Maths D (Fast Track) Year 9 (2 YEARS)

8/14/2019 Maths D (Fast Track) Year 9 (2 YEARS)

1/37

SCHEME OF WORK FOR SPN-21 (MATHEMATICS)YEAR 9 (FAST TRACK)

Content coverage Scope and Development Suggested Activities Resources

1. ALGEBRA 3( 3 weeks)

1.1 Factorisation ofQuadraticTrinomials

)( 2 cbxax ++

Review factorisation by finding the highestcommon factors, grouping and difference of twosquares.

Introduce factorisation ofax2 + b x + c where a = 1.

Proceed to cases where a 1. Writing thex-term as two terms, then perform

factorisation by grouping e.g.

)3(1)3(236235222

++=+=+ xxxxxxxx

)3)(12( += xx .

Extend the concept to factorisation ofax2 + b x y

+ cy2..

Extend to situations where a < 1 (e.g. 8 + 2 x -3 x

2)

Write a few algebraicexpressions and arrangethem according to the typeof factorisation they belongto.

Guide the students to

recognise the pattern ofquadratic trinomials.

Show some expansionse.g. (x+3)( x +2) =x 2 + 5 x+ 6

(x 3)( x 2) =x 2 5x+ 6

(x +3)( x 2) =x 2 +x 6

(x 3)( x +2) =x 2 x 6

Use the idea that expansionis the reverse offactorisation and guide thestudents to observe someimportant patterns.

Trial and Error with CrossMultiplication is a morepowerful method.For more able students, we

should encourage them to

http://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htm

http://www.coolmath.com/algebra/algebra-practice-problems.html

SPN-21 (Interim Stage) Year 9 Fast Track (2 + 2) Page 1 of 37
http://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.coolmath.com/algebra/algebra-practice-problems.htmlhttp://www.coolmath.com/algebra/algebra-practice-problems.htmlhttp://www.coolmath.com/algebra/algebra-practice-problems.htmlhttp://www.coolmath.com/algebra/algebra-practice-problems.htmlhttp://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.mathsteacher.com.au/year10/ch10_factorisation/06_further_quadratic_trinomials/furth.htmhttp://www.coolmath.com/algebra/algebra-practice-problems.htmlhttp://www.coolmath.com/algebra/algebra-practice-problems.htmlhttp://www.coolmath.com/algebra/algebra-practice-problems.html


2/37

just write down the result offactorisation by inspection.




3/37

1.2 Combined

Factorisation

Discuss the method of doing combinedfactorisations

e.g. 4y2 36 ; 2x2 + 6x - 20

Train the students to bealert to see whether a givenexpression can befactorised first by taking outthe common factor.

http://www.coolmath.com/algebra/Algebra2//04_what.htm

1.3 QuadraticEquations

(a) Solving byFactorisation

(b) Solving byTaking

Square Root

Explain that if 0=ba , then either a = 0 or b =0.

Apply the concept to solve quadratic equationsax2 + bx= 0 and ax2 + bx+ c =0 byfactorisation.

Extend to cases of(ax + b) 2 = c where c is not aperfect square.

Show that a quadratic equation of the form

0222 =bxa can be solved by factorisation e. g.

0492

=x gives 0)23)(23( =+ xx ,3

2=x

Show that the product oftwo factors being zeromeans that one of thefactors must be zero is thereasoning behind themethod of solution byfactorisation.Emphasise that there arealways two solutions forquadratic equation withspecial situations where theroots are repeated.

Summarise the key steps:1. Make one side of theequation

to become 0 2. Factorise the equation3. Equate each factor to 0and

solve the two linearequations

Summarise the differentsituations involvingquadratic expression (e.gx 2 -3 x 4) and quadraticequation (e.g. x 2 3x 4 =0).The final answer forx 2 3x4 is (x4)( x +1) whereasthe final answers for x 2 3x4 = 0 are x = 1 or 4

Show the students the

graph of y = x

2

4x +3

http://www.purplemath.com/modules/variant1.htm

http://www.coolmath.com/algebra/Algebra2//04_what.htmhttp://www.coolmath.com/algebra/Algebra2//04_what.htmhttp://www.coolmath.com/algebra/Algebra2//04_what.htmhttp://www.purplemath.com/modules/variant1.htmhttp://www.purplemath.com/modules/variant1.htmhttp://www.purplemath.com/modules/variant1.htmhttp://www.coolmath.com/algebra/Algebra2//04_what.htmhttp://www.coolmath.com/algebra/Algebra2//04_what.htmhttp://www.coolmath.com/algebra/Algebra2//04_what.htmhttp://www.purplemath.com/modules/variant1.htmhttp://www.purplemath.com/modules/variant1.htmhttp://www.purplemath.com/modules/variant1.htm


4/37

or3

2=x or bytaking square root on both sides.

eg 049 2 =x gives 492=x then

.3

2

9

4==x

Extend to cases of(ax + b) 2 = c where c is aperfect square.

and that the solutions ofthe quadratic equationx2 4x + 3 = 0 are thevalues of xwhere thegraph intersects the x-axis. [ solving graphically]




5/37

(c) Solving byusing

QuadraticFormula

Introduce the Quadratic Formula

a

acbbx

2

42 = to find solutions to

02 =++ cbxax .

Solve equations involving use of the quadraticformula.

Show an example of aquadratic equation thatdoes not factorise

e. g. 0242 =+ xx .

Introduce the quadraticformula to solve theequation and remindstudents the need toidentify a, b and c termsfirst and to be careful whenb and c are negativenumbers.

Guide the students to knowwhen this method is to be

used. The clue is when thequestion asks for theanswers to be given to acertain number of decimalplaces. This indicates thatthe expression cannot befactorised and thus have tobe solved using theQuadratic Formula.

Show clearly the correctway to write when b isnegative. (e.g. if b = -3,

then we have (-3) and then(-3) 2 for b2 ,not -32.

Check carefully whether thestudents are able to use thecalculator efficiently or not.

If the question asks foranswers to be rounded offto 2 decimal places, theworking values should berounded off to 3 or moredecimal places.



6/37




7/37

1.4 AlgebraicFractions

(a) Addition andSubtraction

(b) Multiplicationand

Division

Go through the addition and subtraction ofnumeric fractions and stress on the need to findthe LCM of the denominators.

Perform addition and subtraction of algebraicfractions with numerical denominators andfollowed by algebraic denominators.

Emphasise on putting brackets on denominatorsand numerators which are algebraic expressionsbefore simplifying the numerator.

