Further Mathematics Further Mathematics WorkshopWorkshop

Stowupland High SchoolStowupland High School88thth November 2005 November 2005

See next slide for details of how the lesson on See next slide for details of how the lesson on curve sketching was organisedcurve sketching was organised

Lesson followed the slides in this presentationLesson followed the slides in this presentation ““Role play” involved the students in a warm up Role play” involved the students in a warm up

exercise where they modelled, using their arms, the exercise where they modelled, using their arms, the graphsgraphs

Syllabus specification and key words were then Syllabus specification and key words were then covered with explanationcovered with explanation

Sketching the function (slide 7 ) was a teacher led Sketching the function (slide 7 ) was a teacher led explanationexplanation

Card matching activity used resource sheets that Card matching activity used resource sheets that accompany this lessonaccompany this lesson

Students were then given the handout with the Students were then given the handout with the opportunity to make their own notes against the six opportunity to make their own notes against the six card matching graphs, explaining to themselves how card matching graphs, explaining to themselves how the key features were identifiedthe key features were identified

Final activity was in pairs working on the questions at Final activity was in pairs working on the questions at the end of the handoutthe end of the handout

Role PlayRole Play

y = 0y = 0 x = 0x = 0 y = xy = x y = -xy = -x x = -yx = -y y = xy = x2 2

y = - xy = - x22

x = yx = y22

y = xy = x33

y = -xy = -x33

y = sin xy = sin x y = cos xy = cos x y = tan xy = tan x y = siny = sin22x + cosx + cos22xx

More role play……More role play……

G y = xG y = x

M y = 2xM y = 2x G y = xG y = x22

M y = 2xM y = 2x22

G y = xG y = x22

M y = (x-1)M y = (x-1)22

G y = 1/x, x < 0G y = 1/x, x < 0

M y = 1/x, x > 0M y = 1/x, x > 0 G y = 1/xG y = 1/x22, x < 0, x < 0

M y = 1/xM y = 1/x22, x < 0, x < 0

this time in pairs! One person to be George and the other this time in pairs! One person to be George and the other person Mildred.person Mildred.George to always stand in front of Mildred.George to always stand in front of Mildred.

CURVE SKETCHINGTreatment and sketching of graphs of rational functions.

FP1C1 •Be able to sketch the graph of y=f(x) obtaining information about symmetry, asymptotes parallel to the axes, intercepts with the co-ordinate axes, behaviour near x=0 and for numerically large x.

•Be able to ascertain the direction from which a curve approaches an asymptote.

•Be able to use a curve to solve an inequality.

Key wordsKey words

Rational functionRational function A function which can be expressed as N(x)/D(x) where N(x) and A function which can be expressed as N(x)/D(x) where N(x) and

D(x) are both polynomials and D(x) is not the zero polynomial.D(x) are both polynomials and D(x) is not the zero polynomial.

PolynomialPolynomial F(x) = aF(x) = a00+a+a11x+ax+a22xx22+a+a33xx33+…..+a+…..+annxxnn

AsymptoteAsymptote a straight line towards which a curve approaches but a straight line towards which a curve approaches but

ever meetever meet SketchSketch

show axis intersections, asymptotes, and behaviour of show axis intersections, asymptotes, and behaviour of the graph either side of any asymptote.the graph either side of any asymptote.

Sketch the graph y = (3-x)/(2-x)(4-x)Sketch the graph y = (3-x)/(2-x)(4-x)

check where graph crosses axes.check where graph crosses axes. look for vertical asymptotes.look for vertical asymptotes. find behaviour as x approaches infinity.find behaviour as x approaches infinity. consider approach towards asymptotes.consider approach towards asymptotes. check with Autograph or graphical check with Autograph or graphical

calculatorcalculator

Card matching activityCard matching activity

match the six graphs with the six match the six graphs with the six equationsequations

Graph A…..Eqn RGraph A…..Eqn RGraph B…..Eqn WGraph B…..Eqn WGraph C…..Eqn PGraph C…..Eqn PGraph D…..Eqn TGraph D…..Eqn TGraph E…..Eqn QGraph E…..Eqn QGraph F…..Eqn SGraph F…..Eqn S