Overall Physics Syllabus - Hwa Chong Institutionhsmath.wiki.hci.edu.sg/file/view/2015 S3_MA_SOW_(2015... · Web viewTrigonometry III (4.5 wk) Unit 1: Further Trigonometry (1.5 wk)

Hwa Chong Institution Subject/Programme : MathematicsScheme of Work 2015 Level : Secondary 3

Scheme of Work (incorporating GL-Matrix, Differentiation: italicised for SBGE, refer to Appendix for further differentiation in SSMT)

Learning outcomes are statements of what a student should know, understand and/or be able to demonstrate after completion of a process of learning.

OVERVIEWTerm 1 Term 2 Term 3 Term 4

Algebra III (7 wk)Unit 1: Advanced Algebraic Manipulation (1 wk)Unit 2: Surds and Indices (1 wk)Unit 3: Equations and Inequalities (3 wk)Unit 4: Introduction to Relations & Functions (1 wk)Unit 5: Polynomials (1 wk)

Algebra III (3 wk)Unit 5: cont’d Polynomials (1 wk)Unit 6: Exponents & Logarithms (2 wk)

Relations and Functions (2.5 wk)Unit 1: Modulus Functions (0.5 wk)Unit 2: Power Functions + Standard Graphs (1 wk)Unit 3: Parabolas and Circles (1 wk)

Matrices (1 wk)

Coordinate Geometry II ( 3 wk) Unit 1: Points, Lines and Shapes (1.5 wk)Unit 2: Applications of Straight Line Graphs (1.5 wk)

Trigonometry III (4.5 wk)Unit 1: Further Trigonometry (1.5 wk)Unit 2: Trigonometric Functions (2 wk)Unit 3: Simple Trigonometric Identities and Equations (1 wk)

Revision (2 wk)

2 Class Tests (10%) (Topic are subjected to changes)

Term 1 Test 1: Week 4(a) Sec 2 Algebra(b) Advanced Algebraic Manipulation

Term 1 Test 2: Week 7(a) Surds and Indices(b) Equations and Inequalities

2 Class Tests (10%)(Topic are subjected to changes)

Term 2 Test 3: Week 4(a) Polynomials + Functions

Term 2 Test 4: Week 7(a) Exponents and Logarithms(b) Modulus Functions(c) Power Functions + Standard Graphs

2 Class Tests (10%)(Topic are subjected to changes)

Term 3 Test 5: Week 4(a) Parabolas and Circles(b) Points, lines and shapes(c) Applications of straight line graphs

Term 3 Test 6: Week 7(a) Further Trigonometry(b) Trigonometric Functions

EOY Exam (70%)All Topics

Time Allocated Content/Learning Outcomes Parallel Curriculum Learning Activities Assessment/Feedback ResourcesIn Weeks/hrs Core: Connection: Suggestions: Suggestions: Online Resources:

1

Hwa Chong Institution Subject/Programme : MathematicsScheme of Work 2015 Level : Secondary 3Time Allocated Content/Learning Outcomes Parallel Curriculum Learning Activities Assessment/Feedback Resources1 week

Time frameTerm 1 Week 1

Algebra III (Unit 1/Advanced Algebraic Manipulation)

At the end of the topic, students will be able to:1. Expand and factorise more complex

algebraic expressions.2. Change the subject of a formula with

involvement of roots.3. Add and subtract algebraic fractions

with quadratic denominators leaving answer as a single fraction in simplest

form. (e.g. or

)4. Solving equations involving rational

fractions.5. Manipulate algebraic formulae:

.6. Expand the expressions: (x y)3 and

(a+b+c)3.7. Extend expansion and factorisation

techniques to expressions where the basic unit of unknown is of higher power (x4-9), (x4-5x2+6).

8. Extend expansion and factorization to

expressions - SMTP

Apply the identities learnt in Sec 2

to more advanced problems.

Compare and contrast the concepts of equal identity/equality in Maths vs humanities.

Blended/Cooperative Learning:Students to be organized in expert groups, each group to watch one of the suggested videos and share with rest of group on what they learnt about the algebraic identities.

Experiential Learning:Use algebraic manipulatives to visualise algebraic identities of order 3.

