MODULE 5 Geometril resoning 1 OBJECTIVES This module is for study y n individul teher or group of tehers. It: looks t pprohes to developing pupils visulistion nd geometril resoning skills; onsiders progression towrds geometri proof. CONTENT The module is in five prts. 1 Introdution 2 Conventions, definitions nd derived properties 3 Deriving properties 4 Looking t lesson on geometril resoning 5 Summry RESOURCES Essentil Your personl file for inserting resoure sheets nd mking notes s you work through the tivities in this module The Frmework for tehing mthemtis: Yers 7, 8 nd 9 Video sequene 3, Yer 8 geometry lesson, from the DVD ompnying this module, nd DVD plyer The resoure sheets t the end of this module: 5 5 5 5d 5e Visulistion tivities Conventions nd definitions Deriving properties More derivtions Bol s lesson 5f Exmples from Ntionl Curriulum tests for Key Stges 2 nd 3 5g Summry nd further tion on Module 5 Desirle Yer 9 geometril resoning: mini-pk www.stndrds.dfes.gov.uk/midins/keystge3/y9_geometriresoning.pdf Tehing nd lerning geometry 11 19, joint report from the Royl Soiety nd the Joint Mthemtil Counil, ville t: www.roylso..uk/templtes/sttements/sttementdetils.fm?sttementid=154 The QCA Mthemtis glossry for tehers in Key Stges 1 to 4, ville t www.q.org.uk/ges3-14/downlods/glossry.pdf STUDY TIME Allow pproximtely 90 minutes. 1 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Prt 1 1 Introdution As pupils move through Key Stges 1 nd 2, they progress from minly prtil nd experientil pproh to shpe nd spe to more strutured, formlised pproh. The min im in Key Stge 3 is for pupils to develop their knowledge of geometril ides nd use it to support geometril resoning. This pproh lys the foundtions for more forml pproh to geometril proof in Key Stge 4. (A mthemtil proof involves estlishing the truth of sttement y rigorous logil rgument.) Geometril resoning nd proof hve hd greter emphsis in the Ntionl Curriulum sine 2000 thn previously. This emphsis is refleted in the Frmework for tehing mthemtis: Yers 7, 8 nd 9. Geometril resoning is the fous for this nd the next module. 2 Try the two tivities on Resoure 5, Visulistion tivities. 3 The two tivities on Resoure 5 im to develop visulistion, geometril resoning nd justifition. These nd similr tivities mke useful orl nd mentl strters for geometry lessons. Sine there is often more thn one wy of justifying the result, you n sk pupils to ompre their justifitions y desriing them to prtner or to the whole lss. Compre your rguments with those elow. Midpoints P A Q D O B S C R ABCD is squre with n re tht is hlf the re of squre PQRS. A justifition: AC nd BD re the perpendiulr isetors of the sides of the squre PQRS. They re equl in length nd iset eh other t right ngles. Sine AC nd BD re the digonls of ABCD, it follows tht ABCD is squre. The eight tringles formed y the onstrution lines re ongruent nd so equl in re (two sides nd the inluded right ngle re equl). It follows tht the re of squre ABCD, whih is formed from four of the tringles, is hlf the re of squre PQRS. Chnging shpes The digonl of the retngle retes two identil right-ngled tringles. These n e used to mke five possile shpes in ddition to the originl retngle: two different prllelogrms, two different isoseles tringles nd kite. 2 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
x y y x y x x y In these two shpes, it n e shown tht opposite sides re equl nd opposite ngles re equl, mking them prllelogrms. In these two shpes, it n e shown tht two sides re equl, nd tht the third side is stright line, mking them isoseles tringles. (It is neessry to prove tht the third side is stright line, otherwise the shpe would e qudrilterl with two djent sides equl.) In this shpe, it n e shown tht two pirs of djent sides re equl, mking it kite. 4 Now look in more detil t spets of geometril resoning in the Frmework for tehing mthemtis: Yers 7, 8 nd 9. Study the tehing progrmmes for Yers 7, 8 nd 9, Frmework setion 3, pges 7, 9 nd 11. As you study the tehing progrmmes, jot down in your personl file some of the words tht illustrte the emphsis on geometril resoning. As well s severl referenes to explining resoning, there re ojetives in the Yer 9 tehing