Maths D (Normal Track) Year 10 (3 YEARS)

8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)

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SCHEME OF WORK FOR SPN-21 (MATHEMATICS)YEAR 10 NORMAL TRACK (2 + 3)

Content coverage Scope and Development Suggested Activities Resources

1. SYMMETRY(2 week)

1.1 Line Symmetry Introduce the idea of symmetry of plane

figures 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 workon symmetry

http://www.bbc.co.uk/schools/gcsebitesize/maths/shape/symmetryrev2.shtml hasinteractivedemonstrations andinformation aboutsymmetry

SPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 1 of 23
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.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


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1.2 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 statethe 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 the

figures.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.


1.3 SymmetricalProperties

of RegularPolygons

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

Give materials to studentsto design shapes with

- specified number of

lines of symmetrySPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 2 of 23


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Find the lines of symmetry, the centre and the

order of rotational symmetry of the regularpolygons.

- specified order ofrotational symmetry

An example is this figurewith order of rotational

symmetry =6

1.4 Symmetry in Solids

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 the

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



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2. PROPERTIES OFCIRCLES

(4 weeks)

2.1 SymmetryProperties of

Circles

2.2 Angles Propertiesof 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 theradius of the circle at point of contact,

(d) two tangents from an external point to acircle are equal in length,

(e) the angle between two tangents drawn froman 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, congruent

triangle and the exteriorangle 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 theangles 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,

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


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both angles must besubtended by the samechord (usually the chord isnot drawn).Emphasize that in cyclicquadrilateral all the fourvertices of the quadrilateral

touches the circumference ofthe circle.


3. TRIGONOMETRY(6 weeks)

3.1 Solutions of Right-

angledTriangles

Review trigonometric ratios of sine, cosine and

tangent (SOH, CAH, TOA) and Pythagorastheorem 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

3. 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.

3.3 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 = 7cm. 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.html

Maps from around theworld athttp://www.theodora.com/maps/abc_world_maps.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


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c = 10 cm.

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

Cab sin2

1.

Use the formula to solve related problems.

3.5 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 anddistances from a givenplace, or according to theirbearings from two differentplaces.


3.6 Three Dimensional

Problems

Identify right angles in diagrams of 3-D

objects (e.g. prisms, pyramids, wedges etc). From the 3-D diagram, draw right-angled

triangles usinghorizontal and vertical lines instead of slant

lines as seenfrom 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

horizontal. Include problems on finding 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


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greatest angle of elevation.




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4. MENSURATION(3 weeks)

4.1 Perimeter and Area(a) Perimeter and

Area

of CommonFigures

(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, circumference

and 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 figure

along 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

shaded) region and subtract

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


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all those which are holes(usually unshaded).




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4.2 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 is

1/3 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


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5. SIMPLECONSTRUCTIONSAND LOCI (3weeks)

14.1 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

5.2 Scale Drawing Read 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


12/23

5.3 Locus Use 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


6. MATRICES(3 weeks)

6.1 Introduction andBasic

Definition

Define matrix (plural matrices) as a rectangulararray of elements (usually numbers) arranged inrows and columns.

Explain that a matrix with m rows and n columnsis said to have order m x n (read as m by n).

Define thedifferent types of matrices: row

Introduce matrix bydisplaying information inthe form of matrices ofdifferent orders.For examples :

a) The marks of twostudents in English,

Science and History:

http://www.sosmath.com/matrix/matrix0/matrix0.htmlhas introduction tomatrix algebra.

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.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.htmlhttp://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.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.html


13/23

matrix, column matrix, square matrix, diagonalmatrix, null matrix, identity matrix or unit matrixand equal matrix.

Student A obtained 70marks for English,87 marks for Scienceand 56 marks for History.Student B obtained 72marks for English, 80marks for Science and 70marks for History.

78 07 2

58 77 0or

70

80

72

56

87

70

b) The sales of a

department store for 2items on 2 successivedays:Thursday : 10 bags, 12

belts;Friday : 8 bags, 5

belts.

58

1210or

512

810

Explain briefly how theSPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 13 of 23


14/23

matrix is formed and whateach row and columnrepresent.


6.2 Matrix Addition,Subtraction andMultiplication by a

Scalar

Showthe addition and subtraction of twomatrices.

Show the multiplication of a matrix by a scalarquantity.

When doing subtraction,give strong emphasis thatthe minus sign should notbe touched whenmultiplying the scalar of thesecond matrix with theelements of that matrix. Forexample,

51

512

14

32=

102

102

14

32

A common mistake at thisstep is

102

102

14

32

6.3 MatrixMultiplication

Explain the technique of the multiplication of twomatrices.Emphasize that two matrices can onlybe multiplied when the number of columns in thefirst matrix is the same as the number of rows inthe second matrix.

Show the results that AB BA.(except for multiplication by identity matrixwhere IA = AI).

Use real life example toshow the logic ofmultiplying row withcolumn. You may use theexample stated above. Thatis considering the sales of adepartment store for the 2items on 2 successive days.In addition, let the price ofthe bag be $8 per piece andthe belt at $3 per piece.



