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Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
OVERVIEW
Term 1 Term 2 Term 3 Term 4
Advanced Algebraic Manipulation (1 wk) Simultaneous Equations (0.5 wk) Indices & Surds (1 wk) Quadratic Functions & Inequalities (3 wks)
Matrices (Self-Study)
Polynomial (2 wks)
Log and Exponential Functions (2.5 wks)
Trigonometry (3 wks)
Variation (Self-Study)
Trigonometry (2 wks)
Coordinate Geometry (1.5 wks)
Linear Law (1.5 wks)
Relation & Function (0.5 wk)
Modulus Function (1 wk)
Standard Graphs (1 wk)
Revision
2 Class Test 2 Class Tests 2 Class Tests EOY Exam
Curriculum of Core Curriculum of Connection Curriculum of
Practice Curriculum of
Identity
Term 1 (1 Week)
Advanced Algebraic Manipulation
expand and factorize more complex algebraic expressions.
change the subject of a formula
add and subtract algebraic fractions with quadratic denominators leaving answer as a single fraction in simplest form
solving equations involving rational fractions Additional algebraic formulae:
3 3 2 2( )( )a b a b a ab b
3 3 2 2( )( )a b a b a ab b
Optional for Mainstream SMO types of Questions
Connections: Apply change of subject for scientific formulae-conversion between Fehranheit and Celsius Optional for SBGE
( 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
Derivation of quadratic formula by completing the square.
Life of mathematicians
Processes (include pedagogies and indicate infusion of
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
ICT)
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Environment Face to face & Department wiki. Online Discussion
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 1 (Self-Study)
Unit 2 : Matrices
present information in the form of a matrix of any order,
define equal, zero, identity matrices,
find unknowns in equal matrices,
perform addition and subtraction on matrices of same order, perform scalar multiplication,
perform matrix multiplication on small order matrices,
find determinant of a 2 2 matrix,
understand singular and non-singular matrices,
find the inverse of a 2 2 non-singular matrix by formula,
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
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.
Self-study
Explore history of matrices
explore use of matrices in transformation: Eigen values and Eigen vectors for the purpose of orthogonalization
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
Environment Face to face & Department wiki. Online
Discussion
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 1 (0.5 Week)
Unit 3 : Simultaneous Equations
solve linear and non-linear simultaneous equations
discuss the geometrical significance of the algebraic solution of simultaneous equations with the use of suitable IT tools,
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),
solve word problems involving linear and non-linear equations.
express a pair of simultaneous linear equations in matrix form and solving the equations by inverse matrix method.
Optional for Mainstream SMO types of Questions
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
Link to inequalities and graphs
Optimization Problem - Linear
Programming
Online independent study
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
Term 1 (3 Weeks)
Unit 4 : Quadratic Equations & Inequalities
solve quadratic equations (1) by factorization, (2) by completing the square, (3) by formula.
understand relationships between the roots and coefficients of a quadratic equation
deduce equations of parabolic curves with x-intercepts/turning points/y-intercepts given.
Utility of discriminant.
conditions for 2ax bx c to be always positive/negative
solve intersection problems between a line and curve.
purposeful transformation of quadratic expressions among the 3 forms.
find the maximum or minimum value by completing the square.
sketching graphs of quadratic functions given in the 3 forms.
solve quadratic inequalities using algebraic and graphical methods, & representing the solution on the number line.
Optional for Mainstream
Solving inequalities of the form f ( )
0g( )
x
x where f(x) and g(x) are quadratic
expressions that are either factorizable or always positive.
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)
Derivation of quadratic formula –expose students to algebraic manipulation experienced by practitioner.
Explore History
Diophantus, Al-Kharizmi, Abu Kamil Babylonian tablets (Math Through the Ages)
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 1 (2 Weeks)
Unit 5: Polynomial & Factoring
Definition of polynomial.
Multiplication and division of polynomial.
Types of equations – identity vs conditional equation.
Equating two equivalent polynomials and then comparing coefficients.
f x Q x D x R x ( ) ( ) ( ) ( )
able to recognize quotient & remainder from a given identity.
