our heritage is expressed as a token of ideal acquaintance as Bengali Language with which we survive on full faith of livelihood and as such every elegiac influence is concerned in achieving the recognition of this day in the world. We can think our own belief that we are created equally in respect of expressing our own tradition, culture and religion which is bedded on our soil, grass, plants, creeper and our dwelling place. We cannot think even for a moment that a boy is treated more or less as a terrorist or miscreant or he is excommunicated at an immature stage. If we lose our faith in our own nationalism, we need to be responsible to build him or her who can give his identity as a Bengali nation. To speak the truth, the 21st February, as a symbol of blaze illumination is our rectitude for which our survival as Bengali nation has been reflected through out the whole world.

Over the course of the lessons, through research, reading tasks and a mock interview with an artist from the period, pupils became increasingly knowledgeable about Impressionism. ‘They developed confidence in describing visual images in French’, commented one teacher, ‘and began to express their opinions – albeit at a simple level – about paintings.’ Pupils’ language work covered, in particular, adjectives, question words and the ‘passé compose’. Pupils’ final presentations were assessed by both the MFL and art departments. To finish, pupils completed a worksheet in French. This consolidated everything covered during the topic and gave pupils the opportunity to reflect on what they had learnt.

In using the theme of societal concept

Staff believes the shift of emphasis had a positive impact on pupils’ learning. One French teacher noted, ‘The focus of learning switched and language became a genuine vehicle for communication. Pupils strove to express themselves effectively on a range of important issues rather than trying to use language structures in order to demonstrate their ability.’ Pupils enjoyed using language to communicate about ‘genuine’ issues and themes. There are a number of key concepts that underpin the study of languages. Pupils need to understand these concepts in order to deepen and broaden their knowledge, skills and understanding.

1.1 Linguistic competence: This is important to learn moral and ethical values in life.

a. It aims at developing the skills of listening, speaking, reading and writing in a range of situations and contexts.

b. It also envisages in applying linguistic knowledge and skills to understand and communicate effectively.

1.2 Knowledge about language

a. It indicates Understanding how a language works and how to manipulate it.

b. It shows recognizing that languages differ but may share common grammatical, syntactical or lexical features.

1.3 Creativity and Modern Technology:

Language is indispensable for learning the various techniques of Modern technology in the field of Economics, Commerce and Science as a tentative flow.

a. It actuates in using familiar language for new purposes and in new contexts.

b. It recoups in using imagination to express thoughts, ideas, experiences and feelings.

1.4 Intercultural understanding

Language opens up in every people and their community of religion, tradition and heritage to live with society, friendship and love.

a. Appreciating the richness and diversity of other cultures.

b. Recognizing that there are different ways of seeing the world, and developing an international outlook.

Aims, values and purposes

Education both influences and reflects the values of our society, and the kind of society we want to be. It is therefore important to recognize a set of common aims, values and purposes that underpin the school curriculum and the work of schools.


Three statutory curriculums aim that children become successful, confident and responsible people.


The curriculum reflects values in our society and these underpin the work that schools do.


The statutory curriculum should establish an entitlement for all children and promote high standards.

The purposes of having a statutory curriculum are:

• to establish an entitlement for all children, regardless of social background, culture, race, gender, differences in ability and disabilities, to develop and apply the knowledge, skills and understanding that will help them become successful learners, confident individuals and responsible citizens

• to establish national standards for children’s performance that can be shared with children, parents, teachers, governors and the public

• to promote continuity and coherence, allowing children to move smoothly between schools and phases of education and providing a foundation for lifelong learning

• To promote public understanding, building confidence in the work of schools and in the quality of compulsory education.

In particular, the curriculum should:

• promote high standards, particularly in literacy, innumeracy and ICT capability

• provide continued entitlement from early years to a coherent, broad and balanced curriculum

• instill in children a positive disposition to learning and a commitment to learn

• promote and pass on essential knowledge, skills and understanding valued by society to the next generation

• be relevant to children and prepare them for the here and now, for the next phase of their education, and for their future

• widen horizons and raise aspirations about the world of work and further and higher education

• make children more aware of, and engaged with, their local, national and international communities

• help children recognize that personal development is essential to wellbeing and success.

