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Maths course exercises Vittoria International School – Torino The circle

Maths course exercises Vittoria International School – Torino The circle in accordo con il Ministero dell’Istruzione, Università, Ricerca e sulla base delle Politiche Linguistiche della Commissione Europea percorso formativo a carattere tematico-linguistico-didattico-metodologico

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Maths course exercises Vittoria International School – Torino The circle

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  1. Maths course exercises Vittoria International School – Torino The circle in accordo con il Ministero dell’Istruzione, Università, Ricerca e sulla base delle Politiche Linguistiche della Commissione Europea percorso formativo a carattere tematico-linguistico-didattico-metodologico scuola secondaria di secondo grado a cura di Serenella Iacino

  2. Indice modulo clil • Strategies – Before • Prerequisites • LinkingtoPreviousKnowledge and Predicting • GlossaryItalian/English • Strategies – During • Video • Keywordsriferite al video attraverso esercitazioni mirate • ConceptualMap • Strategies – After • Exercises: • - Multiple Choice • - Matching • - True / False • - Cloze or Completion • - Flow Chart • - Think and Discuss • Summary and SummaryQuestions • Web Referencesdi approfondimento come input interattivi per test orali e • scritti e per esercitazioni basate sul ProblemSolving • Answersheets

  3. 1 Strategies Before Prerequisities Maths the prerequisites are • Equation of first degree • Equation of second degree • System of first degree • System of second degree • System of forth degree • Equation of a straight line • Equation of a set of straight line The circle

  4. Strategiesbefore LinkingtoPreviousKnowledge and Predicting • Do youknow the definitionofGeometric Locus? • Howmanyconditions do youneedto solve a system ofequations? • Do youknow the equationof a set ofstraightlinespassingthrough a point ? • Are youableto solve a system of first, second, or fourthdegree? • Whendoes a straightline pass through a point P ( x ; y ) P P P ( x ; y )? P P

  5. StrategiesBefore GlossaryItalian/English asse - axis asse radicale - radicalaxis centro - centre circonferenza - circle circonferenza concentrica - concentriccircle circonferenza degenere - degenerate circle coefficiente angolare - angle coefficient combinazione lineare - linearcombination conica - conicsection curva - curve determinante - determinant discriminante - discriminant distanze - distance equazione - equation

  6. Strategies Before Glossary Italian/English esterna - external fascio di circonferenze - set of circles luogo geometrico - geometric locus punti base - base points Radici coincidenti - coincident roots radici distinte - distinct roots radici immaginarie - imaginary roots raggio - radius retta - straight line retta normale – normal line secante - secant tangente - tangent

  7. StrategiesDuring Keywords • angle coefficient • axis • base points • centre • circle • set of circles • coincident roots • concentric • conic section • curve • degenerate circle • determinant • discriminant • distance

  8. Strategies During Keywords Key Words • Circle the odd one out Geometric locus, centre, radius, secant, tangent, external, straight line, triangle, equation, discriminant, radica axis, function, base points. • Circle the odd one out Curve, conic section, determinant, distance, axis, coincident roots, imaginary roots, distinct, sin function, normal line, set of circles, angle coefficient, linear combination, degenerate circle, concentric.

  9. Strategies During Conceptual Map base points inside In which Set of circles another circle The circle secant Depends on constants a,b,c defined by Depends on radius tangent straight line external Complete the conceptual map using the following words: tangent external center point P secant radical axis

  10. Multiple Choice Strategies After What is the radius of the circle having this equation? x² + y² – 4 x – 2 y – 3 = 0 4 2 2 2 2

  11. Strategies After • Multiple Choice Which of the following equations doesn’t represent a circle? x² + y² – 2 x – 4 y – 7 = 0 x² + y² – 2 x – 4 y + 7 = 0 x² + y² – 2 x – 4 y – 2 = 0 x² + y² – 2 x – 4 y + 3 = 0

  12. Strategies After Multiple Choice What is the radius of the circle having centre C (3;2) and passing through the point P(2;3)? None of the above 2 5 3

  13. Strategies After Multiple Choice Whichof the followingconditionsexpresses the passageof the circlex² + y² + a x + b y + c = 0through the pointP (1; - 1)? a – b – c = 2 a + b + c = - 2 a – b – c = - 2 a – b + c = - 2