- Formative assessment: Pop Quiz

- Alternative formative assessment: Students to present on what they have learnt from the videos to the rest of the class

- Summative assessment: Term 1 Test 1

Special Factoringhttp://www.purplemath.com/modules/specfact2.htm

Geometrical Interpretation of

http://www.youtube.com/watch?v=5x4gJPchSiYGeometrical Interpretation of

http://www.youtube.com/watch?v=9RHJt0GXLcY

In Weeks/hrs Core: Connection: Suggestions: Suggestions: Online Resources:2

http://www.youtube.com/watch?v=5x4gJPchSiY



Hwa Chong Institution Subject/Programme : MathematicsScheme of Work 2015 Level : Secondary 3Time Allocated Content/Learning Outcomes Parallel Curriculum Learning Activities Assessment/Feedback Resources1 week


Algebra III (Unit 5/Indices and Surds)*solving in integer solutions (rational to be discussed in unit 3*)At the end of the topic, students will be able to:1. Recognise surds and understand that

surds can be written in index form, or with radical signs and state the rules of surds

2. Perform arithmetical operations (addition, subtraction, and multiplication) on expressions involving simple surds in the numerator using properties of real numbers

3. Rationalise fractions involving surds in the denominator (without being instructed)

4. Solve problem sums involving surds5. Solve equations involving surds.6. Understand that by squaring an

equation involving surds, extraneous solutions may be introduced and the solutions need to be checked to ensure they still fulfil the original equation

7. Solve equations involving indices including substitution method

8. Recognize/identify/plot (different bases) basic graphs of exponential functions (for connection to population growth, compound i/r and etc), e.g. y=2x,

9. Solve challenging equations involving surds (surds within surds) - SMTP

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).

Justify the existence of irrational numbers.

Relate the operations of surds and rationalisation of denominator to the three algebraic identities learnt in Sec 2 (e.g.

).

Justify the use of exponential graphs to population growth, radioactivity decay, half-life, compound interest etc.

Practice:Using graphing tools or software such as Microsoft Excel to model bacteria growth using exponential graph.

Collaborative Learning: Investigate how squaring an equation would lead to extraneous solutions.

- Formative assessment: Pop Quiz

- Alternative formative assessment: Open Ended Tasks on Surds (One Equals to Zero and Other Mathematical Surprises Pg 12-13, 18 - 22)

- Alternative formative assessment: Mathematical Modeling on Marshall Cavendish Pg 57


History of Surdshttp://www.mathsisgoodforyou.com/AS/surds.htm

Hotel Infinityhttp://www.mathsisgoodforyou.com/artefacts/hilberthotel.htm

Print Resources:- Additional

Mathematics 360 by Marshall Cavendish Chapter 2

In Weeks/hrs3 week

Time frame

Core:Algebra III (Unit 3/Equations and Inequalities)

At the end of the topic, students will be able

Connection:Explore the use of equations and inequalities in business problems, physics and decision making

Suggestions:Experiential Learning:Use a spreadsheet or graphing software to

Suggestions:- Formative

assessment: Pop Quiz

Online Resources:Simultaneous Equationshttp://www.youtube.com/

3

http://www.youtube.com/watch?v=SZ4x-HzhaKo


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



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



Hwa Chong Institution Subject/Programme : MathematicsScheme of Work 2015 Level : Secondary 3Time Allocated Content/Learning Outcomes Parallel Curriculum Learning Activities Assessment/Feedback ResourcesTerm 1 Week 3 to 5

to:1. Solve quadratic equations using (1)

factorization (2) completing the square (3) formula [emphasis on completing the square where the x2 coeff is not 1. – working is essential]

2. Understand that the quadratic formula is derived from the quadratic equation form by completing the square method

3. Solve linear and non-linear simultaneous equations

4. Discuss the geometrical significance of the algebraic solution of simultaneous equations with the use of suitable IT tools,

5. 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),

6. Solve word problems involving linear and non-linear equations.

7. Relationships between the roots and coefficients of a quadratic equation

8. Understand that different discriminant give rise to different types of solutions for a quadratic equation:(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

9. Apply conditions for to be always positive (or always negative)

10.Transform to the form

Explore biographies of Mathematicians such as Diophantus, Al-Kharizmi, Abu Kamil Babylonian tablets and reflect on how these mathematicians pursue their passion and their role in fulfilling societal needs at that time.

Practice:Model the trajectory path of an object in the air or the stopping distance of a car using quadratic function.