Revise multiplication and division of arithmetic

fractions. Explain the method of multiplying algebraic

fractions.

Explain the method of division involving algebraicfractions.

Caution on the commonsign mistakes whenexpanding bracket with a

before the bracket.(e.g. for 2(x +3), somestudents may give as 2x +6

Emphasise on theappropriate way to writethe expression. E.g. (x 4)2should be written as 2(x 4). Similarly, (2x)(4y) shouldbe simplified to 8xy.

Use the numerical fractionsto recall the main idea,focussing on cancellationbetween numerators anddenominators.

For letter with powers,encourage the students touse the rules of indices.Do not encourage thestudents to expandbrackets as brackets will

help in final simplification.

Guide students to transferthe techniques tomultiplying and dividingalgebraic fractions thatrequire no factorisation.Multiply and dividealgebraic fractions thatrequire factorisation of thenumerator and or thedenominator.



8/37




9/37

1.5 Word Problems

Revise solving simple algebraic equations e. g. 3

2x= 7.

Identify key words and use key words totranslate word problems into an algebraicexpression or equation e. g. is (means = ),older than, increase, twice, etc.

Introduce symbols/letters to represent theunknown quantities and translate word problemsinto quadratic equations or algebraic fractionalequations, algebraic expression or equation.

Solve the algebraic equations and interpret thesolutions obtained. For examples, refer to PastO level Questions : Jun.2002/Paper2/Qs.10, Nov.2000/Paper2/Qs.11,Nov.2002/Paper2/Qs.7.

Start with situations whichare easier for students tovisualise. For e.g. the

comparison of the ages ofsome people (common keywords: older than, youngerthan etc). Introduce keywords like twice, half, total,average, three times etc ineach situation.

Some students may find ithard to stop at situation likex +3. Very common to findthis to be simplified to 3x

(Teachers must then stressthat in Maths x +3 is apossible answers, so isx 3etc).

Begin with problemsinvolving quadraticequations, then introducealgebraic equations withnumerical denominatorsand with three terms or lessthat are reducible to linear

equations in one unknown,

e. g. 16

5

4

13=

+ xx

3 x= 12

Provide examples with avariety of simple fractionalequations.Proceed with exampleswhere the denominator is inalgebraic form and are

reducible to linearSPN-21 (Interim Stage) Year 9 Fast Track (2 + 2) Page 9 of 37


10/37

equations in one unknownor to quadratic equations,e.g.

1

3

x

x

x

x

3

12 = 4




11/37

2. VARIATIONS(1 week)

2.1 Direct Variation

Define direct variation as the relationshipwhereby one quantity increases as the otherquantity increases in direct proportion and viceversa.

Express a direct variation in the form of anequation involving two variables and use theequation to find unknown quantities.

Begin work on variation byusing an example such as

the price of a piece of cloth,c being proportional to its

length, l, ie. lc .Show that the variables arein direct proportion, i.e.

,

2

1

2

1k

l

l

c

c== constant of

proportionality).Show how to form theequation connecting the

variables: c = kl.Relate the equation to y =mx and explain that thegraph ofc against lis astraight line passingthrough the origin.

Explanations andexamples of word

problems involvingdirect and inversevariations athttp://regentsprep.org/regents/math/variation/pracdirect.htm

http://www.ex.ac.uk/cimt/mepres/allgcse/bkc15.pdfsection 15.5onwards

2.2 Inverse Variation Define inverse variation as the relationshipwhereby one quantity increases as the otherquantity decreases and vice versa.

Explain that the two quantities are inverselyproportional to each other.

Express an inverse variation in the form of anequation involving two variables and use theequation to find unknown quantities.

Give an example thatrelates to the Less men,more share concept suchas If 12 men were tocomplete a job in 10 days,how long will it take tocomplete the job if 6 menwork on it? Show that thevariables are in inverse

proportion, i.e.1

2

6

12

2

1 ==n

n

while

.2

1

20

10

2

1 ==d

d

Explain that the product n xdis a constant.

Show that if n variesSPN-21 (Interim Stage) Year 9 Fast Track (2 + 2) Page 11 of 37
http://regentsprep.org/regents/math/variation/pracdirect.htmhttp://regentsprep.org/regents/math/variation/pracdirect.htmhttp://regentsprep.org/regents/math/variation/pracdirect.htmhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc15.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc15.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc15.pdfhttp://regentsprep.org/regents/math/variation/pracdirect.htmhttp://regentsprep.org/regents/math/variation/pracdirect.htmhttp://regentsprep.org/regents/math/variation/pracdirect.htmhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc15.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc15.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc15.pdf


12/37

inversely as d, in notation,

dn

1 , then

d

kn = ,

where k is a constant.


3. COORDINATEGEOMETRY 2(REVISIT)

(1 weeks)

3.1 Formulae forDistance,

Mid-point andGradient

Review formulae for distance, mid-point andgradient and solve problems that require the useof these formulae including finding one end-pointof a line segment given the midpoint and one

other end-point.

[Remark : Covered in Topic 7 in Year 2]

http://www.mathsisfun.com/equation_of_line.

html

http://www.mathsnet.net/asa2/2004/c2.html#

2

3.2 Gradient andParallel Lines

Solve problems on finding the gradient of astraight line and of lines parallel to a givenstraight line.

3.3 Equationof a Straight Line

Find the equation of the line when given

- the gradient and they-intercept,

- one point and the gradient,

- two points,

- one point and the equation of a parallel line,

- a diagram of a triangle or quadrilateral.

Introduce the use of

)11 ( xxmyy =

where ),( 11 yx is any point

on the lineand m is thegradient.

http://www.mathsisfun.com/equation_of_line.htmlhttp://www.mathsisfun.com/equation_of_line.htmlhttp://www.mathsisfun.com/equation_of_line.htmlhttp://www.mathsnet.net/asa2/2004/c2.html#2http://www.mathsnet.net/asa2/2004/c2.html#2http://www.mathsnet.net/asa2/2004/c2.html#2http://www.mathsisfun.com/equation_of_line.htmlhttp://www.mathsisfun.com/equation_of_line.htmlhttp://www.mathsisfun.com/equation_of_line.htmlhttp://www.mathsnet.net/asa2/2004/c2.html#2http://www.mathsnet.net/asa2/2004/c2.html#2http://www.mathsnet.net/asa2/2004/c2.html#2


13/37

3.4 Miscellaneous Problems

Solve miscellaneous problems including findingcoordinates of intersection points, the unknown xand y-coordinates, area of a triangle, etc.

