overview - Hwa Chong Institution

Hwa Chong Institution

Scheme of Work 2015

Subject/Programme : Mathematics

Level : Secondary 3

Scheme of Work (incorporating GL-Matrix, Differentiation: italicised for SBGE, refer to Appendix for further differentiation in SSMT )

Learning outcomes are statements of what a student should know, understand and/or be able to demonstrate after completion of a process of learning.

OVERVIEW

Term 1

Algebra III (7 wk)

Unit 1: Advanced Algebraic Manipulation (1 wk)

Unit 2: Surds and Indices (1 wk)

Unit 3: Equations and Inequalities (3 wk)

Unit 4: Relations, Functions (Part 1) (1 wk)

Unit 5: Polynomials (1 wk)

Unit 4: Matrices (1 wk)

2 Class Tests (10%)

Term 1 Test 1: Week 3

(a) Sec 2 Algebra

(b) Advanced Algebraic Manipulation


(a) Surds and Inequalities

(b) Equations and Inequalities

Term 2

Algebra III (3 wk)

Unit 5: Surds and Indices (1 wk)

Unit 5: cont’d Polynomials (1 wk)

Unit 6: Logarithms (2 wk)

Relations and Functions I (2.5 wk)

Unit 1: Relations, Functions and Modulus

Functions (Part 2) (0.5 wk)

Unit 2: Power Functions + Standard

Graphs (1 wk)

Unit 3: Parabolas and Circles (1 wk)

Coordinate Geometry II (1.5 wk)

Unit 1: Points, Lines and Slopes (1.5 wk)

Coordinate Geometry II (1 wk)

Unit 2: Direct and Inverse Proportions (1 wk)

Matrix (1 wk) – HBL




(a) Polynomials + Functions


(a) Surds and Indices and Logarithms

(b) Functions (Modulus)

(c) Power Functions + Standard Graphs

Term 3

Matrix (1 wk) – cont’d from HBL


Coordinate Geometry II (1.5 wk)

Unit 2: Applications of Straight Line

Graphs (1.5 wk)

Trigonometry III (4.5 wk)

Unit 1: Further Trigonometry (1.5 wk)

Unit 2: Trigonometric Functions (2 wk)

Unit 3: Simple Trigonometric Identities and

Equations (1 wk)



(a) Parabolas and Circles

(b) Coordinates Geometry

(c) Applications of straight line graphs


(a) Further Trigonometry

(b) Trigonometric Functions

Term 4

Revision (2 wk)

EOY Exam (70%)

All Topics

1



Time Allocated

In Weeks/hrs

1 week

Time frame

Term 1 Week 1

Content/Learning Outcomes

Core:

Algebra III (Unit 1/Advanced Algebraic

Manipulation)

At the end of the topic, students will be able to:

1.

Expand and factorise more complex algebraic expressions.

2.

Change the subject of a formula with involvement of roots.

3.

Add and subtract algebraic fractions with quadratic denominators leaving answer as a single fraction in simplest form. (e.g. x

 x x

2  y

2 x

1

 y

 x

1

 y

or

2



 x 2 x x



2



6 x



16

) x



2

4.

Solving equations involving rational fractions.

5.

Manipulate algebraic formulae: a a

3

3

  

  

)(

)(

2

2

  ab b

2

2

)

)

  .

6.

Expand the expressions: (x

 y) 3 and

(a+b+c) 3 .

7.

Extend expansion and factorisation techniques to expressions where the basic unit of unknown is of higher power

(x 4 -9), (x 4 -5x 2 +6).

8.

Extend expansion and factorization to expressions a n  b n - SMTP



Parallel Curriculum

Connection:

Apply the identities learnt in Sec 2 a a

2

2



 b

2

2  ab



( a b

2





)(

( a



 b b

)

)

2 a

2 

2

 2 

( a

 b )

2 to more advanced problems.

Compare and contrast the concepts of equal identity/equality in Maths vs humanities.

Learning Activities

Suggestions:

Blended/Cooperative

Learning:

Students to be organized in expert groups, each group to watch one of the suggested videos and share with rest of group on what they learnt about the algebraic identities.

Experiential Learning:

Use algebraic manipulatives to visualise algebraic identities of order 3.

Assessment/Feedback

Suggestions:

Formative assessment: Pop

Quiz

Alternative formative assessment:

Students to present on what they have learnt from the videos to the rest of the class

Summative assessment: Term 1

Test 1

Resources

Online Resources:

Special Factoring http://www.purplemath.c

om/modules/specfact2.

htm

Geometrical

Interpretation of a

3  b

3 http://www.youtube.com

/watch?v=5x4gJPchSiY

Geometrical

Interpretation of a 3

 b 3 http://www.youtube.com

/watch?v=9RHJt0GXLc

Y

Print Resources:

-

2



Time Allocated Content/Learning Outcomes

In Weeks/hrs

0.5 week

1 week

Time frame

Term 1 Week 2

Core:

Algebra III (Unit 5/Indices and Surds)

*solving in integer solutions (rational to be discussed in unit 3*)


1.

Recognise surds and understand that surds can be written in index form, or with radical signs and state the rules of surds

2.

Perform arithmetical operations

(addition, subtraction, and multiplication) on expressions involving simple surds in the numerator using properties of real numbers

3.

Rationalise fractions involving surds in the denominator (without being instructed)

4.