(a) investigate the relationship between the number of points of intersection and the nature of solutions of a pair of simultaneous equations, one linear and one quadratic.

(b) explain how the roots of the equation

are related to the sign of

.(c) show graphically why

there are no real solutions to a quadratic equation

when

is negative.(d) investigate how the

positions of the graph

vary with the sign of

, and describe the graph

when .(e) Examine the solution of

a quadratic equation and that of its related quadratic inequality

(e.g.

and ), and describe both solutions and their relationship.

- Alternative formative assessment: Mathematical Modelling Task on Marshall Cavendish Pg 33

- Alternative formative assessment: Open Ended Tasks on Quadratics (One Equals to Zero and Other Mathematical Surprises Pg 8 – 11, 31 - 35)

- Performance Task – Paper Helicopter ASMS


watch?v=SZ4x-HzhaKoSimultaneous Equations and Intersections of Graphshttp://www.purplemath.com/modules/syseqgen.htm



- Math Through the Ages

- New Syllabus Mathematics 7th Edition by ShingleeChapter 1

4

http://www.purplemath.com/modules/syseqgen.htm




Hwa Chong Institution Subject/Programme : MathematicsScheme of Work 2015 Level : Secondary 3Time Allocated Content/Learning Outcomes Parallel Curriculum Learning Activities Assessment/Feedback Resources

and use it to (i) sketch the graph

11.Transform to the form

and use it to (i) sketch the graph

12.Understand that quadratic function can be expressed in many forms. The values of a, b, and c or h and k or p and q gives different information about the quadratic function.

13.Solve quadratic inequalities, and representing the solution on the number line

14. Solve inequalities of the form

.15. Discuss the difference between the

parabola vs catenary – SMTPIn Weeks/hrs1 week


Core:Algebra III (Unit 4/Relations and Functions)At the end of the topic, students will be able to:1. Define the terms function (one-to-one,

many-to-one), relation (one-to-one, many-to-one, many-to-many, one-to many), domain, range and image.

2. Illustrate a relation using the arrow diagram.

3. Find the expression for inverse function.

4. Acquire the concept and skills needed for composite functions and their inverses – SMTP

Connection:Understand that sometimes it is possible to model data from a real world situation with a linear equation.

Understand that the operations (adding and subtracting functions) within the domain of the functions involved, are similar to that of real numbers.

Suggestions:Experiential Learning:Match graphs with functions (Maths Resource)




Summative assessment: Term 2 Test 3



In Weeks/hrs2 week

Time frame

Algebra III (Unit 5/Polynomials)

At the end of the topic, students will be able to:

Connection:Make connections between division of

polynomial and division of whole

Suggestions:

Experiential Learning:Use a spreadsheet or




Mathematics 360 by Marshall Cavendish

5

Hwa Chong Institution Subject/Programme : MathematicsScheme of Work 2015 Level : Secondary 3Time Allocated Content/Learning Outcomes Parallel Curriculum Learning Activities Assessment/Feedback ResourcesTerm 1 Week 8 to Week 9 to Term 2 Week 1

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

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 theorem11. Use factor theorem to solve for

unknowns in polynomial.12. Apply factor theorem to factorise cubic

expressions and solve cubic equations (including unknown constants to test for understanding instead of using the calculator).

13. Understand that the degree of a polynomial equation tells them about the number of roots that the equation has

14. Sketch cubic graphs and understand that there is always 1 real root and conditions for possible solutions (1) 3 real distinct roots, (2) 1 real root, 2 repeated roots, (3) 1 real root, 2 imaginary roots (4) 3 repeated roots

15. Sketch polynomial graphs in product form of any order and using it to solve polynomial inequalities in product form.(guided SMTP + SBGE)

number, and express the division

algorithm as

.

Make connections between the Fundamental theorem of algebra (prime factors) and the factor theorem.

Explore biographies of Mathematicians Tartaglia Vs Cardano and reflect on how these mathematicians pursue their passion and their role in fulfilling societal needs at that time.

Practice:Relate cubic equations to design of roller coasters (consideration of max allowed speed) and link to integrated resorts.

