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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Topic Learning Outcomes Resources/Activities Time
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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Topic Learning Outcomes Resources/Activities Time
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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Topic Learning Outcomes Resources/Activities Time
(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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Topic Learning Outcomes Resources/Activities Time
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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Topic Learning Outcomes Resources/Activities Time
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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Topic Learning Outcomes Resources/Activities Time
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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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
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Topic Learning Outcomes Resources/Activities Time
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