Know the Division Algorithm (long division)
define remainder theorem and know its limitation.
apply reminder theorem to solve for unknowns in polynomial.
able to revert back to the division algorithm to find the quotients & the remainder when the divisor is non- linear.
define factor theorem
use factor theorem to solve for unknowns in polynomial.
apply factor theorem to factorise cubic expressions and solve cubic equations, Optional for Mainstream SMO types of Questions
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
Antique question on
1. Trisecting an angle
2. Doubling the cube
3. Constructing a regular heptagon
Tartaglia Vs Cardano in
solving of cubic equations.
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)
Self-exploration of polynomial of higher degree.
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 1 (1 Week)
Unit 6: Indices and Surds
reduce indicial expressions into simple algebraic form
recognise surds and state the rules of surds
perform arithmetical operations (addition, subtraction, and multiplication) on expressions involving simple surds in the numerator
rationalise fractions involving surds in the denominator
solve problem sums involving surds
solve equations involving surds.
Optional for Mainstream SMO types of Questions
History of Surds
http://www.mathsisgoodforyou.co
m/AS/surds.htm
Hotel Infinity
http://www.mathsisgoodforyou.co
m/artefacts/hilberthotel.htm
Financial ideas
Use of scientific notation for real life data.
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
Term 2 2.5 Weeks
Unit 1: Logarithmic and Exponential Functions
know functions , , log , lna
x xa e x x and their graphs
know equivalence of logx
ay a x y
show that log 1a a and log 1 0a for any 0a and 1a
understand and apply Laws of logarithms: (1) product (2) quotient (3) power (4) change-of-base laws
solve equations involving exponential and logarithmic functions Optional for Mainstream SMO types of Questions
Richter Scales and Earthquakes, Decibel Systems
Use graphing tools to explore growth and decay phenomena
model real-life problems using exponential functions, such as the half-life function and heat cooling function.
Invention of Logarithms
Log tables
Use of Semi-log graph in scientific research, pH scale
Caring Thinking Module:
How students can help in face of natural disasters
NE: Sichuan Earthquake 2008 .Effects of Indonesian Tremors to Singapore
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 2 2 weeks
Unit 2 : Triangle Trigonometry
solve triangles through Sine Rule & Cosine rule
formula for area of triangle
Leaning tower of Pisa http://www.clarku.edu/~djoyce/trig/apps.html
Ambiguous
cases –
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
know the concept of bearings
solve 2D, 3D problems
computing angles of elevation and depression, shortest distance, maximum angle elevation.
Optional for Mainstream SMO types of Questions
Applications of Trigonometry –
geography and astronomy,
physics and Engineering
Clinometer, theodolite
FerrisWheels around the world
Flight of an aeroplane: cosine
rule
treasure hunt.
Use Iphone to
locate
treasures
given bearing.
Noon day
project
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 2 2 week
Unit 3: Circle Trigonometry
know the concept of unit circle
six trigonometric functions for angles of any magnitude (in degrees)
Relate 2 2cos sin 1x x to
Pythagoras theorem.
Applications of Trigonometry – geography and astronomy,
Historical development of trigonometry – from circle trigonometry to
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
know principal values of 1 1 1sin ,cos , tanx x x
know the exact values of the trigonometric functions for special angles (0, 30,
45, 60, 90, 180,…)
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
solve simple trigonometric equations
Proofs of simple trigo identities
Simplify trigo expressions
Optional for Mainstream SMO types of Questions
physics and Engineering
Foreshortening in perspective drawing
Leaning tower of Pisa http://www.clarku.edu/~djoyce/trig/apps.html
triangle trigonometry.
(astronomy)
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 2 1 week
Unit 4: Graphs of Trigonometric functions
amplitude, periodicity and symmetries related to sine and cosine functions
graphs of sin , cosy a bx c y a bx c
Connections:
modeling of natural phenomena –tides, heart beat, music etc.
Applications of Trigonometry – geography and astronomy,
Graphs of
( )siny f x x
sin( ( ))y a f x wh
ere f(x) can be 1/x,
Group work to submit Mathemati
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
sin , cosx x
y a c y a cb b
tany a bx
physics and Engineering http://www.clarku.edu/~djoyce/trig/apps.html Application of trigonometric
functions in daily life applications eg. Singapore flyer
x^2, e^x, sqrt(x) and relate to real life examples of sound waves with such patterns.
cal modeling.
