S3_MA_SOW_(2013)

Hwa Chong Institution (High School Section)

Subject: Integrated Mathematics Level: Sec 3 IP/SMTP

2013 Scheme of Work

OVERVIEW

Term 1 Term 2 Term 3 Term 4

Algebra III* (7 wk)

Unit 1: Advanced Algebraic Manipulation* (1 wk)

Unit 2: Matrices (0.5 wk)

Unit 3: Simultaneous Equations (0.5 wk)

Unit 4: Solving Quadratic Equations and Inequalities*

(3 wk)

Unit 5: Remainder & Factor Theorem (2 wk)

Indices III* (2.5 wk)

Unit 1: Indices and Surds* (0.5 wk)

Unit 2: Logarithmic and Exponential

Functions* (2 wk)

Trigonometry III (4.5 wk)

Unit 1: Triangle Trigonometry (1.5 wk)

Unit 2: Trigonometric Ratios (1 wk)

Unit 3: Trigonometric Identities and

Equations (1 wk)

Unit 4: Basic Graphs of Trigonometric

Functions (1 wk)

Coordinate Geometry II* (3 wk)

Unit 1: Coordinate Geometry* (1.5 wk)

Unit 2: Linear Law (1.5 wk)

Functions I* (4 wk)

Unit 1: Relations, Functions and

Modulus Functions (2 wk)

Unit 2: Standard Graphs* (1 wk)

Unit 3: Variation (1 wk)

Revision (2 wk)

2 Class Tests 2 Class Tests 2 Class Tests EOY Exam

Schedule Curriculum of Core Curriculum of Connection Curriculum of Practice

Curriculum of Identity

Term 1

(1 week) Topic: Algebra III*

Unit 1: Advanced Algebraic

Manipulation*

1. Expand and factorize more complex

algebraic expressions.

2. Change the subject of a formula

3. Add and subtract algebraic fractions

with quadratic denominators leaving

answer as a single fraction in

simplest form

4. Solving equations involving rational

fractions

5. Manipulate algebraic

formulae:*3 3 2 2( )( )a b a b a ab b

3 3 2 2( )( )a b a b a ab b

Connections:

Apply change of subject for scientific formulae-conversion

between Fehranheit and Celsius

( 1)1 2 3 ...

2

n nn

( 1) ( 1)( 2)1 3 6 ...

2 6

n n n n n

2 2 2 2 ( 1)(2 1)1 2 3 ...

6

n n nn

3 3 3 3 21 2 3 ... (1 2 3 ... )n n

Practice:

Derivation of quadratic formula by

completing the square.

Identity:

Life of mathematicians



2013 Scheme of Work

Term 1

(0.5 week) Unit 2: Matrices

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.

Connections:

Use of matrix in solving system of equations.

Matrices & transformations

Learning Experiences:

Students should have the opportunities to:

(LE O/3-4/N9a)

Justify if two matrices can be multiplied by checking the

orders of the matrices.

(LE O/3-4/N9b)

Discuss some applications of matrix multiplication, e.g.

decoding messages and transformation matrices for movie

making.

Practice:

Use of matrices in crytography

Optimization problem

Hill Cipher

Computer Security/encrypt

Harry Potter & cryptography

Use of Excel in Crytography

Using GC to input matrices and to

compute inverse matrices – simplify

decoding process.

Identity:

Self-study

Explore history of matrices

Explore use of matrices in

transformation: Eigen values and

Eigen vectors for the purpose of

orthogonalization

Term 1

(0.5 week) Unit 3: Simultaneous Equations

1. Solve linear and non-linear

simultaneous equations

2. Discuss the geometrical significance

of the algebraic solution of

simultaneous equations with the use

of suitable IT tools,

3. Discuss the number of solutions of a

pair of simultaneous linear and non-

linear equations (i.e. there may be 2

solutions, 1 solution or no solution),

4. Solve word problems involving

linear and non-linear equations.

