Hwa Chong Institution (High School Section)

Subject: Integrated Mathematics Level: Sec 3 IP/SBGE

2011 Scheme of Work

Curriculum of Core / Curriculum of Connection / Curriculum of Practice / Curriculum of Identity
Term 1
(Self-Study) / Unit 1 : Circular Measure
1.  understand and define angles in radian
2.  find the arc length and area of sector/segment
3.  solve problem sums involving arc length, sector and segment. / Connections:
Stedian (SI unit for angle), Gradian (angle unit), Degree
History of radian and why it is preferred over degrees / Practice:
·  Jobs that require angle measurement, knowledge of circle / ·  Self-exploration through self-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
Environment / Face to face & Department wiki. / Online Discussion
Resources
Curriculum of Core / Curriculum of Connection / Curriculum of Practice / Curriculum of Identity
Term 1
(5 weeks) / Unit 2 : Further Trigonometric Identities
At the end of the unit, students should be able to
1*derive (optional for mainstream)
- compound angle formulae
- double angle formulae
- factor formulae
- R-formulae
2. Use of the above formulae to
(a) simplify trigonometric expressions
(b) to solve trigonometrical equations in a
given interval
(c) prove trigonoemtric identities
3. use of radian measure of angle. / Connections
History of Trigonometric Functions
: http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html / Practice:
¨  Jobs that involve trigonometric identities
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
Environment / Face to face & Department wiki. / Online Discussion
Resources
Curriculum of Core / Curriculum of Connection / Curriculum of Practice / Curriculum of Identity
Term 1
(1 Week) / Unit 3 : Binomial Theorem
1.  construct the Pascal’s triangle (via tossing coins) and use it to expand (a + b)n, especially for smaller values of n.
2.  use the Binomial Theorem to expand (a + b)n for positive integer n.
3.  use of the general term Tr + 1 = an – rbr to find a particular term in the expansion of (a + b)n.
know the notation . / Connections:
Use tossing coins and algebraic approach to create the Pascal Triangle.
Blaise Pascal, Yang Hui, Omar Khayyám, Pingala
Plinko game, Sierpinski triangle
Application in physics, integration (calculus) / Practice:
Jobs that use polynomials / 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
Term 1 & Term 2
(7 Weeks) / Unit 4.1 Differentiation
Techniques of Differentiation (1 w eek)
1.  derive the gradient function of simple polynomials etc.
2.  differentiate standard functions (for any rational n) together with constant multiples and sums & differences of these functions.
3.  use the notation,,, .
4.  differentiate usning chain rule, product rule & quotient rule.
Unit 4.2: Applications of differentiation (3 weeks)
At the end of the unit, students should be able to:
1.  find equations of tangent and normal using.
2.  apply differentiation to solve problems involving related rates of change
3.  explain increasing/decreasing functions using
4.  apply differentiation to find stationary points
Unit 4.3: Further differentiation (3 weeks)
At the end of the unit, students should be able to:
*Optional for mainstream
1.  Extend differentiation from first principle to derive first derivative of sine, cosine and tangent functions.
2.  Extend differentiation from first principle to derive first derivative logarithmic (general base) and exponential functions.
3.  Find the derivatives of trigonometric functions.
4.  Find the derivatives of exponential and logarithmic functions (base e).
5.  Apply the above to all the problems in Calculus I, Unit 2. /
Connections:
History of Calculus, or from Wikipedia
History of Integration, or from this pdf file
As the world turns around : http://www.karlscalculus.org/calc7_1.htm
l / Practice:
Jobs that require calculus
What is calculus used for?
Illuminations Cardiac Output /
Affective Education
· Trace the history of the development of calculus and the contributions made by various mathematicians and their impact to the development of mankind.
·Comment on the controversy between Newton and Leibniz.
·From the readings on the mathematicians, comment on the ways they go about creating mathematics and the aspects of their lives that has the impacts you the most
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) / Topic : Probability
1.  Understand the properties of probability
2.  Calculate the probability of a single event
3.  Calculate combined events probabilities with the help of probability diagrams and probability trees
4.  Distinguish between independent and mutually exclusive events
5.  Understand probability trees and use them to illustrate probability problems.
6.  Problems involving set symbols.
7.  Relate probability to binomial theorem.
Optional for mainstream :
8.  Calculate advanced probability problems combining combinatorics methods of counting
9.  Understand and apply conditional probability
10.  Illustrate law of total probability using Venn diagram
Introduction on Binomial Probability and the Binomial Formula / Connections:
Illuminations
·  Birthday paradox
·  Exploration with chance
·  Forest Fire Simulation
·  Monty Hall Game
·  Random Drawing Tool
·  Adjustable Spinner
·  / Practice:
·  Jobs that use probability or this site
·  Video: What makes you mad – Mall Parking
How to ask an embarrassing question in survey / Integrated resorts in 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 3
(0.5 Weeks)
align with BSC / Unit 1: Basic IntegrationAt the end of the unit, students should be able to:
1.  understand integration as a reverse process of differentiation.
2.  determine indefinite integrals of sums of terms with powers of x including .
Unit 1.1 Partial Fractions
At the end of the unit, students should be able to
1.  decompose a rational expression into partial fractions,
2.  perform long division on improper rational expressions before expressing the proper rational expressions as partial fractions,
3.  using “cover-up” rule to determine the unknown constants
4.  include cases where denominator is of the form , and
Unit 2: Further Integration
At the end of the unit, students should be able to:
1.  Integrate functions of the form, , , , , where a, b and n are real.
2.  Recall how to decompose proper rational functions in partial fraction where the denominator is of the form
, or
Apply partial fractions to simple rational expressions and then integrate Integral involving the use of double angle formulae .
Unit 3: Applications of integration
At the end of the unit, students should be able to:
1  understand and define definite integrals.
2  evaluate definite integrals.
3  Find area under curve by
(a)  estimation by drawing rectangles / trapeziums.
(b)  Definite integral.
Unit 4: Kinematics
At the end of the unit, students should be able to:
1.  apply differentiation and integration to kinematics problems involving
·  Displacement / total distance travelled
·  Velocity
·  Accel. of a particle moving in a straight line with variable.
Use and sketch the x-t and v-t graphs (Optional for mainstream) /
Connections:
History of Calculus, or from Wikipedia
History of Integration, or from this pdf file
/ Practice:
Jobs that require calculus
What is calculus used for?
·  Illuminations Cardiac Output /
Affective Education
· Trace the history of the development of calculus and the contributions made by various mathematicians and their impact to the development of mankind.
·Comment on the controversy between Newton and Leibniz.
·From the readings on the mathematicians, comment on the ways they go about creating mathematics and the aspects of their lives that has the impacts you the most
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 2/3
(Self-Study) / Unit 1: Vector (June/September Self-study)

