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8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
1/23
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.shtml8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
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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.
Content coverage Scope and Development Suggested Activities Resources
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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Content coverage Scope and Development Suggested Activities Resources
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.
SPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 4 of 23
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.html8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
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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.
Content coverage Scope and Development Suggested Activities Resources
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
SPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 5 of 23
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.html8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
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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.
Content coverage Scope and Development Suggested Activities Resources
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
SPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 6 of 23
http://nrch.maths.org/public/leg.phphttp://nrch.maths.org/public/leg.phphttp://nrch.maths.org/public/leg.phphttp://nrch.maths.org/public/leg.php8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
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greatest angle of elevation.
Content coverage Scope and Development Suggested Activities Resources
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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
SPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 8 of 23
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.shtml8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
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all those which are holes(usually unshaded).
Content coverage Scope and Development Suggested Activities Resources
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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
SPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 10 of 23
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.shtml8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
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Content coverage Scope and Development Suggested Activities Resources
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
SPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 11 of 23
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.pdf8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
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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
Content coverage Scope and Development Suggested Activities Resources
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.
SPN-21 (Interim Stage) Year 10 Normal Track (2 + 3) Page 12 of 23
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.html8/14/2019 Maths D (Normal Track) Year 10 (3 YEARS)
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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
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matrix is formed and whateach row and columnrepresent.
Content coverage Scope and Development Suggested Activities Resources
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.
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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.
Content coverage Scope and Development Suggested Activities Resources
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.
Content coverage Scope and Development Suggested Activities Resources
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)?
Content coverage Scope and Development Suggested Activities Resources
7. TRANSFORMATIONS(7 weeks)
7.1 Translation Introduce translation as a transformation that
Explain that http://www.bbc.co.uk/s
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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.
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Content coverage Scope and Development Suggested Activities Resources
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.
Content coverage Scope and Development Suggested Activities Resources
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).
Content coverage Scope and Development Suggested Activities Resources
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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