Proportionality Theorems

6 Key excerpts on "Proportionality Theorems"

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  eBook - ePub

  Big Ideas in Mathematics

  Yearbook 2019, Association of Mathematics Educators

  • Tin Lam Toh, Joseph B W Yeo(Authors)
  • 2019(Publication Date)
  • WSPC

    (Publisher)

  At the secondary level, proportionality is applied in topics across all three content strands. The students make use of proportion when they read scales on a Cartesian plane and find the gradient of a linear function under Number and Algebra. Then under Measurement and Geometry, they encounter proportionality in similarity of figures, scale drawing, arc length and area of a sector of a circle, and basic trigonometric ratios. Proportionality is also manifested in Statistics and Probability when the students learn statistical graphs such as pie chart, bar chart and histogram with equal class width, as well as in computing the probability of an event happening.

  As the previous two paragraphs make clear, the notion of proportionality is a major idea in the learning of mathematics, with wide applications in several mathematics topics across different content strands. It is therefore not surprising to see proportionality being regarded as a big idea in mathematics education.

  4Teaching towards Proportionality

  4.1Determining proportionality

  Secondary school students in Singapore are generally competent in solving word problems on proportion. Consider the 2013 GCE O Level question which stated that the frequency of a note produced by a string is proportional to the square root of the string’s tension. The question also provided the information that the string produces a note with a frequency of 360 Hertz when the tension is 64 Newtons. Students were asked to find an equation connecting the frequency and tension first, then use it to compute the tension in the string when a note of 540 Hertz is produced. The examiners’ report indicated that many students were able to correctly write the relationship between the variables and answer the computation question well (Cambridge International Examinations, 2014). A research study on proportional reasoning involving 14-year-old students in Singapore has reported similar findings as well (Ramakrishnan, 2016). Despite the students’ success in procedural fluency, do they really have a good grasp of proportionality? In Ramakrishnan’s study, the students were found to correctly describe two variables that were in an inverse proportion relationship in one question, but in a different question, not many were able to identify the graphical representation of two variables in an inverse proportion relationship. It can thus be suggested that many students, although adept in computing a quantity in a proportion question, still somewhat cannot reason and figure out the graphical representation of inverse proportion relationship. In a separate study on high school students’ proportional reasoning by McLaughlin (2003), the students were found to be competent in solving most of the proportion problems but they did not understand what they were doing. It is therefore reasonable to require students to gain a good grounding in proportionality when it is introduced at the lower secondary level. Otherwise, they may face obstacles in understanding mathematics taught at a higher level (Langrall & Swafford, 2000).

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  eBook - ePub

  Teaching Fractions and Ratios for Understanding

  Essential Content Knowledge and Instructional Strategies for Teachers

  • Susan J. Lamon(Author)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  Proportional reasoning is one of the best indicators that a student has attained understanding of rational numbers and related multiplicative concepts. While, on one hand, it is a measure of one’s understanding of elementary mathematical ideas, it is, on the other, part of the foundation for more complex concepts. For this reason, I find it useful to distinguish proportional reasoning from the larger, more encompassing, concept of proportionality. Proportionality plays a role in applications dominated by physical principles—topics such as mechanical advantage, force, the physics of lenses, the physics of sound, just to name a few. Proportional reasoning, as this book uses the term, is a prerequisite for understanding contexts and applications based on proportionality.

  Clearly, many people who have not developed their proportional reasoning ability have been able to compensate by using rules in algebra, geometry, and trigonometry courses, but, in the end, the rules are a poor substitute for sense-making. They are unprepared for real applications in statistics, biology, geography, or physics, where important, foundational principles rely on proportionality. This is unfortunate at a time when an ever-increasing number of professions rely on mathematics directly or use mathematical modeling to increase efficiency, to save lives, to save money, or to make important decisions.

  For the purposes of this book, proportional reasoning will refer to the ability to scale up and down in appropriate situations and to supply justifications for assertions made about relationships in situations involving simple direct proportions and inverse proportions. In colloquial terms, proportional reasoning is reasoning up and down in situations in which there exists an invariant (constant or unchanging) relationship between two quantities that are linked and varying together. As the word reasoning implies, it requires argumentation and explanation beyond the use of symbols

  a b

  =

  c d

  ⋅

  In this chapter, we will examine some problems to get a sense of what it means to reason proportionally. We will also look at a framework that was used to facilitate proportional reasoning in four-year longitudinal studies with children from the time they began fraction instruction in grade 3 until they finished grade 6.

