Add_maths_(Normal Track) Sow_year 10 (310807) v2 - 3 years

Topic

Learning Outcomes

Resources/Activities

Time

1 BINOMIAL EXPANSIONS



Identify a binomial as an algebraic expression that contains two terms. (1+ b)n for n = 0, 1, 2, 3, 4 Write out expansions for (1+ and 5 and show that the binomial coefficients, when arranged, form the Pascal’s Triangle.



Use the notations n!, and nC r or   .



1.1  The Binomial Expansion of  ( 1 + b) n where n is a positive integer

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

1.2 The Binomial Expansion of  ( a + b) n

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 n   r    n   n! Evaluate nC r using the formula   r    = (n − r )!    

or by

using the calculator directly. Write out the expansion of  (1 + b)n using the general term nC r r br .

Perform expansion of  (a + b)n and (ax + b)n by applying the binomial theorem ( a + b)



r !

n

=a

n

New Additional Mathematics Chapter 14 Additional Mathematics Chapter 12

4 weeks

http://mathforum.org/dr.math/ faq/faq.pascal.triangle.html www.acts.tinet.ie/introduction tothebinom_674.html

www.themathpage.com/aPre Calc/binomial-theorem.htm

www.acts.tinet.ie/ Bino

 n   n−1  n   n−2 2  n   n−r  r  mialtheorem ... +  ... + +     a b +  2    a b + ...  r    a b + ... 1            

State the properties of the expansion of 

 n   n −r 

( a +b) n

such as the general term is      a b , the number  r    of terms is n + 1 and the sum of powers of  a and b in each term is n. 

r 

 n      

n −r  r  Use the general term  r    a b or list out the terms

in the expansion of  (px + q)(ax + b)n to find a specific ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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

term. Evaluate unknowns in the given expansions.

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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Topic

Learning Outcomes

Resources/Activities

Time

Define 1 radian as the angle subtended at the centre of a circle by an arc equal in length to the radius. State the relationship between an angle in radians and in degrees.

New Additional Mathematics Chapter 12 Additional Mathematics Chapter 97

3 weeks

New Additional Mathematics Chapters 10 and 11 Additional Mathematics Chapter 10

7 weeks

2 CIRCULAR MEASURE 2.1 Radian Measure





2.2 Arc Length and Area of a Sector



State that Arc Length s = r θ   and Area of Sector = 1 2 1 2

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2.3 Problems Related to Circular Measure





2 r  θ  , where θ   is in radians or Area of Sector =

rs .

Find the arc length and area of sector. Solve problems involving finding the arc length, area of sector, chord length, area of segment and angle of a sector. Solve problems involving circular measure including the use of geometry and trigonometry.

3 TRIGONOMETRY  3.1 Trigonometric 3.1 Trigonometric Ratios

Define the three basic trigonometric functions of sine, cosine and tangent. Evaluate the three basic trigonometric functions of acute angles in right-angled triangles with two sides given. Find the exact values of trigonometric functions of special angles of 30 o, 45o and 60o (useful to know but not compulsory to memorise).

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ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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3.2 General Angles and  Trigonometric Ratios of Any Angle

Determine the location of any angle in the four quadrants and hence determine the sign of the trigonometric functions in the four quadrants using S A . 

www.acts.tinet.ie/tri gonometry_645.html

T C

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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Topic

Learning Outcomes

Resources/Activities

Time

Determine the trigonometric function of any angle by expressing it in terms of its basic/principal angle and writing the correct sign.



Solve basic trigonometric equations by identifying the quadrant the angle  x lies in, the basic angle and the value of  x in the required interval. 

3.3 Graphs of the Sine, Cosine and Tangent Functions

Sketch the graph of the sine, cosine and tangent functions for the domain in degrees or in radians in terms of  π  . State the properties of the sine, cosine and tangent functions in terms of its range, maximum and minimum values. State the amplitude and periodicity of the graphs and know the relationship between graphs of  y  y = sin x and  y = 2 sin x , between  y = sin x and  y = sin 2x. Draw and use the graphs of  y= a sin(bx) + c, y= c,  y= a 





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3.4 Reciprocal of   Trigonometric Functions

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

Use Graphmatica software or graphic calculator to study the properties of the graphs of the Sine, Cosine and Tangent Functions

Determine the number of solutions to trigonometric equations in a given interval by using the graphical method.



Define secant, cosecant and cotangent as reciprocals of cosine, sine and tangent functions. Evaluate expressions and solve simple equations involving the three reciprocal functions.

