Scheme of Work 2012
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SMK RAJA PEREMPUAN, IPOH, PERAK
SCHEME OF WORK 2012 ADDITIONAL MATHEMATICS FORM 4
01/01/2012 SUGGESTED LEARNING TEACHING AND WEEK LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES 1 (4/1/12- ORIENTATION WEEK 6/1/12) 2-3 A2. LEARNING AREA: QUADRATIC EQUATIONS (9/1/12- 20/1/12) Students will be taught to 1. Understand the Use graphic calculator 1.1 Recognise quadratic equation and quadratic concept of or computer software express it in general form. equation quadratic equation such as the and its roots. Geometer’s Sketchpad 1.2 Determine whether a given value is the Question for 1.2 (b) is general form and spreadsheet to root of a quadratic equation by given in the form explore the concept of a) substitution ( x a )( x b ) 0 a, b root quadratic equations. b) inspection. are numerical values. Students will be Students will be able to: Substitution taught to: 1.3 Determine roots of a quadratic inspection equation by trial and improvement method. trial and improvement method 2. Understand the 2.1 Determine the roots of a quadratic Discuss when concept of equation by ( x p )( x q ) 0, factorisation quadratic a) factorisation; hence x p o or equations b) completing the square x q 0 . Include case c) using the formula when p q Derivation of formula for 2.1c is not required.
2.2 Form a quadratic equation from given If x=p and x=q are the completing the 2 roots. roots, then the quadratic square equation is ( x p )( x q ) 0 , that is x 2 ( p q )x pq 0 Involve the use of: b and a c a where and are rooof the quadratic equation ax 2 bx c 0 Students will be Students will be able to: taught to:
3. Understand and 3.1 Determine the types of roots of b 2 4ac 0 discriminant use the conditions quadratic equations from the value of b 2 4ac 0 for quadratic b 2 4ac real roots equations to have b 2 4ac 0 a) two different roots 3.2 Solve problems involving b 2 4ac , Explain that “no roots” b) two equal roots in quadratic equations to means “no real roots” c) no roots a) find an unknown value b) derive a relation 4-5 (25/1/12- A3. LEARNING AREA: QUADRATIC FUNCTIONS 3/2/11) quadratic function 1. Understand the Use graphing 1.1 Recognise quadratic functions. concept of calculator or computer tabulated values quadratic functions software such as 1.2 Plot quadratic functons graphs and their graphs. Geometer’s Sketchpad a) based on given tabulated values axis of symmetry to explore the graphs b) by tabulating values, based on given of quadratic functions. functions parabola 3 Use examples of 1.3 Recognise shapes of graphs of Discuss cases where everyday situations to quadratic functions. a 0 and a 0 for introduce graphs of f ( x ) ax 2 bx c 0 quadratic functions. 1.4 Relate the position of quadratic functions graphs with types of roots for f ( x ) 0 Students will be Students will be able to: taught to: Emphasise the marking of maxmum or minimum 2. Find maximum Use graphic calculator 2.1 Determine the maximum or minimum point and tw other points maximum point and minimum or dynamic geometry value of a quadratic function by on the graphs drawn or by values of quadratic software such as the completing the square. finding the axis of minimum point functions. Geometer’s Sketchpad symmetry and the to explore the graphs intersection with the y- completing the of quadratic functions. axis. square
Use graphing axis of symmetry 3. Sketch graphs of calculator or dynamc 3.1 Sketch quadratic function graphs by Determine other points by quadratic gemetry software such determining the maximum or finding the intersection sketch functions. as the Geometer’s minimum point and two other points. with the x-axis (if it Sketchpad to reinforce exists) intersetion the understanding of graphs of quadratic vertical line functions. quadratic Use graphing calculator or dynamic inequality 4. Understand and geometry software 4.1. Determine the ranges of values of x that Emphasise on sketching use the concept of such as the satisfies quadratic inequalities. graphs and use number range quadratic Geometer’s Sketchpad lines when necessary inequalities. to reinforce the number line understanding of graphs of quadratic functions
4 SUGGESTED LEARNING TEACHING AND WEEK LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES 6-7 (8/2/11- LEARNING AREA: SIMULTANEOUS EQUATION 17/2/11)
Students will be Use graphic calculator Students will be able to: Limit non-linear Simultaneous taught to: or dynamic geometry 1.1 Solve simultaneous equations using the equations up to second equations 1. Solve software such as the substitution method. degree only. simultaneous Geometer’s Sketchpad intersecton equations in two to explore the concept 1.2 Solve simultaneous equations unknowns one of simultaneous involving the real life situations. substitution method linear equation equations. and one non- linear equation. Use examples in real- life situations such as area, perimeter and others. 7-10 (20/2/11- A1. LEARNING AREA: FUNCTIONS 9/3/11)
Students will be Use pictures, role-play Students will be able to: Discuss the idea of set function taught to: and computer software and introduce set to introduce the notation. relation 1 Understand the concept of relations 1.1 Represent a relation using concept of a) Arrow diagrams object relations b) Ordered pairs c) Graphs image
1.2 Identify domain, codomain,object range image and range of a relation. domain 1.3 Classify a relation shown on a mapped
5 diagrams as: one to one, many to one, codomain one to many or many to many relation. Represent functions using map 2. Understand the 2.1 Recognise functions as a special arrow diagrams, ordered concept of relation. pairs of graphs. ordered pair functions e.g. f: x 2x arrow diagrams 2.2 Express functions using function f (x)= 2x notation. “function f maps x to 2x”.