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 2 Self Study (to be assessed in Term 2 or 3)
Unit 1: Variations
understand and apply direct variation
sketch straight line graphs illustrating direct variations
understand and apply inverse variation in word problems
sketch reciprocal graphs illustrating inverse variations
understand and apply part variation in word problems
sketch graphs to show part variations
investigate effect of different proportionality constants in the graphs, understand and formulate joint variation in word problems
solve challenging problems involving different types of variations.
Explore variations in science experiments.
Solve real life problems involving variation.
Self-development through self-study
Processes (include pedagogies and indicate infusion of
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
ICT)
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 3 (1.5 week)
Unit 2: Coordinate Geometry
Given coordinates of two points calculate
mid-point
distance
gradient
prove squares, rectangles, parallelograms and other standard polygons
understand and solve problems involving collinear points
understand gradient of a perpendicular line using the relationship 1 2 1m m
identify equations of parallel or perpendicular lines
formulate equations of lines passing through a given point and parallel or perpendicular to another given line
find equation of perpendicular bisector between two points
find the area of rectilinear figure given its vertices(Shoelace Formula)
Optional for Mainstream SMO types of Questions
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
two parallel lines
two perpendicular lines
Links with Geography, Descartes and Coordinate System http://www.bookrags.com/research/descartes-and-his-coordinate-system-mmat-02/ How to locate epicenter by solving 3 circle equations (detected from 3 stations.)
Find centre of a broken circular wheel in archaeological studies.
discuss other ways of finding area of rectilinear figures.
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products Assignments/worksheets/skills check Discussion
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
(Assignments/ Tests/)
Textbook exercises Online Quizzes Class Tests
thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 3 (1.5 week)
Unit 3: Linear Law
determine a linear relation based on experimental results of two non-linearly related quantities
convert non-linear equations into linear form
derive the relationship between two variables given the straight line graphs
determine unknowns in relations using experimental data by applying linear law to obtain straight line graphs
understand independent and dependent variables
understand and identify outliers or incorrect readings
Expected to plot linear graph given set of experimental data. (with no scale given)
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)
Presentation of the use of linear law in their science practical.
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Curriculum of
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
Practice Identity
Term 3 (0.5 week)
Unit 4: Relations & Functions
illustrate a relation using the arrow diagram
define the terms domain, range and image
find the expression for inverse function. Optional for Mainstream SMO types of Questions
Relate it to real life data.
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 3 (1 week)
Unit 5 : Modulus Functions.
Know x and the graph of ( )f x where ( )f x is linear, quadratic or
trigonometric
Solve equations involving modulus functions,
Optional for Mainstream SMO types of Questions
Processes Problem Solving
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
(include pedagogies and indicate infusion of ICT)
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 3 (1 week)
Unit 6: Standard graphs
accurately sketch the standard graphs
power function
2 3ny ax where n
exponential function
xy ka growth,
xy ka decay
parabolic 2y kx
2 2 2( ) ( )x a y b r
2 2 2 2 0x y fx gx h
perform simple transformation of standard graphs Optional for Mainstream
find the equation of the circle passingthrough three given points
intersection of two circles
Sketch polynomial graphs in product form of any order and using it to solve
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
human-made debris in orbit to
develop an understanding for
functions and modeling at
http://illuminations.nctm.org/le
ssonplans/9-
12/debris/index.html
Students develop and analyse
exponential model for the
behaviour of light passing
through water at
http://illuminations.nctm.org/le
ssonplans/9-
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)
Hwa Chong Institution (High School Section)
Subject: Integrated Mathematics Level: Sec 3 IP/SBGE/SMTP
2012 Scheme of Work
polynomial inequalities in product form.
Step functions and greatest integer functions.
12/light/index.html .
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving
Concept teaching and assessment (via online concept quiz.)
Self-directed independent study
Drill & Practice
Products (Assignments/ Tests/)
Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests
Discussion thread on wiki:
Environment Face to face & Department wiki. Online group discussions
Resources
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 (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%)