There is no denying the fact that there are some events in the past history of Bangladesh which play a vital role to identify as a nation of separate entity bedded on heroic deeds of Bengali nationalism. It is the magnificence news of the peoples of the entire world that the real heroes of freedom have laid down their lives for the sake of the self-esteem of mother tongue which is exceptional in view of languages of the world. The blood-shed history for the dignity of mother tongue is the first and foremost event in Bangladesh which needs to be memorized decades after decades. We express our gratitude for their treasured contribution of Bengali language when on earth predominantly the 21st February is carried out as an International Mother Tongue day each year with honor and high stature. The 21st February is a red-letter day in the history of our mother tongue which is also a very remarkable day in the sense that we have been able to establish our mother tongue as our state language. It is our glory and brainwave that we have realized sovereignty from the movement of this day. We believe that we could not accomplish our freedom if 21st February was not embryonic in 1952. Due to the movement of this day, we have shown our demonstration against the rulers of the then Pakistan and a such suffice it to say that the 21st February, as a symbol of blaze elucidation is our rectitude for which our survival as Bengali nation has been point toward through out the whole world. In this day some young persons of our country have volunteered to create resistance against the conspiracy of our mother tongue. They have stepped up the movement by degrees and being uncontroversial, the then rulers have marched into them and in due course they had shot them dead. This is such a faction where our heroes have laid down their lives for the cause of prominent deportment of our mother tongue. In the entire world, such incomparable movement has never been taken place in the whole world.

In view of the above it is evident that in learning and experiencing language based on education, there exists particular ways in which language is the means by which theological meaning has powerful impact on human behavior and culture. The very existence of language is proof that a human lives in relation to others. The words emitted from the mouth are not merely pictures of the world, but in fact words are part of the world. They constitute the realities that indicate human endeavors. For example, building, traveling, playing, and fighting are human practices that require a mutual understanding of rules between participants. And even a religious life could not be practiced alone. After all, it is evident that anyone can be self-critical, but faith requires an acknowledgment and confession of sins to those who have injured and to those sweethearts to us. Wittgenstein, a world class philosopher established the inadequacy of language understood in modern terms as representational.

In view of the above it is evident that disputing the notion that language is private knowledge preventing the speaker from relevant action, he has promulgated that language as the means to go on in meaningful relation to others. Like a city that we learn to navigate, the grammar of language indicates how to understand the thoughts as well as convictions of others and how we relate to them. Language games, identifies that the reality of life with others may be reflected as foundation of language which lies in a “depth of understanding, interdependence, and shared practice”.

Kh. Atiar Rahman is a prolific author as well as a poet who has started writing articles, poems, stories and novel from his school life. He was a brilliant student. His main theme of writing is bedded on Literature, soil, nature, science and history. He was born in the former district of Kushtia.

Marked and Unmarked Terms in the English Language


In binary oppositions:

Marked and unmarked terms are frequently being used in binary oppositions. It means that a term is not equal in its weight, but the one (unmarked) is neutral or more positive in contrast to the other term. As Geoffery leech observes, where there is a contrast between two or more terms, tenses or cases, one of them is marked if it have some extra ‘affix’ in contrast to the unmarked one which does not contain any marker. For example the cat is an unmarked and neutral term while cats is marked with a suffix -s, similarly actor is an unmarked term while actress is a marked term with an affix -ess, also polite is a positive term in contrast to its negative term ‘impolite’. In general the plural of nouns in English language are marked term (books) in comparison to the singular (book). In French language the feminine is generally marked and the masculine is unmarked term for instance petit in contrast to petite; however, in English if sex is marked it is done lexically.i.e. by giving special words to one sex and none for the other one, for example word duck is a female term which is unmarked while maleness is marked by drake which is absent in duck and this word gives services for the whole specie. Moreover in the pronouns opposite marking is being observed, that is male as an unmarked term and female term as marked one. For example,

One in HIS senses would not do a thing like that (unmarked)

One in HER senses would not do a thing like that (marked by femaleness)

It is the male sex who is marked because the first statement could refer to either gender, but the second one will specify it for femaleness.