Explore mathematical questions crafted using Chinese poetic verses in Arithmetic in Nine Sections.

graphing software to(a) 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

- Alternative formative assessment: Concept Map on the behavior of roots for Quadratic vs Cubic functions



Chapter 3

In Weeks/hrs2 week

Core:Algebra III (Unit 6/Exponents

Connection:Relate the solution of the equation

Suggestions:Experiential Learning:


Online Resources:Richter Scale

6


Time frameTerm 2 Week 2 to Week 3

&Logarithms)

At the end of the topic, students will be able to:

1. Know functions and their graphs.

2. Know equivalence of

and that they are inverse functions. [Students must

also be aware and prove , e.g 10lg a = a]

3. Show that and

for any and .4. Understand and apply Laws of

logarithms:(1) product (2) quotient (3) power (4) change-of-base laws.

5. Understand that logarithmic laws can be used to simplify, solve exponential expressions/equations and vice versa.

6. Solve logarithmic and exponential equations.

7. Discuss the use of exponential and logarithmic functions to other disciplines via word problems (e.g. pH value, Richter scale of earthquakes, decibel scale for sound intensity, radioactive decay, population growth). Refer to Add Maths textbook (Scatter plot), pg 189 for e.g.]

8. Sketch and understand the properties of logarithmic graphs vs exponential (of different bases) graphs. (aware the existence of asymptotes and indicate the intercepts and. Understand the properties of the 2 graphs exp/log - inverse of each other)

9. Solve challenging questions involving

to the graph to verify the existence of the solutions or to justify that the solution does not exist.

Trace the history of logarithms and appreciate the complexity of the logarithmic tables and how they have been programmed in calculators, computers.

Model real-life problems using exponential functions, such as the half-life function and heat and cooling function.

Use a spreadsheet or graphing software to(a) investigate the

characteristics of exponential and logarithmic graphs.

(b) display real-world data graphically and model with an appropriate exponential or logarithmic function.

Collaborative Learning:(a) Students to investigate

the cause and effect earthquakes that happened in the last 5 years for e.g. Sichuan Earthquake 2008, China, Tohoko Earthquake 2011, Japan, Christchurch Earthquake, New Zealand. Students to reflect on how they could help in the face of such natural disasters.

(b) Students to investigate the impact of Indonesian Tremors to Singapore.




http://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_properties/v/richter-scale

Logarithms in the Real Worldhttp://www.youtube.com/watch?hl=en-GB&v=3oZPPIVC8MU&gl=SG



7

http://www.youtube.com/watch?hl=en-GB&v=3oZPPIVC8MU&gl=SG







exponential and logarithmic functions.In Weeks/hrs0.5 week


Core:Functions I(Functions and Modulus Functions)

1. Know and sketch the graph (reflection on the x-axis – no other

transformation yet) of where

is linear, quadratic, exponential/logarithmic or trigonometric (trigo in the later chapter).

2. Solve equations involving modulus functions (methods/skills – regions/squaring/substitution).

3. Understand that 4. Understand the difference between

and

Connection:Understand that an absolute value of x is its distance from 0 and the absolute of f(x) is its distance from the line y = 0.

Understand that the operations (adding and subtracting functions) within the domain of the functions involved, are similar to that of real numbers.

Suggestions:Experiential Learning:Match graphs with functions (Maths Rm Resource)





Print Resources:Additional Mathematics 360 by Marshall Cavendish Chapter 4

In Weeks/hrs0.5 week1 week

Time frameTerm 2 Week 5/6

Core:Functions I (Standard Graphs + Transformation)

At the end of the topic, students will be able to:1. Sketch graphs illustrating direct and

indirect proportionality2. Sketch the power functions y=axn,

n = -2,-1,0,1,2,3 … or is rational. E.g.s

3. Identify the basic function and perform simple transformation (translation, reflection, stretch) of standard graphs including exponential, logarithmic,

Connection:Identify the basic unit and appreciate that new products are created as a result of successive transformations for product design, architecture

Compare transformations to point, line, rotational symmetry.

Suggestions:Experiential Learning:Use a spreadsheet or graphing software to(a) explore the

characteristics of the various functions.

(b) display real-world data and match it with appropriate functions (regression).

Collaborative Learning:Work in groups to match and justify sketches of graphs with their respective functions.



- Performance Task: 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



- Shinglee Textbook New Syllabus Mathematics 7th Edition Chapter 5

8

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




power, trigonometric functions (trigo to be discussed later). For graphs with asymptotes, students can relate the movements of the asymptotes to certain transformations

4. Introduce the notion of ,

5. Describe the transformations (at most 3 successive) that the functions have undergone and find the original function from resultant transformed function or vice versa.

