Yearly Plan – Additional Mathematics Form 4 (2009)

Yearly Plan – Additional Mathematics Form 4 (2009)

 

 

Week No	Learning Objectives Pupils will be taught to.....	Learning Outcomes Pupils will be able to…	No of Periods	Suggested Teaching & Learning activities/Learning Skills/Values	Points to Note
Topic/Learning Area Al : FUNction --- 3 weeks First Term
2   5-9/1/09	1. Understand the concept of relations.	1.1 Represent relations using arrow diagrams ordered pairs graphs Identify domain, co domain, object, image and range of a relation. 1.3 Classify a relation shown on a mapped diagram as: one to one, many to one, one to many or many to many relation.	1               1	Use pictures, role-play and computer software to introduce the concept of relations.     Skill : Interpretation, observe connection between domain, co domain, object, image and range of a relation.	Discuss the idea of set and introduce set notation.
	2.    Understand the concept of functions.	2.1 Recognise functions as a special relation..   2.2 Express functions using function notation.   2.3 Determine domain, object, image and range of a function. 2.4 Determine the image of a function given the object and vice versa.	1         1	Give examples of finding images given the object and vice versa. Given f : x ® 4x – x2. Find image of 5. Given function h : x ® 3x – 12. Find object with image = 0.   Use graphing calculators and computer software to explore the image of functions.	Represent functions using arrow diagrams, ordered pairs or graphs, e.g.   "" is read as "function f maps x to 2x". ""is read as "2x is the image of x under the function f". Include examples of functions that are not mathematically based. Examples of functions include algebraic (linear and quadratic), trigonometric and absolute value. Define and sketch absolute value functions.
3   12-17/1/09	3. Understand the concept of composite functions.	3.1 Determine composition of two functions.     3.2 Determine the image of composite functions given the object and vice versa   3.3 Determine one of the functions in a given composite function given the other related function.	1         1         2	Use arrow diagrams or algebraic method to determine composite functions. Give examples of finding images given the object and vice versa for composite functions   For example : Given f : x ® 3x – 4. Find ff(2), range of value of x if ff(x) > 8.   Give examples for finding a function when the composite function is given and one other function is also given.   Example : Given f : x® 2x – 1. find function g if The composite function fg is given as fg : x ® 7 – 6x composite function gf is given as gf : x ® 5/2x.	Involve algebraic functions only.       Images of composite functions include a range of values. (Limit to linear composite functions). Define composite functions
4   19-23/1/09	4. Understand the concept of inverse functions.	4.1 Find the object by inverse mapping given its image and function.   4.2 Determine inverse functions using algebra.   4.3 Determine and state the condition for existence of an inverse function Additional Exercises	1   1       1 1	Use sketches of graphs to show the relationship between a function and its inverse.   Examples : Given f: x, find	Limit to algebraic functions. Exclude inverse of composite functions.         Emphasise that the inverse of a function is not necessarily a function.
5   26-30/1/09		PUBLIC HOLIDAY (CHINESE NEW YEAR)
Topic A2 : Quadratic Equations ---3 weeks
6   2-6/2/09	1. Understand the concept of quadratic equations and their roots.	Recognise a quadratic equation and express it in general form.     1. 2 Determine whether a given value is the root of a quadratic equation by substitution; inspection.   1.3 Determine roots of quadratic equations by trial and improvement method.	1             1	Use graphing calculators or computer software such as the Geometer's Sketchpad and spreadsheet to explore the concept of quadratic equations           Values : Logical thinking Skills : seeing connection, using trial and error methods	Questions for 1..2(b) are given in the form of ; a and b are numerical values.
7   9-14/2/09	2. Understand the concept of quadratic equations.	2.1 Determine the roots of a quadratic equation by factorisation; completing the square c) using the formula.     2.2 Form a quadratic equation from given roots.	1   1       2	If x = p and x = q are the roots, then the quadratic equation is , that is . Involve the use of: and where α and β are roots of the quadratic equation   Skills : Mental process, trial and error	Discuss when , hence or . Include cases when p = q.   Derivation of formula for 2.1c is not required.