Solve problem sums involving surds

5.

Solve equations involving surds.

6.

Understand that by squaring an equation involving surds, extraneous solutions may be introduced and the solutions need to be checked to ensure they still fulfil the original equation

7.

Solve equations involving indices including substitution method

8.

Sketch Recognize/identify/plot (different bases) basic graphs of exponential functions (for connection to population growth, compound i/r and etc), e.g. y=2 x , y



P (1

 i ) n

9.

Solve challenging equations involving

surds (surds within surds) - SMTP

Core: In Weeks/hrs


Connection:

Make sense of numbers in surd form and recognise that the quadratic formula gives the real roots of quadratic equations in various forms

(integer, rational number and conjugate surds).

Justify the existence of irrational numbers.

Relate the operations of surds and rationalisation of denominator to the three algebraic identities learnt in Sec

2 (e.g.

(2



5) , (3



5)(3



5) ).

Justify the use of exponential graphs to population growth, radioactivity decay, half-life, compound interest etc.

Practice:

Using graphing tools or software such as Microsoft Excel to model bacteria growth using exponential graph.

Connection:





Resources

Suggestions:

Collaborative Learning:

Investigate how squaring an equation would lead to extraneous solutions.

Suggestions:

Suggestions:


Quiz

Alternative formative assessment: Open

Ended Tasks on

Surds (One Equals to Zero and Other

Mathematical

Surprises Pg 12-13,

18 - 22)


Mathematical

Modeling on

Marshall Cavendish

Pg 57


Test 3

Suggestions:


History of Surds http://www.mathsisgood

foryou.com/AS/surds.ht

m

Hotel Infinity http://www.mathsisgood

foryou.com/artefacts/hil berthotel.htm


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 2

(2)(3)(4)(5)

Ex2.1 (Surds)

Qn 10, 11, 12, 14, 15,

18, 20, 22

*check whether (6) is present in Q15

(7)

Ex 2.2 (Indices)

Qn 5, 7, 8, 9, 11, 13, 14

(7)

Ex 2.3 (Equations involving indices)

Qn 7, 9, 10, 12, 13

(8)

Ex 2.4 (Exponential

Functions)

Qn 7, 9, 12


3



Time Allocated

3 week

Time frame

Term 1 Week 3 to

6


Algebra III (Unit 3/Equations and Inequalities)


1.

Solve quadratic equations using (1) factorization (2) completing the square

(3) formula [emphasis on completing the square where the x 2 coeff is not 1.

– working is essential]

2.

Understand that the quadratic formula is derived from the quadratic equation form by completing the square method

3.

Solve linear and non-linear simultaneous equations

4.

Discuss the geometrical significance of the algebraic solution of simultaneous equations with the use of suitable IT tools,

5.

Discuss the number of solutions of a pair of simultaneous linear and nonlinear equations (i.e. there may be 2 solutions, 1 solution or no solution),

6.

Solve word problems involving linear and non-linear equations.

7.

Relationships between the roots and coefficients of a quadratic equation

8.

Understand that different discriminant give rise to different types of solutions for a quadratic equation:

(a) two real roots

(b) two equal roots

(c) no real roots and related conditions for a given line to:

(a) intersect a given curve

(b) be a tangent to a given curve

(c) not intersect a given curve


Explore the use of equations and inequalities in business problems, physics and decision making

Explore biographies of

Mathematicians such as Diophantus,

Al-Kharizmi, Abu Kamil Babylonian tablets and reflect on how these mathematicians pursue their passion and their role in fulfilling societal needs at that time.

Practice:

Model the trajectory path of an object in the air or the stopping distance of a car using quadratic function.



Use a spreadsheet or graphing software to

(a) investigate the relationship between the number of points of intersection and the nature of solutions of a pair of simultaneous equations, one linear and one quadratic.

(b) explain how the roots of the equation ax

2   

0

are related to the sign of b

2 

4 ac

.

(c) show graphically why there are no real solutions to a quadratic equation ax 2

 bx c 0 when b

2 

4 ac

is negative.

(d) investigate how the positions of the graph

 2   vary with the sign of b 2



4 ac

, and describe the graph when b 2



4 ac



0 .

(e) Examine the solution of a quadratic equation and that of its related quadratic inequality

(e.g.

4

2 x x 5 0 and

4 x 2 x 5 0

), and describe both



Quiz


Mathematical

Modelling Task on

Marshall Cavendish

Pg 33


Ended Tasks on

Quadratics (One

Equals to Zero and

Other Mathematical

Surprises Pg 8 – 11,

31 - 35)

- Performance Task –

Paper Helicopter

ASMS


Test 2



Resources

Simultaneous Equations http://www.youtube.com

/watch?v=SZ4x-HzhaKo

Simultaneous Equations and Intersections of

Graphs http://www.purplemath.c

om/modules/syseqgen.

htm


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 1

Math Through the

Ages

Additional

Mathematics 360 by

Marshall Cavendish

Chapter 1

Compiled by Mr Zong

LX

(3)(6)

Ex 1.1 (Simultaneous

Equations)

Qn 6, 8, 10, 11, 13

(7)

Ex 1.2 (Sum and

Product of Roots)

Qn 8, 9, 10, 13, 15, 19,

21

(8)(9)

Ex 1.3 (Discriminant,

Roots and x-intercepts)

4




9.

