Teaching parametric equations using graphing technology

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2 Teaching parameric equaions using graphing echnology The session will sar by looking a problems which help sudens o see ha parameric equaions are no here o make life difficul bu are imporan and give rise o some beauiful curves. Then we will look a ways in which graphing echnology (calculaors, GeoGebra and Auograph) can help sudens make sense of parameric curves for hemselves. Suiable for all eachers who have augh parameric equaions in A level Mahemaics.

3 Teaching parameric equaions using graphing echnology Wich of Agnesi and cycloids: learn he calculaor funcionaliy wih some nice mahs. Cards aciviy: hinking how calculaors can deepen undersanding, supporing learning in class. 07 SAMs quesions: hinking how calculaors can suppor exam echnique. Inroducing paramerics using GeoGebra. A few invesigaions.

4 The Wich of Agnesi Maria Agnesi, hps://

5 The Wich of Agnesi There has been much argumen over he reason why he curve is called a 'wich'. In 78 Guido Grandi (67-74, an Ialian Jesui who worked on geomery and hydraulics) gave i he Lain name 'versoria' which means 'rope ha urns a sail' and he so named i because of is shape. Grandi gave he Ialian 'versiera' for he Lain 'versoria' and indeed Agnesi quie correcly saes in her book ha he curve was called 'la versiera'. John Colson ranslaed Agnesi's Insiuzioni analiiche ad uso della giovenù ialiana ino English [and] misook 'la versiera' for 'l'aversiera' which means 'he wich' or 'he she-devil'. hp://www-hisory.mcs.s-and.ac.uk/

6 Cycloids x at asint y a a cost hps://

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12 hps://

13 Abou MEI Regisered chariy commied o improving mahemaics educaion Independen UK curriculum developmen bo We offer coninuing professional developmen courses, provide specialis uiion for sudens and work wih employers o enhance mahemaical skills in he workplace We also pioneer he developmen of innovaive eaching and learning resources

14 MEI Conference 07 Teaching parameric equaions using graphing echnology Bernard Murphy

15 The Wich of Agnesi and he Casio fx-cg0 The circle has radius and cenre 0,. The poin A moves along he line y. The line OA, which makes an angle T wih he y -axis, mees he circle a B. Lines BP and PA are parallel o he axes. Wha is he locus of P? Firs rese your calculaor o facory seings:pazywqd. Add a new Graphs screen: p5. Se he graph ype o Parameric: ee 3. Draw he graph x an T, y cos T : kfl(jf)slu 4. Experimen wih he V-Window Le o se appropriae ranges for X, Y and T. Afer each one use l and end wihd. 5. By eliminaing he parameer, find he Caresian equaion of his curve. See if you re righ by enering his on your calculaor, firs seing he graph ype o Caresian: deq

16 Cycloids A wheel rolls along a fla surface wihou slipping. A poin P is on he circumference of he wheel. Wha is he equaion of he pah of P? Le he poin P on he rim sar a he origin, O. The diagram shows he posiion of he wheel afer i has rolled a disance o he righ. Since here is no slipping, he horizonal disance moved along he road, OA, mus be he same as he disance PA on he wheel rim. OA = PA = at (where T is measured in radians). When C has coordinaes at, a, P has coordinaes at asin T, a a cost. So he pah of P can be described by parameric equaions: x at asin T, y a a cost Invesigae he locus of oher poins along he radius.

17 x d y 3 6 d, gives he poin (3,3) A, 6 x 4 y ln d d, gives he poin (4,0) A, d x x d 3 y d d y 3, gives he poin (,) 3 A, d x x d y d, gives he poin (,) A, x 4 4 d y d, gives he poin (5,) A, d x 6 x 3 3 d y d 3, gives he poin (4,) A, d x 3 x 4 4 d y d, gives he poin (5,) A, d x 3

18 Tangens o parameric curves (Adaped from a resource from hp:// ). Add a new Graphs screen: p5. Se Derivaive: On and Angle: Radians: LpNNNNNNqNNNNwd 3. Se he graph Type o Parameric: ee (You migh wan o delee exising funcions here.) 4. Use V-Window o se he range of T o Tmin: -3, Tmax: 3, pich: 0.05 LeNNNNNNNn3l3l0.05ld 5. Draw he graph x y 3,. X=T², Y=T³: fslf^3lu 6. Add he angen a he poin (Skech > Tangen): rw Use!/$ o move he posiion of he poin on he curve or choose a value using f. Quesions Wha is he relaionship beween d y, d x and d y d d? Can you verify his algebraically for some oher parameric curves? Problem (Check your answer by ploing he graph and he angen on your calculaor) Find he coordinaes of he poins on he curve x cos, y sin, 0 for which he angen o he curve is parallel o he x axis. Furher Tasks Explore how you can find he equaion of he angen o a parameric curve a a poin. Describe how o find he angen o a parameric curve ha passes hrough a specific poin ha is no on he curve.

19 Firs rese your calculaor o facory seings:pazywqd. Add a new Graphs screen: p5. Se he graph ype o Parameric: ee 3 3. Draw he graph x, y 4 7 : f-l4f-7+3mflu 4. Experimen wih he V-Window Le o se appropriae ranges for X, Y and T. Afer each one hi l and end wih d. 5. Se he graph ype o Caresian deqand eiher eliminae he parameer o find he Caresian equaion of his curve or experimen using he modify command: (fs+aff+ag)m(f+)ly 6. Using he cursors change he values of A and B o view he resuling graph.

20 . Add a new Graphs screen: p5. Se Derivaive: On and Angle: Radians: LpNNNNNNqNNNNwd 3. Se he graph Type o Parameric: ee (You migh wan o delee exising funcions here.) 4. Use V-Window o se appropriae ranges for x, y and T using Le he N cursor, l afer each enry and d o finish. 5. Ener x cos, y sin and u o draw. 6. Add he angen a he poin (Skech > Tangen): Lrw and use!/$ o move he posiion of he poin on he curve or choose a value using f.

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22 Invesigaion Imagine wo poins, P and Q, on a uni circle. They sar from,0 and go a consan speeds in he same direcion round he circle bu Q goes wice as fas as P. Now hink abou he chord PQ. Is lengh is changing coninuously. Wha pah is followed by he midpoin, M, of PQ? Wha if he raio of he speeds of P and Q was differen? Wha if P and Q wen in differen direcions? Wha pah is followed by he poin N on he chord where PN PQ? 3 Poin P is on he line y. Invesigaion A circle cenred on P ouches he uni circle as shown. The radius PQ is parallel o he x axis. As P moves along he line y, wha is he pah raced ou by poin Q? Invesigaion 3 Using your graphing calculaor, invesigae curves wih parameric equaions x 3cos cos k, y 3sin sin k For which values of k do you ge a dimple?...a cusp?...crossover poins? Find he coordinaes of poins where he angen is parallel o an axis. Wha oher properies could you explore?

23 x x x 4 x 4 x x 3 x 4 y y 3 y ln y y 3 y y

24 d d 3 d 4 d d 4 d d d d d 3 d d 3 d 6 d

25 A, A, d x 3 3 A, d x A, 6 A, d x A, d x 3 A, d x 6, gives he poin (,), gives he poin (4,0), gives he poin (,), gives he poin (3,3), gives he poin (5,), gives he poin (4,), gives he poin (5,)