Ensure that the studentsare able to state that thecoordinates of the x-intercept of a line is (x, 0)and y-intercept of a line

is (0,y).


4. GRAPHS OFFUNCTIONS

(2 weeks)

4. 1 Constructing atable of

values and drawinga smooth curve

Construct tables of values for functions of theformy = a xn where n = 2, 1, 0, 1, 2, 3.

Calculate the unknown y-value in the table ofvalues for a given equation

Explain the techniques of drawing graphs,stressing the importance of using the givenscale and to plot points accurately, then draw asmooth curve through all the points.

Get students to recognise the basic shapes ofthese graphs and sketch them.

Interpret graphs of linear, quadratic, cubic,reciprocal and exponential functions.

Begin with n = 0, 1 andshow that they are straightlines. Proceed to n = 2 forthe parabola, then n = 3 forthe cubic function before n= 1 for the hyperbola andn = 2.

Discuss the basic propertiesof the graphs for thedifferent values ofn. Advise

students to memorise thebasic shapes so that theycan sketch the graphseasily.

http://www.coolmath.c

om/algebra/PreCalc/01MoreGraphing/01_lovegraphs.htm

http://www.coolmath.com/algebra/Algebra1/11Quadratics/07_introgr

aphing.htm

http://www.mathsisfun.com/graph/index.html

4. 2 Finding the valuesof

variables from agraph

Determine from the graph the value ofy, giventhe value ofxand vice versa, includingmaximum and minimum values.

Explain that any function inthe formy = a xn + chasay-intercept ofc as it is atranslation of the graph ofy= a xnupwards by c units.Thex-intercept can befound by solvingy= 0.

http://www.coolmath.com/algebra/PreCalc/01MoreGraphing/01_lovegraphs.htmhttp://www.coolmath.com/algebra/PreCalc/01MoreGraphing/01_lovegraphs.htmhttp://www.coolmath.com/algebra/PreCalc/01MoreGraphing/01_lovegraphs.htmhttp://www.coolmath.com/algebra/PreCalc/01MoreGraphing/01_lovegraphs.htmhttp://www.coolmath.com/algebra/Algebra1/11Quadratics/07_intrographing.htmhttp://www.coolmath.com/algebra/Algebra1/11Quadratics/07_intrographing.htmhttp://www.coolmath.com/algebra/Algebra1/11Quadratics/07_intrographing.htmhttp://www.coolmath.com/algebra/Algebra1/11Quadratics/07_intrographing.htmhttp://www.coolmath.com/algebra/PreCalc/01MoreGraphing/01_lovegraphs.htmhttp://www.coolmath.com/algebra/PreCalc/01MoreGraphing/01_lovegraphs.htmhttp://www.coolmath.com/algebra/PreCalc/01MoreGraphing/01_lovegraphs.htmhttp://www.coolmath.com/algebra/Algebra1/11Quadratics/07_intrographing.htmhttp://www.coolmath.com/algebra/Algebra1/11Quadratics/07_intrographing.htmhttp://www.coolmath.com/algebra/Algebra1/11Quadratics/07_intrographing.htm


14/37

4. 3 Gradient of acurve

Estimate the gradient of a curve by drawing atangent to the curve at the given point andexplain that a tangent to a curve is a line thatjust touches the graph at that given point.

4. 4 Solve equations

by graphicalmethod

Solve the equations of the form (i) f(x) =g(x),(ii) f(x) = a (where a is a constant), using thegraph drawn.

4.5 Graphs in practicalsituations

Extend the skill in graph drawing to somepractical situations (E.g. between height andtime, profit and numbers of book printed etc)

Solve related problem using the graph drawn.

Content coverage Scope and Development Suggested Activities Resources5. INEQUALITIES

(1 weeks)

5.1 Meaning andsymbols

Define the symbols used in inequalities :

means greater than, means less than

or equal to and means greater than or equalto.

Compare the size of two numbers using thesymbols .

Use the number line to aidin the understanding of theinequality symbols.Get students to read, e.g.x> 3 asxis greater than 3.

http://home.xnet.com/~fidler/triton/math/review/mat085/linIneqone

/ineql.htm

http://home.xnet.com/~fidler/triton/math/review/mat085/linIneqone/ineql.htmhttp://home.xnet.com/~fidler/triton/math/review/mat085/linIneqone/ineql.htmhttp://home.xnet.com/~fidler/triton/math/review/mat085/linIneqone/ineql.htmhttp://home.xnet.com/~fidler/triton/math/review/mat085/linIneqone/ineql.htmhttp://home.xnet.com/~fidler/triton/math/review/mat085/linIneqone/ineql.htmhttp://home.xnet.com/~fidler/triton/math/review/mat085/linIneqone/ineql.htmhttp://home.xnet.com/~fidler/triton/math/review/mat085/linIneqone/ineql.htm


15/37

http://www.coolmath.com/

algebra/Algebra1/08System2x2/06_inequaliti

es.htm

5.2 Solve LinearInequality

List the values of a linear inequality such as 1x ,

2x , 32 use a circle or

dotted vertical line tomark the end point whereas for or use a

dot or solidvertical line to mark the end point).

Solve linear inequalities in one variable.

Solve simultaneous linear inequalities in onevariable.

Determine the possible solutions or solution setof a given inequality under various conditions.

Find the least and greatest sum, difference,product and quotient of two variables given in

two separate inequalities. (include their squares)

Caution students aboutfinding the greatest or leastvalues ofx2where x isgiven as a range thatstretches from negative topositive e.g. 35 x .Thegreatest value ofx2is 25and not 9 and the least value

is 0.Also for 35 x , theinequality for x2 is

2592 x .

5.3 GraphicalRepresentation ofInequalities

Review sketching of straight lines and writingequations for lines in a given diagram.

Remind students about the convention in usingsolid and dotted lines and indicate by a sketch,the region defined by an inequality (usually byshading the unwanted region).

Write the inequalities which define a region(usually unshaded) where the equations of theboundary lines are given or not given.

Explain how to obtain the region defined by asystem of linear inequalities.

Determine the maximum or minimum ofax +byfor a defined region by evaluating the expressionat the vertices of the polygon formed by theregion.

Caution the students toread the questions carefullyas it is not always right toshade the unwanted region sometimes the wantedregion should be shaded.


http://www.coolmath.com/http://www.coolmath.com/http://www.coolmath.com/http://www.coolmath.com/


16/37

6. LIMITS OFACCURACY( week)

6. 1 SignificantFigures,

Decimal Placesand

Estimation(Revisit)

Review the technique of rounding off numbers to

the required accuracies.

