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All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning.

The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them. Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics.

According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.

As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.

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The Pythagoreans are credited with the first proof of the Pythagorean theorem , [38] though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers. Though he made no specific technical mathematical discoveries, Aristotle —c. The diagram accompanies Book II, Proposition 5.

Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. Euclid also wrote extensively on other subjects, such as conic sections , optics , spherical geometry , and mechanics, but only half of his writings survive.

Archimedes c. Apollonius of Perga c. AD 90— , a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years.

Following a period of stagnation after Ptolemy, the period between and AD is sometimes referred to as the "Silver Age" of Greek mathematics. His main work was the Arithmetica, a collection of algebraic problems dealing with exact solutions to determinate and indeterminate equations.

He is known for his hexagon theorem and centroid theorem , as well as the Pappus configuration and Pappus graph. His Collection is a major source of knowledge on Greek mathematics as most of it has survived.

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The first woman mathematician recorded by history was Hypatia of Alexandria AD — The New Math method was the topic of one of Tom Lehrer 's most popular parody songs, with his introductory remarks to the song: " The problems can range from simple word problems to problems from international mathematics competitions such as the International Mathematical Olympiad. Problem solving is used as a means to build new mathematical knowledge, typically by building on students' prior understandings.

Recreational mathematics : Mathematical problems that are fun can motivate students to learn mathematics and can increase enjoyment of mathematics. Relational approach: Uses class topics to solve everyday problems and relates the topic to current events. Rote learning : the teaching of mathematical results, definitions and concepts by repetition and memorisation typically without meaning or supported by mathematical reasoning.

A derisory term is drill and kill. In traditional education , rote learning is used to teach multiplication tables , definitions, formulas, and other aspects of mathematics. Content and age levels[ edit ] Different levels of mathematics are taught at different ages and in somewhat different sequences in different countries. Sometimes a class may be taught at an earlier age than typical as a special or honors class. Elementary mathematics in most countries is taught in a similar fashion, though there are differences.

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Most countries tend to cover fewer topics in greater depth than in the United States. Mathematics in most other countries and in a few U. Students in science-oriented curricula typically study differential calculus and trigonometry at age 16—17 and integral calculus , complex numbers , analytic geometry , exponential and logarithmic functions , and infinite series in their final year of secondary school. Probability and statistics may be taught in secondary education classes.

Science and engineering students in colleges and universities may be required to take multivariable calculus , differential equations , and linear algebra.

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Applied mathematics is also used in specific majors; for example, civil engineers may be required to study fluid mechanics , [17] while "math for computer science" might include graph theory , permutation , probability, and proofs.

Standards[ edit ] Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to, realistic for, and considered socially appropriate for their pupils. In modern times, there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In England , for example, standards for mathematics education are set as part of the National Curriculum for England, [19] while Scotland maintains its own educational system.

Ma summarised the research of others who found, based on nationwide data, that students with higher scores on standardised mathematics tests had taken more mathematics courses in high school.

This led some states to require three years of mathematics instead of two. In , they released Curriculum Focal Points, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards are enforced as American states and Canadian provinces choose.

A US state's adoption of the Common Core State Standards in mathematics is at the discretion of the state, and is not mandated by the federal government. The MCTM also offers membership opportunities to teachers and future teachers so they can stay up to date on the changes in math educational standards. Please help rewrite this section from a descriptive, neutral point of view , and remove advice or instruction. April Learn how and when to remove this template message "Robust, useful theories of classroom teaching do not yet exist".

The following results are examples of some of the current findings in the field of mathematics education: Important results [24] One of the strongest results in recent research is that the most important feature in effective teaching is giving students "opportunity to learn". Teachers can set expectations, time, kinds of tasks, questions, acceptable answers, and type of discussions that will influence students' opportunity to learn.

This must involve both skill efficiency and conceptual understanding. Conceptual understanding [24] Two of the most important features of teaching in the promotion of conceptual understanding are attending explicitly to concepts and allowing students to struggle with important mathematics.

Both of these features have been confirmed through a wide variety of studies. Explicit attention to concepts involves making connections between facts, procedures and ideas. This is often seen as one of the strong points in mathematics teaching in East Asian countries, where teachers typically devote about half of their time to making connections.

At the other extreme is the U. Deliberate, productive struggle with mathematical ideas refers to the fact that when students exert effort with important mathematical ideas, even if this struggle initially involves confusion and errors, the end result is greater learning. This has been shown to be true whether the struggle is due to challenging, well-implemented teaching, or due to faulty teaching the students must struggle to make sense of.

Formative assessment [26] Formative assessment is both the best and cheapest way to boost student achievement, student engagement and teacher professional satisfaction. Results surpass those of reducing class size or increasing teachers' content knowledge.

Effective assessment is based on clarifying what students should know, creating appropriate activities to obtain the evidence needed, giving good feedback, encouraging students to take control of their learning and letting students be resources for one another.

Homework [27] Homework which leads students to practice past lessons or prepare future lessons are more effective than those going over today's lesson. Students benefit from feedback.

Students with learning disabilities or low motivation may profit from rewards. For younger children, homework helps simple skills, but not broader measures of achievement. Students with difficulties [27] Students with genuine difficulties unrelated to motivation or past instruction struggle with basic facts , answer impulsively, struggle with mental representations, have poor number sense and have poor short-term memory.

Techniques that have been found productive for helping such students include peer-assisted learning, explicit teaching with visual aids, instruction informed by formative assessment and encouraging students to think aloud. Algebraic reasoning [27] It is important for elementary school children to spend a long time learning to express algebraic properties without symbols before learning algebraic notation. When learning symbols, many students believe letters always represent unknowns and struggle with the concept of variable.

They prefer arithmetic reasoning to algebraic equations for solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns.

Students often have trouble with the minus sign and understand the equals sign to mean "the answer is Quantitative research includes studies that use inferential statistics to answer specific questions, such as whether a certain teaching method gives significantly better results than the status quo.Fast Food Objective: On completion of the lesson the student will know the formal units of measurement for volume and capacity for bigger objects.

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There are no assumptions that the reader already knows a piece of content, and every explanation is very clear and easy to understand. On completion of the lesson the student will be able to understand the probability of simple events. On completion of the lesson the student will be able to use written methods to solve subtraction problems involving 4 and 5 digit whole numbers.

This led some states to require three years of mathematics instead of two. Students will increase their vocabulary and develop their reading skills by reading, in which the present continuous tense is used.

Effective assessment is based on clarifying what students should know, creating appropriate activities to obtain the evidence needed, giving good feedback, encouraging students to take control of their learning and letting students be resources for one another.

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