  14. StrategiesAfter Multiple Choice Giventwocircleshaving the equations and the radicalaxisis the straightline: x² + y² + 2x - 2y - 4 = 0 x² + y² - 2x - 4 = 0, X + 3Y = 0 2X - Y + 1= 0 It doesn’t exist 2X – Y= 0

  15. Strategies After Matching Match the constants a, b, c, and the pictures C a b = 0 C C b c = 0 1 3 2 c a = 0 d a = b = 0 e a = c = 0 C C C f b = c = 0 4 6 5

  16. Strategies After Matching Match the equations and the pictures 4 2 C -1 3 -2 4 C 2 -1 C A C B -2 -4 x² + y² - 2x + 2y - 8 = 0 1 2x² + 2y² - 4x + y - 6 = 0 2 2x² + 2y² - 2x - 7y - 4 = 0 3

  17. 1 2 3 • StrategiesAfter • Matching Link the sentence to the right picture if the distance between the centres is less than the sum of the radii A C C’ if the distance between the centres is greater than the sum of the radii B C C’ if the distance between the centres is equal to the sum of the radii C C C’

  18. Strategies After True/False • If a circle passes through two points A and B, its centre belongs to the perpendicular straight line to AB and passes through the middle-point of AB T F • The equation represents for each a = 0 a circle having the centre on the Y axis. x² + y² + a x = 0 T F

  19. Strategies After True/False • If the distance of a straight line from the centre of the circle having equation is 3, then the straight line is a tangent to the circle. x² + y² - 4x - 5 = 0 T F • If a straight line r is a tangent to a circle in the point P, then the centre of the circle belongs to the straight line passing through the point P and perpendicular to r. T F

  20. Strategies After True/False • The set ofcircles, generatedbycircles • and hastwo base points x² + y² = 4 x² + y² - 4x = 0 T F • The radical axis of two esternal circles is not perpendicular to the line joining their centres T F • If two circles are tangent within then the sum of their radii equals the distance between the centres F T

  21. Strategies After Completion Exercise • Complete these sentences • A straight-line is secant to the circle if……………………………………………………………………………………………………………………………………….…………………………………………………………………………………. • In a set of non concentric circles, the radical axis is considered …………………………………………………………………………………………………………………………………………..………………….………………….…………………………………………. • The base points of the set of circles are points for which ………………………………………………………………………………………………………………………………………………………………………………………………………………………………

  22. StrategiesAfter • CompletionExercise • Complete these sentences • If the straight line having equation is tangent to the circle having equation then the ………………………. of the equation solving the system between the circle and the line is equal to ……………………………………………………………………………………………………………………………………………………….………………….………………………………………………… • If the distance between the centres of two circles is greater than the sum of the radii, then the circles ………………….………………….…………………………………………………………………………………………………………………………………………………………………………………….. y = mx + q x² + y² + ax + by + c = 0

  23. Strategies After Flow Chart Complete the flow chart in order to determine the equation of a straight line tangent to a circle in its point start P ( x ; y ) P P Defining a system Equation of the 2nd degree Set of straight lines Discriminant is equal to 0 end Equation of the circle

  24. StrategiesAfter Thinkand Discuss The followingactivity can beperformed in a written or oralform. The teacherwillchoose the modality, depending on the ability (writing or speaking ) thatneedstobedeveloped. The contexts in which the task willbepresentedto the students are: • The studentiswritinganarticleabout the useof the circle in the modernArchitecture; • The studentispreparingforaninterview on a local TV about the useof the circle in the Architectureof the Renaissance. The studentshould: • Writeanarticleabout the useof the circle in the romanbuildings: the Pantheon; • Prepare the article or the debate, outlining the mainpointsof the argument, on the basisofwhathasbeenstudied; • If the writtenactivityis the modalitychosenby the teacher, the studentshouldprovide a writtenarticle, indicating the target ofreaderstowhom the articleisaddressed and the typeof magazine / newspaper / school magazine where the articlewouldbepublished; • If the oralactivityis the modalitychosenby the teacher, the studentshouldpresenthispointofview on the topicsto the wholeclass and a debatecould start at the end ofhispresentation.