progrmme, nd in the Yer 9 extension progrmme, whih rely on firly sophistited thought, suh s: distinguish etween onventions, definitions nd derived properties; distinguish etween prtil demonstrtion nd proof (Yer 9 extension). This module fouses in prtiulr on the progression through Key Stge 3 tht uilds up to these ojetives in Yer 9. Prt 2 1 Conventions, definitions nd derived properties Study the exmples of onventions nd definitions on Resoure 5, Conventions nd definitions. Are you fmilir with these? Other properties of ngles nd shpes n e derived from these definitions. For exmple, it is possile to prove, rther thn define, tht vertilly opposite ngles re equl or tht lternte ngles re equl. 2 Sue Wring, in her ook Cn you prove it?, pulished y The Mthemtil Assoition, identifies four possile stges for pupils s they work towrds forml proof: 3 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Stge 1 Stge 2 Stge 3 Stge 4 Convine yourself (mentl justifition) Convine friend (orl justifition) Convine pen-friend (informl written justifition) Convine your mthemtis teher (more forml written justifition) These four stges re illustrted elow in reltion to proof tht vertilly opposite ngles re equl. Stge 1 Stge 1 involves onvining yourself. For exmple, you might think: Those two ngles re on stright line nd so re those two ngles. So I n tke tht ig ngle wy from oth those stright ngles nd the two remining little ngles must e equl. Stge 2 Stge 2 involves onvining friend. For exmple, you might sy: The ngle mrked with irle nd the ngle mrked with squre dd up to 180. The sme is true for the ngle mrked with ross nd the one mrked with the squre. So this ngle (point to the irle) must equl this one (point to the ross). Stge 3 Stge 3 involves n informl written justifition, whih might go like this. + = 180 + = 180 So = Stge 4 Stge 4 involves forml written justifition, whih might go like this. x y w z x + y = 180 (ngles on stright line) y + z = 180 (ngles on stright line) x + y = y + z, giving x = z. 4 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
3 Now study the exmples in the supplement of exmples, Frmework setion 4, pges 178 179. As you look t the exmples, think out the differenes etween onventions, definitions nd derived properties. Distinguishing etween demonstrtion nd proof is in the Key Stge 4 progrmme of study nd is in the Yer 9 extension tehing progrmme exemplified, in itlis, in the lst prgrph of pge 179. Now study the exmples on pges 180 201. These illustrte the progression from informl explntion nd justifition to forml proof. As you work through the exmples, mke note in your personl file of ny tht would e useful to explore with the lsses tht you teh. Prt 3 1 Deriving properties Do the prolems on Resoure 5, Deriving properties. You re given some definitions of geometri properties nd, from these, must dedue some further geometri properties. Aim to produe stge 4 forml proof wherever possile. 2 Here re some possile rguments tht n e used for the prolems on Resoure 5. 1 Tke ny pir of lternte ngles, for exmple nd. = (vertilly opposite ngles) = (orresponding ngles) = So lternte ngles etween prllel lines re equl. 2 d Tke ny pir of opposite ngles, for exmple nd. = (orresponding ngles) = (lternte ngles) = So the opposite ngles of prllelogrm re equl. 3 d Extend one side of the tringle nd onstrut line through one vertex prllel to the opposite side, s shown. = (lternte ngles) 5 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
= d (orresponding ngles) + = + d So the exterior ngle of tringle equls the sum of the interior opposite ngles. 4 d e d = (lternte ngles) e = (lternte ngles) d + + e = + + = 180 (ngles on stright line) So the sum of the interior ngles of tringle is 180. 