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- explorat ion 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 (Assignmen ts/ Tests/) Assignments/worksheets/skills check Textbook exercises Online Quizzes Class Tests Environmen t 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) Connections History of Trigonometric Functions : http://www-history.mcs.st- andrews.ac.uk/HistTopics/Tri Practice: Jobs that involve trigonometric identities
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 radian2. find the arc length and area of sector/segment3. 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
Environment Face to face & Department wiki. Online Discussion
ResourcesCurriculum of Core Curriculum of Connection Curriculum of
PracticeCurriculum of
Identity
Term 1 (5 weeks) Unit 2 : Further Trigonometric Identities
At the end of the unit, students should be able to1*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
ConnectionsHistory of Trigonometric Functions: http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html
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, PingalaPlinko game, Sierpinski triangleApplication 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
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 change3. explain increasing/decreasing functions using 4. apply differentiation to find stationary points
Unit 4.3: Further differentiatio n (3 weeks) At the end of the unit, students should be able to:*Optional for mainstream1. 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).
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.html
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
5. Apply the above to all the problems in Calculus I, Unit 2. 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
Environment Face to face & Department wiki. Online group
discussionsResources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 2(Self-Study)
Topic : Probability 1. Understand the properties of probability2. Calculate the probability of a single event3. Calculate combined events probabilities with the help of probability diagrams and
probability trees4. Distinguish between independent and mutually exclusive events
Connections:Illuminations Birthday paradox Exploration with chance Forest Fire Simulation Monty Hall Game
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 counting9. Understand and apply conditional probability10. Illustrate law of total probability using Venn diagramIntroduction on Binomial Probability and the Binomial Formula
Random Drawing Tool Adjustable Spinner
– Mall ParkingHow to ask an embarrassing question in survey
Processes (include pedagogies and indicate infusion of ICT)
Problem Solving Concept teaching and assessment (via online concept quiz.) Self-directed independent study Drill & Practice
Environment Face to face & Department wiki. Online group
discussionsResources
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 FractionsAt the end of the unit, students should be able to1. 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
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
4. include cases where denominator is of the form ,
and
Unit 2: Further IntegrationAt 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 integrationAt 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: KinematicsAt 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 xt and vt graphs (Optional for mainstream)
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
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 hereInvestigation: 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 problemsMan goes to the movies:http://plus.maths.org/issue42/features/lasenby/index.htmlUnderstanding Turbulence:http://plus.maths.org/issue1/turb/index.htmlIlluminations Airplane storm chaserCar 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 GSPFor 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
Environment Face to face & Department wiki. Online group
discussionsResources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 22 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,…)
Relate to Pythagoras theorem.
Applications of Trigonometry – geography and astronomy, physics and Engineering
Foreshortening in perspective drawing
Historical development of trigonometry – from circle trigonometry to triangle trigonometry.
Environment Face to face & Department wiki. Online group
discussionsResources
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 ICT)
Problem Solving Concept teaching and assessment (via online concept quiz.) Self-directed independent study Drill & Practice
Environment Face to face & Department wiki. Online group
discussionsResources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of Identity
Term 3(1 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.
Environment Face to face & Department wiki. Online group
discussionsResources
Curriculum of Core Curriculum of Connection Curriculum of Practice
Curriculum of 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.
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
Environment Face to face & Department wiki. Online group
discussionsResources
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
exponential function growth, decay
parabolic
perform simple transformation of standard graphs
relate the graph of to
to
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/lessonplans/9-12/debris/index.html
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)
SMTP, optional for IP find the equation of the circle passing through three given points intersection of two circles Sketch polynomial graphs in product form of any order and using it to solve
polynomial inequalities in product form. Step functions and greatest integer functions.
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
.Processes (include pedagogies and indicate infusion of ICT)
Problem Solving Concept teaching and assessment (via online concept quiz.) Self-directed independent study Drill & Practice
Environment Face to face & Department wiki. Online group
discussionsResources
Suggested Reading Lists
(1) Writing Math Research Papers – A Guide for Students and Instructors by Rpbert 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
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