maths d (fast track) year 10 (2 years)

8/14/2019 Maths D (Fast Track) Year 10 (2 YEARS)

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SCHEME OF WORK FOR SPN-21 (MATHEMATICS)

YEAR 10 FAST TRACK (2+2)

Content coverage Scope and Development Suggested activities Resources

1. MATRICES(2 weeks)

1.1 Introduction andBasic

Definition

Define matrix (plural matrices) as arectangular array of elements(usually numbers) arranged in rowsand columns.

Explain that a matrix with m rowsand n columns is said to have orderm x n (read as m by n).

Define thedifferent types ofmatrices: row matrix, column matrix,square matrix, diagonal matrix, nullmatrix, identity matrix or unit matrixand equal matrix.

Introduce matrix by displayinginformation in the form of matrices ofdifferent orders.For examples :a) The marks of two students in

English, Science and History:Student A obtained 70 marks for

English,87 marks for Science and 56 marks

for History. Student B obtained 72

marks for English, 80 marks forScience and 70 marks for History.

78 07 2

58 77 0or

70

80

72

56

87

70

b) The sales of a department store for2 items on 2 successive days:Thursday : 10 bags, 12 belts;Friday : 8 bags, 5 belts.

58

1210or

512

810

Explain briefly how the matrix isformed and what each row and

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

SPN-21 (Interim Stage) Year 10Fast Track (2 + 2) Page 1 of 27
http://www.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.htmlhttp://www.sosmath.com/matrix/matrix0/matrix0.html


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column represent.

1.2 Matrix Addition,Subtraction andMultiplication by a

Scalar

Showthe addition and subtraction oftwo matrices.

Show the multiplication of a matrixby a scalar quantity.

When doing subtraction, give strongemphasis that the minus sign shouldnot be touched when multiplying thescalar of the second matrix with the

elements of that matrix. For example,

51

512

14

32=

102

102

14

32

A common mistake at this step is

102

102

14

32




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1.3 Matrix Multiplication Explain the technique of the

multiplication of two matrices.Emphasize that two matrices canonly be multiplied when the numberof columns in the first matrix is the

same as the number of rows in thesecond matrix.

Show the results that AB BA.(except for multiplication by

identity matrixwhere IA = AI).

Use real life example to show the logicof multiplying row with column. Youmay use the example stated above.That is considering the sales of adepartment store for the 2 items on 2

successive days. In addition, let theprice of the bag be $8 per piece andthe belt at $3 per piece.

Present the above information inmatrix form. Explain clearly how tocalculate the total amount of moneyreceived by the store for the two dayssales.

Explain how the row in the first matrixis related to the column in the secondmatrix so that it can be multiplied.

Hence, generalize the technique andproceed to show the technique ofmultiplication of two (2 x 2)matrices and matrices of differentorders:

(a) Label the rows of the first matrixR1, R2 etc and the columns of thesecond matrix C1, C2 etc and thencalculate R1C1, R1C2 etc outsidethe main step. After multiplying allthe rows and columns, write down

all the products follow the row andcolumn numbers in the resultantmatrix.

(b) Making summary Row x Column.(c) Stress on the importance ofcorrect order for

the answer.



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1.4 Matrix Equations Solve matrix equation where the

unknowns are elements.

Solve matrix equation where theunknown is a matrix.


1.5 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

whose determinant is non-zero andsingular matrix as matrix whosedeterminant is zero and it has noinverse.

Show the method of finding theinverse of a non- singular matrix.

(A 1 =

ac

bd

Adet

1).

Solve problems with given value ofdeterminant and find the unknownelement in the matrix.

Find unknown element in matrix whichhas no inverse.

Caution students on the commonmistake of using + instead of when calculating determinant becausesometimes they can get mixed up withthe procedure in doing multiplication

of matrices.



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1.6 Identity Matrix Explain that an identity matrix, I is a

square matrix whose elements in theprincipal diagonal are 1 and theother elements are zero. e.g. I =

1001 ,

100

010

001

.