Apply conditions for ax

2

 bx

 c to be always positive (or always negative)

10.

Transform ax

2  bx

 c to the form

( ) 2

 k and use it to (i) sketch

In Weeks/hrs

1 week

Time frame

Term 1 Week 7 the graph

11.

Transform

(



)( ax

2

  to the form



) and use it to (i) sketch the graph

12.

Understand that quadratic function can be expressed in many forms. The values of a, b, and c or h and k or p and q gives different information about the quadratic function.

13.

Solve quadratic equations using (1) factorization (2) completing the square

(3) formula

14.

Understand that the quadratic formula is derived from the quadratic equation form by completing the square method

15.

Conditions for ax

2   to be always positive (or always negative)

16.

Solve quadratic inequalities, and representing the solution on the number line

17.

Solve inequalities of the form

(

(





)(





)

)



0

.

18.

Discuss the difference between the parabola vs catenary – SMTP

Core:

Algebra III (Unit 4/Relations and Functions)


1.

Define the terms function (one-to-one, many-to-one), relation (one-to-one,


Connection:

Understand that sometimes it is possible to model data from a real world situation with a linear equation.

Understand that the operations

Learning Activities solutions and their relationship.



Assessment/Feedback Resources

Qn 8, 11, 12, 13, 14,

16, 18

(16)

Ex 1.4 (Quadratic

Inequalities)

Qn 3, 7, 10, 12, 14, 17,

18

*no clear mentioning of

(1), (2), (4), (5), (10),

(11), (12), (16 – number line)

Suggestions:


Match graphs with functions

(Maths Rm Resource)

Suggestions:


Quiz



Additional

Mathematics 360 by

Marshall Cavendish

Chapter 4

5



Time Allocated

In Weeks/hrs

2 week

Time frame

Term 1 Week 8 to

Week 9 to Term 2

Week 1

Content/Learning Outcomes many-to-one, many-to-many, one-to many), domain, range and image.

2.

Illustrate a relation using the arrow diagram.

3.

Find the expression for inverse function.

4.

Acquire the concept and skills needed for composite functions and their inverses – SMTP

Algebra III (Unit 5/Polynomials)


1.

Definition of polynomial.

2.

Multiplication and division of polynomial.

3.

Types of equations – identity vs conditional equation.

4.

Equating two equivalent polynomials and then comparing coefficients f ( )



( ) ( )



( )

5.

Able to recognize quotient & remainder from a given identity.

6.

Know the Division Algorithm (long division)

7.

Define remainder theorem and know its limitation.

8.

Apply reminder theorem to solve for unknowns in polynomial.

9.

Able to revert back to the division algorithm to find the quotient and the remainder when the divisor is non-linear

10.

Define factor theorem

11.

Use factor theorem to solve for unknowns in polynomial.

12.

Apply factor theorem to factorise cubic expressions and solve cubic equations

(including unknown constants to test for understanding instead of using the


(adding and subtracting functions) within the domain of the functions involved, are similar to that of real numbers.





Mathematical

Modeling on Marshall

Cavendish Pg 236


Test 3

Resources

Connection:

Make connections between division of polynomial and division of whole number, and express the division algorithm as

( )



( x

 a Q x



R ) ( )

.

Make connections between the

Fundamental theorem of algebra

(prime factors) and the factor theorem.

Explore biographies of

Mathematicians Tartaglia Vs Cardano and reflect on how these mathematicians pursue their passion and their role in fulfilling societal needs at that time.

Practice:

Relate cubic equations to design of roller coasters (consideration of max allowed speed) and link to integrated resorts.

Explore mathematical questions crafted using Chinese poetic verses in

Arithmetic in Nine Sections.

Suggestions:



(a) investigate the graph of a cubic polynomial and discuss

(i) the linear factors of the polynomial and the number of real roots; and

(ii) the number of real roots of the related cubic equation, with reference to the points of intersection with the x-axis

Suggestions:


Quiz


Concept Map on the behavior of roots for

Quadratic vs Cubic functions


Mathematical

Modeling on

Marshall Cavendish

Pg 89


Test 3


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 3

Complied by Ms Pek

RH

(1)(2)(4)

Ex. 3.1 (Polynomials and Identities)

Q7, 9 , 11

(1)(4)(5)(6)(7)(8)

Ex. 3.2 (Division of

Polynomials)

Q5(b), 7, 8, 9, 12 (by using Long Division) ,

13 H.O.T.

(8)(9)

Ex.3.3 (The Remainder

Theorem)

Q5, 6, 7, 8, 11

(10)(11)

Ex 3.4 (The Factor

Theorem)

6



Time Allocated Content/Learning Outcomes calculator).

13.

Understand that the degree of a polynomial equation tells them about the number of roots that the equation has

14.

Sketch cubic graphs and understand that there is always 1 real root and conditions for possible solutions (1) 3 real distinct roots, (2) 1 real root, 2 repeated roots, (3) 1 real root, 2 imaginary roots (4) 3 repeated roots

15.

Sketch polynomial graphs in product form of any order and using it to solve polynomial inequalities in product form.(guided SMTP + SBGE)

In Weeks/hrs

2 week

Time frame

Term 2 Week 2 to

Week 3

Core:

Algebra III (Unit 6/Logarithms)


1.

Know functions a x

, e , log a x , ln x and their graphs.

2.