3 Now try the prolems on Resoure 5d, More derivtions. To find the sum S of the interior ngles of n n-sided polygon, identify point O inside the polygon. Join O to eh of the verties of the polygon, forming n tringles. S n e regrded s the sum T of the ngles of ll n tringles, less the sum A of the ngles round point O. Sine A is 360 or 4 right ngles, S n e lulted in two wys: sine eh of the n tringles hs n ngle sum of 180, T is 180n, nd S = 180n 360 degrees, whih is (n 2) 180 ; sine eh of the n tringles hs n ngle sum of 2 right ngles, T is 2n right ngles, nd S = 2n 4 right ngles. Prt 4 1 Looking t lesson on geometril resoning In this prt of the module, you will hve hne to onsider the use of ICT in developing geometril resoning. First study the exmples on pges 180 181 of the supplement of exmples, Frmework setion 4. These exmples refer to the use of ette sheets for n overhed projetor or use of omputer softwre. As you study the exmples, think out the reltive dvntges of one medium over the other. Note in your personl file ny exmples tht would e useful to explore with the lsses tht you teh. 2 Get redy to wth Video sequene 3, Yer 8 geometry lesson. The teher is Bol. Bol is using dynmi geometry softwre to egin to develop her pupils ides of proof. In the first prt of the video sequene, Bol disusses with the lss how to nme ngles nd demonstrtes the equlity of vertilly opposite ngles. In the seond prt, whih is muh lter in the lesson, she questions pupils out their proof tht the sum of the ngles of tringle is 180. As you wth the video, onsider the usefulness of dynmi geometry softwre, fousing on the questions on Resoure 5e, Bol s lesson. The video sequene lsts out 5 minutes. When you hve finished wthing, spend few minutes ompleting the notes you hve mde on Resoure 5e. Then think out how Bol s pproh ompres with wht you would hve done. 6 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
3 Study the geometri prolems in the exmples on using nd pplying mthemtis in the supplement of exmples, Frmework setion 4, pges 14 17. These inlude exmples tht mke use of omputer softwre or ette sheets. They lso inlude more exmples in whih properties of shpes hve to e derived y geometril resoning. Add to the notes in your personl file one or more prolems tht would e useful to offer to the lsses tht you teh. Prt 5 1 Summry It is importnt tht Key Stge 3 pupils ppreite the differenes etween geometri onventions, definitions nd derived properties. As they use nd pply their developing knowledge of geometril properties, pupils in Key Stge 3 should move from informl justifitions of their rguments to more forml written proofs. 2 Look t Resoure 5f, Exmples from Ntionl Curriulum tests. Wht definitions would pupils need to know in order to nswer the questions? For eh question, think out the kinds of informl or forml rguments tht you would expet pupils to give to justify their resoning. 3 Look k over the notes you hve mde during this module. Hve you identified wht you my need to onsider nd dopt in your plnning nd tehing of geometry? Use Resoure 5g, Summry nd further tion on Module 5, to list key points you hve lerned, points to follow up in further study, modifitions you will mke to your plnning or tehing, nd points to disuss with your hed of deprtment. 4 You my find it interesting to red the rtile y Pul Chmers on Tehing Pythgors theorem from the Septemer 1999 issue of Mthemtis in Shool, pulished y The Mthemtil Assoition, 259 London Rod, Leiester LE2 3BE. If you re interested in reding more out the tehing of geometry in seondry shools, red Tehing nd lerning geometry 11 19, joint report from the Royl Soiety nd the Joint Mthemtil Counil. This report reitertes the entrlity of geometry to the mthemtis urriulum nd how importnt it is tht this rnh of the sujet should not e negleted. Appendix 9: Proof why nd wht? is of prtiulr interest. You n downlod the report from www.roylso..uk/templtes/sttements/sttementdetils.fm?sttementid=154. You ould lso downlod nd look t the Yer 9 geometril resoning: mini-pk from www.stndrds.dfes.gov.uk/midins/keystge3/y9_geometriresoning.pdf. 7 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Resoure 5 Visulistion tivities The first prt of eh of these tivities should e rried out without ny drwing. 1 MIDPOINTS Imgine squre. Join the midpoints of eh pir of djent sides. Wht is the insried shpe? How does the re of the insried shpe relte to the re of the originl squre? Now justify your resoning. Drw sketh nd use informl lnguge if you wish. [ontinued on the next pge] 8 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