Show using examples the propertiesthat

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

I.

1.7 Application ofMatrices

Show how to place data into matrixform and interpret elements in amatrix as related to the giveninformation.

Show how to solve the problems andhence interpret the results.

Recall the example given in section1.3.

To interpret the result of multiplicationof two matrices, guide the students totell what is the quantity in the firstmatrix (R1) and what is the quantity inthe second matrix (C1) and whenthese two quantities (R1 and C1) aremultiplied, what do we obtain?Also in situations where there aremore than one element in each row ofthe first matrix, what do we obtainwhen the products are added (i.e.R1C1+ R2 C2 etc)?




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2. TRANSFORMATIONS(5 weeks)

2.1 Translation Introduce translation as atransformation that moves all

objects through a fixed distance in afixed direction.

Show examples where students haveto find images of the figures whengiven a translation in a diagram ordescription in words.

Describe fully in words thetranslation given in a diagram by

stating the translation vector

k

h.

Explain that transformations act uponobject points would change them (in

terms of position) into image points.When an object figure is transformedinto an image figure, there could bechanges in the shape and size of theimage. The transformations oftranslation, reflection and rotation areisometric as they do not cause anychanges in shape or size i.e. theobjects and images are congruent.

http://www.bbc.co.uk/schools/gcsebitesize/m

aths/shape/transformationsrev1.shtml

2.2 Reflection

Introduce reflection as atransformation that reflects anobject point in the line of reflectiononto its image point. Discussproperties of reflection in terms ofthe object distance equals the imagedistance and the line of reflection isperpendicular to the line joining theobject point and the image point.

Show examples where students haveto draw the images for individualpoints when given a line of

reflection. Focus on the x- andy-axes, lines parallel to the axes,y = xandy = x.

Extend the concept to figures andshow that if ABC is labelled in theclockwise direction, then the image,

111 CBA will be in the

anticlockwise direction and viceversa.

Given a point P and its image P1 on a

Relate reflection to study of reflectionof light in science as the sameproperties apply especially theconcept of lateral inversion.

This property is important as it helpsstudents to distinguish between areflection and a rotation when askedto describe a transformation.

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.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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diagram, explain that the line ofreflection is actually theperpendicular bisector of the line PP1.and describe the reflection fully bystating the equation of the line ofreflection.


2.3 Rotation Introduce rotation as a

transformation that moves an objectpoint through a fixed angle about acentre of rotation in a certaindirection.

Show examples where students haveto draw the images for figures undera given rotation. Focus on rotationsof multiples of 90.

Given a diagram showing an objectand its image, explain that thecentre of rotation is the point ofintersection of the perpendicularbisectors of two lines, each joiningone object point to its image point.

Stress that a rotation must bedescribed fully by stating the centre

of rotation, the angle and direction(except 180o rotation) it movesthrough.

Show that ABC and its image111 CBA are labelled in the same

sense which distinguishes a rotationfrom a reflection.



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2.4 Enlargement Introduce enlargement as a

transformation that changes theposition of an object point from acentre of enlargement by a scalefactor k.

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

Draw images for objects given thedescription of the enlargement.

Show that when k > 1, the image is

enlarged and when k < 1, the

image is reduced and introduce theconcept that

2factor)(scaleobjectofarea

imageofarea= in

relation to similar figures.

Given a diagram showing an objectand its image, explain that the centreof enlargement, C, is the point ofintersection of the two lines, eachjoining one object point P to its imagepoint P1 and the scale factor,

CP

CPk 1= .

Stress that an enlargement must bedescribed fully by stating the centre of

enlargement and its scale factor(positive or negative).

Introduce enlargement as atransformation that is not isometricand the size of the figure changes butthe shape remains the shape. Thismeans that the object and image are

similar.

Use the work on similar figures to linkto enlargement.Derive the ratio for similar trianglesand relate it to the scale factor ofenlargement

Show that an enlargement of scalefactor kwill produce an areaenlargement of scale factor k2 andvolume scale factor ofk3.