Know equivalence of y

 a x x log a y and that they are inverse functions. [Students must also be aware and prove e.g 10 lg a = a] n log n a  a ,

3.

Show that log a a



1

and log 1 0 for any a



0

and a



1 .

4.

Understand and apply Laws of logarithms:


Connection:

Relate the solution of the equation f ( )



0

to the graph y

 f ( )

to verify the existence of the solutions or to justify that the solution does not exist.

Trace the history of logarithms and appreciate the complexity of the logarithmic tables and how they have been programmed in calculators, computers.

Model real-life problems using exponential functions, such as the half-life function and heat and cooling function.


Suggestions:



(a) investigate the characteristics of exponential and logarithmic graphs.

(b) display real-world data graphically and model with an appropriate exponential or logarithmic function.


(a) Students to investigate the cause and effect earthquakes that




Suggestions:


Quiz


Ended Tasks on

Logarithmic

Functions (One

Equals to Zero and

Other Mathematical

Surprises Pg 23 –

24, 27 - 30)

Alternative formative

Resources

Q5, 7, 10, 12, 13

(14)

Activity 3C

Investigate the connections between linear factors, the number of intersections of the graph of a cubic polynomial with the xaxis

(12)(14)

Ex 3.5 (Cubic

Polynomials and Eqns)

Q5c, 6, 7, 9, 10, 13, 14,

15, 16, 17(i)-(iv) H.O.T

*no clear mentioning of

(3) and (13)


Richter Scale http://www.khanacade

my.org/math/algebra/l ogarithmstutorial/logarithm_pro perties/v/richter-scale

Logarithms in the Real

World http://www.youtube.co

m/watch?hl=en-

GB&v=3oZPPIVC8MU

&gl=SG


Additional

Mathematics 360 by

7




(1) product (2) quotient (3) power (4) change-of-base laws.

5.

Understand that logarithmic laws can be used to simplify, solve exponential expressions/equations and vice versa.

6.

Solve logarithmic and exponential equations.

7.

Discuss the use of exponential and logarithmic functions to other disciplines via word problems (e.g. pH value,

Richter scale of earthquakes, decibel scale for sound intensity, radioactive decay, population growth). Refer to Add

Maths textbook (Scatter plot), pg 189 for e.g.]

8.

Sketch and understand the properties of logarithmic graphs vs exponential (of different bases) graphs. (aware the existence of asymptotes and indicate the intercepts and. Understand the properties of the 2 graphs exp/log - inverse of each other)

9.

Solve challenging questions involving

exponential and logarithmic functions.

Parallel Curriculum Learning Activities happened in the last 5 years for e.g. Sichuan

Earthquake 2008,

China, Tohoko

Earthquake 2011,

Japan, Christchurch

Earthquake, New

Zealand. Students to reflect on how they could help in the face of such natural disasters.

(b) Students to investigate the impact of

Indonesian Tremors to

Singapore.



Assessment/Feedback assessment:

Mathematical

Modeling on

Marshall Cavendish

Pg 189


Test 4

Resources

Marshall Cavendish

Chapter 2

Compiled by Ms PekRH

2)(5)(6)

Ex 7.1 (Introduction to

Logarithms)

Q10(d), 11, 13(c), (d),

14(a), (c), 15(a), 16,

17(b), 18(b), 19, 21

(4)(5)

Ex.7.2 (Laws of

Logarithms)

Q6(b), 8, 9, 12(a), (c),

14, 15, 17, 18

(6)

Ex 7.3 (Logarithmic

Equations)

Q5, 7(a), (c), 8(b), 9(a),

(b), (d), 10, 11, 12, 15

(6 – in the form a x  b )

Ex 7.4 (Logarithmic

Equations of the form a x  b )

Q5, 6, 7(d), 8(c), 9(b),

(d), 10

(8)

Activity 7A (pg 181)

Investigate the characteristics of a logarithmic graph

8




In Weeks/hrs

1 0.5 week

Time frame

Term 2 Week 5

In Weeks/hrs

0.5 week

1 week

Time frame

Core:

Functions I

(Functions and Modulus Functions) (Part 2)

1.

Know x and sketch the graph

(reflection on the x-axis – no other transformation yet) of f ( ) where f ( ) is linear, quadratic, exponential/logarithmic or trigonometric

(trigo in the later chapter).

2.

Solve equations involving modulus functions (methods/skills – regions/squaring/substitution).

3.

Understand that x

2  x

4.

Understand the difference between f ( ) and f( )

Core:

Functions I (Unit 2/Power Functions +

Standard Graphs + Transformation)

At the end of the topic, students will be able


Connection:

Understand that an absolute value of

x is its distance from 0 and the absolute of f(x) is its distance from the line y = 0.

Understand that the operations

(adding and subtracting functions) within the domain of the functions involved, are similar to that of real numbers.


Suggestions:


Match graphs with functions

(Maths Rm Resource)




(7)(8)

Ex 7.5 (Graphs and

Applications of

Logarithmic Fns)

Q7, 8, 9, 12, 13

*(1), (3) no clear mentioning

Disclaimer: solving eqns and rejecting answers might touch on (3)


Additional Mathematics

360 by Marshall

Cavendish Chapter 4

Suggestions:


Quiz


Mathematical


Cavendish Pg 236


Test 4

Connection:

Identify the basic unit and appreciate that new products are created as a result of successive transformations for product design, architecture

Suggestions:



(a) explore the

Suggestions:


Quiz


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 4

9



Time Allocated

Term 2 Week 5/6

Content/Learning Outcomes to:

1.