8   16-20/2/09	3. Understand and use the conditions for quadratic equations to have a) two different roots; b) two equal roots; c) no roots.    a)dua punca berbeza;	3.1 Determine types of roots of quadratic equations from the value of .   3.2 Solve problems involving in quadratic equations to: a) find an unknown value; b) derive a relation.   Additional Exercises	2       2           2	Giving quadratic equations with the following conditions : , and ask pupils to find out the type of roots the equation has in each case.   Values: Making conclusion, connection and comparison	Explain that "no roots" means "no real roots".
Topic A3 : Quadratics Functions---3 weeks
9   23-27/2/09	1. Understand the concept of quadratic functions and their graphs.	1.1 Recognise quadratic functions	1	1) Use graphing calculators or computer software such as Geometer's Sketchpad to explore the graphs of quadratic functions. f(x) = ax2 + bx + c f(x) = ax2 + bx f(x) = ax2 + c * pedagogy : Constructivism Skills : making comparison & making conclusion
		1.2 Plot quadratic function graphs:      a)based on given tabulated values; b) by tabulating values based on given functions.	2	1) Use examples of everyday situations to introduce graphs of quadratic functions.   Contextual learning
		1.3 Recognise shapes of graphs of quadratic functions.	1		Discuss the form of graph if a > 0 and a < 0 for   Explain the term parabola.
10   2-6/3/09		1.4 Relate the position of quadratic function graphs with types of roots for .	2	Recall the type of roots if : b2 – 4ac > 0 b2 – 4ac < 0 b2 – 4ac = 0	Relate the type of roots with the position of the graphs.
	2. Find the maximum and minimum values of quadratic functions.	2.1 Determine the maximum or minimum value of a quadratic function by completing the square.	2	Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the graphs of quadratic functions   Skills : mental process , interpretation	Students be reminded of the steps involved in completing square and how to deduce maximum or minimum value from the function and also the corresponding values of x.
11   9-13/3/09	3. Sketch graphs of quadratic functions.	3.1 Sketch quadratic function graphs by determining the maximum or minimum point and two other points.	2	Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to reinforce the understanding of graphs of quadratic functions. Steps to sketch quadratic graphs: a) Determining the form"È" or "Ç" b) finding maximum or minimum point and axis of symmetry. c) finding the intercept with x-axis and y-axis. d) plot all points e) write the equation of the axis of symmetry	Emphasise the marking of maximum or minimum point and two other points on the graphs drawn or by finding the axis of symmetry and the intersection with the y-axis. Determine other points by finding the intersection with the x-axis (if it exists).
	4. Understand and use the concept of quadratic inequalities.	4.1 Determine the ranges of values of x that satisfies quadratic inequalities.	2	Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the concept of quadratic inequalities.	Emphasise on sketching graphs and use of number lines when necessary.
12   16-20/3/09		SCHOOL HOLIDAY
Topic A4: Simultaneous equations---2 weeks
13   23-27/3/09	1. Solve simultaneous equations in two unknowns: one linear equation and one non-linear equation.	1.1 Solve simultaneous equations using the substitution method.	4	Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the concept of simultaneous equations. Value: systematic Skills: interpretation of mathematical problem	Limit non-linear equations up to second degree only.
14   30-3/4/09		1.2Solve simultaneous equations involving real-life situations.       Additional Exercises	2           2	Use examples in real-life situations such as area, perimeter and others.   Pedagogy: Contextual Learning Values : Connection between mathematics and other subjects
Topic G1. Coordinate Geometry---5 weeks
15   6-10/4/09	1. Find distance between two points.	Find the distance between two points, using formula	1	Skill : Use of formula	Use the Pythagoras' Theorem to find the formula for distance between two points.
	2.    Understand the concept of division of line segments	2.1Find the midpoint of two given points.   2.2Find the coordinates of a point that divides a line according to a given ratio m : n.	1   2	Skill : Use of formula Value : Accurate & neat work	Limit to cases where m and n are positive. Derivation of the formula is not required.
