Add Maths Year 10

8/14/2019 Add Maths Year 10

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Topic Learning Outcomes Resources/Activities Time

1 BINOMIAL

EXPANSIONS

1.1The BinomialExpansion of

(1 + b) n where n is a

positive integer

1.2 The BinomialExpansion of

( a + b) n

Identify a binomial as an algebraic expression thatcontains two terms.

Write out expansions for (1+ b)nfor n = 0, 1, 2, 3, 4and 5 and show that the binomial coefficients, whenarranged, form the Pascals Triangle.

Use the notations n!, and nCr orn

r

.

Evaluate nCr using the formula !)!(

!

rrn

n

r

n

=

or by

using the calculator directly.

Write out the expansion of(1 + b)n using the generalterm nCrbr.

Perform expansion of(a + b)n and (ax + b)nby applyingthe binomial theorem

......21

)(221 rrnnnnn ba

r

nba

nba

naba ++

++

+

+=+

State the properties of the expansion ofnba )( +

such as the general term isrrn

bar

n

, the number

of terms is n + 1 and the sum of powers ofa and bin each term is n.

Use the general termrrn

bar

n

or list out the terms

in the expansion of(px + q)(ax + b)n to find a specificterm.

New AdditionalMathematics Chapter14AdditionalMathematics Chapter12

http://mathforum.org/dr.math/faq/faq.pascal.triangle.html

www.acts.tinet.ie/introduction

tothebinom_674.html

www.themathpage.com/aPreCalc/binomial-theorem.htm

www.acts.tinet.ie/Binomialtheorem

4 weeks

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http://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://www.acts.tinet.ie/introductiontothebinom_674.htmlhttp://www.acts.tinet.ie/introductiontothebinom_674.htmlhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.acts.tinet.ie/Binomialtheoremhttp://www.acts.tinet.ie/Binomialtheoremhttp://www.acts.tinet.ie/Binomialtheoremhttp://www.acts.tinet.ie/Binomialtheoremhttp://www.acts.tinet.ie/Binomialtheoremhttp://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://www.acts.tinet.ie/introductiontothebinom_674.htmlhttp://www.acts.tinet.ie/introductiontothebinom_674.htmlhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.themathpage.com/aPreCalc/binomial-theorem.htmhttp://www.acts.tinet.ie/Binomialtheoremhttp://www.acts.tinet.ie/Binomialtheorem


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Evaluate unknowns in the given expansions.

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2 CIRCULAR MEASURE

2.1 Radian Measure

2.2 Arc Length and Areaof a

Sector

2.3 Problems Related toCircular Measure

Define 1 radian as the angle subtended at the

centre of a circle by an arc equal in length to theradius.

State the relationship between an angle in radiansand in degrees.

State that Arc Length s = r and Area of Sector =

2

2

1r , where is in radians or Area of Sector =

rs2

1.

Find the arc length and area of sector. Solve problems involving finding the arc length,

area of sector, chord length, area of segment andangle of a sector.

Solve problems involving circular measure includingthe use of geometry and trigonometry.

New Additional

Mathematics Chapter12Additional MathematicsChapter 97

3 weeks

3 TRIGONOMETRY

3.1Trigonometric Ratios

3.2 General Angles and

Define the three basic trigonometric functionsof sine, cosine and tangent.

Evaluate the three basic trigonometric functionsof acute angles in right-angled triangles with twosides given.

Find the exact values of trigonometric functionsof special angles of 30o, 45o and 60o (useful toknow but not compulsory to memorise).

New AdditionalMathematics Chapters10 and 11AdditionalMathematics Chapter10

7 weeks

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Trigonometric Ratiosof Any

Angle Determine the location of any angle in the fourquadrants and hence determine the sign of thetrigonometric functions in the four quadrants usingS A .