F(x)= 2x is read as “2x is the image of x under the function f”.
Students will be Students will be able to: taught to: Include the examples of functions that are not mathematically based.
2.3 Determine domain, object, image and Examples of functions range of a function. include algeraic (linear and quadratic), trigonometric and absolute value.
Define and sketch absolute value functions.
Use graphic calculator 2.4 Determine image of a function given and computer software the object and vice versa. to explore the image of function.
3. Understand the Use arrow diagrams or 3.1 Determine composition of two Involve algebraic composite function concept of algebraic method to functions. functions only. composite determine composite
6 functions. functions. 3.2 Determine image of composite Images of composite inverse mapping functions given the object and vice functions include a range versa. of values. (Limit to linear composite functions). 3.3 Determine one of the functions in a given composite function given the other related function.
SUGGESTED LEARNING TEACHING AND WEEK LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES
Students will be Students will be able to: taught to:
4. Understand the Use sketch of graphs 4.1 Find object by inverse mapping given Limit to algebraic concept of inverse to show the its image and function. functions. fuctions. relationship between a function and its 4.2 Determine inverse function using Exclude inverse of inverse algebra. composite functions.
4.3 Determine and state the condition for Emphasis that inverse of existence of an inverse function a function is not necessarily a function
9 (28/2/2012 USBF 1 - 1/3/2012
7 (10/3/12- Cuti Pertengahan penggal I 18/3/12) SUGGESTED LEARNING TEACHING AND WEEK LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES 12-14 (19/3/12- A5. LEARNING AREA: INDICES AND LOGARITHMS 6/4/12)
Students will be taught Students will be able to: to: 1. Understand and use Use examples of 1.1 Find the value of numbers given in the base the concept of real –life situations form of: indices and laws of to introduce the a) integer indices integer indices indices to solve concept of indices. b) fractional indices problems fractional indices Use computer 1.2 Use law of indices to findthe values of sofeware such as numbers in index form that are index form the spreadsheet to multiplied, divided or raised to a enchance the power. raised to a power understanding of indices. 1.3 Use laws of indices to simplify Discuss zero index and law of indices algebraic expressions negative indices.
2. Understand and use Use scientific 2.1 Express equation in index form and Explain definition of the concept of calculator to vice versa. lagorithm: logarithms to solve x enchance the N a ;log a N x problems understanding the with a.0, 1 concept of Emphasise that: logarithm. log a 1 0;log a a 1
8 SUGGESTED LEARNING TEACHING AND WEEK LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES Students will be taught Students will be able to: to: 2.2 Find logarithm of a number. Emphasise that: index form a) logarithm of negative numbers is undefined; logarithm form b) logarithm of zero is undefined. logarithm
Discuss cases where the undefined given number is in: a) index form b) numerical form.
2.3 Find logarithm of numbers by using Discuss laws of laws of logarithms. logarithm
2.4 Simplify logarithmic expressions to the simplest form.
3. Understand and use 3.1 Find the logarithm of a number by Discuss: the change of base changing the base of the logarithm to a 1 log b of logarithm to suitable base. a log a solve the problems. b 3.2 Solve problems involving the change of base and laws of logarithm
9 SUGGESTED WEEK LEARNING TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES Students will be taught Students will be able to: to: Equations that involve 4.1 Solve equations involving indices. indices and logarithms 4. Solve equations are limited to equations involving indices with single solution only. and logarithms Solve equations indices by: a) Comparison of indices and bases; b) Using logarithms 4.2 Solve equations involving logarithms.