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  eBook - ePub

  A First Course in Geometry

  • Edward T Walsh(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  similar.

       Eudoxus’ theory of proportion provides the tools we will use to explore and refine the similarity relation. The notion of similarity will then lead us to a proof of what is known as the Pythagorean Theorem. How ironic that the discovery that was most disturbing to Pythagoras should lead to a deft verification of the theorem for which he is most honored !

       Applications of the ideas developed in this chapter are incorporated into the scale models used by industrial designers, the process of photographic enlargement, and a host of other activities.

  6.2  RATIO AND PROPORTION

       Before proceeding with our development of the geometric notion of similarity, we pause for a brief review of some of the facts about ratio and proportion that you studied in arithmetic and algebra.

       DEFINITION 6.1 Let x and y be real numbers, y ≠ 0. The ratio of x to y is the number

       Example 1 Let x = 3 and y = 4. Then the ratio of x to y is and the ratio of y to x is .

       DEFINITION 6.2 Any list of numbers such that it can be determined which number is first, second, third, and so on, is called a sequence.

       Example 2 The elements of the set of natural numbers: 1, 2, 3, 4, . . . may be considered as a sequence.

       Example 3 Given ΔABC and ΔDEF, the lists

                           AB, BC, AC

  and

                           DE, EF, DF

       are both sequences.

       You should notice that a sequence may be infinite, as in Example 2 , or it may be finite as in Example 3 .

       Example 4 Suppose we are given the two sequences

                           3, 4, 5

       and

                           12, 16, 20

       Each member of the second sequence is four times the corresponding entry in the first sequence, or equivalently,

       We also might write

       Such sequences are said to be proportional.

       DEFINITION 6.3 Let a1 , a2 , a3 , . . . and b1 b2 , b3 , . . . be sequences of positive numbers. If

  then the sequences are proportional. Any such statement of equality is called a proportion

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  The Metaphysics of the Pythagorean Theorem

  Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos out of Right Triangles

  • Robert Hahn(Author)
  • 2017(Publication Date)
  • SUNY Press

    (Publisher)

  However, the route that led Thales to this insight may be, I will argue in the next chapter, that the grasping of a continued proportion of three straight lines is part of understanding the cosmos as unfolding or growing (or collapsing) in a regular and orderly way; the discovery of this pattern, this order, had as its visual proof the areal equivalences created by the line lengths—the rectangle and square. Connectedly, this is how Thales came to discover the areal relation among the sides of a right triangle when he searched further to discover the relation among the sides of the two similar right triangles into which the perpendicular partitioned the hypotenuse. The right triangle contains within itself these dissectible relations.

  Figure 1.47.

  The idea of the geometrical mean in a continuous proportion—a relation among three proportional line lengths, and consequently areas of figures constructed on them—is a principle of organic growth. It is my thesis, to be developed in the next chapter, that the discovery of the mean proportional—and hence continuous proportion—provided an answer to a question that looms behind Thales’s doctrine that everything is ὕδωρ, how the cosmos grows, how the right triangle expands into different shapes, and hence different appearances.

  In Euclid, the idea that we know mathematically as “squaring”—which is a term of geometrical algebra, and not part of Euclid’s technical parlance, nor the parlance of the early Greek mathematicians or philosophers—is presented as “in duplicate ratio” (ἐν διπλασίονι λόγῳ); the expression is employed to describe a geometrical fact about the relation between side lengths and the areas of figures drawn on them, not a process or operation to be performed. Let us try to clarify this ancient Greek idea. A duplicate ratio is not a geometric version of an operation, algebraic or otherwise. It refers to a ratio obtained from a certain proportion: it refers to a state of affairs rather than a procedure. Thus, if you have a proportion with three terms, A, B, and C so that B is the middle term (that is, A : B :: B : C), then A : C is the duplicate ratio of A : B. The chief geometrical fact is that if two rectilinear figures, for example triangles, are similar, then the ratio of their areas is the duplicate ratio of their corresponding sides (“is the duplicate ratio” not “is produced by duplicating the ratio”!). Indeed, when we consider the proof of VI.19, below, we begin with similar triangles with corresponding sides BC and EF; next we bring forth a third proportional BG, that is, a line such that BC : EF :: EF : BG; then we show that the ratio of the areas (via VI.15) is the same as BC : BG. For Euclid, then, the duplicate ratio simply is so; it is not produced by an operation on the sides like squaring.19