Connections:

Using geogebra to investigate the relationship between the

nature of solutions of a pair of SE (one linear & one non-

linear) and the number of points of intersection.

Application in physics, decision making (linear algebra),

business problems

Practice:

Optimization Problem

Linear Programming

Identity:

Online independent study



2013 Scheme of Work

Term 1

(3 week) Unit 4: Quadratic Equations and

Inequalities*

1. Conditions for a quadratic equation

to have:

(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

2. Conditions for 2ax bx c to be

always positive (or always negative)

3. Relationships between the roots and

coefficients of a quadratic equation

4. Solving quadratic inequalities, and

representing the solution on the

number line

5. Solve inequalities of the form*

( )( )

0( )( )

x a x b

x c x d

.

Connections:

Satellite dishes, solar cooker, headlights, whispering

gallery, medical device – to blast kidney stone.

Parabola Vs Catenary

iPhone game,

Winter Olympic (need Calculus)

Spaceship & Star (NCTM)



(LE O/3-4/N7a)

Explain why there are no real solutions to a quadratic

equation 2 0ax bx c when

2 4b ac is negative.

(LE A/A1a)

Explain how the roots of the equation 2 0ax bx c

are related to the sign of 2 4b ac .

(LE A/A1b)

Transform 2ax bx c to the form

2( )a x h k and

use it to (i) sketch the graph;

and (ii) deduce the quadratic formula.

(LE A/A1c)

Use a graphing software to investigate how the positions of

the graph2y ax bx c vary with the sign of

2 4b ac , and describe the graph when 2 4 0b ac .

(LE A/A1d)

Use a graphing software to investigate the relationship

Practice:

Derivation of quadratic formula –

expose students to algebraic

manipulation experienced by

practitioner.

Identity:

Explore History

Diophantus, Al-Kharizmi, Abu

Kamil Babylonian tablets

(Math Through the Ages)



2013 Scheme of Work

between the number of points of intersection and the nature

of solutions of a pair of simultaneous equations, one

linear and one quadratic.

(LE A/A1e)

Examine the solution of a quadratic equation and that of its

related quadratic inequality (e.g. 24 5 0x x 0 and

24 5 0x x ), and describe both solutions and their

relationship.

Term 1

(2 week) Unit 5: Remainder & Factor Theorem

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 x Q x D x R x ( ) ( ) ( ) ( )

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,

Connections:

Connect polynomial division to number division algorithm

Use of factor theorem in curve-sketching

Relate cubic equations to design of roller coasters

(consideration of max allowed speed) and link to integrated

resorts.

Arithmetic in Nine Sections,

Fundamental theorem of algebra

Ancient questions on

1. Trisecting an angle

2. Doubling the cube

3. Constructing a regular heptagon

Tartaglia Vs Cardano in solving of cubic equations.



(LE A/A3a)

Make connections between division of polynomial and

division of whole number, and express the division

algorithm as ( ) ( ) ( )P x x a Q x R .

(LE A/A3b)

Practice:

Use graphing tools to explore

characteristics of cubic curves.

Using a sheet of A4 size page to

display the mathematics of

polynomials. (Refer to Panpac

Textbook Pg 78)

Identity:

Self-exploration of polynomial of

higher degree.

http://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art



2013 Scheme of Work

Associate the remainder and factor theorems with the

division algorithm.

(LE A/A3c)

Use a graphing software to 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

Term 2

(1 week) Topic: Indices III*

Unit 1: Indices and Surds*

1. Recognise surds and state the rules

of surds

2. Perform arithmetical operations

(addition, subtraction, and

multiplication) on expressions

involving simple surds in the

numerator

3. Rationalise fractions involving surds

in the denominator

4. Solve problem sums involving surds

5. Solve equations involving surds.

6. Solve challenging equations

involving surds.*

Connections:

History of Surds

http://www.mathsisgoodforyou.com/AS/surds.htm

Hotel Infinity

http://www.mathsisgoodforyou.com/artefacts/hilberthotel.ht

m

Financial ideas



(LE A/A2a)

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).