6. 2 Upper and LowerBounds Introduce the concept of absolute error as

2

1

smallest division of a measuring instrument,

and so a measurementx is written as (xabsolute error) where (x absolute error) is the

lower bound and (x + absolute error) is theupper bound.

Give appropriate upper and lower bounds fordata given to specified accuracy (e.g. measuredlengths).

Discuss the ideas of greatest and least values ofsum, difference, product and quotient.

Introduce the idea of lower bound and upperbound of a basic quantity from various types ofstatements. For example,(a) 8.6 cm measured correct to the nearest 0.1

cm,(b) 8300 correct to the nearest hundred

Show the common ways of expressing allpossible values of the given quantity ( l = 8.60

0.05 cm or 8.55 cm l < 8.65 cm) and themethod of obtaining the lower bound and upperbound from these expressions.

Apply lower bound or upper bound of basicquantities to find the least possible and thegreatest possible perimeter, area, volume, etc.

Use straight forwardexamples to determine upperand lower bounds of data.For example, a length, l,measured using an ordinaryruler as 3 cm (to the nearestmillimetre) has an absolute

error of 2

10.1 cm = 0.05

cm. This gives a

measurement of (3.00

0.05) cm which has a lowerbound of 2.95 cm and anupper bound of 3.05 cm.Show that this informationcan be written usinginequality signs e.g. 2.95 cm

l < 3.05 cm.

Investigate upper and lowerbounds for quantitiescalculated from givenformulae by specifying theaccuracy of the input data.

Discuss further exampleson lower and upper boundwhich includes:

5.62 correct to 3 sig.

figures,SPN-21 (Interim Stage) Year 9 Fast Track (2 + 2) Page 16 of 37


17/37

24.9 correct to 1 decimalplace.In each example, promptthe students to state anypossible value which givesthe stated value after

rounding off according tothe accuracy stated.Lead them to arrive at thelowest possible value (lowerbound) and the largestpossible value (upperbound).




18/37

7. TRAVEL GRAPHS (1 weeks)

7. 1 Distance-timegraph

Draw and interpret qualitatively distance-timegraphs (horizontal line stationary, sloping line

uniform speed, convex curve speed isdecreasing and concave curve speed isincreasing).

State that the gradient of the graph is the rate ofchange of the distance with respect to time, i.e.speed.

Use the formula Average Speed =

takentimeTotal

travelleddistanceTotal.

Solve problems involving distance time graphs.

Present a distance-timegraph and have students to

create their own storydescribing the journeyrepresented by the graph.Proceed to introduce theterms constant or uniformspeed, stationary, forwardjourney and returnedjourney.Extend to distance-timegraph with non-uniformspeed, which is a curve.

http://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtml

http://www.regentsprep.org/Regents/physics/

phys-topic.cfm?Course=PHYS&TopchicCode=01a

7. 2 Speed- time

Graph

Draw and interpret qualitatively speed-timegraphs (horizontal line constant speed, slopingline uniform acceleration or retardation, curve non-uniform acceleration or retardation).

State that the gradient of the graph is the rate ofchange of the speed with respect to time, i.e.acceleration (positive gradient) or retardation(negative gradient).

State that the distance travelled is equal to thearea under a speed-time graph and use it tosolve related problems.

Solve problems on the speed-time graphincluding finding the speed at a particular time.

Sketch the distance-time or acceleration-timegraph from the given speed-time graph.

Discuss that if the rate isconstant, the speed-timegraph will be a straight linewhose gradient is:- positive, when the speedisincreasing (accelerating)

uniformly,- negative, when the speedisdecreasing (retarding)

uniformly,- zero, when the speed isconstant(i.e. no acceleration).

Extend the discussion tothe speed-time graph whichaccelerate or decelerate

http://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev5.shtml


19/37

with non-uniform speeds. Inthis case the graph shows acurve.




20/37

8. FUNCTIONNOTATION

(1 week)

8.1 Introduction of

Functionand Evaluation of

f(x)

Explain the meaning of a function as a

relationship that maps an element of one setonto one and only one element in another set.

Explain that for an objectx, the image ofxunderfunction f is f(x) and introduce the domain as theset of objects and the range as the set of images.

Emphasize that there are two ways to indicatethe function notation i. e. f(x) = 3x 5read as fofxis equal to 3x 5and f : x 3x 5as f maps xonto 3x 5.

Find the image of a function by evaluating f(x).

Introduce a function, using

diagrams, as a one-onemapping or many-onemapping.Show that afunction has a one-to-onemapping or many-to-onemapping (i.e. it has exactlyone image only).Evaluate f(x) for specificvalues of x, describing thefunctions using f(x) notationand mapping notation.

Connect this to y = 3x 5.Here we say that y is thefunction of x and f(x) = 3x 5 is the same as y = 3x 5.

http://www.bbc.co.uk/

education/asguru/maths/13pure/02functions/index.shtmlhas some work oninverse functions.

Also search forfunctions athttp://www.learn.co.uk/

8.2 Finding andEvaluating

Inverse Function

Explain the meaning of an inverse function andthe notation used to represent an inversefunction.

Explain the method of finding an expression forthe inverse function and evaluate the inversefunction at a given value ofx.

Explain the method of evaluating an inversefunction without having to find the expression forthe inverse function first

e. g. To find )4(1f given the function f(x) =

3x 5, we let

)4(1f =x.The solution can then be found by

solvingf(x) = 4,giving 3x 5 = 4 , then x= 3.

Introduce the inversefunction as an operationwhich undoes the effect ofa function i.e.when f : x y, then

xyf :1 or when f( x) =

y, then xyf = )(1 .

Point out that only one-to-one function has an inverse.

Evaluate simple inversefunctions for specificvalues, describing thefunctions using f-1(x)notation and mappingnotation.

http://www.bbc.co.uk/education/asguru/maths/13pure/02functions/index.shtmlhttp://www.bbc.co.uk/education/asguru/maths/13pure/02functions/index.shtmlhttp://www.bbc.co.uk/education/asguru/maths/13pure/02functions/index.shtmlhttp://www.bbc.co.uk/education/asguru/maths/13pure/02functions/index.shtmlhttp://www.learn.co.uk/http://www.learn.co.uk/http://www.bbc.co.uk/education/asguru/maths/13pure/02functions/index.shtmlhttp://www.bbc.co.uk/education/asguru/maths/13pure/02functions/index.shtmlhttp://www.bbc.co.uk/education/asguru/maths/13pure/02functions/index.shtmlhttp://www.learn.co.uk/http://www.learn.co.uk/


21/37

8.3 Solving EquationsInvolving Functions

Solve equations involving functions using giveninformation e.g. given f(x) = 4, f(x) = g(x) , etc.


