Ratio - Wikipedia
It is impossible to trace the origin of the concept of ratio (A rational number may be expressed as the quotient of two integers.) Euclid collected the results appearing in the Elements from earlier sources. FIGURE Relationship between PQ and Treatment and Belt size. Historically, the problem is dealt with by a method called ANCOVA, from ANalysis of of X and Y differences that expressed the relationship between two interval- level Sum of Squares in PQ into components resulting from Belt, Treatment, and error. An ordered pair, commonly known as a point, has two components which are the x and y coordinates. This is an Main Ideas and Ways How to Write or Represent Relations. As long as the We can show it in a table, plot it on the xy- axis, and express it using a mapping diagram. Relation in . But there's a little problem.
Second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.
- Research problem and the most frequent mistakes in forming it
The first two definitions say that a part of a quantity is another quantity that "measures" it and conversely, a multiple of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a part of a quantity meaning aliquot part is a part that, when multiplied by an integer greater than one, gives the quantity.
Euclid does not define the term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity measures the second. Note that these definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII. Definition 3 describes what a ratio is in a general way.
It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists when there is a multiple of each that exceeds the other.
This condition is known as the Archimedes property. Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but Euclid did not accept the existence of the quotients of incommensurate, so such a definition would have been meaningless to him. Thus, a more subtle definition is needed where quantities involved are not measured directly to one another.
Though it may not be possible to assign a rational value to a ratio, it is possible to compare a ratio with a rational number. Euclid's definition of equality can be stated as that two ratios are equal when they behave identically with respect to being less than, equal to, or greater than any rational number.
In modern notation this says that given quantities p, q, r and s, then p: There is a remarkable similarity between this definition and the theory of Dedekind cuts used in the modern definition of irrational numbers. Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5.
In modern notation it says that given quantities p, q, r and s, then p: As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms p, q and r to be in proportion when p: This is extended to 4 terms p, q, r and s as p: Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions.
Equations for proportional relationships (video) | Khan Academy
The cause can be found only in things which are absolutely clear — when we know it from the outside but not because of data. Causal relationships can be verified by experiments. What is the dependency between the result of an entrance test and the final result of studies at university? What is the relationship between child drug addiction and the socio-economic status of the family?
Experiment with two groups of persons is carried out. Statistics — finding the significance of differences: Is nondirective educational style more effective in creating positive opinions of students on a teacher than a directive one? Most research topics allow us to form research problems of all three types.
What kind of praise do teachers use? Scientific hypotheses may be formed only for relative and causal research problems. Since hypothesis is defined as a statement about a relationship between two variables. The easiest one time, means is descriptive research problem.
Then it is followed by relative and only then by causal. On the other hand, causal problem is of the greatest value in pedagogical theory. It is then followed by relative and only then by descriptive problem. Variables A variable is a phenomenon, quality, condition or agent which is being explored; e.
It is a unit of exploration which can get different values which must be defined. For instance, gender is of two values male — female ; for our purposes, marital statues can be only of two values single — marriedat a different time, however, it can be of four values divorced, widowedstill at a different place even more separated. Variables must be turned into operations — defined operatively — so that they could be measured, found and observed. For instance, the ability to speak a foreign language can be defined as a result of a known test, interest in history as a number of questions asked by a student in a history lesson, the number of books on history read by a student or a membership in a history interest club.
It is advisable to differentiate between two main kinds of variables; it is important also for statistical data assessment. With measurable variables, it is possible to define number, degree of a certain phenomenon or quality. Variables get values within a certain range between — worse; more — less; sooner — later. Categorical variables cannot be quantified; they only can be divided into classes, categories.
They can be dichotomy variables sex: Independent variable — it is a cause of a change in the other variable. Dependent variable — it is the one which changes being affected by a different variable.
It is dependent on the one which affects it independent variable. Hypotheses The research problem forms the basic focus of the research; however, it does not communicate sufficient information to direct the research.
Writing proportional equations from tables
Therefore, hypotheses are needed as they are more specific. Hypotheses divide the research problem into smaller parts, they control the whole structure of the research, and they are validated or invalidated.
H2 Elementary school students have lesser knowledge in a subject which is taught by a teacher with the non-directive teaching style than in a subject taught by a teacher with the directive style.
Hypothesis is a scientific presupposition; it is drawn from a theory which requires a lot of reading and thinking. It is thus no any presupposition. Rarely is it based on personal experience and general knowledge; this happens only when nothing has been known in such matter so far and the paper presents the first action concerning the issue Hypotheses broaden our knowledge and learning — they test parts of a theory empirically.
On the basis of new findings, theories can be broadened or modified. Research is time- effort- and money-consuming. Therefore, it is important to define such hypotheses which are worth the effort, i. The most valuable are creative hypotheses which are able to advance our knowledge. Hypothesis controls the research; therefore, it is not possible to start with collecting data and form hypotheses only during collecting or even when it is finished!
Of course, there is a kind of research — descriptive, exploration — which is absolutely correct and still it does not relate variables to anything and thus it does not work with hypotheses as they were described above.
Such research is used mainly when the theory about the given issue is poor and there is nothing to stick to, the problem is only being mapped and own theory is being created. In such a case, hypotheses are formed only after making the theory a system. Forming hypotheses To form hypotheses correctly, it is necessary to carefully follow three basic requirements rules for formulating hypotheses violating of the rules is the most frequent cause of mistakes: Hypotheses are statements and they must be formulated as declarative sentences.
They must not be confused with research question problem.
Hypotheses must express a relationship of at least two variables. Such relationship between two phenomena must be clearly and explicitly expressed. It is good to compare and verify variables: Hypothesis must have the ability to be tested.How to SUPER CLEAN your Engine Bay
It must be possible to validate or invalidate a hypothesis. Variables must have the ability to be measured or classified age: Schoolchildren in the second grade like school education more than students schoolchildren in the fifth grade.
The more the teacher praises the students, the more the students learn. The more cohesive the group is, the bigger is its influence on individual members.
Correct examples the hypotheses expresses the relationship of two variables: Girls perform better in language tests than boys. Two levels of one variable — sex Authoritative style of raising children develops creativity less than democratic style.
Authoritative and creativity — would be measured with a test The most frequent mistakes in forming hypotheses: Formulation is too complicated and long.