Sketch graphs illustrating direct and indirect proportionality

2.

Sketch the power functions y=ax n ,

n = -2,-1,0,1,2,3 … or is rational. E.g.s y

 ax

2

3 (0 1), y



5

(



1)

In Weeks/hrs

1 week

Time frame

Term 2 Week 7/8 and y

 ax



3

2 or ax



5

9 ( n



0)

3.

Identify the basic function and perform simple transformation (translation, reflection, stretch) of standard graphs including exponential, logarithmic, power, trigonometric functions (trigo to be discussed later). For graphs with asymptotes, students can relate the movements of the asymptotes to certain transformations

4.

Introduce the notion of y

 f( )

, y

  f( bx )

 d or y a f( x c ) d

5.

Describe the transformations (at most 3 successive) that the functions have undergone and find the original function from resultant transformed function or vice versa.

6.

Understand that 180 o rotation about the origin as a successive reflections along the x and y axes

7.

Explore the different ways of transformation given the original and

Core:

Functions I (Unit 3/Parabolas and Circles)

At the end of the topic, students will be able to: end functions.


Compare transformations to point, line, rotational symmetry.

Connection:

Compare equation of circle with

Pythagoras’ Theorem.

Relate the concept of loci of points

Learning Activities characteristics of the various functions.

(b) display real-world data and match it with appropriate functions

(regression).


Work in groups to match and justify sketches of graphs with their respective functions.




Performance Task:

Students examine the problem of space-pollution caused by humanmade debris in orbit to develop an understanding for functions and modeling at http://illuminations.nc

tm.org/lessonplans/9

-12/debris/index.html


Test 4

Resources

Shinglee Textbook

New Syllabus

Mathematics 6 th

Edition Chapter 7

Suggestions:



Suggestions:


Quiz


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 9

10




1.

relate the graph of



 y

2  x

to y

 to y

 x

2 x

 and that they are inverse functions of each other.

2.

sketch the graphs of

(a) parabolic y

2  kx

(b)

( x a )

2

( y b )

2  r

2 .

3.

Perform simple transformation of these graphs.

4.

Derive and define the equation of a circle with centre (a, b) and radius r using the Pythagoras’ theorem, and the special case when the centre is at the origin.

5.

Transform general form of equation of circle to

(



)

2 y b )

2  r

2 by completing the square method.

6.

Compare the three cases where line meets circle (1) 2 distinct roots, (2) 2 repeated roots, (3) imaginary roots.

7.

Find the intersection of two circles.

8.

Find the equation of the circle passing through three given points. (move to coord geometry – perp bisectors)

Parallel Curriculum equidistant from a fixed point to equation of circle.

Recognise concept of circles as a special class of ellipses.

Explore the use of parabolas in other discicplines (e.g. sciences parabolic motion) and in the real world.

Discuss how to solve geometry problems involving intersection of a parabola/circle and a straight line.

Practice:

Determine centre of a broken circular wheel in archaeological studies

Find epicentre by solving 3 circle equations, detected from 3 satellite stations.


(a) explore the characteristics of the various functions.

(b) investigate the graph of y

2  kx when k varies.

(c) display real-world data and match it with appropriate functions

(regression).


(a) Work in groups to match and justify sketches of graphs with their respective functions.





Test 5

Resources

Compiled by Mr

ChenZH

(1b)(2a)

Ex 9.1 (Graphs of

Parabolas of the Form y

2  kx )

Q1

Q2. Sketch, on the same diagram, the graph of

(i) h : x x

2

for the domain 0  x

 2,

(ii) h



1

 5.

for the domain 1  x



1

Ex 9.1 (Graphs of

Parabolas of the Form y

2  kx )

Q11 (modified)

(i) same as (i)

(ii) same as (ii)

(iii) Write down the coordinates of the

11




In Weeks/hrs

1 week

Time frame

Term 2 Week 10

[in Sec 2 SOW from 2014 onwards]

Core:

Coordinate Geometry II (Unit 2/Direct and

Inverse Proportion)

1.

Understand and apply direct proportion

2.

Sketch straight line graphs illustrating direct proportions

3.

Understand and apply inverse proportion in word problems

4.

Sketch reciprocal graphs illustrating


Connection:

Understand the meaning of ratio, i.e. if the variables are related (i.e. same kind), the ratio represent the rate and will not change.

Represent variations graphically to illustrate how x and y are related.

Link to real life scenarios such as


Suggestions:

Blended Learning:

Engage in homebased learning, followed by classroom discussion.

Inquiry Learning:

Work in groups and design real life problems involving



Assessment/Feedback Resources vertex, C, of the curve.

Hence, find the area of triangle ABC.

(2b)(5)

Ex 9.2 (Coordinate

Geometry of Circles)

Q17 (modified)

(i) Sketch the circle x 2 + y 2 +2x – 2y – 3 = 0 and state its radius.

(ii) Original question.

(6)

Revision Ex 5

QA5 n or B5

(7)

Q6. Find the points of intersection of two circles x 2  y 2  

and x

2  y

2  .