9. ARITHMETIC(3 weeks)

9.1 More on H.C.F. andL.C.M. Review common factors and common multiples.

Give students practice on word problemsinvolving HCF / LCM, for example, refer to pastO level questions : Nov.1998/Paper1/Qs.5 ,Nov.1999/Paper1/Qs.10.

9.2 Squares, squareroots,

cubes and cuberoots of

numbers

Evaluating without using calculator, square rootfor perfect squares and non-perfect squares.

Example 1

Find (a)4

12 , (b) 0009.0 , (c) 12100 .

Example 2

Given 808.127.3[ = , 718.57.32 = ,

evaluate (i) 3270 , (ii) 00327.0 .

Find the cube root for cubic numbers.



22/37

9.3 Directed numbers Use directed numbers in practical situations such

as in temperature change and tide levels.

Use number line to aidaddition and subtraction ofpositive and negativenumbers. Illustrate by usingpractical examples, e.g.temperature change andtide levels.

Stress that when findingdifference or change,always take the highervalue subtract the lowervalue.

http://www.ex.ac.uk/cimt/mepres/allgcse/bkb10.pdfhas work ondirected numbers.

Weather statistics forover 16000 cities athttp://www.weatherbase.com/

9.4 Time Calculate time in terms of the 12-hour and 24-hour clock: read clocks, dials and timetables.

Convert between hours, minutes and seconds.

Find the sum and differences of times.

Let students practise infinding information fromtime schedules. Calculatetime difference, departure

time, arrival time and timetaken for a plane/train totravel from one place toanother.

Suggestion: Instead ofthinking of a clock as around thing, it is easier tosee the relationship ofstarting time, duration oftime and finishing time if

we think of it as a straightline. This is especiallyuseful in situation wherethe finishing time is on thenext day.

Use locally-publishedtimetables e.g. forbuses.

Practice usingtimetables is athttp://www.ex.ac.uk/cimt/mepres/allgcse/bkb8.pdf.

Explain the idea on local time and the terms

used, (e.g. BSB is 8 hoursahead

of

Ensure that the studentsare able to convert betweenhours, minutes and secondsbefore finding the sum anddifference of times.

Caution the students to

http://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motio

nrev2.shtml

http://www.ex.ac.uk/cimt/mepres/allgcse/bkb10.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkb10.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkb10.pdfhttp://www.weatherbase.com/http://www.weatherbase.com/http://www.ex.ac.uk/cimt/mepres/allgcse/bhttp://www.ex.ac.uk/cimt/mepres/allgcse/bhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev2.shtmlhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkb10.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkb10.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkb10.pdfhttp://www.weatherbase.com/http://www.weatherbase.com/http://www.ex.ac.uk/cimt/mepres/allgcse/bhttp://www.ex.ac.uk/cimt/mepres/allgcse/bhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/physics/forces_and_motion/representing_motionrev2.shtml


23/37

London) and the method of finding local time.write the final answer fortime correctly (e.g. arrivaltime is 13 45 and not 13hours 45 minutes. Flighttime is 8 hours 30 minutesand not 08 30).

http://www.financefreak.com/

9.5 Financial Transaction andPercentage

Solve word problems involving financial transactions(review cost price, selling price, discounts, profit,loss, hire purchase, simple interest andcommissions).Examples of O Level questions :Nov

2000/P2/Q1,Nov 2002/P2/Q6

Solve word problems involving percentages calculate a given percentage of a quantity;express one quantity as a percentage of

another; percentage increase or decrease;calculations involving reverse percentages.


10. CONGRUENCEAND

SIMILARITY(2 weeks)

10.1 Congruence Meaning of congruent figures.

Understand and apply the tests for congruenttriangles (SSS, SAS, ASA or AAS and RHS)

Solve problems and give simple explanationsinvolving congruent triangles.

Discuss the conditions forcongruent triangles.

http://www.coolmath.com/congruent.html

http://regentsprep.org/Regents/math/congrue

n/Ttriangles.htm

http://www.gcseguide.co.uk/

http://www.financefreak.com/http://www.financefreak.com/http://www.coolmath.com/http://www.coolmath.com/http://www.coolmath.com/http://regentsprep.org/%20Regents/math/congruen/Ttriangles.htmhttp://regentsprep.org/%20Regents/math/congruen/Ttriangles.htmhttp://regentsprep.org/%20Regents/math/congruen/Ttriangles.htmhttp://www.gcseguide.co.uk/%20similar_triangles.htmhttp://www.gcseguide.co.uk/%20similar_triangles.htmhttp://www.financefreak.com/http://www.financefreak.com/http://www.coolmath.com/http://www.coolmath.com/http://regentsprep.org/%20Regents/math/congruen/Ttriangles.htmhttp://regentsprep.org/%20Regents/math/congruen/Ttriangles.htmhttp://regentsprep.org/%20Regents/math/congruen/Ttriangles.htmhttp://www.gcseguide.co.uk/%20similar_triangles.htmhttp://www.gcseguide.co.uk/%20similar_triangles.htm


24/37

similar_triangles.htm

http://regentsprep.org/Regents/math/similar/

Lstrategy.htm

http://www.ex.ac.uk/cimt/

mepres/book9/y9s14os.pdf

http://www.ex.ac.uk/cimt/mepres/book9/y9s14os.pdf

http://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/areaandv

olumerev1shtml

10.2 Similarity Meaning of similar figures.

Understand and apply the tests for similartriangles.Test 1: Two pairs of corresponding angles are

equal.

Test 2: Corresponding sides are in proportion.Test 3: Two pairs of sides are in the same ratio

and their included angles are equal.Solve problems and give simple explanationsinvolving similarity.

Use the fact thatcorresponding sides are inthe same ratio to calculatethe length of an unknownside.

10.3 Areas of SimilarPlane

Figures

When two figures are similar, the ratio of theareas= (the ratio of the corresponding lengths)2 i.e.

2

2

1

2

1

=

l

l

A

A.

Use this relationship to solve problems onareas of similar plane figures.

Note that other thencorresponding length, thecorresponding height, orsides can also be used.

10.4 Surface Areas andVolumes of Similar

Solids

For two geometricallysimilar solids,Ratio of the surface area = (ratio of the

corresponding lengths)2 i.e.

