Proportionality Theorems

5 Key excerpts on "Proportionality Theorems"

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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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  A Companion to Ancient Philosophy

  • Mary Louise Gill, Pierre Pellegrin, Mary Louise Gill, Pierre Pellegrin, Pierre Pellegrin, Mary Louise Gill, Pierre Pellegrin(Authors)
  • 2009(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  And the derivation of the laws of proportion from the definition is in a class with Peano’s reduction of classical mathematics to number theory or Frege’s reduction of number theory to logic. But after Book V the definition is invoked only once, in the first proposition of Book VI; otherwise all that matters are the laws which are applied in the 701 greek mathematics to the time of euclid proof of particular propositions, as in the examples we saw on pages 694 –700. There is no reason to doubt that many of those propositions were first used independently of anything like Book V and perhaps not proved on the basis of any theory of proportion at all. A definition of proportionality in Aristotle The first proposition of Book VI says: VI,1 Triangles and parallelograms which are under the same height are to one another as their bases. Euclid derives the parallelogram case from the triangle case, using the fact (I,41) that a parallelogram with the same base and height as a triangle is twice its size. For the triangles Euclid imagines ACB and ACD arranged as in Figure 35.14 with BCD extended in either direction. One multiplies, e.g., CD by marking off segments DK, KL, etc. on the extension BDL of BD, and because triangles with equal bases and heights are equal (I,38) one multiplies triangle ACD by connecting KA, LA, etc. Euclid assumes what he could easily prove, namely that of triangles with the same height the one with the greater base is greater. From this and I,38 it follows that if the multiple of CD is greater than (equal to, less than) the multiple of BC, the multiple of triangle ACD is greater than (equal to, less than) the multiple of triangle ACB. This theorem is of considerable interest because in the Topics (VIII.3, 158b24 –35) Aristotle says that the version for parallelograms is immediately evident when the definition of same ratio is stated, the definition apparently being that a is to b as c is to d if and only if a,b and c,d have the same antanairesis .

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  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  For example, from geometry we have the theorem: The altitude drawn to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse. What does that mean? When the means (the two inside terms) of a proportion are equal, as in the term b is called the mean proportional between a and c. Solving for b, we get 47 The mean proportional b is also called the geometric mean between a and c, because a, b, and c form a geometric progression (a series of numbers in which each term is obtained by multiplying the previous term by the same quantity). ◆◆◆ Example 8: Find the mean proportional between 3 and 12. Solution: From Eq. 47, ◆◆◆ Returning to our triangle, Fig. 17–2, and with our definition of mean proportional, we can interpret the theorem to read, Try to prove this theorem by using similar triangles. We will use the mean proportional in a later chapter, to insert a geometric mean between two given terms. AD h  h DB b   23(12)  6 b   2ac Mean Proportional b 2  ac a : b  b : c  2x (x  3) z  4x 2 (x  3) 2x x  3 2x  z 4x 2 x   13 2 2x  13 5x  10  3x  3 5(x  2)  3(x  1) x  2 3  x  1 5 . C h B A D Section 1 ◆ Ratio and Proportion 495 Applications All the remaining sections in this chapter will use ratio and proportion to solve problems from technology. We will give a few simple applications here. ◆◆◆ Example 9: Gear Ratio: The ratio of the speeds of two gears, Fig. 17–3, is found from the ratio of the number of teeth in each, with the smaller gear always turning faster than the larger. Find the speed N of gear A, having 44 teeth, if gear B has 12 teeth and rotates at Solution: We set up the proportion from which ◆◆◆ ◆◆◆ Example 10: Wire Resistance: The resistance of a wire to the flow of current is proportional to its length. If a 15.0 ft length of wire has a resistance of 0.500 , find the resistance of 12.0 ft of the same wire.

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  Helping Children Learn Mathematics

  • Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  They found that the majority of the children could give correct answers, and that those who struggled did not realise that the problems involved ratio and proportion, did not have sufficient understanding of geometry, or had difficulties working with proportion and geometry or measurement within the problems (p. 76). Avan and Isiksal-Bostan provided a specific example where a child tried to use additive thinking to solve a scaling problem involving length (p. 77), highlighting how some children’s difficulties to progress from additive thinking to multiplicative thinking can impact. Using a zoom in or out on a computer screen or when photocopying both illustrate proportions between the original and its new image. The concepts of ratio and proportion can be naturally connected to geometry problems, as similarity is based on proportions. Two figures are similar if their respective sides are in the same ratio (i.e. proportional). Thus, all squares are similar, but all rectangles are not. Figure 13.9 shows drawings of similar rectangles. Notice that in each of the five rectangles, the ratio between the vertical side and the horizontal size is constant. For example, in the smallest rectangle, the ratio is 2:6 and in the largest rectangle the ratio is 6:18. The rectangles in figure 13.9(b) show that a single line passes through the vertex of each of these rectangles, and this demonstrates that the slope of the diagonals (i.e. the ratio of the vertical height to the horizontal length) of each of these rectangles is the same. This is visible from figures 13.9(a) and 13.9(b). The ratio table shown in figure 13.9(c) shows these lengths and the equivalence of each of the ordered pairs shown. The pattern can be extended in the graphs as well as in the ratio table. An examination of these ratios, such as 2:6, 3:9 or 5:15, confirms that each of them is equivalent, and the resulting gradient is 1 3 .

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  Calculus for The Life Sciences

  • Sebastian J. Schreiber, Karl J. Smith, Wayne M. Getz(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Moreover, for any measurement of length L (e.g., height, arm length, chest circum- ference), surface area S (e.g., palm surface area, cross-sectional area of a muscle), and mass M (e.g., mass of a hair or the entire body), the relationships S ∝ L 2 and M ∝ L 3 continue to hold. To work with proportionality relationships, we need to remember a few basic rules. Essentially these rules have this effect: we can treat a proportionality symbol for manipulative purposes like an equality sign. Rules of Proportionality.  Transitive property: If x ∝ y and y ∝ z , then x ∝ z .  Power-to-root property: If y ∝ x b with b = 0, then x ∝ y 1/b .  General transitive property: If x ∝ y b and y ∝ z c , then x ∝ z bc . Example 4 Rules of proportionality Demonstrate that proportionality satisfies the properties listed in the box above. Solution Transitive property: Since x ∝ y, then there exists a constant a > 0 such that x = ay. Since y ∝ z , then there exists a constant b > 0 such that y = bz . Therefore, x = ay = a(bz ) = (ab)z This equality implies that x ∝ z with proportionality constant ab. Power-to-root property: If y ∝ x b , then there exists a constant a > 0 such that y = ax b . Solving for x in terms of y yields x =  y a  1/b = a −1/b y 1/b Hence, x ∝ y 1/b with proportionality constant a −1/b > 0. General transitive property: This property is really just a simple extension of the transitive property, but it is easily demonstrated directly. If x ∝ y b and y ∝ z c , then there exist a 1 > 0 and a 2 > 0 such that x = a 1 y b and y = a 2 z c . Therefore, x = a 1 (a 2 z c ) b = a 1 a b 2 z bc . Hence, x ∝ z bc with proportionality constant a 1 a b 2 . 42 Chapter 1 Modeling with Functions Example 5 Dangers of getting wet To understand the dangers of getting wet, it is reasonable to assume that the mass, W, of the water on your body of mass M after getting wet is proportional to the surface area, S, of your body. a. For cubical critters find the value of b such that W ∝ M b .

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