T C

www.acts.tinet.ie/trigonometry_645.html

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http://www.acts.tinet.ie/trigonometry_645.htmlhttp://www.acts.tinet.ie/trigonometry_645.htmlhttp://www.acts.tinet.ie/trigonometry_645.htmlhttp://www.acts.tinet.ie/trigonometry_645.html


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3.3 Graphs of the Sine,Cosine

and TangentFunctions

3.4 Reciprocal ofTrigonometric

Functions

3.5 Simple TrigonometricIdentities

3.6 Trigonometric

Determine the trigonometric function of anyangle by expressing it in terms of itsbasic/principal angle and writing the correct sign.

Solve basic trigonometric equations byidentifying the quadrant the anglex lies in, thebasic angle and the value ofx in the requiredinterval.

Sketch the graph of the sine, cosine andtangent functions for the domain in degrees or inradians in terms of .

State the properties of the sine, cosine andtangent functions in terms of its range, maximum

and minimum values. State the amplitude and periodicity of thegraphs and know the relationship between graphsofy = sin x andy = 2 sin x, between

y = sin x andy = sin 2x.

Draw and use the graphs ofy= a sin(bx) + c,y= acos(bx) + c,

y = a tan(bx) + c where a, b and c are constants.

Determine the number of solutions totrigonometric equations in a given interval byusing the graphical method.

Define secant, cosecant and cotangent asreciprocals of cosine, sine and tangent functions.

Evaluate expressions and solve simpleequations involving the three reciprocal functions.

Use Graphmaticasoftware or graphiccalculator to study theproperties of thegraphs of the Sine,Cosineand Tangent Functions

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Equations State and use the identities

A

AA

cos

sintan and

A

AA

sin

coscot .

Apply the identities sin2A + cos2A 1,sec2A 1 +tan2A , cosec2A 1 + cot

2Ato prove other simple

trigonometric identities.

Apply the above identities to solvetrigonometric equations by

(i) reducing to the basic form e.g. sin x = k,sin(ax + b) = k,

(ii) factorisation,

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(iii) substitution using one of the identities.

Solve trigonometric equations where the anglesare in radians.

4 STRAIGHT LINEGRAPHS /

LINEAR LAW

4.1 Expressyin terms ofx

4.2 Determination ofUnknown

Constants From theStraight

Line

4.3 Equations of the Typey = axn

andy = Abx

Expressyin terms ofxfor a given graph of astraight line by writingY = mX + c.

Determine theXand Yterms in the equation Y =mX + c.

Tabulate values and draw the line of best fit to

determine the gradient and Y-intercept of the graph. Determine unknown constants by calculating the

gradient and intercept of the transformed graph.

Transform equations which require the use of lg xorln xanddetermine the unknown constants by calculating

the gradient orthe Y-intercept of the transformed graph.

New AdditionalMathematics Chapter 8Additional MathematicsChapter 8

3 weeks

5 MATRICES

5.1 RepresentInformation as a

Matrix

Display information in the form of a matrix.

Interpret the data in a given matrix.

Know the terms order, elements or entries, rowand column of a matrix.

Recognise a row matrix, column matrix, zero or

New AdditionalMathematics Chapter6Additional MathematicsChapter 14

3 weeks

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null matrix, square matrix and identity matrix.

Know that two matrices are equal if they havethe same order and if their correspondingelements are equal.

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5.2 Addition, Subtractionand Scalar

Multiplication of

Matrices

5.3 Multiplication ofMatrices

5.4 Determinant andInverse of a

2 2 Matrix

Add matrices of the same order by adding theircorresponding elements;

Know properties of matrix addition:IfA, B and O are of the same order, where O is anull matrix,

1. A + O = A

2. A + B = B + A (commutative)

3. A + (B + C) = (A + B) + C (associative).

Subtract matrices of the same order by subtractingtheir corresponding elements.

Calculate the product of a scalar quantity and amatrix by multiplying each element in the matrix by

the scalar quantity

=

kdkc

kbka

dc

bak .

Find the product of two matrices.

Know the properties of matrix multiplication:

1. AB BA (not commutative)

2. A (BC) = ( AB )C (associative);

3.

22

22

dc

ba

dc

ba

dc

ba.

solve problems involving the calculation of the sumand product of two matrices.