2

2

1

2

1

=

l

l

SA

SA.

Ratio of the volumes = (ratio of the

corresponding lengths)3

i.e.

3

2

1

2

1

= ll

V

V

.

Use these relationships to solve problems onsurface areas and volumes of geometricallysimilar solids.

Find in terms of the

surface area and volume ofspheres of radius 1 cm and 2cm and compare the results.Try with 3 cm and 5 cm radiiand compare the results ofthese two circles with thecircle of radius 1cm.Show how to relate some

situations to length or area orvolume. E.g. Price of drink ina container is proportional tothe volume of the drink.Cost of painting the surfaceof a container is proportionalto its area etc.


http://www.gcseguide.co.uk/%20similar_triangles.htmhttp://regentsprep.org/%20Regents/math/similar/Lstrategy.htmhttp://regentsprep.org/%20Regents/math/similar/Lstrategy.htmhttp://regentsprep.org/%20Regents/math/similar/Lstrategy.htmhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.bbc.co.uk/%20schools/gcsebitesize/maths/shapeih/areaandvolumerev1shtmlhttp://www.bbc.co.uk/%20schools/gcsebitesize/maths/shapeih/areaandvolumerev1shtmlhttp://www.bbc.co.uk/%20schools/gcsebitesize/maths/shapeih/areaandvolumerev1shtmlhttp://www.bbc.co.uk/%20schools/gcsebitesize/maths/shapeih/areaandvolumerev1shtmlhttp://www.gcseguide.co.uk/%20similar_triangles.htmhttp://regentsprep.org/%20Regents/math/similar/Lstrategy.htmhttp://regentsprep.org/%20Regents/math/similar/Lstrategy.htmhttp://regentsprep.org/%20Regents/math/similar/Lstrategy.htmhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.ex.ac.uk/cimt/%20mepres/book9/y9s14os.pdfhttp://www.bbc.co.uk/%20schools/gcsebitesize/maths/shapeih/areaandvolumerev1shtmlhttp://www.bbc.co.uk/%20schools/gcsebitesize/maths/shapeih/areaandvolumerev1shtmlhttp://www.bbc.co.uk/%20schools/gcsebitesize/maths/shapeih/areaandvolumerev1shtml


25/37

10.5 Scales and Mapproblems

Interpret the scale 1 : n as 1 cm on the map isequivalent to n cm on the ground.

Calculate the actual distance between twoplaces on a map, given its scale.

Explain the ideas of linear scale, area scale

and volume scale and the method of obtainingone scale from the other.the ratio of the areas = (the ratio of the


2

2

1

2

1

=

l

l

A

A,

Ratio of the surface area = (ratio of the


2

2

1

2

1

=

l

l

SA

SA,

Ratio of the volumes = (ratio of the


3

2

1

2

1

=ll

VV .

e.g. Linear scale is 1cm : 5 km,

Area scale is 1cm 2 : 25 km 2 and

volume scale is 1cm 3 :125 km 3 .

Explain the method of changing units (linear,area and volume units).

Calculate the distance on a map given the

scale and actual distance.

Measure the dimensions ofthe classroom, includingdoors and windows. Use asuitable scale, draw a planof the classroom on paper.Then calculate the area ofthe classroom floor and thevolume of the classroom.

Require students to bringtheir atlas or geographybook and apply theirknowledge on maps andscales to find the actualdistance between towns.

http://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtml

http://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/geography/geogskills/geogskillsmapsrev1.shtml


26/37


11. SYMMETRY

(1 week)

11.1 Line Symmetry Introduce the idea of symmetry of planefigures in general using practical examples likepaper folding, mirror images, live examplesfrom nature such as leaves and flowers, models,etc.

Recognise symmetrical figures, identify thelines of symmetry and determine the number oflines of symmetry.

Complete the missing part of a figure, givenits line(s) of symmetry.

Guide students to discover that a circle hasan infinite number of lines of symmetry.

Use paper cuttings andfoldings to demonstratethat certain shapes havelines of symmetry whereasothers may not have any.Get students to use papersand scissors to designshapes that have one lineof symmetry and othersthat have more lines of

symmetry.Select students cut-outsand paste them on a chartshowing the shapes and thenumber of lines ofsymmetry.

http://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhas useful work

on symmetry

http://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryr

ev2.shtmlhasinteractive

demonstrations and

http://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtmlhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtml


27/37

information aboutsymmetry

11.2 RotationalSymmetry

Introduce the idea of rotational symmetry.

Recognise figures which possess rotationalsymmetry and identify figures that have norotational symmetry.

Determine the centre of rotation and state

the order of rotational symmetry for givenfigures, shapes and logos.

Give examples of point of symmetry, notingthat the centre of rotational symmetry is a pointof symmetry if the order of rotational symmetryof the figure is a multiple of 2.

Discuss the symmetric properties ofequilateral and isosceles triangles, square,rectangle, rhombus, parallelogram, trapeziumand kite.

Introduce the idea ofrotation by demonstrationusing a teaching aid. Arotational symmetry boardcan be made as follows:

1. Draw on a manila card:rectangle, equilateraltriangle, square,rhombus, regularpentagon, parallelogram,isosceles triangle,scalene triangle andtrapezium.

2. Draw the same figureson

another manila card ofdifferent

colour and cut out thefigures.3. Secure the cut-outs overtheir

respective figures on thebig card

(Step 1) using pinsthrough the

centre of rotation.4. Rotate the cut-outs oneby one

and explain the idea ofrotational

symmetry. Note the cutouts

rotate about the fixedpoint called

the centre of rotation.




28/37

11.3 SymmetricalProperties

of RegularPolygons

Discuss line symmetry and rotational symmetryproperties of the regular polygons: equilateraltriangle, square and other regular polygons.

Find the lines of symmetry, the centre and theorder of rotational symmetry of the regularpolygons.

Give materials to studentsto design shapes with

- specified number oflines of symmetry

- specified order ofrotational symmetry

An example is this figurewith order of rotationalsymmetry

=6

11.4 Symmetry inSolids

Introduce the idea of symmetry of solids ingeneral using models such as cubes, cuboids,cylinders, cones and pyramids, etc.

Recognise symmetry with respect to a plane.

Explain the technique to identify an axis ofrotational symmetry of a solid with its respectiveorder of rotational symmetry.

Discuss solids with an infinite number ofplane symmetry such as spheres, cylinders, etc.