16   13-17/4/09	3    Find areas of polygons.	3.1 Find the area of a triangle based on the area of specific geometrical shapes.   3.2 Find the area of a triangle by using formula.   3.3 Find the area of a quadrilateral using formula.	1           1	Values : Systematic & neat   Skills : use of formula , recognise relationship and patterns	Limit to numerical values. Emphasise the relationship between the sign of the value for area obtained with the order of the vertices used.   Derivation of the formula: is not required. Emphasise that when the area of polygon is zero, the given points are collinear.
17   20-24/4/09	4     Understand and use the concept of equation of a straight line.	Determine the x-intercept and the y-intercept of a line. Find the gradient of a straight line that passes through two points.   Find the gradient of a straight line using the x-intercept and y-intercept   4.4 Find the equation of a straight line given: gradient and one point; two points; x-intercept and y-intercept. 4.5Find the gradient and the intercepts of a straight line given the equation.   4.6Change the equation of a straight line to the general form 4.7Find the point of intersection of two lines.	1       1         2             1         1	Use dynamic geometry software such as the Geometer's Sketchpad to explore the concept of equation of a straight line.   Skills : drawing relevant diagrams, using formula, recognising relationship, compare and contrast.   Values : Neat & systematic Pedagogy: contextual learning                             Finding point of intersection of two lines by solving simultaneous equations	Answers for learning outcomes 4.4(a) and 4.4(b) must be stated in the simplest form. Involve changing the equation into gradient and intercept form
18   27-30/4/09                                       19   4-8/5/09	5.    Understand and use the concept of parallel and perpendicular lines.	5.1 Determine whether two straight lines are parallel when the gradients of both lines are known and vice versa.   5.2 Find the equation of a straight line that passes through a fixed point and parallel to a given line.       5.3 Determine whether two straight lines are perpendicular when the gradients of both lines are known and vice versa. 5.4 Determine the equation of a straight line that passes through a fixed point and perpendicular to a given line. 5.5 Solve problems involving equations of straight lines.	1             1           1           1       2	Use examples of real-life situations to explore parallel and perpendicular lines.   Skill: Use of formula; making comparison               Students to be exposed to SPM exam type of questions.     Values : hard work, cooperative   Pedagogy : Mastery learning	Emphasise that for parallel lines: .                       Emphasise that for perpendicular lines . Derivation of is not required.
	6    Understand and use the concept of equation of locus involving distance between two points.	6.1 Find the equation of locus that satisfies the condition if: a)the distance of a moving point from a fixed point is constant; b) the ratio of the distances of a moving point from two fixed points is constant     6.2 Solve problems involving loci.	1             1         2	Use examples of real-life situations to explore equation of locus involving distance between two points. Use graphic calculators and dynamic geometry software such as the Geometer's Sketchpad to explore the concept of parallel and perpendicular lines.   Value : Patience, hard working Pedagogy: contextual learning Skill : drawing relevant diagrams
Topic T1: Circular Measures---3 weeks
20   11-15/5/09	1. Understand the concept of radian.	1.1 Convert measurements in radians to degrees and vice versa.	1	Use dynamic geometry software such as the Geometer's Sketchpad to explore the concept of circular measure. Students measure angle subtended at the centre by an arc length equal the length of radius. Repeat with different radius. Skill : contextual learning Value : Accurate, making conclusion.	Discuss the definition of one radian. "rad" is the abbreviation of radian. Include measurements in radians expressed in terms of π.   p rad = 1800
	2. Understand and use the concept of length of arc of a circle to solve problems.        bulatan	2.1 Determine: length of arc; radius; and angle subtended at the centre of a circle based on given information.	2	Use examples of real-life situations to explore circular measure. Derivation of S = j θ by use of ratio or by deduction using definition of radian. Skill : Making conclusion or deduction, application of formula	Major and minor arc lengths discussed   Emphasize that the angle must be in radian. Students can also use formula S= when the angle given is in degree
21   18-22/5/09		Find perimeter of segments of circles. Solve problems involving lengths of arcs.	1     1
22   25-29/5/09	3. Understand and use the concept of area of sector of a circle to solve problems	3.1 Determine the: a) area of sector; b)radius; and c)angle subtended at the centre of a circle based on given information.   3.2 Find the area of segments of circles.   3.3 Solve problems involving areas of sectors.	2             2     2	Deriving the formula L= ½ j2 θ Using ratio Skill : drawing relevant diagrams , recognising relationship & making conclusion Value : Systematic & logical	Emphasize that the angle must be in radian. Area of major sektor need to be discussed Students can also use formula L= if the angle given is in degree.