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3.5 Simple Trigonometric Identities

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ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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3.6 Trigonometric Equations



State and use the identities tan  A ≡

cot  A ≡

cos  A sin sin  A

sin sin  A cos  A

and

.

Apply the identities sin2 A + cos2 A ≡ 1, sec2 A ≡ 1 + tan  A , cosec2 A ≡ 1 + cot2 A to prove other simple trigonometric identities. 

2

Apply the above identities to solve trigonometric equations by (i) reducing to the basic form e.g. sin x = k, sin(ax + b) = k, (ii) factorisation, 

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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Topic

Learning Outcomes

Resources/Activities

Time

New Additional Mathematics Chapter 8 Additional Mathematics Chapter 8

3 weeks

(iii) substitution using one of the identities. Solve trigonometric equations where the angles are in radians. 

4 STRAIGHT LINE GRAPHS / LINEAR LAW



4.1 Express y  Express y in in terms of   x  

4.2 Determination of  Unknown Constants From the Straight Line

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

Express y  Express  y in in terms of  x for  x for a given graph of a straight line by writing Y = mX + c. c. Determine the X  the X and and Y terms Y terms in the equation Y = mX + c. c.  Tabulate values and draw the line of best fit to determine the gradient and Y -intercept -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 x or x or ln x and x and determine the unknown constants by calculating the gradient or the Y -intercept -intercept of the transformed graph. 

4.3 Equations of the Type  y = ax n and y and y = Ab x 

5

MATRICES

5.1 Represent Information 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, row and column of a matrix.

  

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

New Additional Mathematics Chapter 6 Additional Mathematics

3 weeks

7

Recognise a row matrix, column matrix, zero or null matrix, square matrix and identity matrix. Know that two matrices are equal if they have the same order and if their corresponding elements are equal.



Chapter 14



ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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Topic

5.2 Addition, Subtraction and Scalar Multiplication of  Matrices

Learning Outcomes 







 a  c

 

b  

 ka   =    d    kc

kb  

 .

kd    

Find the product of two matrices. Know the properties of matrix multiplication: 1. AB ≠ BA (not commutative) 2. A (BC) = ( AB )C (associative);

3. 

Time

Add matrices of the same order by adding their corresponding elements; Know properties of matrix addition: If A, B and O are of the same order, where O is a null 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 subtracting their corresponding elements. Calculate the product of a scalar quantity and a matrix by multiplying each element in the matrix by the scalar quantity k  

5.3 Multiplication of  Matrices

Resources/Activities

 a   c

b   a

  d     c

 a 2  2   ≠ d   c     b  

b2 d 

2

     .  

solve problems involving the calculation of the sum and product of two matrices.

5.4 Determinant and Inverse of a 2 2 Matrix

×

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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

find the determinant of a 2x2 matrix M =

denoted by det M or





ab Mo r c d 

 a b    c d 

,

.

Know that a matrix with zero determinant is called a singular matrix and it does not have an inverse. Find the inverse of a non-singular matrix.

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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Topic

5.4 Determinant and Inverse of a 2 2 Matrix (Continued)

×

5.5 Solving Simultaneous Equations by a Matrix Method

5.6 Word Problems Involving Matrices

Learning Outcomes

Time

New Additional Mathematics Chapter 15 Additional Mathematics Chapter 15 and 16

3 weeks

Know the properties of inverse matrix and identity matrix: MM-1 = I and M-1M = I IA = A and AI = A. Use the above properties to solve a matrix equation. 







6

Resources/Activities

Write a given pair of simultaneous equations in the form of matrix equation and solve using the matrix method.

Form matrices to represent the information given in a table or from the description of a real life situation. Solve related problems and interpret the results.

DIFFERENTIATION

6.1 The Gradient Function



Define the gradient at any point on a curve as the gradient 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 of  ax n is nax  x-1 .



Use the notations   f  ' ( x ),



Know that if  y =  y = k ( a constant),



  

dy dx

. dy dx

= 0.

6.2 Function of a Function ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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(Composite Function)



State that the derivative of composite function is given by the Chain Rule

dy dx

= dy × du , and solve du

dx

problems related to composite functions.

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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Topic

Learning Outcomes

6.3 Product of Two Functions

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



6.5 Equations of Tangent and Normal



d  dx

(uv )

=v

du dx

+u

dv dx

.