15-17 (9/4/12- G1. LEARNING AREA: COORDINATE GEOMETRY 27/4/12) Students will be taught Students will be able to: to: Use Pythgoras’ theorem distance 1. Find distance Use examples of 1.1 Find distance between two points using to find the formula for between two points. real life situations formula. distance between two midpoint to find the distance points. between two coordinates points. ratio 2. Understand the 2.1 Find midpoint of two given points. Limit to cases where m concept of division and n are positive. of a line segment. 2.2 Find coordinates of a point that divides a line according to a given ratio m: n Derivation of the formula nx mx ny my 2 , 1 2 m n m n is not required. 10 SUGGESTED WEEK LEARNING TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES Students will be able to: Use dynamic geometry software such as the 3.1 Find area of a triangle based on the area Geometer’s of spesific geometrical shapes. Limit to numerical Sketchpad to values. explore the concept of area of Emphasise the area polygons relationship between the sign of the value for area polygon 1 x1 x2 x3 x1 obtained with the order for 2 y1 y2 y3 y1 of the vertices used. geometrical shape substitution of Derivation of the 3.Find areas of quadrilateral coordinates into 3.2 Find area of triangle y using formula. formula: polygons. the formula. 1 x y x y x y vertex 1 2 2 3 3 1 2 x y x y x y 2 1 3 2 1 3 vertices is not required. 3.3 Find area of a quadrilateral using clockwise formula. Emphasise that when the area of polygon is the anticlockwise zero, the given points are collinear.
11 SUGGESTED LEARNING TEACHING AND WEEK LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES
Students will be taught Students will be able to: to: Answers for learning 21 outcomes 4.4 (a) and 4.4 21/5/12- 4. Understand Use dynamic 4.1 Determine the x intercept of a line. (b) must be stated in the modulus 25/5/12 and use the concept of geometry software simplest form. equation of a straight such as the Geoetre’s 4.2 Find the gradient of line that passes Involve changing the collinear line Sketchpad to explore through two points. equation into gradient the concept of equation of a straight 4.3 Find the gradient of a straight line using and intercept form. x-intercept line. the x-intercept. Emphasise that for y-intercept parallel lines :
m1 m2 gradient 4.4 Find the equation of a straight line given: Emphasise that for straight line a) gradient and one point. perpendicular lines m m 1 general form b) two points 1 2 intersection m m 1 c) x-intercept and y-intercept Derivation of 1 2 is not required gradient form 4.5 Find the gradient and the intercepts of a straight line given the equation. intercept form
4.6 Change the equation of a straight line to the general form.
4.7 Find the point of intersection of two lines
12 SUGGESTED LEARNING TEACHING AND WEEK LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES
Students will be taught Students will be able to: to:
Emphasise that for 5.1 Determine whether two staright lines perpendicular lines parallel
5. Understand and use Use examples of are paralel when gradients of both lines m1m2 1 the concept of real life situations are known and vice versa. perpendicular parallel and to explore parallel Derivation of m m 1 perpendicular lines. 5.2 Find the equation of a straight line that 1 2 and perpendicular is not required lines. passes through a fixed point and parallel to a given line Use graphic calculator and 5.3 Detremine whether two straight lines dynamic geometry are perpendicular lins are known and software such as vice versa. Geometer’s Sketchpad to 5.4 Determine the equation of a straight explore the line that passes through a fixed point concept of parallel and perendicular to a given line. and perpendicular line 5.5 Solve problems involving equations of a straight lines.
13 SUGGESTED LEARNING TEACHING AND WEEK LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES Students will be able Use examples of Students will be able to: Derivation of the equation of locus to: real life situations formula: to explore equation 6.1 Find the equation of locus that satisfies moving points 6. Understand and the condition if: of locs involving 1 x1 y2 x2 y3 x3 y1 use the concept of a) The distance of a moving point from loci distance between 2 x y x y x y equation of locus two points. a fixed point constant. 2 1 3 2 1 3 involving the Use graphic b)The ratio of the distances of a moving is not required. distance between calculator and point from two fixed points is constant two points dynamic geometry Emphasise that when the software such as area of polygon is the the Geometer’s zero, the given points are Sketchpad to 6.2 Solve problems involving loci. collinear. explore the concept of paralel and perpendicular Emphasise that for lines. perpendicular lines m1m2 1
Derivation of m1m2 1 is not required
19-20 MID YEAR EXAMINATION (7/5/12- 18/5/12)
14 LEARNING SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARY WEEK OBJECTIVES LEARNING ACTIVITIES
24 (11/6/12- S1.LEARNING AREA: STATISTICS 22/6/12) Students will be taught Students will be able to: to:
1. Understand and use Use scientific calculators, 1.1 Calculate mean of Discuss grouped data and measures of central the concept of graphing calulators and ungrouped data. ungrouped data. tendency measures of central spreadsheets to explore 1.2 Determine mode of tendency to solve measures of central tendency. ungrouped data. mean problems. 1.3. Determine median of ungrouped data. mode Students collect data from real 1.4 Determine modal calss of Involve uniform class life situations to investigate grouped data from the intervals only. median measures of central tendency frequency distribution table.