In polar oppositions:

The same kind of marked/unmarked distinction is observed in polar oppositions as well (having two poles) good/bad, rich/poor, day/night, low/high, short/long and we prefer to measure things by the mean of length rather than the shortness. We would rather ask how long this cloth, than how short this cloth is, or how high this building is instead of how low this building is. Because the former will give a neutral expression which mean it could be long or short, while in latter we are left with only one possibility of being short. It does not only rely on the scale of measurement but can also be used in such cases,

How WELL does she speak French? Very poorly

How BADLY does she speak French? Like a native

The first statement is neutral and different from the second one which is marked in this context thus the answer is completely different.

Markedness can be defined as the relationship between the form and meaning. If there is a contrast of two different forms on a single dimension the unmarked one would be neutral one and could be applied on the whole dimension rather than a specific aspect of it. It could be argued that this phenomenon is due the negative-positive inherent to the semantic opposition itself. Normally the unmarked one is considered positive while the marked one is taken a negative term for instance, happy/unhappy, complete/incomplete, stable/unstable; however, in some cases there is an invisible element of negation, like it is easy to define dead by not alive than alive by not dead.

Polyyanna hypothesis:

The detailed explanation of markedness is given on the basis of psychological or experiential ground for which some psycholinguists have given a so called hypothesis called “Pollyanna hypothesis” according to which people tend to think more positively towards life and pay more heed to brighter side of life which provides an argument for associating good with ‘unmarked’ terms and bad with ‘marked’ suffixes and prefixes.

In relative opposition:

There is also a chance of bias in relative oppositions but it is better to call this ‘dominance’ instead of ‘markedness’ for instance in parent/child, front/behind, right/wrong the first term seems to be more dominant than the other one, thus we prefer to place the dominant term before (parent-child) or maybe giving one name to both terms using dominant one (ownership). Markedness and dominance seems to have variation in strength but it deeply depends upon the psychological basis. There is no logical significance in giving symbols to these terms of oppositions. The distinction between ‘dead’ and ‘alive could be given equal logical explanation as +dead/-dead as by -live/+dead because both of these are logically equivalent. This shows that the unmarked term has gained the discrimination of + and upward arrow while the dominant term of a opposition has gained the right arrow.
But the distinguishing term for the marked term is never omitted and the neutralization of the opposition is still indicated (oparent, oright, ogood etc)

Ruth Kempson rule:

To account for lexical ambiguities due to markedness Ruth has given a rule. For this rule we can take dog and bitch as an example.

If a) there are two words W1 and W2 having meanings m1 and m2, and m1 differs from m2 only in having an extra feature -X

And if b) there is no word like W3 with meaning m3 and m2 differs m3 in having an extra feature of +X

It means that m3 is an additional meaning of W1. (m2 and m3 are co-hyponyms of m3 and thus W1 is an unmarked term). This rule accounts for all the ambiguities having first term as more general containing an extra feature while the second one as more specific one. There is also an explanation for other type of ambiguities, such as it is a tautology to say that a calf is a young cow, but on the other hand it is not the tautology to say that this is a cow not a calf. This is how ambiguity through same words is created. There can also be some of the hierarchical structures for the same word.

Reasons to Learn German – Why Learn German?

Germany’s economy has been growing rapidly, reaching far beyond the Rhine. Such growth has motivated the country into one of the world’s strongest economies. It’s in the top 10 for import and export partners for both the U.S. and the U.K.

If you’re thinking about learning a new language, German Language is the excellent choice to learn. German is the most common native language. It is spoken by more than 120 million people in the world and it’s an official language in Austria, Switzerland, Liechtenstein, and Belgium. It’s a recognized minority language in Russia, Kazakhstan, Namibia and the southern tip of Brazil.

Considerable German-speaking minorities are also found in Australia, South Africa, Bolivia, Paraguay, and Argentina.

Below are the five most interesting reasons why German should take pride of place on your skills list.

Reasons to Learn the German Language
Knowing German creates business opportunities.

Germany’s economic strength increases business opportunities.Multinational business opportunities exist in the European Union and in the Eastern European countries, After Russian German is the second most spoken language throughout the world.Top MNC Companies like BMW, Daimler, SAP, Infineon, Siemens, Bosch, BASF, Lufthansa and many others companies need international partners. The Japanese, who have the 2nd most powerful economy in the world, understand the business advantages that a knowledge of German will bring them: 68% of Japanese students study German.