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2.5 Shear

Introduce shear as a transformationthat moves an object point parallelto a line called the invariant line (x-axis or y-axis).

Stress that points on invariant linedo not move under a shear.

Give the definition of shear factorand show how to apply the definitionto locate the position of the imagepoint.(Caution on situations where the

object point ison the negative region of the

invariant line and

also where the shear factor isnegative).

Given a shear and a figure (e.g.triangle), draw and label the imageof the figure.

Recognise a shear by its properties,i.e. changing in shape but not insize.

Given an object figure and its imagefigure, describe a shear completely(the description must include theword shear, the invariant line andthe shear factor).

Stack up some books (same height))on the table. Use a ruler and apply ahorizontal shear force to the books.Indicate the three obvious effects:

(i) the book on the table does not

move. Use this effect to explainthe meaning of invariant line.

(ii) all the books movement areparallel to the table top. Use thiseffect to explain that a shearmoves points parallel to theinvariant line.

(iii) The higher the books height,the more it moves. Use thiseffect to explain the definition ofshear factor.

To show that size does not changeunder a shear, apply the formula forarea of triangle (1/2 base

height) on both the object andimage (this is a good revision to findthe area of a triangle when it is drawnon a grid).

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

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

chools/gcsebitesize/maths/shape/transformationsrev1.shtml

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2.6 Stretch

Introduce stretch as atransformation that moves an objectpoint perpendicular to a line calledthe invariant line (x-axis or y-axis).

Stress that points on the invariantline do not move under a stretch.

Give the definition of stretch factorand show how to apply the definitionto locate the position of the imagepoint.

Given a stretch and a figure (e.g.triangle), draw and label the image.

Recognize a stretch by itsproperties. A stretch changes boththe shape and size (the object canbecome bigger or smaller) of theobject.

Given an object figure and its imagefigure, describe a stretch completely(the description must include theword stretch, the invariant line andthe stretch factor).

Use a geoboard and rubber bands toshow a stretch. Indicate the threeeffects:

(i) All points on the invariant linedo not move,

(ii) every point movesperpendicular to the invariantline,

(iii) the amount of movement ofany point depends on itsdistance from the invariant line.

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


2.7 CombinedTransformation

Explain the notation used for singletransformation (e.g. T(A) is theimage of A under the Translation, T).

Explain the notation used forcombined transformation (e.g. ET(A)is the image of point A under thetranslation ,T followed by the

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Enlargement, E).

Given an object figure and acombined transformation, eitherexpressed in notation or in words,draw and label the image figure.

2.8 Use of Matrix inTransformations

Use the idea that a transformationmaps an object to an image toestablish the quantitativerelationship (Matrix) (Object) =(Image), except for Translation is(Matrix) + (Object)= (Image).

Represent the object as a matrixwith x-coordinates as theelements in the first row andy-coordinates as the elements in thesecond row.

Use the results of the multiplicationof (Matrix) (Object) to indicatethe coordinates of the various imagepoints corresponding to each objectpoint.

Given a transformation represented

by a matrix and a figure, find thecoordinates of the image points anddraw and label the image.

Write down a matrix whichrepresents a given transformation.

Review the method of multiplying twomatrices.

Extend the idea of (Matrix) (Object)

= (Image) and the idea of M

10

01

= M to show that the matrix

representing a given transformationcan be obtained by mapping the point(1, 0) and (0, 1) to their respectiveimages under that transformation. Theelements of the matrix are thecoordinates of the images in thatorder.