1.5 Find mode from histogram. ungrouped data
1.6. Calculate mean of grouped frequency data. distribution table
1.7 Calculate median of Derivation of the median modal class grouped data from the formula is not required. cumulative frequency unifor class interval distribution table. Ogive is also known as histogram 1.8 Estimate median of cumulative frequency grouped data from an curve. midpoint ogive.
15 SUGGESTED WEEK LEARNING TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARY OBJECTIVES LEARNING ACTIVITIES
Students will be able to: cumulative frequency
1.9 Determine the effects on mode, median, Involve grouped and distribution table and mean for a set of data when: ungrouped data a) Each set of data is changed ogive uniformly. b) Extreme values exist. range c) Certain data is addedor removed. interquartile measures of 1.10 Determine the most suitable dispersion measure of central tendency for extreme value given data. lower boundary
Students will be taught Students will be able to: to:
2. Understand and use 2.1 Find the range of ungrouped data. the concept of 2.2 Find the interqurtile range of measures of ungrouped data. dispersion to solve 2.3 Find the range of grouped data. Determine upper and problems. 2.4 Find the interquartile range of grouped lower quartiles by using data from the cumulative frequency the first principle. standard deviation table. 2.5 Determine the interquartile range of class interval grouped data from an ogive. 2.6 Determine the variance of upper quartile a) ungrouped data b) grouped data. lower quartile 2.7 Determine standard deviation of a) ungrouped data variance
16 b) grouped data 2.8 Determine the effects on range, interquartile range variance and standard deviation for a set of data when a) each data is chaged uniformly b) extreme values exist Emphasise that c) certain data is added ir removed comparison between two 2.9 Compare the measures of central sets of data using only tendency and dispersion between two measures of cenral sets of data. tendency is not sufficient. 21-22 (26/5/12- FIRST SEMESTER BREAK 10/6/12)
26 (25/6/12- T1. LEARNING AREA: CIRCULAR MEASURES 29/6/12)
Students will be taught Use dynamic Students will be able to: to: geometry software such as 1. Understand the Geometer’s 1.1 Convert measurements in radians to Discuss the defination of radian concept of radian. Sketchpad to degrees and vice versa. one radian. explore the degree concept of circular “rad” s the abbreviation of measure radian.
2. Undestand and use Use examples of 2.1 Determine: the concept of real life situations a) length of arc circle length of arc of a to explore circular b) radius and Include measurements in circle to solve measure. c) angle subtended at the centre of a radians expressed in terms perimeter problems circle of . based on given information. segment
2.2 Find perimeter of segments of circles. area
17 2.3 Solve problems involving length of arcs.
Students will be taught Students will be able to: to:
3. Understand the 3.1 Determine concept of area of a) area of sector sector of a circle to b) radius and solve problems. c) anle subtended at the centre of a circles based on given information
3.2 Find area of segments of circle.
3.3 Solve problems involving area of sectors.
27-28 (2/7/12- AST1.LEARNING AREA: SOLUTION OF TRIANGLES 13/7/12) Students will be taught Students will be able to: to:
1. Understand and use Use dynamic 1.1 Verify sine rule. sine rule the concept of sine geometry software rule to solve such as the 1.2 Use sine rule to find unknown sides or Include obtuse-angled acute-angled triangle problems. Geometer’s angles of triangle. triangles. Sketchpad to obtuse-angled explore the sine 1.3 Find unknown sides and angles of a triangle rule. triangle in an ambiguous case. Use examples of ambiguous rel life situations to 1.4 Solve problems involving the sine rule. explore the sine rule. 18 2. Understand and use Use dynamic 2.1 Verify cosine rule. Include obtuse-anged cosine rule the concept of geometry software triangles. cosine rule to solve such as the 2.2 Use cosine rule to find unknown sides problems. Geometer’s or angles of a triangle. Sketchpad to explore the cosine 2.3 Solve problems involving cosine rule. rule. Use examplesof 2.4 Solve problems involving sine and real life situations cosine rules. to explore the cosine rule.
