hwa chong institution (high school section) subject...

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



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



Drill & Practice





2012 Scheme of Work

Environment Face to face & Department wiki. Online

Discussion

Resources



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.


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


Problem Solving



Drill & Practice



Discussion thread on wiki:

Environment Face to face & Department wiki. Online group discussions

Resources





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)


Problem Solving



Drill & Practice





Resources



2012 Scheme of Work



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.


Problem Solving



Drill & Practice





Resources

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



2012 Scheme of Work



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.


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.


Problem Solving



Drill & Practice





Resources



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

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

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

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



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


Problem Solving



Drill & Practice





Resources



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 –

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




2012 Scheme of Work

know the concept of bearings

solve 2D, 3D problems

computing angles of elevation and depression, shortest distance, maximum angle elevation.


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


Problem Solving



Drill & Practice





Resources



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



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


physics and Engineering

Foreshortening in perspective drawing

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

triangle trigonometry.

(astronomy)


Problem Solving



Drill & Practice





Resources



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





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.


Problem Solving



Drill & Practice





Resources



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



Drill & Practice





2012 Scheme of Work

ICT)





Resources



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)


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.


Problem Solving



Drill & Practice

Products Assignments/worksheets/skills check Discussion

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





2012 Scheme of Work

(Assignments/ Tests/)

Textbook exercises Online Quizzes Class Tests

thread on wiki:


Resources



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.


Problem Solving



Drill & Practice





Resources

Curriculum of Core Curriculum of Connection Curriculum of Curriculum of



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.


Problem Solving



Drill & Practice





Resources



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,


Processes Problem Solving



2012 Scheme of Work

(include pedagogies and indicate infusion of ICT)



Drill & Practice





Resources



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)

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



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




2012 Scheme of Work

polynomial inequalities in product form.

Step functions and greatest integer functions.

12/light/index.html .


Problem Solving



Drill & Practice





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%)


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