Ask the students toconstruct the prisms toenable them to see thesymmetry properties more

easily. Cut the solids intotwo equal parts and identifythe plane of symmetry.Give examples of solidswith no plane symmetrysuch as irregular solids.



29/37


12. PROPERTIES OFCIRCLES(2 weeks)

12.1 SymmetryProperties of

Circles

12.2 AnglesProperties of Circles

Identify the terms circumference, radius,diameter, chord, segment (major and minor),

sector, arc and semicircle.

Use the following symmetry properties ofcircles to calculate unknown sides and anglesand give simple explanations:

(a) equal chords are equidistant from the centre,

(b) the perpendicular bisector of a chord passesthrough the centre,

(c) a tangent to a circle is perpendicular to the

radius of the circle at point of contact,(d) two tangents from an external point to a

circle are equal in length,(e) the angle between two tangents drawn from

an external point to a circle is bisected bythe line through the external point and thecentre of the circle.

Identify and use the following anglesproperties of circles to calculate the unknownangles and give simple explanations:(a) angle at the centre is twice angle at the

circumference,

(b) angle in semicircle is equal to 90,

(c) angle in the same segment are equal,

(d) angles in opposite segments (or oppositeangles of a cyclic quadrilateral) add up to180,

(e) external angle of a cyclic quadrilateral isequal to the opposite interior angle,

(f) angles in alternate segments are equal,

Let the students explore theproperties of chords andtangents by drawingdiagrams and cut out.Measure the lengths andangles to see therelationships and hencegeneralize the properties.(Use the properties ofisosceles triangles, congruenttriangle and the exterior

angle to a triangle, etc.)Have students paste all thecut out circles onto their notebooks.Explain the term tangent asthe line which touches thecircle at only one point. Makestudents practise drawingtangents.

Let students explore the

angles properties of circlesby using diagrams. Requirestudents to measure theangles or use paper cut outto compare the angle sizeand their relationship. Hencegeneralize the properties.

Caution: for the correct pairon angle at the centre, angleat the circumference andangle in the same segment,both angles must be

http://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/circles2hirev10.shtml

Sections 3.8 and 3.9 ofhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdf

There are interactiveinvestigations aboutthe angle properties athttp://teachers.henrico.k12.va.us/math/rd03/GeometryActs/CircleAngle01.html

DiscoveringMathematics 3A, Unit6.

http://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/circles2hirev10.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/circles2hirev10.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/circles2hirev10.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/circles2hirev10.shtmlhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://teachers.henrico.k12.va.us/math/rd03/GeometryActs/CircleAngle01.htmlhttp://teachers.henrico.k12.va.us/math/rd03/GeometryActs/CircleAngle01.htmlhttp://teachers.henrico.k12.va.us/math/rd03/GeometryActs/CircleAngle01.htmlhttp://teachers.henrico.k12.va.us/math/rd03/GeometryActs/CircleAngle01.htmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/circles2hirev10.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/circles2hirev10.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/circles2hirev10.shtmlhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://teachers.henrico.k12.va.us/math/rd03/GeometryActs/CircleAngle01.htmlhttp://teachers.henrico.k12.va.us/math/rd03/GeometryActs/CircleAngle01.htmlhttp://teachers.henrico.k12.va.us/math/rd03/GeometryActs/CircleAngle01.html


30/37

subtended by the samechord (usually the chord isnot drawn).Emphasize that in cyclicquadrilateral all the fourvertices of the quadrilateraltouches the circumference ofthe circle.


13. TRIGONOMETRY(4 weeks)

13.1 Solutions of Right-angled

Triangles

Review trigonometric ratios of sine, cosine andtangent (SOH, CAH, TOA) and Pythagoras

theorem and use them to find the unknownangles or sides in a given right-angledtriangle.

http://www.mathsnet.net/asa2/2004/c2.html#

4

http://www.waldomaths.com/SinRule1NL.jsp

13.2 Sine Rule State the sine rule.

Use the sine rule to solve non right-angled

triangles.

Draw triangle ABC withAB = 6 cm, BC = 7 cm andCA= 8 cm. Measure anglesA, B and C. Calculate (i)

C

AB

sin, (ii)

A

BC

sinand (iii)

B

CA

sin.

Repeat the above activityusingAB= 10.6 cm, BC =7.2 cm and CA = 9.3 cm.

13.3 Cosine Rule State the cosine rule.

Use the cosine rule to solve non right-angled

triangles.

Point out the situations when sine rule and

cosine rule should be used.

Draw triangle ABC with a =8 cm, b = 6 cm and c = 7

cm. Measure C .Calculate (i) Cos C and

(ii)ab

cba

2

222 +. Repeat

the above activity using a =6.5 cm, b = 8.5 cm and

http://www.sailingissues.com/navcourse4.ht

ml

Maps from around theworld at

http://www.theodora.com/maps/abc_world_m

aps.html

http://www.sailingissues.com/navcourse4.htmlhttp://www.sailingissues.com/navcourse4.htmlhttp://www.sailingissues.com/navcourse4.htmlhttp://www.theodora.com/maps/abc_world_maps.htmlhttp://www.theodora.com/maps/abc_world_maps.htmlhttp://www.theodora.com/maps/abc_world_maps.htmlhttp://www.sailingissues.com/navcourse4.htmlhttp://www.sailingissues.com/navcourse4.htmlhttp://www.sailingissues.com/navcourse4.htmlhttp://www.theodora.com/maps/abc_world_maps.htmlhttp://www.theodora.com/maps/abc_world_maps.htmlhttp://www.theodora.com/maps/abc_world_maps.html


31/37

c = 10 cm.

13.4 Area of Triangle State the formula of the area of triangle =

Cab sin2

1.

Use the formula to solve related problems.

13.5 Bearings Find the bearing of a point from another point(always measure clockwise from the north line

and the bearing must be stated in three digits). Recall the angle properties of parallel lines,angles at a point and angle properties of triangleand use these properties to solve problems onbearings.

Solve trigonometric problems (includeproblems incorporating speed, distance andtime).

Identify places according totheir bearings and

distances from a givenplace, or according to theirbearings from two differentplaces.


13.6 Three

DimensionalProblems

Identify right angles in diagrams of 3-D

objects (e.g. prisms, pyramids, wedges etc).

From the 3-D diagram, draw right-angledtriangles using

horizontal and vertical lines instead of slantlines as seen

from the 3-D diagram.

Use the right-angled triangles drawn to solve theproblems.