If you’re looking for employment in the US, knowing German can give you good advantages. Most of the top Companies in the US could choose the employees with German literacy over equally qualified candidates.

Germany is an economic powerhouse
German is not only an interesting option for academics but also those in the business world. Germany is the biggest economy in European and the fourth largest Worldwide. It is home to numerous international corporations and on the front line of new Technologies.

German is the Most Spoken Native Language in EU

Worldwide, German is the eleventh most widely spoken language. Considering that Germany is a country with a population of about 90 million people and 95 million people speak German as a first language, meaning there are around as many speakers of German as a second language as there are native speakers. Talking of Native speakers German is the native language of Germany, Austria and the decent chunk of Switzerland. Many people in Eastern Europe choose to study German as a second language. Outside the European Union, it’s the third most taught Foreign Language. As a result, German is the largest number of native speakers in the European Union.

German is the gateway to a world-class higher education

In the year 2011, Germany was the fourth most known destination for all students from abroad with more than a quarter million foreigners being enrolled in German Universities. One of the top reason why German has such a high standing in the science Community is the fact that German’s Universities have an excellent International reputation.

The German structure for higher education increases a number of universities with a very low or no tuition fee. No wonder scholars and researchers are assembling there! Learning German to save on student debt sounds like a pretty good ROI. You Can Attend University in Germany for $0.00

Who says you have to spend a money to get a decent education?

In Germany? Tuition is free at most universities. Some institutions charge around €500 a semester. And there’s no sacrifice in quality that comes with that. Many German universities are in the world’s top 100.

German has an enormous cultural heritage

There’s no getting away from the fact that a large percentage of the world’s most impressive achievements were first to come up within the German language.

Musically, German can lay claim to most of the classical greats, including Beethoven, Bach, Handel, Mozart, and Brahms. Everything from music to science and literature to opera has intense roots in this rich and flexible language. On the science front, there’s the top famous scientist ever to live, Einstein, but also contemporary scientists who are making huge contributions to our current lives. For example, a German was recently responsible for discovering the newest elements on the Periodic Table.

German is a very distinctive language

There are many languages that have Germanic roots, none are quite as distinctive as German itself. One of the main reasons for this is the language’s common use of extremely long compound words. Today, the length of these is ‘rindfleischetikettierungsüberwachungsaufgabenübertragungsgesetz’, an imposing 64 characters long – which translates into English as “beef labelling supervision duty assignment law”. Try slipping that into the conversation! German is a living language, and as one of the only languages in the world that authorize such outrageously lengthy compound words.

English – A Language Close to My Heart

English has always been a subject close to my heart! Even though I have no qualms in accepting that I never scored above 60% in the language in school.

There is a funny notion in our heads; rather an assumption: if one speaks flawless English (any other language for that matter), we conveniently assume that the person would have scored really well on that language in school as well. This is rather a shock for people who don’t realize that speaking and writing skills may not necessarily be at the same level for an individual.

I know of people who struggle to articulate simple thoughts in the language; however, they exceed expectations when you ask them to pen those thoughts down.

More often than not, I believe that your schooling plays a crucial role in your comfort with a certain language. For example: In my case, the only exposure I ever got in English was at school. Just that much exposure was enough for me to be fluent in the language.

But then again, how about the grammatical knowledge of the language? I was fluent and had no issues whatsoever in expressing or articulating my thoughts comfortably. However, after I stopped studying grammar in school; I lost touch with theoretical knowledge of English grammar and; therefore, was unable to identify various parts of speech in a sentence.

Once I entered college, I still wasn’t sure what I wanted to do with my life. I was quite unsure of the career that I would like to like to opt for. After completing my graduation I worked in a few places just to gather experience – again without any clarity of where I would want to go in terms of my career. In of these random jobs I came across a person who identified the skill of being a good English Language Trainer in me. He advised me to make my career in training. His judgement only came from the knowledge that I could speak well. I took his advice and started applying for jobs where my language skills could be used. Finally, I landed a job as a Voice Coach. During the interview they specifically asked me if I was comfortable with theoretical aspect of English Language – to which my answer was a definite ‘NO’.

I thought I would not get a call from the firm. However, they called me a few days later saying that they like my confidence and command on spoken English; however, grammar is something that I will have to work on. After I joined I had to undergo a Train the Trainer session with experienced trainers – This session helped me to understand the nuances of grammar.