Suggestions:

Activities

Using the existing

Science experiments and demonstrate the use of variation(s)

Formative

assessment:

(1a), (3), (4) no clear mentioning

Online resources:

Graphs and Proportion

– Higher: http://www.bbc.co.uk/sc hools/gcsebitesize/math s/algebra/proportionhire v1.shtml

Proportions/Variations: http://www.onlinemathle

12



Time Allocated Content/Learning Outcomes inverse proportions

5.

Understand and apply part variation in word problems

6.

Sketch graphs to show part variations

7.

Investigate effect of different proportionality constants in the graphs,

8.

Understand and formulate joint variation in word problems

9.

Solve challenging problems involving different types of variations.

In Weeks/hrs

1.5 week

Time frame

Term 3 Week 8 to

Week 9

Core:

Coordinate Geometry III (Unit 1/Points, Lines and Slopes)


1.

Given coordinates of two points calculate, revise

(a) mid-point

(b) distance

(c) gradient.

2.

Prove squares, rectangles, parallelograms and other standard polygons.

3.

Understand and solve problems involving collinear points.

4.

Understand that they can compare the slopes of two lines and determine if the lines are parallel or perpendicular.

5.

Understand gradient of a perpendicular line using the relationship m m

2

 

1

6.

Identify equations of parallel or perpendicular lines.

7.

Find the equation of the circle passing through three given points. (using

Parallel Curriculum household bills, work-rate problems, phone bills etc.

Practice:

Explore the application of variations in science experiments (e.g. Hooke’s

Law, Boyle’s Law and etc)

Discuss the impact of variations in

Sciences.

Learning Activities variation(s).

Connection:

Relate gradient to tangent ratio of the angle of inclination between the line and the positive direction of the x-axis

Deduce the relationship between the gradient of (a) two parallel lines, (b) two perpendicular lines.

Synthesise the different methods to find area of triangles learnt (1)

1

Shoelace method, (2) bh and

2 extend these to finding the area of polygons.

Relate the concept of foot of perpendicular to finding shortest distance from point to line.

Relate estimation of gradient of tangent for speed-time graph to instantaneous speed in kinematics.

Suggestions:


Explore and discuss ways of finding the area of a triangle (or polygon) with given vertices.

Discuss other ways of finding area of rectilinear figures.


Pop Quiz

Alternative formative

assessment:

Using the existing

Science experiments and demonstrate the use of variation(s)


Test 5

Suggestions:


Quiz

Alternative formative assessment:

Concept Map of properties of lines in

Coordinate Geometry

Alternative formative assessment:

Mathematical Modeling on Marshall Cavendish

Pg 162

Summative assessment:

Term 3 Test 5



Resources arning.com/proportions.

html

Variation: http://www.themathpage

.com/alg/variation.htm


New Syllabus

Mathematics 2 7 th

Edition by Shinglee

Chapter 1


Descartes and

Coordinate System http://www.bookrags.co

m/research/descartesand-his-coordinatesystem-mmat-02/


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 6

Shinglee Textbook

Compiled by Mr Lim BH

(1a)(1b)(2)

Ex 6.1 (Mid-point of

Line Segment)

Q7,12, 13, 17

(1c)(2)(3)(4)(6)

Ex 6.2 (Parallel Lines)

Q1a, 9, 13

13



Time Allocated Content/Learning Outcomes perpendicular bisectors)

8.

Formulate equations of lines passing through a given point and parallel or perpendicular to another given line.

9.

Find equation of perpendicular bisector between two points.

10.

Find the area of rectilinear figure given its vertices(Shoelace Formula).

11.

Estimation of the gradient of a curve by drawing a tangent

Parallel Curriculum Learning Activities




(4)(5)(6)(8)(9)

Ex 6.3 (Perpendicular

Lines)

Q1, 4(ii), 12, 17, 18

(10)

Ex 6.4 (Areas of

Triangles and

Quadrilaterials)

Q1b, 4b, 7, 13

(11)

Question: The diagram shows the graphs of y

 x

2  x



3 and y



1

2 x



2

.

(a)Use the graph to solve the equations:

(i) x

2 

2



3



0

(ii)

2 x

2  x



10



0

(b)By drawing a tangent, find the

gradient of at the point



3, 0



.

In Weeks/hrs Core: Connection: Suggestions: Suggestions: Online Resources:

14



Time Allocated

1 week

Time frame

Term 3 Week 1

In Weeks/hrs

1.5 week

Time frame

Term 3 Week 2 to

3


Algebra III (Unit 4/Matrices)

At the end of the topic, students will be able to:

1.

Present information in the form of a matrix of any order,

2.

Define equal, zero, identity matrices.

3.

Find unknowns in equal matrices.

4.

Perform addition and subtraction on matrices of same order, perform scalar multiplication.

5.

Perform matrix multiplication on small order matrices.

6.

Find determinant of a 2  2 matrix,

7.

Understand singular and non-singular matrices,

8.

Find the inverse of a 2  2 non-singular matrix by formula,

9.

Express a pair of simultaneous linear equations in matrix form and solving the equations by inverse matrix method.

10.

Solve word problems involving the sum and product of matrices and interpret the data in the given or computed matrices

Core:

Coordinate Geometry III (Unit 2/Applications of Straight Line Graphs)


1.

Distinguish between linear and nonlinear relationships.

2.

Determine a linear relation based on experimental results of two non-linearly related quantities.


Contrast the matrices operations with algebraic operations (e.g. AX = B is non-commutative whereas ax = b is commutative).