6. Understand that 180o rotation about the origin as a successive reflections along the x and y axes

7. Explore the different ways of transformation given the original and end functions.


In Weeks/hrs1 week

Time frameTerm 2 Week 7/8

Core:Functions I (Parabolas and Circles)

At the end of the topic, students will be able to:1. relate the graph of

to

to and that they are inverse functions of each other.

2. sketch the graphs of

(a) parabolic

(b) .3. Perform simple transformation of these

graphs.4. Derive and define the equation of a

circle with centre (a, b) and radius r using the Pythagoras’ theorem, and the

Connection:Compare equation of circle with Pythagoras’ Theorem.

Relate the concept of loci of points equidistant from a fixed point to equation of circle.

Recognise concept of circles as a special class of ellipses.

Explore the use of parabolas in other discicplines (e.g. sciences parabolic motion) and in the real world.

Discuss how to solve geometry problems involving intersection of a parabola/circle and a straight line.

Suggestions:

Experiential Learning:Use a spreadsheet or graphing software to(a) explore the

characteristics of the various functions.

(b) investigate the graph of

when varies.

(c) display real-world data and match it with appropriate functions (regression).

Collaborative Learning:(a) Work in groups to

match and justify






9


special case when the centre is at the origin.

5. Transform general form of equation of

circle to by completing the square method.

6. Compare the three cases where line meets circle (1) 2 distinct roots, (2) 2 repeated roots, (3) imaginary roots.

7. Find the intersection of two circles.

Practice:Determine centre of a broken circular wheel in archaeological studies

Find epicentre by solving 3 circle equations, detected from 3 satellite stations.

sketches of graphs with their respective functions.

In Weeks/hrs1 week


Core:Algebra III (Matrices)

At the end of the topic, students will be able to: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.

10. Solve word problems involving the sum and product of matrices and interpret the data in the given or computed matrices

Connection:Contrast the matrices operations with algebraic operations (e.g. AX = B is non-commutative whereas ax = b is commutative).

Practice:Create encrypted messages using matrices and decrypt messages using matrix operations.

Investigate the use of matrices in operation research

Discuss some applications of matrix multiplication, e.g. transformation matrices for movie making.

Discuss how the idea of matrices is being used in spreadsheets and how these programs are useful in their everyday lives.

Suggestions:

Experiential Learning:Use a graphing calculatorto input matrices and to compute inverse matrices – simplify decoding process.

Collaborative Learning:Students to get into groups and justify if two matrices can be multiplied by checking the orders of the matrices.



- Alternative formative assessment: Students to encode and decode using shift transformations (refer to NSA lesson plan) and present their work in an oral presentation


Online Resources:Matrices Khan Academyhttp://www.khanacademy.org/math/algebra/algebra-matricesNSA lesson plan on encoding and decodinghttp://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/matrices_secret_weapon.pdf

In Weeks/hrs1.5 week

Core:Coordinate Geometry III (Unit 1/Points, Lines and Shapes)

Connection:Relate gradient to tangent ratio of the angle of inclination between the line

Suggestions:

Collaborative Learning:


assessment: Pop

Online Resources:Descartes and Coordinate System

10

Hwa Chong Institution Subject/Programme : MathematicsScheme of Work 2015 Level : Secondary 3Time Allocated Content/Learning Outcomes Parallel Curriculum Learning Activities Assessment/Feedback ResourcesTime frameTerm 3 Week 1 to Week 2

At the end of the topic, students will be able to: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 that they can compare the slopes of two lines and determine if the lines are parallel or perpendicular.

5. Understand gradient of a perpendicular

line using the relationship 6. Identify equations of parallel or

perpendicular lines.7. Find the equation of the circle passing

through three given points. (using perpendicular bisectors)

8. Formulate equations of lines passing through a given point and parallel or perpendicular to another given line.

9. Find equation of perpendicular bisector between two points.

10. Find the area of rectilinear figure given its vertices(Shoelace Formula).

11. Estimation of the gradient of a curve by drawing a tangent

and the positive direction of the x-axis

Deduce the relationship between the gradient of (a) two parallel lines, (b) two perpendicular lines.