This is when I asked myself – ‘Is it really important for one to learn English Grammar to be able to speak well?’. My answer was – ‘NO’. However, if one wishes to make a career in the field, then certainly along with spoken fluency, one must also get acquainted with the foundation of the language. Theoretical knowledge of grammar helps trainers to offer explanations to students when they get stuck. It facilitates quick and logical learning in students.

Latin: Reasons Why It Will Survive Another 100 Years

The dead language- Latin is a classical language belonging to the Italic branch of the Indo-European languages (Wikipedia). It’s the official language of the Holy See, the working language of the Roman Rota and its public journal the Acta Apostolicae Sedis. Latin has transcended throughout history to become the most influential language birthing several major ones like Spanish, Portuguese, Italian, French and English. Latin is immune from frequent changes experienced by other living languages whose word’s meaning are manipulated and distorted. It has metamorphosed through centuries from the; Old- Classical-Vulgar-Medieval-Renaissance-New and to the recent Contemporary Latin. From famous quotes& phrases, science & amp; arts, law and literature, Latin has had profound influence throughout history of mankind, ensuring its survival to the next millennia.

From countries, institutions, military organizations, movies, and mass media, the language influence is felt across all sectors in our society. (Portus cale)- warm harbor in Latin is where Portugal derived its name; similarly Egypt is from (Aegyptus)- meaning the land below Aegean Sea in Latin, and Switzerland Latin name is (Confoederatio Helvetica )and adopting its short form Helvetia on her coins& stamps. (A mari usque ad mare)- From sea to sea is Canada’s official motto, with U.S state of Missouri adopting (Salus populi suprema lex esto) – the health of the people should be the highest law as its state motto, and state of West Virginia being (Montani semper liberi) – Mountaineers are always free. The Royal Air Force of Britain( Per ardua ad astra)- Through adversity/struggle to the stars is its official motto, United States Marine Corp adopting the phrase (Semper fidelis)- Always faithful, and Harvard University’s (Veritas) meaning truth who was a goddess of truth, daughter of Saturn & mother of virtue. Also Movies like the award winning Passion of Christ infusing it for a more realistic feel, with films having a Latin sub-title and websites, T.V & Radio shows & magazines done entirely on the language.

(Ad impossibilia nemo tenetur)- No one is obligated to perform the impossible, one of the Latin terms you’ll probably hear lawyers use before a judge and law students must master. When the Roman Empire fell, the conquered regions under her were already accustomed to their laws, language & culture and hence continued using it. In forming their own set of laws in resolving disputes, Latin became the language of choice for those studying law and became the basis on which it was practiced. With the advent of major languages like English, Spanish, & French, Latin was less used and eventually scrapped but remained heavily used in law schools and lawyers for their terms and phrases. Some of the common Latin terms that comes to mind like; (Actus non facit reum nisi mens sit rea)- The act does not make a person guilty unless the mind is also guilty, commonly used in defense of an accused, (Actore non probante, reus absolvitur)- When the plaintiff does not prove his case, the defendant is absolved, a term denoting the burden of proof is upon the plaintiff, (Animus confidenti)- Intention to confess, (Amicus Curiae)- A friend of the court, and (Aberratio ictus)- Mistake in the blow meaning an error where the wrong person gets hurt. Whichever country you’re from, you’ll never miss the use of such terms in law, highlighting its continued effect and importance on the legal profession worldwide.

Science continues to borrow heavily from Latin especially in coining new words for the International scientific vocabulary (I.S.F) – It comprises of scientific & specialized words whose language of origin may or may not be certain, but which are in current use in several modern languages (Wikipedia). Its Translingual meaning Latin cuts across modern languages worldwide from; English, Russian, French, Swedish to Japanese, Thai, Kiswahili and Hebrew. Hence a word like Femur- Thigh bone remains the same when used in any modern language, and it’s interoperable. Binomial nomenclature used by scientists in naming of plants & animals uses Latin with the main intention of helping those clueless about Classical languages to better understand and remember such scientific names like; Apis mellifera- Honey Bee. There’re myriad of common Latin names & terms you’ll bound to encounter like; Tibia- Shin bone, Fibula- Leg bone, Fetus- Foetus (Unborn Baby), Citrus aurantium- Bitter orange, Eubalaena Autris– Southern Right Whale, and Eptesicus Brasiliensis – Brazilian Brown Bat.