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/MatrixTrans

Notes1.htm

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


http://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.mathsfiles.com/excel/MatrixTransNotes1.htmhttp://www.mathsfiles.com/excel/MatrixTransNotes1.htmhttp://www.mathsfiles.com/excel/MatrixTransNotes1.htmhttp://www.uz.ac.zw/science/maths/zimaths/73/sheila.htmlhttp://www.uz.ac.zw/science/maths/zimaths/73/sheila.htmlhttp://www.uz.ac.zw/science/maths/zimaths/73/sheila.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.math.lsu.edu/~verrill/teaching/linearalgebra/linalg/linalg5.htmlhttp://www.mathsfiles.com/excel/MatrixTransNotes1.htmhttp://www.mathsfiles.com/excel/MatrixTransNotes1.htmhttp://www.mathsfiles.com/excel/MatrixTransNotes1.htmhttp://www.uz.ac.zw/science/maths/zimaths/73/sheila.htmlhttp://www.uz.ac.zw/science/maths/zimaths/73/sheila.htmlhttp://www.uz.ac.zw/science/maths/zimaths/73/sheila.html


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3. VECTORS (2 weeks)

3.1 Representation of

Vectors

Define a vector as a quantity which

has both magnitude and direction.

Show the representation of vectorsin diagram and how to write andname a vector. Students areexpected to write vector a by

underlining,a

.

Show the representation of thevalue of a vector using column

vectors, a =

y

xor

=

y

xAB .

Describe what each element in thecolumn vector represents.

Show how to draw the vector on agrid when given a column vector andvice versa.

Use 3.1 and 3.2 to explain the basicconcepts on vectors. Then applythese basic concepts in the threetypes of questions with reference tosections 3.3, 3.4 and 3.5.

Explain why it is important to includedirection when stating a quantity. Forexample, when directing a tourist tothe taxi station which is about 50m tothe right of the junction.

Link vector to some vector quantitiesin Physics like velocity, acceleration,force etc.

Draw a few vectors and use thediagrams to guide the students towrite down and say out each vectorcorrectly.

Explain that BAAB = but

A B

BA

Stress that

AB should be read as

vector AB and not merely AB .

Similarly, por p should be read asvector pand not merelyp.

Tell the students that p and p are the

same and it is easier to write p as pin their work. Check that they do notwritep as vector p.

When finding the column vectorrepresenting a vector drawn on a grid,it is important to stress on thesystematic procedure, i.e. from theinitial point, move left or right firstfollowed by up or down to reach thefinal point so that the two values will

http://www.bbc.co.uk/schools/gcsebitesize/m

aths/datahandlingfi/probabilityrev1.shtml

http://standards.nctm.org/document/examples/chap7/7.1/part2.htm has interactive workabout vectors sums.

Go tohttp://www.standards.nctm.org/and click the

search button to findresources on othertopics from site.

http://www.ex.ac.uk/cimt/mepres/allgcse/bkc19.pdfis a chapterabout vectors

http://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://standards.nctm.org/document/examples/chap7/7.1/part2.htmhttp://standards.nctm.org/document/examples/chap7/7.1/part2.htmhttp://standards.nctm.org/document/examples/chap7/7.1/part2.htmhttp://standards.nctm.org/document/examples/chap7/7.1/part2.htmhttp://www.standards.nctm.org/andhttp://www.standards.nctm.org/andhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc19.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc19.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc19.pdfhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://standards.nctm.org/document/examples/chap7/7.1/part2.htmhttp://standards.nctm.org/document/examples/chap7/7.1/part2.htmhttp://standards.nctm.org/document/examples/chap7/7.1/part2.htmhttp://www.standards.nctm.org/andhttp://www.standards.nctm.org/andhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc19.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc19.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bkc19.pdf


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be written at the correct positions.


Show how to write a vector, in termsof the given vector, which is parallel(a) and equal (equivalent vector),(b) but not equal,(c) but opposite.



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3.2 Vector Addition andScalar Multiplicationof Vectors

Explain how to express a vector asthe addition and subtraction(equivalent to addition of negativevector) of two or more vectors.

Explain triangle law and parallelogramlaw.

Use the idea of routes for an effectiveexplanation on vector addition.For example, to go fromA to B isthe same as fromA to C andthen from C to B. So

+= CBACAB .