Practice:

Use of scientific notation for real life

data.

Term 2

(2 week) Unit 2: Logarithmic and Exponential

Functions*

1. Know functions

,e , log ,lna

x xa x x and their

graphs

2. Know equivalence of

logx

ay a x y

Connections:

Richter Scales and Earthquakes, Decibel Systems

Use graphing tools to explore growth and decay phenomena



(LE A/A5a)

Practice:

Model real-life problems using

exponential functions, such as the

half-life function and heat cooling

function.

Invention of Logarithms

Log tables

Identity:

Caring Thinking Module:

How students can help in face of

natural disasters

NE:

Sichuan Earthquake 2008

Effects of Indonesian Tremors to

http://www.mathsisgoodforyou.com/AS/surds.htm

http://www.mathsisgoodforyou.com/artefacts/hilberthotel.htm

http://www.mathsisgoodforyou.com/artefacts/hilberthotel.htm



2013 Scheme of Work

3. Show that log 1a a and

log 1 0a for any 0a and

1a

4. Understand and apply Laws of

logarithms:

(1) product (2) quotient (3) power (4)

change-of-base laws

5. Solve equations involving

exponential and logarithmic

functions

6. Solve challenging equations

involving exponential and

logarithmic functions.*

Use a graphing software to explore the characteristics of

various functions.

(LE A/A5b)

Relate the solution of the equation f ( ) 0x to the graph

f ( )y x to verify the existence of the solutions or to

justify that the solution does not exist.

(LE A/A5c)

Use a graphing software to display real-world data

graphically and match it with an appropriate function.

(LE A/A5d)

Relate the exponential and logarithmic functions to sciences

(e.g. pH value, Richter scale of earthquakes, decibel scale

for sound intensity, radioactive decay, population growth).

Use of Semi-log graph in scientific

research, pH scale

Singapore

Term 2

(1.5 week) Topic: Trigonometry III

Unit 1: Triangle Trigonometry

1. Solve triangles through Sine Rule &

Cosine rule

2. Formula for area of triangle

3. know the concept of bearings

4. Solve 2D, 3D problems

5. Compue angles of elevation and

depression, shortest distance,

maximum angle elevation.

Connections:

Leaning tower of Pisa

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

Applications of Trigonometry –geography and astronomy,

physics and Engineering

Clinometer, theodolite

Ferris Wheels around the world

Flight of an aeroplane: cosine rule



(LE O/3-4/G4a)

Visualise height, north direction, right-angled triangle, etc.

from 2D drawings of 3D situations.

(LE O/3-4/G4b)

Use the sine and cosine rules to articulate the relationships

Practice:

Ambiguous cases – treasure hunt.

Use Iphone to locate treasures given

bearing.

Noon day project




2013 Scheme of Work

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

Term 2

(1 week) Unit 2: Trigonometric Ratios

Equations

1. Know the concept of unit circle

2. Know the six trigonometric

functions for angles of any

magnitude (in degrees)

3. Know principal values of 1 1 1sin ,cos , tanx x x

4. Know the exact values of the

trigonometric functions for special

angles (0, 30, 45, 60, 90,

180,…)

Connections:

Relate 2 2cos sin 1x x to Pythagoras theorem.

Applications of Trigonometry – geography and astronomy,


Foreshortening in perspective drawing

Leaning tower of Pisa




(LE A/G1a)

Discuss the relationships between sin A, cos A and tan A,

with respect to the line segments related to a unit circle.

(LE A/G1d)

Relate 1 1 1sin ,cos , tanx x x

to the sine, cosine and

tangent functions respectively (e.g. 1sin x

is an angle

whose sine is x, and the principal value of 1 1

sin2

is -

30° or 6

Practice:

Historical development of

trigonometry – from circle

trigonometry to triangle trigonometry.