Veni vidi vici– “I came, I saw, I conquered” Julius Caesar wrote to Amantius in Rome after a decisive victory against Pharnaces II of Pontus during the Battle of Zela fought on August 2, 47 BC in Zile, present day Turkey with historians, philosophers interpreting the phrase to mean anything is achievable if we’re focused and determined on any objective we make. After Pharnaces defeated one of Caesars’s legate at the Battle of Nicopolis, he committed atrocious acts against the captured soldiers and Roman civilians. When Caesar heard about it he declared war against Pharnaces and met him in Zile, a small hilltop in Northern Turkey, he made a surprise attack against Caesar as he was pitching tent on the hilltop creating confusion among his troops and gained ground. But Caesar’s legionnaires quickly regrouped, organized themselves and went on the offensive routing Pharnaces army of approximately 20,000 against Caesar’s 10,960. Denoting Caesar’s clear purpose of defeating Pharnaces, despite suffering early setbacks he remained focused by quickly regrouping, organizing the troops and achieved his goal by defeating Pharnaces. Such Latin phrases are still an inspiration, applicable & relevant to this present day with institutions, individuals, Governments worldwide using them as their motto for motivation. Like; Appius Claudius Caecus’ phrase- Faber est suae quisque fortunae( Every man is the artisan of his fortune), Petronius- Manus manum lavat( The favor for a favor or One hand washes the other), Horace-Vitanda est improba siren desidia( One must avoid that wicked temptress, Laziness) and Caesar’s timeless phrase- Alea iacta est (The die has been cast).

Despite being an Ancient language, it has survived for centuries to birth several modern major languages e.g. English, French& Spanish, scientists worldwide still use it in naming of living organisms, and it’s the official language of the Holy See, head of a billion Catholics worldwide. Latin is still applicable & relevant in these modern times by forming a core part in the practice of legal profession worldwide and inspiring us politically, socially & economically with its timeless quotes, phrases & sayings.

That’s why it will survive for another hundred years.

How to Pass the IELTS

Start your IELTS Pass journey

1. Start studying English – this will help you learn the fundamentals needed for the IELTS test. You cannot go into the exam with a low level of English but a broad knowledge of the test – you will need a combination of these two in order to score highly.

2. Learn about the IELTS test – Study the test-taking strategies and skills needed before going into the exam. That way you will know how to best use your time and can plan accordingly. There are plenty of infographics and blog posts online which will help you with this.

3. Watch films or read books – get familiar with both the pace and cadence of English speech and written texts in order to sound as natural as possible in your Writing and Speaking. These skills will help you pass at least 50% of your exam. Once you are familiar with these you can move onto Listening (which can also be practised through watching films) and then finally Reading which can be practised through reading the news or reading both fiction and non-fiction.

4. Study IELTS skills and strategies needed – these skills and strategies are integral to being able to monitor and use your time effectively as well as being able to predict what is coming next, how to answer a long Reading or difficult Listening passage.

5. Take IELTS practice tests included to perfect your new test-taking skills and strategies – take the official IELTS practice tests once you have learnt all the skills needed.

6. Learn from your mistakes and improve your English – learn from the feedback given by your IELTS teacher/trainer and learn from where you can improve.

7. Take the IELTS and get your perfect score – take the IELTS after you have studied the test-taking skills and strategies and get your perfect score!

A quick summary:

Learn the format of the productive skills and learn the strategies needed

You need to know how to write various letter/essay types, understand what you need to achieve in each Speaking part including introducing yourself, speaking on a topic for a set period of time and how to give yourself time to think and know how to describe graphs in the Academic Writing.


Once you understand these skills and how to use them, you need to practise them in controlled tests and learn from your mistakes. You will need an experienced IELTS trainer or someone familiar with how to get a high score on the IELTS and with a high, if not native, level of English.

Get Feedback

Feedback on these productive skills is vital. You need to learn from what you did well and what you can improve. Speak to a teacher or examiner and keep trying!

Visit IELTS Pass today and take one of our best-selling courses, study test-taking skills, speak to qualified IELTS trainers and download free books and resources at https://www.ieltspass.com