2 CHANGING SHAPES Imgine (non-squre) retngle. Cut it long one of its digonls so tht you hve two shpes. Cll these shpes A nd B. Visulise the different shpes you n mke from A nd B y mthing sides of the sme length. Now sketh eh of your new shpes nd write its nme. For eh shpe, stte the geometril fts you re using to justify your nswer. 9 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Resoure 5 Conventions nd definitions CONVENTIONS Lelling A A A B B C B t C Nottion tringle ngle therefore // is prllel to DEFINITIONS Corresponding ngles lie on the sme side of trnsversl nd on orresponding sides of pir of lines. If the two lines re prllel, the orresponding ngles re equl. An exterior ngle of polygon is the ngle etween side nd n extension of n djent side. In this exmple, ACD is n exterior ngle of ABC. A B C D Perpendiulr lines interset t right ngles. 10 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Resoure 5 Deriving properties You re given the following fts: the ngles on stright line re supplementry, i.e. they dd up to 180 ; orresponding ngles re equl; vertilly opposite ngles re equl; opposite sides of prllelogrm re prllel. Use some or ll of these fts, nd onstrutions where neessry, to prove the following in the order 1, 2, 3, 4. 1 Alternte ngles re equl. 2 Opposite ngles of prllelogrm re equl. 11 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
3 The exterior ngle of tringle is equl to the sum of the interior opposites. 4 The ngles of tringle dd up to 180. 12 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Resoure 5d More derivtions Look t the Yer 9 exmples on pge 183 of the supplement of exmples, Frmework setion 4. Derive the formul for the sum of the internl ngles of n n-sided polygon s (n 2) 180. Think of n lterntive rgument tht would led to (2n 4) right ngles. 13 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Resoure 5e Bol s lesson While wthing the short video extrts of Bol working with her Yer 8 lss, onsider nd mke notes on the questions elow. How does the dynmi geometry softwre filitte demonstrtion of given ft? How does the dynmi geometry softwre filitte proof of geometril property? Wht other spets of geometril resoning ould e enhned y giving pupils the opportunity to move lines nd shpes in this wy? 14 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Resoure 5f Exmples from Ntionl Curriulum tests These exmples re tken from the Ntionl Curriulum tests for Key Stges 2 nd 3. LEVEL 5 1 Here is n equilterl tringle inside retngle. x 12 Not to sle Clulte the vlue of ngle x. Show your working. 2 Look t this digrm. Not to sle x 30 y Clulte the size of ngle x nd ngle y. Show your working. 3 Tringle ABC is equilterl. A x 108 Not to sle B C Clulte the size of ngle x. Show your working. 4 The digrm shows retngle. D C Not drwn urtely 65 35 A K B Work out the size of ngle. Show your working. 15 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
LEVEL 6 1 F is the entre of regulr pentgon. F x 72 Work out the vlue of ngle x. Explin how you worked out your nswer. 2 The digrm shows two shded equilterl tringles. y Not to sle x 65 40 55 Clulte the size of ngle x nd the size of ngle y. 3 The shpe elow hs three identil white tiles nd three identil grey tiles. The sides of eh tile re ll the sme length. Opposite sides of eh tile re prllel. One of the ngles is 70. k 70 m Not to sle Clulte the size of ngle k nd ngle m. Show your working. LEVEL 7 1 A retngle just touhes n equilterl tringle so tht ABC is stright line. A Not drwn urtely B D E C Show tht tringle BDE is isoseles. 16 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
Resoure 5g Summry nd further tion on Module 5 Look k over the notes you hve mde during this module. Identify the most importnt things to onsider nd modify in your plnning nd tehing of geometry. List two or three key points tht you hve lerned. List two or three points to follow up in further study. List two or three modifitions tht you will mke to your plnning or tehing of geometry. 17 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004
List the most importnt points tht you wnt to disuss with your hed of deprtment, or ny further tions you will tke s result of ompleting this module. 18 KS3 Strtegy Mthemtis study modules Study module 5 DfES 0156-2004 G Crown opyright 2004