The Pythagorean Theorem Tile Set

Transcription

1 The Pythgoren Theorem Tile Set Guide & Ativities Creted y Drin Beigie Didx Edution 395 Min Street Rowley, MA DIDAX 201 #211503

2 1. Introdution The Pythgoren Theorem sttes tht in right tringle the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. The hypotenuse is the side opposite the right ngle nd the legs re the sides djent to the right ngle = 2 This theorem is nmed fter the Greek philosopher nd mthemtiin Pythgors (. 570 BC. 95 BC). Pythgors is trditionlly redited with the disovery nd proof of this theorem, lthough it is often sserted tht knowledge of this theorem existed efore Pythgors. There re multitude of proofs of the Pythgoren Theorem, nd there is long nd rih mthemtil history ehind the vrious proofs of this theorem. Generlly, these proofs require resonly sophistited understnding of lger nd geometry nd re thus not very essile to younger students. There exist, however, some proofs of the Pythgoren Theorem tht involve simple rerrngement of shpes. For these proofs y rerrngement, the geometri resoning is intuitive nd the lgeri resoning is miniml. These Pythgoren Theorem Tiles provide hnd-held puzzle tht gives students n opportunity to disover nd visulize the resoning ehind one of the lssi nd most essile proofs y rerrngement for the Pythgoren Theorem. We enourge tehers to use this mnipultive with sense of explortion nd disovery. Setion 2 outlines n tivity with the mnipultive tht strives to strike the pproprite lne etween rigor nd disovery, pproprite for grdes 6 nd ove. Setion 3 provides n exmple solution for the tivity. Setions nd 5 illustrte nd pply the Pythgoren Theorem. Setion 6 ontrsts the Pythgoren Theorem with the results for otuse nd ute tringles. 2 Pythgoren Theorem Tile Set didx.om

3 2. Using Pythgoren Theorem Tiles Strting Fts 1. A right tringle is tringle with right ngle (90 degrees). 2. In right tringle, the side opposite the right ngle is lled the hypotenuse, with length. The sides djent to the right ngle re lled the legs, with lengths nd. 3. The re of squre with side length s is s 2.. Two ojets re ongruent if they re equl in size nd shpe. Mterils for the Ativity 1. Eight ongruent right tringle tiles, eh with side lengths,, nd. 2. Three squre tiles: one with side length, one with side length, one with side length. The squre res re 2, 2, 2, respetively. 3. Two empty ongruent squre frmes, with side lengths +. Note tht the squre tiles of side length,, nd fit long the sides of the right tringle tile, s shown. didx.om Pythgoren Theorem Tile Set 3

4 Ativity 1. Ple tringle tiles, the squre tile with re 2, nd the squre tile with re 2 in one of the empty squre frmes. 2. Ple the remining tringle tiles nd the squre tile with re 2 in the other empty squre frme. 3. Sketh your results.. Wht do you onlude out the reltionship etween 2, 2, nd 2? Explin. 3. Solutions to Ativity Questions 3 nd 3. Sketh your results.. Wht do you onlude out the reltionship etween 2, 2, nd 2? Explin. Let A tringle e the re of ny one of the ongruent right tringles. Sine the squre frmes re ongruent, their res re equl. This mens A tringle = A tringle + 2. Sutrting A tringle from eh side of the eqution gives = 2, whih is sttement of the Pythgoren Theorem. Note how this solution does not require knowledge of the tringle res. Pythgoren Theorem Tile Set didx.om

5 . Illustrting the Pythgoren Theorem Speifi right tringles re often used to illustrte the vlidity of the Pythgoren Theorem. A lssi exmple is 3,, 5 tringle = = = 25 Other ommon exmples inlude 5, 12, 13 tringle ( = 169), n 8, 15, 17 tringle ( = 289), nd pproprite multiples of ny of the ove tringle side lengths (e.g., 6, 8, 10 tringle insted of 3,, 5 tringle). These tringles re onvenient illustrtions of the Pythgoren Theorem euse ll three side lengths re whole numers. Hving three whole numers stisfy = 2 is speil ourrene lled Pythgoren triple. Usully, when two side lengths of right tringle re whole numers, the third side length is not whole numer, nd this more typil sitution is illustrted in the next setion. didx.om Pythgoren Theorem Tile Set 5

6 5. Applying the Pythgoren Theorem The Pythgoren Theorem llows one to solve for missing length in right tringle. In exmple 1 the hypotenuse is unknown, nd in exmple 2 leg is unknown x 3 x The ility to solve for missing length in right tringle is very useful, sine it llows for mesurement long n ritrry diretion in the Crtesin plne. For exmple, the length of line segment MP is not ovious sine the line segment is not prllel to ny grid lines. y P x -2 M - 6 Pythgoren Theorem Tile Set didx.om

7 However, the Pythgoren Theorem omes to the resue y envisioning line segment MP s the hypotenuse of the right tringle MNP. Sine the legs of the right tringle, MN nd NP, re prllel to the grid lines, their lengths re esily determined y simply ounting unit lengths long the grid lines. Applying the Pythgoren Theorem then llows one to determine the length L of the hypotenuse MP. y P x -2 M - N Muh of sientifi nlysis involves quntities tht hve diretion: fore, veloity, displement, nd so on. So the ility to determine mgnitude long n ritrry diretion is ornerstone of mthemtil nlysis in the physil sienes. It is interesting to note tht quntities like 5, 176, nd 89 need to e rounded, sine the numers re irrtionl nd thus their deiml representtions ontinue forever without repetition. Historilly, the pperne of irrtionl numers in right tringles nd other geometri ontexts led to roder understnding of numers, their lssifition, nd their properties. didx.om Pythgoren Theorem Tile Set 7

8 6. Comprison with Otuse nd Aute Tringles The Pythgoren Theorem is so pervsive tht students sometime either forget tht the theorem only pplies to right tringles or re unwre of the reltionship etween side lengths when tringle is not right tringle. We summrize the results elow for ute, right, nd otuse tringles. For ll tringles, represents the longest of the three sides, while nd represent the lengths of the remining two sides (in either order). The res of the squres tthed to eh side help one visulize the reltionships stted elow. If = 2, then the tringle is right (this is the onverse of the Pythgoren Theorem). If > 2, then the tringle is ute (ll ngles re less thn 90º). If < 2, then the tringle is otuse (one ngle is greter thn 90º). 8 Pythgoren Theorem Tile Set didx.om