DYNAMISCHE GEOMETRIE

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1 DYNAMISCHE GEOMETRIE ÄHNLICHKEITSGEOMETRIE & MODELLIERUNG PAUL LIBBRECHT PH WEINGARTEN WS CC-BY VON STAUDT KONSTRUKTIONEN

2 Menü Erinnerung: Strahlensatz Längen, Frame Zielartikel Addition, Subtraktion Multiplikation Kehrwert

3 <latexit sha1_base64="4fsrojko6m3gnvzj0s6spst+go0=">aaacb3icbzc7sgnbfibpxlumt1vlcwedxcrsiqinegnjzwtxbjilze5mkygzf2zmhbbsaeor2fio2pokdr6nk2qltfzhwm93zmhm/f7mmvsw9w0ufhaxllekq6w19y3nlxn7515gisduirgprmvdkniwukcxxwkrfhqhhqdnb3g17jcfqjascu/ukkzugpsh8xnbsqouud/xbsbpzwwmq5khc5sdugb1sty1y1bvmgjngzs3zcjv6jpfnv5ekocginaszdu2y <latexit sha1_base64="xpbufdbs/svezcfqp14+o5ks7s0=">aaacchicbzc7sgnbfibpeo3xtmppmxgkvmfxrg2ejdz2iecaqlke2clsmmt2wsysejztbxwvgwsvwx/bzrdxkmyhit8m/hznhm6c34s5k8qyvo2l5zxvtfxcrnfza3tn19zbv5drigh1smqj0fawpjyf1ffmcdqobcwbx2nlg11p6q0hkislwjs1jqkb4ehifeaw0qhnoq4vmelr9sytlevldf3lpkfjq4oewbiq1lro0di5kuguzs/86vyjkgq0virjktu2fss3xuixwmlw7casxp Strahlensatz Nehmen Sie zwei Geraden d und e, ab ein Punkt O seien noch eine Gerade g und h g seien A = c g, B = d g und A =c h und B =d h Dann haben ein proportionales Verhältnis OA OA 0 = OB OB 0 und AB A 0 B 0 = OB OB 0

4 Längen, Frame Geometrie ohne Koordinate? es geht! Sehr gut! Kongruenzen, Polygone, alles! Aber keine Länge, keine Zahlen Heute wählen wir immer ein Frame willkürlich aber 1cm gross

A GEOMETRIC APPROACH TO DEFINING MULTIPLICATION wir folgen P. McLoughlin Artikel http://arxiv.org/abs/1301.6602 arxiv:1301.6602v6 [math.ho] 14 Oct 2013 PETER F.

5 A GEOMETRIC APPROACH TO DEFINING MULTIPLICATION wir folgen P. McLoughlin Artikel arxiv: v6 [math.ho] 14 Oct 2013 PETER F. MCLOUGHLIN AND MARIA DROUJKOVA The set-theoretic construction of the real numbers in the 1870s marked a shift in emphasis from geometric to algebraic reasoning (see [3]). This shift in reasoning may have inadvertently created some problems in the lower-level mathematics curriculum (in the U.S. at least). Let us clarify what we mean by this. Currently, prospective K-12 math teachers and Science, Technology, Engineering, and Mathematics (STEM) students can graduate from university without ever seeing the following: (1) A rigorous definition of multiplication; (2) A proof that two triangles have the same angles if and only if the lengths of their corresponding sides are proportional; (3) A proof regarding the relationship between multiplication and the area of a rectangle. Part of the problem is that multiplication, the area of a rectangle, and similar triangles are not defined independently of each other in the mathematics curriculum. Here is the definition of similar triangles given in a college algebra text: Two triangles are similar if the corresponding angles are equal and the lengths of the corresponding sides are proportional.. Please note this definition is not independent of multiplication (this definition should actually be a Theorem). As the reader can check, many texts define similar triangles this way. If you ask a STEM student or a prospective K-12 math teacher what is the area of a rectangle? they will more than likely say length times width. Again, this definition is dependent on multiplication (we understand that this is more a problem of semantics than mathematics). The intimate relationships between multiplication, the area of a rectangle, and similar triangles are important because integration rests on limits of areas of rectangles and trigonometry rests on the fact that similar triangles have corresponding sides that are proportional. We are suggesting that multiplication, the area of a rectangle, and similar triangles should be defined independently of each other in the mathematics curriculum. After this the intimate relationships between these three things can be proven.in addition, we believe that, all prospective math teachers and STEM students should be exposed to proofs of these intimate relationships before they graduate from university. In this paper we propose a way that this may be accomplished. In this paper we will do the following: (1) show how to geometrically define multiplication, using only basic plane geometry, independently of area and any notion of similar triangles; (2) prove all the properties of multiplication using only the axioms of plane geometry and the geometric definition of multiplication; (3) explain how the geometric definition of multiplication relates to the area of a right triangle (or 1 präsentiert mathematisch komplette Einführung von Zahlenoperationen durch geometrische Konstruktionen beweist Kommutativität, Assoziativität und Distributivität von Multiplikation wir gestalten visuelle Repräsentationen