Extend this using any otherintermediate points.

For example, to go fromA to D,

++= CDBCABAD or

+= CDACAD

As Vector Addition is a very importantconcept, it is important to trainthe students to be able to writedown the statement correctly.One way to achieve this is togive a few complete statementsand guide the students toobserve the pattern. Thenprepare a set of partiallycomplete statements and havethe students complete them.


C

BA

D

CB


15/27


A very common mistake when findingvector using vector addition is to

write the statement under theinfluence of the vectors given.

For example, to findBC with

AB andAC given, some

students may write

+= ACABBC .

One way to cut down the chance of

making this mistake is to leavethe intermediate letter

blank first , i.e. BC = B + C,.

then look at the question tosearch for the other relevantletter. In this case, it is A. Thiswill lead to the correct statement

+= ACBABC . Use the idea of

negative vector, i.e.

= ABBAto

obtain the next statement

+= ACABBC .



16/27

3.3 Column Vector Derive and use the formula for themagnitude of a column vector andexplain the symbol used torepresent the magnitude.

Derive and use the formula for the

gradient of the line represented bya column vector.

Explain and discuss how to use therelationship between the elementsof two column vectors which areparallel

Use Vector Addition together withnegative vector if necessary to findunknown column vector


3.4 Position Vectors Explain that ifP is the position of a

point with respect to the origin,O ina Cartesian plane, then the vector

OP is known as the position vector

ofP.

Explain that given P(x,y), then theposition vector P with respect to O is

=

y

xOP and vice versa.

Stress that position vector must startfrom the origin.

Make generalization that for questionson vector, if coordinates of points areinvolved, then the idea of positionvectors have to be applied.



17/27

3.5 Problem on VectorsRepresented by

Letters

Apply Vector Addition, NegativeVector and Parallel Vectors to findunknown vectors.

Explain that if the point A,B and C

are collinear, then

AB = kBC or

= hACAB or

=nBCAC .

Discuss situations where a vector isdivided into two or more sections(collinear vectors)

(E.g. CBACABAC 2;3

2== ;AB :

CB = 2 : 3)

Train the students to handle situationsinvolving collinear vectors.

Eg. (a) If ,3

2CBAC = thenAC: CB

= 2 :3.

This meansAC= 2 parts and CB= 3

parts andAB = 5 parts. Thus

AC =

5

2

AB ,

CB = 5

3

AB etc


Brief discussion on vector equationlike ap + bq = mp + nq, ifpand q are non-zeros and not parallel,then a = m and b = n.

(b) IfAC=3

2AB, thenAC : AB =

2 : 3.


32

A BC


18/27

Apply the ideas on vector to solveproblem related to area of triangle

This meansAC=2 parts,AB = 3

parts and CB = 1 part. Then

AC =

2CB2

,CB =

3

1

AB etc.



C BA

2 1


19/27

4. SET LANGUAGEAND NOTATION(2 weeks)

4.1 Basic Concepts Define set as a collection of similar

objects. Use set language and set notation to

describe sets and representrelationships between sets asfollows:For examples:

A = {x : xis a natural number}

B = { (x,y) :y = mx +c}

C = {x : a x b}

D = { a, b, c.}

Define the terms finite and infinitesets, empty/null set, equal sets,subsets, universal sets andcomplement of aset.

Understand and use the followingnotations:...is an element of... or

belongs to

.is not an element of.. or does not belong to )

Number of elements in set A n(A)

The empty/null set

Universal set

Complement of setA A

A is a subset ofB A

B

A is a proper subset ofB A

Require students to group sets ofsimilar objects, for examples, a set ofstationery = {pen, pencil, eraser,ruler}, a set of boys = {Ali, Ahmad,John}, etc. Then introduce the termset, element and the number ofelements.Give examples of sets where thedifferent set notations are used using lists of numbers or alphabets.