15/23

Present the aboveinformation in matrix form.Explain clearly how tocalculate the total amountof money received by thestore for the two days sales.

Explain how the row in thefirst matrix is related to thecolumn in the secondmatrix so that it can bemultiplied.


Hence, generalize thetechnique and proceed toshow the technique ofmultiplication of two (2x 2) matrices and matricesof different orders:

(a) Label the rows of thefirst matrix R1, R2 etcand the columns of thesecond matrix C1, C2 etcand then calculate R1C1,R1C2 etc outside the

main step. Aftermultiplying all the rowsand columns, write downall the products followthe row and columnnumbers in the resultantmatrix.

(b) Making summary Rowx Column.

(c) Stress on theimportance of correctorder for the answer.



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6.4 Matrix Equations Solve matrix equation where the unknowns are

elements.

Solve matrix equation where the unknown is amatrix.

6.5 Determinant andInverse

of a 2 x 2 Matrix Define the determinant of a matrix,if

=

dc

baA ,

then det A= bcadA = .

Calculate the determinant of a matrix.

Define non-singular matrix as matrix whosedeterminant is non-zero and singular matrix asmatrix whose determinant is zero and it has no

inverse. Show the method of finding the inverse of a non-

singular matrix.

(A 1 =

ac

bd

Adet

1).

Solve problems with given value of determinant andfind the unknown element in the matrix.

Find unknown element in matrix which has noinverse.

Caution students on thecommon mistake of using+ instead of whencalculating determinantbecause sometimes theycan get mixed up with theprocedure in doingmultiplication of matrices.


6.6 Identity Matrix Explain that an identity matrix, I is a square

matrix whose elements in the principal diagonalare 1 and the other elements are zero. e.g. I =

10

01,

100

010

001

.

Show using examples the properties that

IA = AI = I, AA 1 = I and A 1 A = I.



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6.7 Application ofMatrices

Show how to place data into matrix form andinterpret elements in a matrix as related to thegiven information.

Show how to solve the problems and hence

interpret the results.

Recall the example given insection 1.3.

To interpret the result ofmultiplication of twomatrices, guide thestudents to tell what is thequantity in the first matrix(R1) and what is thequantity in the secondmatrix (C1) and when thesetwo quantities (R1 and C1)are multiplied, what do weobtain?Also in situations wherethere are more than oneelement in each row of thefirst matrix, what do weobtain when the productsare added (i.e. R1C1+ R2C2 etc)?


7. TRANSFORMATIONS(7 weeks)

7.1 Translation Introduce translation as a transformation that

Explain that http://www.bbc.co.uk/s

http://www.bbc.co.uk/schools/gcsebitesize/maths/%20shape/transformationsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/%20shape/transformationsrev1.shtml


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moves all objects through a fixed distance in afixed direction.

Show examples where students have to findimages of the figures when given a translation ina diagram or description in words.

Describe fully in words the translation given in a

diagram by stating the translation vector

k

h .

transformations act uponobject points would changethem (in terms of position)into image points. When anobject figure is transformedinto an image figure, therecould be changes in theshape and size of theimage. The transformationsof translation, reflectionand rotation are isometricas they do not cause anychanges in shape or size i.e.the objects and images arecongruent.

chools/gcsebitesize/maths/shape/transformationsrev1.shtml

7.2 Reflection Introduce reflection as a transformation that

reflects an object point in the line of reflection

onto its image point. Discuss properties ofreflection in terms of the object distance equalsthe image distance and the line of reflection isperpendicular to the line joining the object pointand the image point.

Show examples where students have to draw theimages for individual points when given a line ofreflection. Focus on the x- andy-axes, linesparallel to the axes,y = xandy = x.

Extend the concept to figures and show that ifABC is labelled in the clockwise direction,

then the image, 111 CBA will be in theanticlockwise direction and vice versa.

Given a point P and its image P1 on a diagram,explain that the line of reflection is actually theperpendicular bisector of the line PP1. anddescribe the reflection fully by stating theequation of the line of reflection.

Relate reflection to study ofreflection of light in science

as the same propertiesapply especially theconcept of lateral inversion.

This property is important

as it helps students todistinguish between areflection and a rotationwhen asked to describe atransformation.

http://www.bbc.co.uk/schools/gcsebitesize/maths/%20shape/transformationsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/%20shape/transformationsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/%20shape/transformationsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/%20shape/transformationsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/%20shape/transformationsrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/%20shape/transformationsrev1.shtml


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7.3 Rotation Introduce rotation as a transformation that

moves an object point through a fixed angleabout a centre of rotation in a certain direction.

Show examples where students have to draw theimages for figures under a given rotation. Focuson rotations of multiples of 90.

Given a diagram showing an object and itsimage, explain that the centre of rotation is thepoint of intersection of the perpendicularbisectors of two lines, each joining one objectpoint to its image point.

Stress that a rotation must be described fully bystating the centre of rotation, the angle anddirection (except 180o rotation) it moves through.

Show that ABC and itsimage 111 CBA are

labelled in the same sensewhich distinguishes arotation from a reflection.