6 Addition & Subtraktion sei ein Frame, und zwei Punkte A und B auf der x-axis wo is A+B? wo is A-B? Konstruieren dynamisch Konstruktion von J. Richter-Gebert, CC-BY-SA-NC

7 Addition & Subtraktion sei ein Frame, und zwei Punkte A und B auf der x-axis wo is A+B? wo is A-B? Konstruieren dynamisch Konstruktion von J. Richter-Gebert, CC-BY-SA-NC

8 <latexit sha1_base64="o1bu7rke5ql2qiryqkvm6te85qk=">aaab+nicbvbnt8jaen3if+jxxaoxjctee2mnus8mibdvygkfbbqy3w5hw3a32d0acewvepggxqu/xjv/xgv6upalk7y8n5ozeuhcqnko820vvlbx1jekm6wt7z3dpxu//kbekjhxsgbctgokckocejpqrtqjjcgoggkfw5up33okulhb7/uoix6m+pxgfcntpj5dzhrn7cprxgddhaqdnepzz644vwcguezcnfrajmbp/uqgaqcx4rozpfthdrltj5hufdmykxvtrrkeh6hpoozyfbplj2e3t+cxuuiycwmkazhtf0+muazuka5mz4z0qc16u/e/r5pq6nifu56kmna8xxsldgobp0hakeqcnrszgrck5laib0gire1cjrocu/jymvfoq+dv5+6suqvnartbitgcj8aff6agbketeacdj/amxsgbnbfe Multiplikation Seien ein Frame A und B auf der x-axis wo ist P damit OP = OA OB Strahlensatz benutzen

9 <latexit sha1_base64="o1bu7rke5ql2qiryqkvm6te85qk=">aaab+nicbvbnt8jaen3if+jxxaoxjctee2mnus8mibdvygkfbbqy3w5hw3a32d0acewvepggxqu/xjv/xgv6upalk7y8n5ozeuhcqnko820vvlbx1jekm6wt7z3dpxu//kbekjhxsgbctgokckocejpqrtqjjcgoggkfw5up33okulhb7/uoix6m+pxgfcntpj5dzhrn7cprxgddhaqdnepzz644vwcguezcnfrajmbp/uqgaqcx4rozpfthdrltj5hufdmykxvtrrkeh6hpoozyfbplj2e3t+cxuuiycwmkazhtf0+muazuka5mz4z0qc16u/e/r5pq6nifu56kmna8xxsldgobp0hakeqcnrszgrck5laib0gire1cjrocu/jymvfoq+dv5+6suqvnartbitgcj8aff6agbketeacdj/amxsgbnbfe Multiplikation Seien ein Frame A und B auf der x-axis wo ist P damit OP = OA OB Strahlensatz benutzen

10 <latexit sha1_base64="+3vijdowd1zv6h159a/w0gdijje=">aaab+hicbvbns8naen3ur1q/oh69lbbbu0le1itq68vbkxhbaepzbdbt0s1u2n0ustp/4swdild/ijf/jzs2b60+ghi8n8pmvcbhvgnh+bjkk6tr6xvlzcrw9s7unr1/8khekjhxsgbcdgkkckocejpqrjqjjcgoggkho9vcb4+jvftwbz1jib+jaacrxugbqw/b0+bntiddoafnxvtarfttqlnz5ob/ivuqkijq6tufvvdgnczcy4au6rpoov0msu0xi7nkl1ukqxiebqrrkecxux42v3wgt4wswkhiu1zdufpzikoxupm4mj0x0ko17oxif1431dgvn1gepjpwvfgup Kehrwert Seien ein Frame A auf der x-axis wo ist B damit OA OB =1

11 <latexit sha1_base64="+3vijdowd1zv6h159a/w0gdijje=">aaab+hicbvbns8naen3ur1q/oh69lbbbu0le1itq68vbkxhbaepzbdbt0s1u2n0ustp/4swdild/ijf/jzs2b60+ghi8n8pmvcbhvgnh+bjkk6tr6xvlzcrw9s7unr1/8khekjhxsgbcdgkkckocejpqrjqjjcgoggkho9vcb4+jvftwbz1jib+jaacrxugbqw/b0+bntiddoafnxvtarfttqlnz5ob/ivuqkijq6tufvvdgnczcy4au6rpoov0msu0xi7nkl1ukqxiebqrrkecxux42v3wgt4wswkhiu1zdufpzikoxupm4mj0x0ko17oxif1431dgvn1gepjpwvfgup Kehrwert Seien ein Frame A auf der x-axis wo ist B damit OA OB =1 X 1 x = x 0 1/x = 0.4 f A

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