At the end of the unit, students should be able to

(1) use notations such as .
(2) undersatnd vector as a directed line segments
(3) understand concept of translation by a vector
(4) understand concept of position vector
(5) define the magnitude of a vector as.
(6) express given vectors in terms of coplanar vectors using sum and difference of vectors.
(7) multiply a vector by a scalar.
(8) Solve geometric problems using vector, e.g. computing area ratios, proving parallelogram, collinearity etc.
exclude :
-express a vector in terms of a unit vector
-solving vectors equations with two unknown parameters. / Connections:
History of vectors or here
Investigation: http://illuminations.nctm.org/ActivityDetail.aspx?ID=42
Sine and cosine rules for solving problems involving vectors
Application of vectors in relative velocity problems, mechanics problems
Man goes to the movies:
http://plus.maths.org/issue42/features/lasenby/index.html
Understanding Turbulence:
http://plus.maths.org/issue1/turb/index.html
Illuminations Airplane storm chaser
Car storm chaser / Practice:
Jobs that use vectors / Infusion of NE:
Students would make connection between actual real world data and vectors. Can make use of examples from the Singapore Geography context.
An online learning site using GSP
For those pupils who prefer this method of learning
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
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
(Align with BSC) / Unit 3: Circle Trigonometry

·  know the concept of unit circle

·  six trigonometric functions for angles of any magnitude (in degrees)

·  know principal values of

·  know the exact values of the trigonometric functions for special angles (0°, 30°, 45°, 60°, 90°, 180°,…)

·  use of , ,

,,

·  solve simple trigonometric equations

·  Proofs of simple trigo identities

·  Simplify trigo expressions / ·  Relate to Pythagoras theorem.
·  Applications of Trigonometry – geography and astronomy, physics and Engineering
·  Foreshortening in perspective drawing
·  Leaning tower of Pisa
http://www.clarku.edu/~djoyce/trig/apps.html / ·  Historical development of trigonometry – from circle trigonometry to 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
/ Connections:
·  modeling of natural phenomena –tides, heart beat, music etc.
·  Applications of Trigonometry – geography and astronomy, physics and Engineering
http://www.clarku.edu/~djoyce/trig/apps.html
§  Application of trigonometric functions in daily life applications eg. Singapore flyer / Graphs of where f(x) can be 1/x, x^2, e^x, sqrt(x) and relate to real life examples of sound waves with such patterns. / ·  Group work to submit Mathematical 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/