7.4 Enlargement Introduce enlargement as a transformation thatchanges the position of an object point from acentre of enlargement by a scale factor k.

Show that when k > 0, the image is on the sameside of the centre as the object and when k < 0, theobject and image are on opposite sides.

Draw images for objects given the description of theenlargement.

Show that when k > 1, the image is enlarged

and when k < 1, the image is reduced and

introduce the concept that

2factor)(scale

objectofarea

imageofarea = in relation to

similar figures.

Given a diagram showing an object and its image,explain that the centre of enlargement, C, is thepoint of intersection of the two lines, each joiningone object point P to its image point P1 and the

Introduce enlargement as atransformation that is notisometric and the size ofthe figure changes but theshape remains the shape.This means that the objectand image are similar.

Use the work on similarfigures to link toenlargement.Derive the ratio for similartriangles and relate it to thescale factor of enlargement

Show that an enlargementof scale factor kwillproduce an areaenlargement of scale factor



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scale factor,CP

CPk

1= .

Stress that an enlargement must be described fullyby stating the centre of enlargement and its scalefactor (positive or negative).

k2 and volume scale factorofk3.


7.5 Shear Introduceshear as a transformation that moves an object

point parallel to a line called the invariant line (x-axis or y-axis).

Stressthat points on invariant line do not move under ashear.

Give thedefinition of shear factor and show how to applythe definition to locate the position of the imagepoint.(Caution on situations where the object point ison the negative region of the invariant line andalso where the shear factor is negative).

Given ashear and a figure (e.g. triangle), draw and labelthe image of the figure.

Recognisea shear by its properties, i.e. changing in shapebut not in size.

Given anobject figure and its image figure, describe ashear completely (the description must includethe word shear, the invariant line and the shearfactor).

Stack up some books (sameheight)) on the table. Use aruler and apply a horizontalshear force to the books.Indicate the three obviouseffects:

(i) the book on thetable does not move.Use this effect toexplain the meaningof invariant line.

(ii) all the booksmovement areparallel to the table

top. Use this effect toexplain that a shearmoves points parallelto the invariant line.

(iii) The higher thebooks height, themore it moves. Usethis effect to explainthe definition of shearfactor.

To show that size does notchange under a shear,

http://www.mathsisfun.com/definitions/transformation.html

http://www.bbc.co.uk/schools/gcsebitesize/maths/shape/transformationsrev1.shtml

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apply the formula for areaof triangle (1/2

base height) on both the

object and image (this is agood revision to find thearea of a triangle when it is

drawn on a grid).

7.6 Stretch Introducestretch as a transformation that moves an objectpoint perpendicular to a line called the invariantline (x-axis or y-axis).

Stressthat points on the invariant line do not moveunder a stretch.

Give thedefinition of stretch factor and show how to applythe definition to locate the position of the imagepoint.

Given astretch and a figure (e.g. triangle), draw andlabel the image.

Recognizea stretch by its properties. A stretch changes

Use a geoboard and rubberbands to show a stretch.Indicate the three effects:

(i) All points on theinvariant line do notmove,

(ii) every point movesperpendicular to theinvariant line,

(iii) the amount ofmovement of anypoint depends on itsdistance from theinvariant line.

http://mathworld.wolfram.com/Stretch.html

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both the shape and size (the object can becomebigger or smaller) of the object.

Given anobject figure and its image figure, describe astretch completely (the description must includethe word stretch, the invariant line and the

stretch factor).


7.7 Combined

Transformation Explainthe notation used for single transformation (e.g.T(A) is the image of A under the Translation, T).

Explainthe notation used for combined transformation(e.g. ET(A) is the image of point A under thetranslation ,T followed by the Enlargement, E).

Given anobject figure and a combined transformation,either expressed in notation or in words, drawand label the image figure.

7.8 Use of Matrix inTransformations

Use theidea that a transformation maps an object to animage to establish the quantitative relationship(Matrix) (Object) = (Image), except for

Translation is (Matrix) + (Object)= (Image).

Representthe object as a matrix withx-coordinates as theelements in the first row andy-coordinates as theelements in the second row.

Use theresults of the multiplication of (Matrix)

Review the method ofmultiplying two matrices.

http://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalhttp://www.uz.ac.zw/science/maths/zimaths/73/sheila.html/linalg5.html

http://www.mathsfiles.com/excel/MatrixTransNotes1.htm

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(Object) to indicate the coordinates of thevarious image points corresponding to eachobject point.

Given atransformation represented by a matrix and afigure, find the coordinates of the image pointsand draw and label the image.

Writedown a matrix which represents a giventransformation.

Extend the idea of

(Matrix) (Object) =(Image) and the idea of M

10 01 = M to show that

the matrix representing agiven transformation can beobtained by mapping thepoint (1, 0) and (0, 1) totheir respective imagesunder that transformation.The elements of the matrixare the coordinates of theimages in that order.

http://www.uz.ac.zw/science/maths/zimaths/73/sheila.html

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Documents

Maths D (Normal Track) Year 10 (3 YEARS)