Differentiate the quotient of two functions using the du dv −u v quotient formula d   u   dx dx .   = dx  v   v2 Apply differentiation to gradients, tangents and normals. State that the normal is perpendicular to the tangent and the gradient of the normal is m2 where m1





Time

Differentiate the product of two functions using the product rule

6.4 Quotient of Two Functions

Resources/Activities

= dy dx

=−

http://www.mathsnet.n et/asa2/2004/c15tanm ethod02.html 1

m1

is the gradient of the tangent at a

given point. Find the equation of the tangent and the normal to a curve at a given point. Solve problems related to tangent and normal to a curve.

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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7 APPLICATIONS OF DIFFERENTIATION & HIGHER DERIVATIVES Calculate the rate of change of variables with respect to time. 

7.1 Rates of Change

7.2 Connected Rates of  Change



Determine the connected rates of change

using the Chain Rule

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

dy dt 

=

dy dx

×

dx dt 

New Additional Mathematics Chapter 16 and 17 Additional Mathematics Chapter 16

6 weeks

.

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Topic

7.3 Small Increments and Approximations

Learning Outcomes  x and Determine small changes δ  



rule 

 y δ   δ    x

≈

dy dx

Resources/Activities δ    y

using the

.

Calculate the approximate change and percentage change in y  in  y or or x   x . Percentage change in in y   y  ≈

 y δ   y

×100% .

7.4 Stationary Points and  The Second Derivative 



State that at stationary / turning points,

dy dx

7.5 Practical Maxima and Minima



dx

=0 .

.

Know that as  x increases across a maximum point, the gradient changes from positive to zero to negative which results in a negative rate of change in gradient,



dy

Know that as  x increases across a minimum point, the gradient changes from negative to zero to positive which results in a positive rate of change in gradient,



Time

dy dx

Recognise

. d  dx

(

dy dx

2

)=

d   y dx

2

as the rate of change

of gradient with respect to  x  and is called the second derivative of  y ,  y , and d 2 y if  > 0 , then it is a minimum point, dx 2 d 2 y if  < 0 , then it is a maximum point. dx 2 State the relationship between the sign of the second derivative and the nature of stationary

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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Problems 



points. Solve problems on finding the coordinates of  stationary or turning points and determining the nature of the stationary points. Solve application problems on maximum and minimum values e.g. physical quantities such as area and volume.

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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Topic

Learning Outcomes

Resources/Activities

Time

New Additional Mathematics Chapter 18 and 19 Additional Mathematics Chapter 18

3 weeks

8 DIFFERENTIATION OF TRIGONOMETRIC, LOGARITHMIC AND EXPONENTIAL FUNCTIONS 8.1 Differentiation of   Trigonometric Functions

State the derivative of the basic trigonometric



functions: d  dx d  dx d  dx

8.2 Differentiation of  Logarithmic Functions

(sin  x)

d  dx

(cos  x)

= −sin sin  x and

= sec 2  x .

(a sin bx )

= ab cos bx ,

(a tan bx )

= ab sec 2 bx .

d  dx

(a cos bx )

= −ab sin bx and

Obtain the derivatives of  sin( ax +b) , cos( ax +b) and tan( ax +b) ; sin n  x , cos n  x and tan n  x where a, b and n are constants.

State the derivative of the logarithmic function,



d  dx

(ln (ln  x)

=

1  x

.

Differentiate logarithmic functions using the



general result: 8.3 Differentiation of  Exponential Functions

= cos  x ,

Find the derivatives involving multiple angles :





(tan  x)

d  dx

d  dx

[ln( ax

+ b)] =

d  dx

[ln   f  ( x)]

a ax

+b

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

=

  f  ' ( x)   f  ( x)

, in particular,

.

17

Differentiate exponential functions using the



general result: d  dx

Topic

8.4 Application of  Differentiation Involving the above Functions

 x

(e )

= e x ,

d  dx

(e  f  ( x ) )

d  ax (e ) dx

=  f  ' ( x)e  f  ( x ) , in particular,

= ae ax ,

d  dx

(eax

+b

)

= ae ax+b .

Learning Outcomes 

Resources/Activities

Time

Apply the concept of differentiation of  trigonometric, logarithmic and exponential functions to problems on small increment and approximation, rate of change, stationary point and coordinates geometry.

 Text books 1.

2.

New Additional Mathematics (Ho Soo Thong & Khor Nyak Hiong) Addi Additi tion onal al Math Mathem emat atic ics s (H (H H Heng Heng,, JF Talb Talber ert) t)

ADDITIONAL MATHEMATICS – SPN 21 SCHEME OF WORK (INTERIM STAGE) YEAR 10 (3YR PROGRAMME)

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