Synthesise the different methods to find area of triangles learnt (1)

Shoelace method, (2) and extend these to finding the area of polygons.

Relate the concept of foot of perpendicular to finding shortest distance from point to line.

Relate estimation of gradient of tangent for speed-time graph to instantaneous speed in kinematics.

Explore and discuss ways of finding the area of a triangle (or polygon) with given vertices.

Discuss other ways of finding area of rectilinear figures.

Quiz

Alternative formative assessment:Concept Map of properties of lines in Coordinate Geometry

Alternative formative assessment: Mathematical Modeling on Marshall Cavendish Pg 162

Summative assessment:Term 3 Test 5

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



- Shinglee Textbook New Syllabus Mathematics 7th Edition Chapter 4


Time frameTerm 3 Week 2 to 3

Core:Coordinate Geometry III(Unit 2/Applications of Straight Line Graphs)

At the end of the topic, students will be able to:1. Distinguish between linear and non-linear relationships.2. Determine a linear relation based on

Connection:Justify the use of straight line graph in science experiment (e.g. oscillation of a pendulum (Hooke’s Law), relationship between resistance in circuit (Ohm’s Law), kinematics graphs.

Understand the purpose to convert

Suggestions:

Maths Journal:Use a table to explore some typical transformations of non-linear equation into a linear equation

Inquiry Learning:



- Alternative formative assessment: Mathematical Modeling on Marshall Cavendish



Online resources:- Singapore data(http://data.gov.sg/

11

http://data.gov.sg/home.aspx





experimental results of two non-linearly related quantities.3. Convert non-linear equations into linear form.4. Derive the relationship between two variables given the straight line graphs.5. By applying linear law to obtain straight line graphs using experimental data, determine unknowns by reading from the graphs.6. Understand independent and dependent variables.7. Understand and identify outliers or incorrect readings.8. Expect to plot linear graph given set of experimental data (with no scale given – proper scale).

non-linear graphs to linear ones and how it is used to prove/justify the relationship between variables.

Practice:Model phenomena in Science or real world using an equation. For example, students can conduct simple Science experiments to collect data. Part (a) to find a formula (V = RI) for the resistance of a resistor.

Part (b) to find a formula () for the period of a pendulum.

Use suitable software to find out what happens when the non-linear equation has more than two unknown constants.

Plot the curve of best fit to fit a given set of data directly to decide the relationship between each set of unknowns.

Engage in simple Science experiments to collect data and analyse data using a straight line graph.

Exploratory Activity:Predict population growth using suitable linear function. (Textbook: Pg. 213)

Collaborative Learning:Working in groups, based on the data on water consumption per capita in Singapore for the past 10 years, students will plot graphs to predict future water consumption based on graph plotted using data from past years.

Identify and explain abnormal or inconsistent data.

Explore new water sources for Singapore

Pg 213


home.aspx)

- To convert non-linear relationships to linear form.(http://www.youtube.com/watch?v=pX6WlxP2eok)

- Applications of linear law(http://www.youtube.com/watch?v=Gvb6MLB_x6I)



Core:Trigonometry III (Unit 1/Further Trigonometry)At the end of the topic, students will be able to:1. Prove of Sine Rule and derive (guided)

the Cosine Rule on their own.2. Use the sine and cosine rules to

articulate the relationships 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.

Connection:Explore and discuss applications of Trigonometry to different fields like geography and astronomy, physics and engineering.

Relate concept of sine rule to congruency tests learnt in Sec 2Illustrate the ambiguous case of Sine Rule using construction/GSP and emphasise congruency tests learnt in Sec 2.

Suggestions:

Experiential Learning:Use Clinometer app on iPhone or Android phone to find the angle of elevation or depression of particular buildings

To organise a treasure hunt where treasures are located at different spots as a result of ambiguous case of sine



- Alternative formative assessment: Concept Map on connecting the areas of triangles from various topics for different types of triangles

Online Resources:Leaning tower of Pisahttp://www.clarku.edu/~djoyce/trig/apps.html

Applications of Trigonometryhttp://www.youtube.com/watch?v=wvmU7XKdt3w

Trigonometry in Real

12

http://www.youtube.com/watch?v=wvmU7XKdt3w



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


http://www.youtube.com/watch?v=Gvb6MLB_x6I



http://www.youtube.com/watch?v=pX6WlxP2eok



http://data.gov.sg/home.aspx


3. Solve triangles through Sine Rule (including ambiguous case) & Cosine rule.

4. Use the formula for area of triangle (Heron’s formula).

5. Prove Heron’s formula – SMTP6. Find the perpendicular height of a

triangle using area of triangle (apart from using trigonometry)

7. Know and use the concept of bearings.8. Solve 2D and 3D problems.9. Compute angles of elevation and

depression, shortest distance, maximum angle elevation and understand that shortest horizontal distance would give maximum angle of elevation or depression.