23&24   1-12/6/09		SCHOOL HOLIDAY
Topic A5 : Indices and Logarithms---4 weeks Second Term
25   15-19/6/09	1. Understand and use the concept of indices and laws of indices to solve problems.	1.1 Find the value of numbers given in the form of: integer indices. fractional indices.   Use laws of indices to find the value of numbers in index form that are multiplied, divided or raised to a power.	1         1	Use examples of real-life situations to introduce the concept of indices.   Use computer software such as the spreadsheet to enhance the understanding of indices.   Pedagogy : Constructivism Skill : making inference, use of laws Value : systematic, logical thinking	Discuss zero index and negative indices.
		1.3 Use laws of indices to simplify algebraic expressions	1
	2. Understand and use the concept of logarithms and laws of logarithms to solve problems.	2.1 Express equation in index form to logarithm form and vice versa.   2.2 Find logarithm of a number	1	Use scientific calculators to enhance the understanding of the concept of logarithm. Explain definition of logarithm. N = ax; loga N = x with a > 0, a ≠ 1.   Value : systematic, abide by the laws   Pedagogy:Mastery learning	Emphasise that: loga 1 = 0; loga a = 1.   Emphasise that: a) logarithm of negative numbers is undefined; b) logarithm of zero is undefined. Discuss cases where the given number is in: a) index form b) numerical form.
26   22-27/6/09		2.3 Find logarithm of numbers by using laws of logarithms 2.4 Simplify logarithmic expressions to the simplest form.	2       1	Activities : Demonstration Value : systematic and organised   Skill : recognising pattern and relationship, application of laws	Discuss laws of logarithms
	3 Understand and use the change of base of logarithms to solve problems.	3.1 Find the logarithm of a number by changing the base of the logarithm to a suitable base.	1	Aktivities : Demonstration Pedagogy: Mastery learning , problem solving	Discuss:
27   29-3/7/09		3.2 Solve problems involving the change of base and laws of logarithms.	2	Aktivities : Demonstration Pedagogy: Mastery learning , problem solving.
	4. Solve equations involving indices and logarithms	4.1 Solve equations involving indices.	2	Aktivities : Demonstration Pedagogy: Mastery learning , problem solving.	Equations that involve indices and logarithms are limited to equations with single solution only. Solve equations involving indices by: a) comparison of indices and bases; b) using logarithms.
28   6-10/7/09	.	4.2 Solve equations involving logarithms.   Additional/reinforcement Exercises on this topic	2     2	Values : Systematic & logical thinking
Topic S1: Statistics ---4 Weeks
29   13-17/7/09                                                 30   20-24/7/09	1 Understand and use the concept of measures of central tendency to solve problems.	1.1 Calculate the mean of ungrouped data. 1.2 Determine the mode of ungrouped data. 1.3 Determine the median of ungrouped data 1.4Determine the modal class of grouped data from frequency distribution tables. 1.5 Find the mode from histograms. 1.6 Calculate the mean of grouped data 1.7 Calculate the median of grouped data from cumulative frequency distribution tables.   1.8 Estimate the median of grouped data from an ogive   1.9 Determine the effects on mode, median and mean for a set of data when: each data is changed uniformly; ii) extreme values exist; iii) certain data is added or removed   1.10 Determine the most suitable measure of central tendency for given data.	1                 2           1       1     2                 1	Use scientific calculators, graphing calculators and spreadsheets to explore measures of central tendency.   Students collect data from real-life situations to investigate measures of central tendency. Eg. 1) Length of leaves in school compound 2). Marks for Add maths in the class.   Values : Cooperative; honest , logical thinking Skill : classification, making conclusion   Pedagogy : 1. Contextual learning 2. Constructivism 3. Multiple intelligence   Skills : Classification; observing relationship, course and effect, able to analise and make conclusion	Discuss grouped data and ungrouped data.         Involve uniform class intervals only.               Derivation of the median formula is not required.       Ogive is also known as cumulative frequency curve.                   Involve grouped and ungrouped data
31   27-31/7/09	2. Understand and use the concept of measures of dispersion to solve problems.	2.1 Find the range of ungrouped data.   2.2 Find the interquartile range of ungrouped data.   Find the range of grouped data	1	Activities : 1. Teacher gives real life examples where values of mean, mode adn medium are more or less the same and not sufficient to determine the consistency of the data and that lead to the need of finding measures of dispersion
		Find the interquartile range of grouped data from the cumulative frequency table       2.5 Determine the interquartile range of grouped data from an ogive.   2.6Determine the variance of a)ungrouped data; b)grouped data. 2.7 Determine the standard deviation of: ungrouped data grouped data.	1         1           2	Values : 1. Honest 2. cooperative       Pedagogy : Contextual learning	Determine the upper and lower quartiles by using the first principle.