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  Usefullness of Mathematical Learning

  Usefulness Mathematical Learning

  • Isaac Barrow(Author)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  Euclid; where nothing is trifling, ambiguous, or incomprehensible, but all Things proceed directly, are supported firmly, and the fewest Hypotheses are supposed; and lastly (which is the best Judgment of a most excellent Definition) where Analogies of particular Subjects are derived out of its very Bowels by an immediate and direct Discursus; as will appear to any that consults all its Applications in the Elements. or else where; and particularly the Propositions alledged in the sixth Book.

  But I here note by the Way, as to And, Tacquet’s Method, that, in my Opinion, he would have done far better if he had defined Proportionass by that Mark of Proportionality, which he brings, and in some sort demonstrates; and if by the Help of that Definition he had demonstrated the other Affections, which I judge (as himself professes) he was able to do. For by so doing he would have emulated Euclid and have proceeded in a scientifical Way; he would have avoided most of the Inconveniences we have mentioned, and might have been defended the same Way with Euclid. To which I exhibit this cognate and seemingly easier Affection of Proportionals, on which such Definition depends, viz. Quantities are said to be proportional, when the Antecedents are equally contained respectively in the Equimultiples of the Consequents, or the Consequents in the Equimultiples of the Antecedents; or when Those may be taken just so often from these, or the contrary. From which Definition it would not be difficult to raise the whole Doctrine of Proportionals in a Procedure very like Euclid’s, but in my Opinion by no Means more short and clear. Against which however all may be objected, that has been objected against Euclid’s Method, and perhaps something besides from the Incommensurability of the Quantities, from whence Euclid

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  Gender Justice and Proportionality in India

  Comparative Perspectives

  • Juliette Gregory Duara(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  4    Proportionality Analysis Theoretical Foundations

  I The Origins of Proportionality Analysis

  Proportionality has been perceived as an integral part of justice since antiquity. The Code of Hammurabi with its “grim retaliatory punishments”1 was probably seen to be proportional at the time. For example, a builder whose shoddy construction resulted in the death of the building’s owner was to be put to death as a form of retributive justice. As Charles F. Horne noted, “[w]e can see where the Hebrews learned their law of ‘an eye for an eye’”.2 According to Aharon Barak, the Biblical “‘eye for an eye’ was considered a measured response”.3

  Moving forward in time some millennia, “[b]y 1215 the Magna Carta had already recognized the principle [of proportionality] in writing: ‘For a trivial offense a free man shall be fined only in proportion to the degree of his offense, and for a serious offence correspondingly but not so heavily as to deprive him of his livelihood.’”4 The continuing relevance of proportionality in popular conceptions of justice is evident in the enduring image of Lady Justice holding her scales aloft, as she has from the days of ancient Greece and Rome.

  These examples demonstrate a persistent conceptual connection between proportionality and justice spanning from antiquity to the present. This historic relationship between justice and proportionality serves as a metaphoric foundation for a form of enquiry generally known as Proportionality Analysis (PA), which is a multifaceted method of determining when a statute’s benefit to one right or public good is proportionate to its detriment to another right or public good. In this book PA refers to this distinct, multi-step method of evaluating the merits of the claims made by each side in the presence of a conflict between two rights or between a right and a public good.

  PA as a methodology is a recent development. This chapter commences with a brief history of the emergence of PA before turning to a theoretical exposition of its elements. The chapter then explores and evaluates some of the critiques of PA, before concluding with an argument in favor of India’s adoption of this analytical methodology. While disquisitions on PA can and do consume volumes, the discussion in these pages will be structured around the aspects of PA relevant to a jurisprudence of gender equality amenable to the Indian doctrinal context, as discussed in the previous chapter. One argument developed in this chapter and the following two chapters is that there are elements of PA which remain identifiable as such across jurisdictions, yet that those elements can be and have been adjusted, for example in emphasis or order, to account for local circumstances. In other words, this project seeks to strike a balance between the two potentially conflicting sides of “one of comparative law’s most vigorously contested dilemmas: the question of similarities and differences among legal cultures”.5

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