(astronomy)




2013 Scheme of Work

Term 2

(1 week) Unit 3: Trigonometric Identities and

Equations

1. Use of sin

tancos

AA

A ,

coscot

sin

AA

A ,

2 2cos sin 1x x ,2 21 tan secA A , 2 21 cot cosecA A

2. Solve simple trigonometric

equations

3. Prove simple trigonometric

identities

4. Simplify trigonometric expressions

Term 2

(1 week) Unit 4: Graphs of Trigonometric

functions

1. amplitude, periodicity and

symmetries related to sine and

cosine functions

2. graphs of

sin , cosy a bx c y a bx c

sin , cosx x

y a c y a cb b

tany a bx

Connections:

Modeling of natural phenomena –tides, heartbeat, music

etc.

Applications of Trigonometry – geography and astronomy,



Application of trigonometric functions in daily life

applications eg. Singapore flyer



(LE A/G1b)

Use a graphing software to display the graphs of

trigonometric functions and discuss their behaviour, and

investigate how a graph (e.g. siny a bx c ) changes

when a, b or c varies.

(LE A/G1c)

Relate the sine and cosine functions to sciences (e.g. tides,

Practice:

Graphs of

f ( )siny x x sin(f ( ))y a x

where f ( )x can be 1

x,

2x , x ,

exand relate to real life examples of

sound waves with such patterns.

Identity:

Group work on Mathematical

modeling.




2013 Scheme of Work

Ferris wheel and sound waves).

Term 3

(1.5 week) Topic: Coordinate Geometry III

Unit 1: Coordinate Geometry

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 gradient of a

perpendicular line using the

relationship 1 2 1m m

5. Identify equations of parallel or

perpendicular lines

6. Formulate equations of lines passing

through a given point and parallel or

perpendicular to another given line

7. Find equation of perpendicular

bisector between two points

8. Find the area of rectilinear figure

given its vertices(Shoelace Formula)

Connections:

Relate gradient to tangent of the angle of inclination

between the line and the positive direction of the x-axis and

deduce the relationship between the gradient of (a) two

parallel lines, (b) two perpendicular lines

Links with Geography,

Descartes and Coordinate System

http://www.bookrags.com/research/descartes-and-his-

coordinate-system-mmat-02/



(LE A/G2a)

Relate the gradient of a straight line to the tangent of the

angle between the line and the positive direction of the x-

axis, and deduce the relationship between the gradients of

(i) two parallel lines and (ii) two perpendicular lines.

(LE A/G2b)

Discuss how to solve geometry problems involving finding

(i) the equation of a line perpendicular or parallel to a given

line, (ii) the coordinates of the midpoint of a line segment

(horizontal, vertical and oblique), and (iii) equation of the

perpendicular bisector of a line segment.

(LE A/G2c)

Explore and discuss ways of finding the area of a triangle

(or polygon) with given vertices.

Practice:

Discuss other ways of finding area of

rectilinear figures.

Term 3

(1.5 week) Unit 2: Linear Law

1. Determine a linear relation based

on experimental results of two non-

linearly related quantities

2. Convert non-linear equations into

Connections:

Relate the use of straight line graph to the experience in

science experiment and explain why they do it (e.g.

oscillation of a pendulum, relationship between resistance

in circuit (Ohm’s Law)

Practice:

Presentation of the use of linear law in

their science practical.

http://www.bookrags.com/research/descartes-and-his-coordinate-system-mmat-02/

http://www.bookrags.com/research/descartes-and-his-coordinate-system-mmat-02/



2013 Scheme of Work

linear form

3. Derive the relationship between

two variables given the straight line

graphs

4. Determine unknowns in relations

using experimental data by

applying linear law to obtain

straight line graphs

5. Understand independent and

dependent variables

6. Understand and identify outliers or

incorrect readings

7. Expected to plot linear graph given

set of experimental data. (with no

scale given)



(LE A/G2i)

Explain the use of a straight line graph in a science

experiment (e.g. oscillation of a pendulum, Hooke‟s Law,

Ohm‟s Law).