32   3-7/8/09		2.8 Determine the effects on range, interquartile range, variance and standard deviation for a set of data when: each data is changed uniformly; b) extreme values exist; c) certain data is added or removed. 2.9 Compare measures of central tendency and dispersion between two sets of data.	2                 2	Skills : 1. Compare and contrast 2. Classification 3. Problem Solving 4. Sorting data from small to big   Pedagogy : Contextual learning   Values : Logical thinking	Emphasise that comparison between two sets of data using only measures of central tendency is not sufficient.
Topic AST1: SOLUTION OF TRIANGLES---2 weeks
33   10-14/8/09	1. Understand and use the concept of sine rule to solve problems.	1.1Verify sine rule.   1.2Use sine rule to find unknown sides or angles of a triangle.   1.3Find the unknown sides and angles of a triangle involving ambiguous case 1.4Solve problems involving the sine rule.	1   1         1     1	Use dynamic geometry software such as the Geometer's Sketchpad to explore the sine rule.   Use examples of real-life situations to explore the sine rule.     Skill : Interpretation of problem Value : Accuracy	Include obtuse-angled triangles
34   17-21/8/09	2. Understand and use the concept of cosine rule to solve problems.	2.1 Verify cosine rule. 2.2 Use cosine rule to find unknown sides or angles of a triangle. 2.3 Solve problems involving cosine rule. 2.4Solve problems involving sine and cosine rules	1     1       2	Use dynamic geometry software such as the Geometer's Sketchpad to explore the cosine rule.   Use examples of real-life situations to explore the cosine rule.   Acticities : Demonstration Skill : Interpretation of datas given Value : Accuracy.	Include obtuse-angled triangles
35   24-28/8/09		SCHOOL HOLIDAY
36   1-4/9/09	3. Understand and use the formula for areas of triangles to solve problems.	3.1 Find the areas of triangles using the formula or its equivalent     3.2.Solve problems involving three- dimensional objects.     Additional Exercises	1   2 1	Use dynamic geometry software such as the Geometer's Sketchpad to explore the concept of areas of triangles.   Use dynamic geometry software such as the Geometer's Sketchpad to explore the concept of areas of triangles. Skills : Recognising Relationship Analising data   Use examples of real-life situations to explore area of triangles.   Value : Systematic
Topic ASS1: INDEX NUMBER---1 week
37   7-11/9/09	1. Understand and use the concept of index number to solve problems	Calculate index number. Calculate price index.   Find Q0 or Q 1 given relevant information.	1 1	Use examples of real-life situations to explore index numbers. Skill : Analise, problem solving Value : Systematic	Q0 = Quantity at base time. Q1 = Quantity at specific time.
38   14-18/9/09	2. Understand and use the concept of composite index to solve problems	2.1 Calculate composite index. 2.2 Find index number or weightage given relevant information. 2.3 Solve problems involving index number and composite index. Additional Exercises or past year questions	2       2     2	Use examples of real-life situations to explore composite index. Eg Composite index of share.   Skill : Analise, problem solving Value : Systematic	Explain weightage and composite index.