Familiarize the students withexamples of writing sets using the setbuilder notation and also by listing theelements.

http://assets.cambridge.org/0521539021/sample/0521539021WS.pdf

http://www.mathworld.wolfram.com/VennDiagram.html

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B

A is not a subset ofB A

B

A is not a proper subset ofB A

B




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4.2 Intersection andUnion of Sets

Define the terms intersection andunion of sets by listing the elementsand introduce the symbols used :

Intersection of A and B A B

Union of A and B A B

Give examples on finding thecomplement sets and its number ofelements.

Illustrate clearly, using examples onthe correct way to write the answers.

1) If finding a set, the element(s)must be enclosed with thebrackets.

2) In finding the number ofelements, the answer should bejust a number without anybracket.

(common mistakes : n(A) = {3}; B =0)

When doing set operations, it is betterto list all the elements in each set inthe operation first.

4.3 Venn Diagram Present the set operations usingVenn diagrams and shade thedefined region and vice versa.

Describe set notations in words.

Use Venn diagram to show therelationship between the sets anddiscuss the meaning of the differentregions of the diagram.

Caution students on correct use ofterms and the necessity to write

statements in detail especially incases involving the and symbolse.g. If M = {set of students studyingmathematics} and P = {set ofstudents studying physics},(i) M P is the set of students

studying mathematics or physicsor both mathematics and physics,

(ii) P M means all students

studying physics also studymathematics.



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4.4 Problem Solving Interpret key words like both, and,either, or, neither, nor andnot.

Solve word problems using Venndiagrams to find the number of

elements in a set. Solve word problems involving

finding the maximum and minimumpossible value.

Introduce the study of set as a tool forproblem solving.


5. STATISTICS 2(3 weeks)

5.1 Data Collection andData

Presentation

Review how to organizestatistical data and represent it in

different ways such as bar charts,histograms, pie charts andpictograms.

Present ungrouped data usingfrequency tables and histograms.

Discuss examples where the diagramsgive misleading information e.g. bar

charts that do not start from the originand pictograms that show a 3-D objectenlarged with a factor of 2 torepresent twice a given quantity.

http://www.geohive.com

http://www.mathsisfun.com /data.html

5.2 Mean, Mode andMedian

Review the term mean, modeand median.

Revise the method of finding themean, mode and median forungrouped data

Extend the idea to the finding ofmean, mode and median fromfrequency tables and bar charts

Use statistics to compare the weatherin different parts of the world, e.g. toinvestigate which place has thelargest mean temperature differencebetween summer and winter.

http://www.mathforum.org/trscavo/statistics/comtents.html

http://www.nytimes.co

m/learning/teachers/lessons/20061128tuesday.html

5.3 FrequencyDistribution Table forGrouped DataIncluding FrequencyPolygons

Present grouped data by usingfrequency tables and histogramswith equal intervals

Show a frequency polygonderived from a histogram by joining

Use a simple example to show howdiscrete data can be grouped intoequal classes e.g. investigate thelength of words used in two differentnewspapers and present the findingsusing statistical diagrams.

http://www.geohive.com/http://www.geohive.com/http://www.mathsisfun.com/http://www.mathsisfun.com/http://www.mathforum.org/trscavo/statistics/comtents.htmlhttp://www.mathforum.org/trscavo/statistics/comtents.htmlhttp://www.mathforum.org/trscavo/statistics/comtents.htmlhttp://www.geohive.com/http://www.geohive.com/http://www.mathsisfun.com/http://www.mathsisfun.com/http://www.mathforum.org/trscavo/statistics/comtents.htmlhttp://www.mathforum.org/trscavo/statistics/comtents.htmlhttp://www.mathforum.org/trscavo/statistics/comtents.html


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the midpoints of upper side ofhorizontal rectangles of thehistogram.

Explain that for grouped data,the class with the highest frequency

is the modal class, the class wherethe median lies is identified byfinding where the middle position isand the mean can be calculated

using the formulaf

fxx

= where

fis the frequency andxis the valueof the variable.