Use dynamic Students will be able to: geometry software such as Geometer’s 1 Students will be taught 3.1 Find area of triangles using formula Sketchpad to to: 2 explore the ab sinC or its equvalent. concept of area of 3.Understand and use three–dimensional triangles. the formula for area of 3.2 Solve problems involving three- object triangles to solve Use examples of dimensional objects problems real life situations to explore area of triangles.
19 29-30 INDEX NUMBER (16//7/12- 27/7/12)
Student will be taught Students will be able to to:
1. Understand and use Use examples of the 1.1 Calculate index number the concept of real life situations to index number to explore index 1.2 Calculate price index solve problems. numbers. 1.3 Find Q0 or Q1 given relevant information. 2. Understand and use Use examples of the the concept of real life situations to 2.1 Calculate composite index composite index to explore index solve problems. numbers. 2.2 Find index number or weightage given relevant information.
2.3 Solve problems involving index number and composite index.
7/8/2012- 9/8/2012 USBF 2
33-38 C1. LEARNING AREA: DIFFERENTIATION (13/8/12- 21/9/12) Students will be Students will be able to: Idea of limit to a function taught to: Use graphic calculator 1.1. Determine value of a function when its can be ilustrated using limit 1. Understand and use or dynamic geometry variable approaches a certain value. graps. the concept of software such as 1.2 Find gradient of a chord joining two Concept of first derivative tangent gradients of curve Geometer’s Sketchpad points on a curve. of a function is explained and differentation. to explore the concept 1.3 Find the first derivative of a function as a tangent to a curve can first derivative of differentiation. y=f(x) as gradient of tangent to its be illustrated by using a
20 graph. graph. gradient 1.4 Find the first derivative for polynomial using first principles.
1.5 Deduce the formula for first derivation of function y=f(x) by induction Students will be Students will be able to: taught to: Limit to induction 2. Understand and use 2.1 Determine first derivative of the y ax n ,n are constants, n the concept of first function y ax using formula. n= 1, 2, 3. curve derivative of polynomial 2.2 Determine value of the first derivative Notation of f (x).is fixed point functions to solve n of the function y ax for a given dy problems. equivalent to when value x. dx 2.3 Determine first derivative of a function y=f(x) read as “f prime x”. involving product a) addition or b) subtraction of algebraic terms. quotient 2.4 Determine first derivative of a product of two polynomials.
2.5 Determine first derivative of a quotient of two polynomials. composite function
2.6 Determine first derivative of composite chain rule function using chain rule. normal 2.7 Determine gradient of tangent at a point on a curve. Limit cases in learning outcomes 2.7 -2.9 to rules 2.8 Determine equation of tangent at a introduced in 2.4 -2.6. point on a curve.
2.9 Determine equation of normal at a 21 point on a curve
3.1 Determine coordinates of turning Emphasise the use of first 3. Understand and use Use graphing points of a curve. derivative yo determine turning point the concept of calculator or turning points. maximum and dynamic geometry 3.2 Determine whether a turning point is a minimum point minimum values to software to explore maximum or minimum point. Exclude points of solve problems. the concept of inflexion. maximum point maximum and 3.3 Solve problems involving maximum Limit problems to two minimum values. and minimum values. variables only. Students will be Students will be able to: taught to: Use graphing 4. Understand and use calculator with 4.1 Determine rates of change for related Limit problems to 3 rates of change the concept of rates computer base quantities. variables only. of change to solve ranger to explore problems. the concept of rates of change. 5. Understand use the 5.1 Detemine small changes in quantities. Exclude cases involving approximation concept of small percentage change. changes and 5.2 Determine approximate values using approximation to differentation. solve problems. d 2 y Introduce as 6. Understand and use 6.1 Determine second derivative of dx 2 second derivative the concept of function y=f(x). d dy second derivative to or solve problems. 6.2 Determine whether a turning point dx dx d is maximum or minimum point if a curve f" ( x ) f ' x using the second derivative. dx
34 Cuti Pertengahan 18/8/12 – Penggal ke dua. 26/8/12
22 39-40 (24/9/12- BLOCK TIME TABLE / REVISION FOR FINAL EXAM 5/10/12) 41-42 (8/10/12- FINAL YEAR EXAMINATION 19/10/12) 43-44 (22/10/12 POST-TEST DISCUSSION - 31/10/12) 45 (1/11/12- BLOCK TIME TABLE 9/11/12)
Prepared by: Checked by: Checked by: Verified by:
______( Puan Noraini bt Nordin ) (Pn. Chan May Fye) (Pn. Azizah bt Mat Cha) (Cik Rusnani Sharuddin) Guru Kanan Vokasional & Teknologi Penolong Kanan Pengetua SMK Raja Perempuan SMK Raja Perempuan SMK Raja Perempuan
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