39   21-25/9/09		PUBLIC HOLIDAY (HARI RAYA AIDIL ADHA)
Topic K1 : Differentiation---5 Weeks
40   28-2/10/09	1. Understand and use the concept of gradients of curve and differentiation.	1.1Determine the value of a function when its variable approaches a certain value. 1.2Find the gradient of a chord joining two points on a curve.   1.3Find the first derivative of a function , as the gradient of tangent to its graph.   Find the first derivative of polynomials using the first principles. Deduce the formula for first derivative of the function by induction.	1       1             2	Use graphing calculators or dynamic geometry software such as Geometer's Sketchpad to explore the concept of differentiation.   Skills : Logical Thinking, relationship, application of rules, making inference, making deduction   Pedagogy : Constructivism   Activities : Explanation & demonstration	Idea of limit to a function can be illustrated using graphs. The concept of first derivative of a function is explained as a tangent to a curve can be illustrated using graphs. Limit to ; a, n are constants, n = 1, 2, 3.   Notation of is equivalent to when , read as "f prime x".
41&42   5-16/10/09	2. Understand and use the concept of first derivative of polynomial functions to solve problems.	2.1 Determine the first derivative of the function using formula. 2.2 Determine value of the first derivative of the function for a given value of x.   2.3Determine first derivative of a function involving: addition, or subtraction of algebraic terms. 2.4Determine the first derivative of a product of two polynomials. 2.5 Determine the first derivative of a quotient of two polynomials.   2.6Determine the first derivative of composite function using chain rule.   2.7Determine the gradient of tangent at a point on a curve. 2.8Determine the equation of tangent at a point on a curve.   2.9 Determine the equation of normal at a point on a curve	1       1         1         1   1       1       1             1	Pedagogy : Constructivism Skills : Logical Thinking, relationship, application of rules, making inference, making deduction Value : Logical thinking         Activities : Explanation and demonstration by teacher                               Limit cases in Learning Outcomes 2.7 through 2.9 to rules introduced in 2.4 through 2.6.
43   19-23/10/09	3. Understand and use the concept of maximum and minimum values to solve problems.	3.1 Determine coordinates of turning points of a curve. 3.2 Determine whether a turning point is a maximum or a minimum point.   3.3 Solve problems involving maximum or minimum values.	2           1	Use graphing calculators or dynamic geometry software to explore the concept of maximum and minimum values Pedagogy : Constructivism     Skills : Interpretation of problem ; Application of approprate method/formula	Emphasise the use of first derivative to determine the turning points. Limit problems to two variables only. Exclude points of inflexion.   Limit problems to two variables only
	4. Understand and use the concept of rates of change to solve problems.	4.1 Determine rates of change for related quantities.	1	Use graphing calculators with computer base ranger to explore the concept of rates of change. Skills : Interpretation of problem ; Application of approprate method/formula	Limit problems to 3 variables only.
44   26-30/10/09	5. Understand and use the concept of small changes and approximations to solve problems.	5.1 Determine small changes in quantities 5.2 Determine approximate values using differentiation.	1	Skills : Interpretation of problem ; Application of approprate method/formula	Exclude cases involving percentage change.
	6. Understand and use the concept of second derivative to solve problems.	6.1 Determine the second derivative of . 6.2 Determine whether a turning point is maximum or minimum point of a curve using the second derivative	1       1	Mathematical logic   Value : systematic problem solving	Introduce as or
PROJECT WORK			3 WEEKS OUTSIDE SCHOOL HOURS
	Carry out project work	In carrying out the project work 1.1Define the problem/situation to be studied. State relevant conjectures Use problem solving strategies to solve problems Interpret and discuss results. Draw conclusions and/or make generalisations based on critical evaluation of results. Present systematic and comprehensive written reports.	Use scientific calculators, graphing calculators or computer software to carry out project work.   Students are allowed to carry out project work in groups but written reports must be done individually.   Students should be given opportunity to give oral presentation of their project work.		Emphasise the use of Polya's four-step problem solving process.   Use at least two problem solving strategies.   Emphasise reasoning and effective mathematical communication.