5.4 Histogram WithUnequal ClassWidths/Intervals

Explain that for histograms withunequal widths, frequency density isused as the vertical axis instead offrequency where frequency density =

widthclass

frequencyand that it is the area of

rectangle that represents thefrequency of each class.

Explain that the mean calculated

using the formulaf

fxx

= is only

an estimate asxis the mid-value ormidpoint of the interval, taken torepresent all values in that interval.

Stress the point that the modal class for

http://www.waldomaths. com/Hist1N.jsp

http://www.waldomaths/http://www.waldomaths/http://www.waldomaths/http://www.waldomaths/http://www.waldomaths/


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the histogram with unequal intervals isnot the class with the highest frequencyfrom the frequency table but the classwith the tallest bar (highest frequencydensity on the histogram).

5.5 CumulativeFrequency TableandCumulativeFrequency Curve

Construct and interpret acumulative frequency table fromgiven raw data or from a groupedfrequency table.

Complete a grouped frequencytable given a cumulativefrequencytable.

Plot, draw and interpret cumulativefrequency curve.

Explain that a cumulative frequencycurve shows the number of variableswith a particular value or less.

5.6 Percentiles,Quartiles and

Interquartile Range

Introduce the term percentile and itsmeaning.

Use a cumulative frequency curveto explain percentiles (introduce the25th, 50th, 75th percentiles) and showhow to estimate these from a graph.

Explain that the median is the 50th

percentile, the lower quartile is the25th percentile, the upper quartile isthe 75th percentile.

Show how to find the median,quartiles, percentiles and interquartile range from a cumulativefrequency curve.

Show how to find the frequency fora given range of values

Use a cumulative frequency curve toexplain percentiles and show how to

estimate these from graphs

The words more than or less than inthe question indicates whether theanswer is on the top portion or bottomportion of the vertical axis (cumulativefrequency)




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6. PROBABILITY(2 weeks)

6.1 Definition ofProbability andSample Space

Understand the meaning ofprobability.

Explain the terms experiment, event,outcomes and sample space, S,used in probability.

Introduce elementary ideas ofprobability using familiar context forexamplethe probability of obtainingan odd number when a dice is rolled,obtaining a red ball when a ball ischosen from a bag without looking, ora weather forecaster saying there is a20% chance of rain today.

Discuss probabilities of 0 (event willnever occur) and 1 (event willdefinitely occur), leading to theoutcome that a probability liesbetween these two values.

http://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtml

http://www.mathgoodies.com/lessons/vol6/intro_probability.hmtl isan introductory lessonon probability

http://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfis a chapter onprobability, suitablesections include 5.1and 5.2

www.mathgoodies.com/lessons/toc_vol6.html

http://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdf

Various problemsinvolving probability athttp://www.nrich.maths.org/publc/leg.php

6.2 Simple Probability Define the probability of anevent A occurring as:

outcomespossibleallofnumbertotal

evetofavourableoutcomesofnumber

i.e.)(

)()(

Sn

AnAP = and calculate the

probability of a single event aseither a fraction or a decimal (not aratio).

Explain that an event A notoccurring is denoted by A and theprobability of A is P(A ) = 1 P(A).

Students do an experiment, forexample throwing a coin, say 50times. Ask them to tabulate theresults. Compare their results andthen pool them to obtain largersamples. Go on to calculate theprobability of getting a head and theprobability of getting a tail.