Practice:

Create encrypted messages using matrices and decrypt messages using matrix operations.

Investigate the use of matrices in operation research

Discuss some applications of matrix multiplication, e.g. transformation matrices for movie making.

Discuss how the idea of matrices is being used in spreadsheets and how these programs are useful in their everyday lives.

Connection:

Justify the use of straight line graph in science experiment (e.g. oscillation of a pendulum (Hooke’s Law), relationship between resistance in circuit (Ohm’s Law), kinematics graphs.

Understand the purpose to convert non-linear graphs to linear ones and how it is used to prove/justify the


Experiential Learning:

Use a graphing calculator to input matrices and to compute inverse matrices – simplify decoding process.


Students to get into groups and justify if two matrices can be multiplied by checking the orders of the matrices.



Quiz


Students to encode and decode using shift transformations

(refer to NSA lesson plan) and present their work in an oral presentation


Test 3

Suggestions:

Maths Journal:

Use a table to explore some typical transformations of non-linear equation into a linear equation

Inquiry Learning:

Engage in simple Science experiments to collect data

Suggestions:


Quiz

(c) Alternative formative assessment:

Mathematical

Modeling on

Marshall Cavendish



Resources

Matrices Khan

Academy http://www.khanacadem

y.org/math/algebra/alge bra-matrices

NSA lesson plan on encoding and decoding http://www.nsa.gov/aca demia/_files/collected_l earning/high_school/alg ebra/matrices_secret_w eapon.pdf

Print Resources:

New Syllabus

Mathematics 3 6th

Edition by Shinglee

Chapter 5


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 8

Online resources:

Singapore data

( http://data.gov.sg/hom e.aspx

)

15




3.

Convert non-linear equations into linear form.

4.

Derive the relationship between two variables given the straight line graphs.

5.

By applying linear law to obtain straight line graphs using experimental data, determine unknowns by reading from the graphs.

6.

Understand independent and dependent variables.

7.

Understand and identify outliers or incorrect readings.

8.

Expect to plot linear graph given set of experimental data (with no scale given

– proper scale).

Parallel Curriculum relationship between variables.

Practice:

Model phenomena in Science or real world using an equation. For example, students can conduct simple Science experiments to collect data. Part (a) to find a formula (V =

RI) for the resistance of a resistor.

Part (b) to find a formula

( T = 2 pendulum.

L ) for the period of a

Use suitable software to find out what happens when the non-linear equation has more than two unknown constants.

Plot the curve of best fit to fit a given set of data directly to decide the relationship between each set of unknowns.

Learning Activities and analyse data using a straight line graph.

Exploratory Activity:

Predict population growth using suitable linear function. (Textbook: Pg.

213)


Working in groups, based on the data on water consumption per capita in

Singapore for the past 10 years, students will plot graphs to predict future water consumption based on graph plotted using data from past years.

Identify and explain abnormal or inconsistent data.

Explore new water sources for Singapore




Pg 213


Test 5

Resources

- To convert non-linear relationships to linear form.

( http://www.youtube.co

m/watch?v=pX6WlxP2 eok )

- Applications of linear law

( http://www.youtube.co

m/watch?v=Gvb6MLB_ x6I )


- Additional Maths 360

(Pg. 190 – Pg. 213)

Compiled by Mr LimBH

(1)(3)

Ex 8.1 (Reducing Eqns to Linear Form)

Q1b, 1d, 1f, 11

+ equations involving log

+ consider asking students to sketch y=x 2 -

2x (y against x) v.s. (y/x against x) – to distinguish between linear and non-linear r/s

(4)

Ex 8.1 (Reducing Eqns to Linear Form)

Q8, 9, 10, 14

(2)(5)(6)(7)(8)

16



Time Allocated Content/Learning Outcomes Parallel Curriculum

In Weeks/hrs

1.5 week

Time frame

Term 3 Week 4 to

Week 5

Core:

Trigonometry III (Unit 1/Further

Trigonometry)


1.

Prove of Sine Rule and derive (guided) the Cosine Rule on their own.

2.

Use the sine and cosine rules to articulate the relationships between the sides and angles of a triangle, e.g. the lengths of the sides are proportional to sine of the corresponding angles,

Pythagoras’ theorem is a special case of the cosine rule, etc.

3.

Solve triangles through Sine Rule

(including ambiguous case) & Cosine rule.

4.

Use the formula for area of triangle

(Heron’s formula).

5.

Prove Heron’s formula – SMTP

6.

Find the perpendicular height of a triangle using area of triangle (apart from using trigonometry)

7.

Know and use the concept of bearings.

8.

Solve 2D and 3D problems.

9.

Compute angles of elevation and depression, shortest distance, maximum angle elevation and understand that shortest horizontal distance would give maximum angle of elevation or

Connection:

Explore and discuss applications of

Trigonometry to different fields like geography and astronomy, physics and engineering.

Relate concept of sine rule to congruency tests learnt in Sec 2

Illustrate the ambiguous case of Sine

Rule using construction/GSP and emphasise congruency tests learnt in

Sec 2.

Use circle properties to derive cosine rule.

Practice:

Determinet the actual flight distance

(geodesic path) using cosine rule.

Explore using software to check consistency of theoretical flight path vs actual flight path.