Term 3

(2 week) Topic: Functions I

Unit 1: Relations, Functions and

Modulus Functions

1. Illustrate a relation using the arrow

diagram

2. Define the terms domain, range

and image

3. Find the expression for inverse

function.

4. Know x and the graph of

( )f x where ( )f x is linear,

quadratic or trigonometric

5. Solve equations involving modulus

functions.

Connections:

Relate it to real life data.

Term 3

(1 week) Unit 2: Standard graphs*

1. accurately sketch the standard

graphs

(a) power

functionny ax for

2 3n

Connections:

Relate the graph of

2y x to

2y x

f ( )y x to f ( )x y

Students examine the problem of space-pollution caused by

Practice:

Use graphing tools to explore the

characteristics of various functions

Use graphing tools to display real-

world data and match it with

appropriate functions (regression)



2013 Scheme of Work

(b) exponential function

xy ka growth,

xy ka decay

(c) parabolic 2y kx

(d) 2 2 2( ) ( )x a y b r

2. Perform simple transformation of

standard graphs

3. Estimation of the gradient of a curve

by drawing a tangent

4. Find equation of circle given centre

and radius.

5. Find the equation of the circle

passing through three given points*

6. Find the intersection of two circles*

7. Sketch polynomial graphs in

product form of any order and using

it to solve polynomial inequalities in

product form.*

human-made debris in orbit to develop an understanding for

functions and modeling at

http://illuminations.nctm.org/lessonplans/9-

12/debris/index.html

Students develop and analyse exponential model for the

behaviour of light passing through water at

http://illuminations.nctm.org/lessonplans/9-

12/light/index.html

.

How to locate epicenter by solving 3 circle equations

(detected from 3 stations.)

Find centre of a broken circular wheel in archaeological

studies.



(LE O/3-4/N6a)

Use Graphmatica or other graphing software to explore the

characteristics of various functions.

(LE O/3-4/N6b)

(b) Work in groups to match and justify sketches of graphs

with their respective functions.

(LE A/G2d)

Use a graphing software to investigate the graph of

2y kx when k

(LE A/G2e)

Relate parabolas to examples in sciences and in the real

world.

(LE A/G2f)

Make connections between the graphs of 2y x to

2y x

http://illuminations.nctm.org/lessonplans/9-12/debris/index.html

http://illuminations.nctm.org/lessonplans/9-12/debris/index.html

http://illuminations.nctm.org/lessonplans/9-12/light/index.html

http://illuminations.nctm.org/lessonplans/9-12/light/index.html



2013 Scheme of Work

(LE A/G2g)

Derive 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.

(LE A/G2h)

Discuss how to solve geometry problems involving

intersection of a parabola/circle and a straight line.

Term 3

(1 week) Unit 3: Variations

1. Understand and apply direct

variation

2. Sketch straight line graphs

illustrating direct variations

3. Understand and apply inverse

variation in word problems

4. Sketch reciprocal graphs illustrating

inverse variations

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.

Connections:

Explore variations in science experiments.

Practice:

Solve real life problems involving

variation.

Identity:

Online independent study

Suggested Reading Lists

(1) Writing Math Research Papers – A Guide for Students and Instructors by Robert Gerver

(2) Problem Solving Through Recreational Mathematics by Bonnie Averbach & Orin Chein

(3) Journey through Genius – The great theorems of Mathematics by William Dunham



2013 Scheme of Work

(4) The Genius of Euler – Reflections on his life and work, edited by William Dunham

(5) Mathematical Universe by William Dunham

Tests:

No. of class test per term (Term 1 to 3) : 2 No. of class tests per year: 6

Duration of class test: 1 hour

Format of class test paper: Structured Questions (100%)

Documents

S3_MA_SOW_(2013)