Collect examples of mutually exclusiveoutcomes. Establish and use the factthat the sum of probabilities of all theoutcomes is 1. For example, raining ornot raining, late or not late, win or notwin.

http://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.mathgoodies.com/lessons/vol6/intro_probability.hmtlhttp://www.mathgoodies.com/lessons/vol6/intro_probability.hmtlhttp://www.mathgoodies.com/lessons/vol6/intro_probability.hmtlhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.mathgoodies.com/lessons/toc_vol6.htmlhttp://www.mathgoodies.com/lessons/toc_vol6.htmlhttp://www.mathgoodies.com/lessons/toc_vol6.htmlhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.nrich.maths.org/publc/leg.phphttp://www.nrich.maths.org/publc/leg.phphttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/probabilityrev1.shtmlhttp://www.mathgoodies.com/lessons/vol6/intro_probability.hmtlhttp://www.mathgoodies.com/lessons/vol6/intro_probability.hmtlhttp://www.mathgoodies.com/lessons/vol6/intro_probability.hmtlhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.mathgoodies.com/lessons/toc_vol6.htmlhttp://www.mathgoodies.com/lessons/toc_vol6.htmlhttp://www.mathgoodies.com/lessons/toc_vol6.htmlhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.ex.ac.uk/cimt/mepres/allgcse/bka5.pdfhttp://www.nrich.maths.org/publc/leg.phphttp://www.nrich.maths.org/publc/leg.php


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6.3 Probability of

CombinedEvents

Draw a possibility diagram to

illustrate the outcomes forcombined events and representoutcomes as points on a grid.

Draw a tree diagram toillustrate the outcomes forcombined events where theoutcomes will be written at the end

of branches and probabilities by theside of the branches.

Calculate the probability ofcombined events using a possibilitydiagram or a tree diagram, explainthe concept of addition by relatingto the term or as in union of setsand multiplication by relating to theterm and as in intersection of

sets.

Start with a simple event such as the

score on a spinner; ask for theprobability of the total score being acertain number when the spinner isspun twice. List the outcomes andthen explain that each combinedoutcome can be written as an orderedpair (x, y) and so can be marked aspoints on a grid as a possibilitydiagram.

To introduce tree diagrams, use anexample such as choosing balls at

random from a bag, when there aredifferent numbers of balls of differentcolours. First, do an example wherethe first ball is replaced before asecond ball chosen (independentevents). Later, do an example wherethe first ball is not replaced(dependent events).

6.4 Problems onProbability

Show examples of howprobability may be applied indifferent situations including solvingprobability involving areas andobtaining information fromfrequency tables.



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7. NUMBERPATTERNS

AND SEQUENCES( 1 weeks)

Continue a given number sequence.

Recognise patterns within and acrossdifferent sequences and generalise tosimple algebraic statements(including expressions for the nth

term) relating to such sequences.

Extend the concept to patterns ofshapes.

Define a sequence of numbers. Workwith simple sequence of even, odd,square, triangle or Fibonacci numbers,etc.

Find the term-to-term rule for a

sequence, e.g. the nth term in thesequence 3, 9, 15, 21, 27, . . . .is 6n 3.

Discuss as a class activity thefollowing problem: Square tables areplaced in a row so that 6 people cansit around 3 tables, and so on. Howmany people can sit around n tables?

Various problemsinvolving sequences ofnumbers athttp://nrich.maths.org/public/leg.php

http://www.coolmath.com/algebra/Algebra2/09SequencesSeries/01_what.htm

http://www.waldomaths.com/Linseq1NL.jsp

http://nrich.maths.org/public/leg.phphttp://nrich.maths.org/public/leg.phphttp://www.coolmath.com/algebra/Algebra2/09SequencesSeries/01_what.htmhttp://www.coolmath.com/algebra/Algebra2/09SequencesSeries/01_what.htmhttp://www.coolmath.com/algebra/Algebra2/09SequencesSeries/01_what.htmhttp://www.coolmath.com/algebra/Algebra2/09SequencesSeries/01_what.htmhttp://nrich.maths.org/public/leg.phphttp://nrich.maths.org/public/leg.phphttp://www.coolmath.com/algebra/Algebra2/09SequencesSeries/01_what.htmhttp://www.coolmath.com/algebra/Algebra2/09SequencesSeries/01_what.htmhttp://www.coolmath.com/algebra/Algebra2/09SequencesSeries/01_what.htm

maths d (fast track) year 10 (2 years)

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