Suggestions:


Use Clinometer app on iPhone or Android phone to find the angle of elevation or depression of particular buildings

To organise a treasure hunt where treasures are located at different spots as a result of ambiguous case of sine rule. Students to use

Bearing app on iPhone or

Android phone to locate the treasures.




Suggestions:


Quiz

(d) Alternative formative assessment:

Concept Map on connecting the areas of triangles from various topics for different types of triangles


Test 6

Resources

Ex 8.2 (Linear Law)

Q7, 9, 12 (no scale given), 13 (no scale given)

(6) no clear mentioning


Leaning tower of Pisa http://www.clarku.edu/~ djoyce/trig/apps.html

Applications of

Trigonometry http://www.youtube.com

/watch?v=wvmU7XKdt3 w

Trigonometry in Real

Life http://www.youtube.com

/watch?v=n1A2HqSXtG

I


New Syllabus

Mathematics 3 6 th

Edition by Shinglee

Chapter 11

17



Time Allocated

In Weeks/hrs

2 week

Time frame

Term 3 Week 6 to

Week 8

Content/Learning Outcomes depression.

Core:

Trigonometry III (Unit 2/Trigonometric

Functions)


1.

Know the concept of unit circle.

2.

Know the six trigonometric functions for angles of any magnitude (in degrees or radians) and the reciprocal relationship between trig functions, e.g. sec x and cos x and etc.

3.

Distinguish between

1 sin cos tan sin x



1

, x

1

, cos x

,



1 x

1 x

, tan and



1 x

.

4.

Know principal values (1 to 1 function) of sin



1 x , cos



1 x , tan



1 x

. – SMTP

5.

understand that the properties of inverse functions expand to trigonometric functions.

6.

Know the exact values of the trigonometric functions for special angles

(0  , 30  , 45  , 60  , 90  , 180  and in radians).

7.

Identify amplitude, periodicity (behaviour repeated over intervals of equal length) and symmetries related to sine, cosine and tangent functions.

8.

Understand that they can translate periodic functions in the same way as they translate other functions.

9.

Sketch graphs of y

 a sin bx



,

 a cos

   c

, y

 a tan

 

   c and


Connection:

Discuss the relationships between sin

A, cos A and tan A, with respect to the line segments related to a unit circle.

Relate the use of sine and cosine functions in sciences (e.g. tides,

Ferris wheel and sound waves).

Understand that trigonometric functions, and their compositions gain significance when they are used to model waves and periodic behaviour.

Practice:

Explore the historical development of trigonometry – from circle trigonometry to triangle trigonometry and its impact on the study of astronomy

Model natural phenomena – tides, heartbeat, music etc. using graphs of y

 f ( ) sin x y

 a where f ( ) can be

1

, x

2 , x , x e x and relate to real life examples of sound waves with such patterns.





Suggestions:


Use a Geogebra or GSP to

(a) investigate the relationship of sin A, cos A and tan A with respect to the unit circle.

(b) display the graphs of trigonometric functions and discuss their behaviour, and investigate how a graph

(e.g. y

 a sin bx

 c ) changes when a, b or c varies.

Suggestions:


Quiz


Mathematical


Cavendish Pg 307

Performance Task:

Students to get into groups to find out the different ferris wheels for e.g. Singapore

Flyer around the world and to use sine or cosine to its function


Test 6


Trigonometric Functions and Unit Circle http://www.youtube.com

/watch?v=rrXLl2WTKEc

Applications of

Trigonometry – geography and astronomy, physics and

Engineering http://www.clarku.edu/~ djoyce/trig/apps.html


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 11

18



Time Allocated Content/Learning Outcomes y

 a sin



,

 a cos

 c

,


In Weeks/hrs

1 week

Time frame

Term 3 Week 9 y

 a tan

 c

.

Core:

Trigonometry III (Unit 3/Simple Trigonometric

Identities and Equations)


1.

Understand that the interrelationships amongst the six basic trigonometric functions make it possible to write trigonometric expressions in various equivalent forms.

2.

Derive and relate cos

2 to Pythagoras’ theorem. x

 sin

2 x



1

3.

Use of sin cos

A

A

 tan A

, cos A

 cot A

, sin A cos 2 x

 sin 2 x



1 ,

2

A

 sec

2

A ,

2

A

 cosec

2

A

4.

Simplify trigonometric expressions.

5.

Solve simple trigonometric equations, including the use of trigo identities, in degrees or in radians. (angles can involve more than 1x, e.g. 2x-30 o ]

6.

Prove simple trigonometric identities, including the use of trigo identities.

(rigourous proving, > 5 steps – SMTP)

Connection:

Make connections between solutions obtained from solving trigonometric equations and graphical method.

Discuss similarities and differences in solving algebraic equations and trigonometric equations.





Suggestions:


Quiz


Mathematical


Cavendish Pg 323


Test 6


Additional

Mathematics 360 by

Marshall Cavendish

Chapter 12

19

overview - Hwa Chong Institution

Scheme of Work (incorporating GL-Matrix, Differentiation: italicised for SBGE, refer to Appendix for further differentiation in SSMT )

Learning outcomes are statements of what a student should know, understand and/or be able to demonstrate after completion of a process of learning.

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overview - Hwa Chong Institution

Scheme of Work (incorporating GL-Matrix, Differentiation: italicised for SBGE, refer to Appendix for further differentiation in SSMT )

Learning outcomes are statements of what a student should know, understand and/or be able to demonstrate after completion of a process of learning.

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