  25. StrategiesAfter Summary The circleis a conicsection and isdefinedas the geometric locus of the points P of the Cartesianplanewhich are equidistantfrom a fixedpointcalled “the centre”. In itsequationthere are threeconstants a, b, c and ifwewanttofind the equationof a circleweneedthreeindependentconditions, thatisoneconditionforeachconstant: forexample the coordinatesof the centre, the radius or the coordinatesof a point P forwhich the circlepassesthrough. Accordingto the valuesof the constants a, b, c we can haveparticularcircleswith the centre at the origin, or on the x axis, or on the y axis, or evenpassingthrough the origin. The position of a straightline in relation to a circlemaybesecant, tangent or external, and fromanalgebraicpointofviewwemust solve a system ofequations, onefor the circle, onefor the straightline; fromthis system weobtainanequationofseconddegree, fromwhich, ifwehave the straightlinesecant, tangent or external. Δ > 0, Δ = 0, Δ < 0

  26. The tangents to a circle from a given point P can be real, coincident or imaginary if the point P is outside the circle, on the circumference or within the circle. The relative position of two circles is external, tangent from the outside, secant, tangent within, inside, if the distance between their centres is greater than the sum of the radii, equal to the sum of the radii, less than sum of the radii, equal to the difference between the radii and less than the difference between the radii. From an algebraic point of view we must solve the system of equations of the two circles; from this system we obtain the equation of a straight line called Radical Axis of two circles that is perpendicular to the line joining their centres. From a linear combination between the equations of two circles we obtain the equation of a set of circles that pass through the common points to the circles, called Base Points. The radical axis is considered as a particular circle of this set, having an infinite radius, and is called Degenerate Circle.

  27. StrategiesAfter SummaryQuestions • How do youdefine the circle? • Howmanyconstants do youfind in itsequation? • Howmanyconditions do youneedtodetermineitsequation? • Whichis the position of a straightline in relation to a circle? • What do you solve todetermine the equationof a tangentto a circle? • Whichis the relative position oftwocircles? • How do youobtain the equationof a set ofcircles? • What are the Base Points? • Whatis the RadicalAxis? Write a short abstractof the summary ( max 150 words ) highlighting the mainpointsof the video.

  28. 13 Activities Based on Problem Solving • Solve the following problems on the circle: • 1) Study the set of circles of equation • and find k so that: • the circle passes through the point • the circle has centre equal to • the circle is tangent to the line of equation x = 1 • 2) Given the equations of two circles • : • write the equation of the set of circles determined by them; • find the base points, and the radical axis; • find the circles of the set having radius equal to 5. x² + y² + 4 x + k y – 5 – 3 k= 0 P (1;1) C (-2;3) x² + y² – 3 x + y + 2 = 0 , x² + y² = 1

  29. Web References http://www.visualmathlearning.com/ Website designed to provide parents and classroom teachers with the means to better employ visual imagery. http://www.videomathtutor.com/ This site is intended to help students from secondary school through college. Teachers, other tutors, and parents will also find this site to be very useful. http://www.aaaknow.com/ AAA Math features a comprehensive set of interactive arithmetic lessons. Unlimited practice is available on each topic which allows thorough mastery of the concepts. http://www.icoachmath.com/ iCoachMath offers students the opportunity to engage in meaningful Math learning that leads students to expand their knowledge on their own and explore in detail different areas of mathematics.

  30. Answer sheets Conceptual Map Base points inside tangent In which Set of circles Another circle Radical axis The circle external secant Depends on Constants a,b,c Depends on Defined by radius tangent Straight line Point P external center secant

  31. Answer Sheets Multiple choice: 2 2 ; Matching: 1c; 2e; 3d; 4f; 5b; 6a A3; B1; C2 A3; B1; C2 True or false: T; F; T; T; T; F; F Completion exercise: “It meets the circle in two real different points”; “A degenerate circle”; “All circles pass through”; “Discriminant is zero”; “Are external” Problem solving: 1) The circles of the set are tangent; k=1/2; k=-6; k=0 e k=12 x² + y² – 2 x – 4 y + 7 = 0 ; 2 ; a – b + c = - 2 ; 2x – y= 0 2) (1+k) x² + (1+k) y² – 3 x + y + 2 - k = 0; A(1;0) B(4/5;-3/5); 3x-y-3=0; k=7/6 e k=7/8

  32. AnswerSheets Flow Chart Solution start • Complete the flow chart in order to determine the equation of a straight line tangent to a circle in its point Set of straight lines Equation of the circle P ( x ; y ) P P Making System Equation of the 2nd degree Discriminant is equal to 0 end

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