Solve problems involving angle of elevationand angle of depression, stressing that these areangles between the line of sight and the

Include cases where sine /

cosine rule may be used tosolve 3 D problems

Various problems at

http://nrch.maths.org/public/leg.php

http://nrch.maths.org/public/leg.phphttp://nrch.maths.org/public/leg.phphttp://nrch.maths.org/public/leg.phphttp://nrch.maths.org/public/leg.php


32/37

horizontal. Include problems on finding thegreatest angle of elevation.




33/37

14. MENSURATION(2 weeks)

14.1 Perimeter andArea

(a) Perimeter and

Areaof Common

Figures

(b) Arc Length andArea

of Sector

(c) Perimeter andArea

of CompositeFigures

Review formulae for perimeter and area ofsquares, rectangles, triangles, the area ofparallelograms and trapeziums, circumferenceand area of circles.

Review parts of a circle chord, arc, sectorsand segments.

Show the relation between arc length andcircumference.

Show the relation between the area of sectorand area of circle.

Solve problems involving the perimeter andarea of common figures including the arc lengthand the area of sector of a circle.

Solve problems involving the perimeter andarea of composite figures including finding thearea of a segment.

Revise, usingstraightforward examples,how to calculate the

perimeter and area ofsquares, rectangles andtriangles, the area ofparallelograms andtrapeziums. It may behelpful to show studentshow the area formulae forparallelograms andtrapeziums may beobtained by splitting theminto two triangles.Also, revise the calculation

of circumference and areaof a circle, then, by usingthe concept of directproportion, show how toderive the formula for arclength and sector area.

For perimeter of acomposite figure, start fromany point at the edge of thefigure, go around the figurealong the edge until thestarting point is reached.The perimeter is the sum ofall the sides.For area of a compositefigure, draw dotted lines tosubdivide the compositefigure into common figures.Find the area of eachcommon figure. Add thearea of all common figuresin the filled (usually

Background about theformulae for area and

circumference, and

may be found athttp://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi through theages.htmlRevision site for arcsand sectors athttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtml

http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi%20through%20the%20ages.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi%20through%20the%20ages.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi%20through%20the%20ages.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi%20through%20the%20ages.htmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtmlhttp://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi%20through%20the%20ages.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi%20through%20the%20ages.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi%20through%20the%20ages.htmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/circlesanglesarcsandsectorsrev3.shtml


34/37

shaded) region and subtractall those which are holes(usually unshaded).




35/37

14.2 Surface Area andVolume.

(a) Total SurfaceArea andVolume of

Common Solids

(b) Total SurfaceArea and Volumeof Pyramids,Cones andSpheres

(c) Total SurfaceArea andVolume ofCompositeSolids

Review formulae for surface area and volumeof cubes, cuboids, prisms and cylinders.

Introduce total surface area and volume ofpyramids, cones and spheres.

Solve problems involving the surface areaand volume of cubes, cuboids, prisms, cylinders,pyramids, cones and spheres (formulae will begiven for pyramid, cone and sphere).

Solve problems involving surface area andvolume of various composite solids includingproblems on the mass of an object using therelation that mass = density volume.

Draw the nets of someprisms and construct theprisms. This activity couldbe set as a task to design a

storage container, leadingto the discussion of surfacearea and volume.

Show by usingsand/coloured water therelation between volume ofpyramids and prisms of thesame base area.

Using the same method toshow that volume of cone is1/3 of that of a cylinder ofthe same base.

For composite solids,subdivide it into commonsolids and find the volumeof each of them. Then addor subtract accordingly.

http://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/index.sht

ml

http://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/index.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/index.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/index.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/index.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/index.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/index.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/index.shtml


36/37


15. SIMPLECONSTRUCTIONSAND LOCI (2weeks)

15.1 SimpleConstructions

Construct simple geometrical figures such astriangle or quadrilateral from given data.

Constructangle bisectors, perpendicularbisectors and parallel lines.

Revise on constructing trianglesfrom different data, given threesides, a side and two angles, ortwo sides and an angle. Includealso construction of some othergeometrical figures, such assome quadrilaterals.Give furtherpractice in constructingperpendicular and anglebisectors.

http://www.mathforum.org/library/topics/constructionshas links forteachers aboutconstructions, givingbackground and ideas

15.2 Scale DrawingRead and make scale drawings.

Apply the construction skills tomaking scale drawings, usingsimple scales only. Drawvarious situations to scale andinterpret results, for example,draw a plan of a room to scaleand use it to determine thearea of carpet needed to coverthe floor.

http://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhas work onscale drawings atsection 3.7

http://www.mathforum.org/library/topics/constructionshttp://www.mathforum.org/library/topics/constructionshttp://www.mathforum.org/library/topics/constructionshttp://www.mathforum.org/library/topics/constructionshttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.mathforum.org/library/topics/constructionshttp://www.mathforum.org/library/topics/constructionshttp://www.mathforum.org/library/topics/constructionshttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka3.pdf


37/37

15.3 LocusUse the following loci and the method of intersectingloci:

(a) sets of points in two or three dimensions(i) which are at a given distance from agiven point,

(ii) which are at a given distance from a

given straight line,(iii) which are equidistant from two givenpoints.

(b) sets of points in two dimensions whichare equidistant

from two given intersecting straight lines.

Introduce the idea of locus byusing examples in theclassroom. I want to stay 1m from this chair/ from thiswall. Where can I go?or askstudents to imagine a point

marked at the end of a bladeof the ceiling fan and followits path as the fan moves.Generalise the method tomemorise:One point implies circle,Two points impliesperpendicular bisector,One line implies parallellines,Two intersecting lines impliesangle bisectors.Progress using pencil and

paper to draw accurate scaledrawings to represent loci intwo dimensions.Include examples ofintersecting loci, for example,given a diagram showing thepositions of villages A and B:Ali lives less than 4 km fromvillage A. He lives nearer tovillage B than to village A.Shade the region where Alilives.

http://www.ex.ac.uk./cimt/mepres/allgcse/bkc14.pdf

http://www.ex.ac.uk./cimt/mepres/allgcse/bkc14.pdfhttp://www.ex.ac.uk./cimt/mepres/allgcse/bkc14.pdfhttp://www.ex.ac.uk./cimt/mepres/allgcse/bkc14.pdfhttp://www.ex.ac.uk./cimt/mepres/allgcse/bkc14.pdfhttp://www.ex.ac.uk./cimt/mepres/allgcse/bkc14.pdfhttp://www.ex.ac.uk./cimt/mepres/allgcse/bkc14.pdf

Documents

Maths D (Fast Track) Year 9 (2 YEARS)