Use circle properties to derive cosine rule.

Practice:Determinet the actual flight distance (geodesic path) using cosine rule.

Explore using software to check consistency of theoretical flight path vs actual flight path.

rule. Students to use Bearing app on iPhone or Android phone to locate the treasures.


Lifehttp://www.youtube.com/watch?v=n1A2HqSXtGI

Print Resources:- New Syllabus

Mathematics 3, 7th Edition by Shinglee Chapter 6,7

In Weeks/hrs2 week


Core: Trigonometry III (Unit 2/Trigonometric Functions)

At the end of the topic, students will be able to:1. Know the concept of unit circle.2. Know the six trigonometric functions for

angles of any magnitude (in degrees or radians) and the reciprocal relationship between trig functions, e.g. sec x and cos x and etc.

3. Distinguish between

and

.4. Know principal values (1 to 1 function)

of . – SMTP5. Know the exact values of the

trigonometric functions for special angles (0, 30, 45, 60, 90, 180 and in radians).

Connection:Discuss the relationships between sin A, cos A and tan A, with respect to the line segments related to a unit circle.

Relate the use of sine and cosine functions in sciences (e.g. tides, Ferris wheel and sound waves).

Understand that trigonometric functions, and their compositions gain significance when they are used to model waves and periodic behaviour.

Practice:Explore the historical development of trigonometry – from circle trigonometry to triangle trigonometry and its impact on the study of astronomy

Model natural phenomena – tides, heartbeat, music etc. using graphs of

Suggestions:

Experiential Learning:Use a Geogebra or GSP to(a) investigate the

relationship of sin A, cos A and tan A with respect to the unit circle.

(b) display the graphs of trigonometric functions and discuss their behaviour, and investigate how a graph (e.g.

) changes when a, b or c varies.




- Performance Task: Students to get into groups to find out the different ferris wheels for e.g. Singapore Flyer around the world and to use sine or cosine to its function


Online Resources:Trigonometric Functions and Unit Circlehttp://www.youtube.com/watch?v=rrXLl2WTKEc

Applications of Trigonometry – geography and astronomy, physics and Engineeringhttp://www.clarku.edu/~djoyce/trig/apps.html



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http://www.youtube.com/watch?v=rrXLl2WTKEc



http://www.youtube.com/watch?v=n1A2HqSXtGI




6. Identify amplitude, periodicity (behaviour repeated over intervals of equal length) and symmetries related to sine, cosine and tangent functions.

7. Understand that they can translate periodic functions in the same way as they translate other functions.

8. Sketch graphs of

,

and

,

.

where can be , , ,

and relate to real life examples of sound waves with such patterns.

In Weeks/hrs1 week


Core:Trigonometry III (Unit 3/Simple Trigonometric Identities and Equations)

At the end of the topic, students will be able to:1. Understand that the interrelationships

amongst the six basic trigonometric functions make it possible to write trigonometric expressions in various equivalent forms.

2. Derive and relate to Pythagoras’ theorem.

3. Use of ,

,

,

,

4. Simplify trigonometric expressions.

Connection:Make connections between solutions obtained from solving trigonometric equations and graphical method.

Discuss similarities and differences in solving algebraic equations and trigonometric equations.






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5. Solve simple trigonometric equations, including the use of trigo identities, in degrees. (angles can involve more than 1x, e.g. 2x-30o]

6. Prove simple trigonometric identities, including the use of trigo identities. (rigourous proving, > 5 steps – SMTP)

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Documents

Overall Physics Syllabus - Hwa Chong Institutionhsmath.wiki.hci.edu.sg/file/view/2015 S3_MA_SOW_(2015... · Web viewTrigonometry III (4.5 wk) Unit 1: Further Trigonometry (1.5 wk)