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find the determinant of a 2x2 matrix M =

dc

ba,

denoted by det M or

dc

bao rM .

Know that a matrix with zero determinant is called asingular matrix and it does not have an inverse.

Find the inverse of a non-singular matrix.

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5.4 Determinant andInverse of a

2 2 Matrix(Continued)

5.5 Solving SimultaneousEquations

by a Matrix Method

5.6 Word ProblemsInvolving

Matrices

Know the properties of inverse matrix and identitymatrix:MM-1= I and M-1M = I

IA = A and AI = A.Use the above properties to solve a matrix equation.

Write a given pair of simultaneous equations in theform of matrix equation and solve using the matrixmethod.

Form matrices to represent the information given ina table or from the description of a real lifesituation.

Solve related problems and interpret the results.

6 DIFFERENTIATION

6.1 The Gradient Function

6.2 Function of a Function(Composite Function)

Define the gradient at any point on a curve as thegradient of the tangent to the curve at that point.

Understand a limiting process through an example.

Find the gradient function of a curve.

Understand the idea of a derived function.

State that the derivative ofax

n

is nax

x-1

. Use the notations ( )

dx

dyxf ,' .

Know that ify= k ( a constant),dx

dy= 0.

State that the derivative of composite function is

New AdditionalMathematics Chapter15Additional MathematicsChapter 15 and 16

3 weeks

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given by the Chain Ruledx

du

du

dy

dx

dy= , and solve

problems related to composite functions.

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6.3 Product of TwoFunctions

6.4 Quotient of TwoFunctions

6.5 Equations of Tangentand

Normal

Differentiate the product of two functions using the

product ruledx

dvu

dx

duvuv

dx

d+=)( .

Differentiate the quotient of two functions using the

quotient formula2v

dx

dvu

dx

duv

v

u

dx

d

=

.

Apply differentiation to gradients, tangents andnormals.

State that the normal is perpendicular to the

tangent and the gradient of the normal is1

2

1

m

m =

wheredx

dym =1 is the gradient of the tangent at a

given point.

Find the equation of the tangent and the normal toa curve at a given point.

Solve problems related to tangent and normal to acurve.

http://www.mathsnet.net/asa2/2004/c15tanmethod02.html

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http://www.mathsnet.net/asa2/2004/c15tanmethod02.htmlhttp://www.mathsnet.net/asa2/2004/c15tanmethod02.htmlhttp://www.mathsnet.net/asa2/2004/c15tanmethod02.htmlhttp://www.mathsnet.net/asa2/2004/c15tanmethod02.htmlhttp://www.mathsnet.net/asa2/2004/c15tanmethod02.htmlhttp://www.mathsnet.net/asa2/2004/c15tanmethod02.html


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7 APPLICATIONS OFDIFFERENTIATION &HIGHER

DERIVATIVES

7.1 Rates of Change

7.2 Connected Rates ofChange

Calculate the rate of change of variables withrespect to time.

Determine the connected rates of change

using the Chain Ruledt

dx

dx

dy

dt

dy= .

New AdditionalMathematics Chapter

16 and 17Additional MathematicsChapter 16

6 weeks

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7.3 Small Increments andApproximations

7.4 Stationary Points andThe Second Derivative

7.5 Practical Maxima andMinima

Problems

Determine small changes x and y using the

ruledx

dy

x

y

.

Calculate the approximate change andpercentage change inyorx.

Percentage change iny %100y

y.

State that at stationary / turning points, 0=dx

dy.

Know that asx increases across a minimumpoint, the gradient changes from negative to zero topositive which results in a positive rate of change in

gradient, .dx

dy

Know that asx increases across a maximumpoint, the gradient changes from positive to zero tonegative which results in a negative rate of change

in gradient, .dx

dy

Recognise2

2

)(dx

yd

dx

dy

dx

d= as the rate of change

of gradient with respect tox and is called thesecond derivative ofy, and

if 02

2

>dxyd

, then it is a minimum point,

if 02

2

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Add Maths Year 10