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YEARLY TEACHING PLAN FOR ADDITIONAL MATHEMATICS FORM 4

LEARNING OBJECTIVESStudents will be taught to…

LEARNING OUTCOMESStudents will be able to…

SUGGESTED TEACHING AND LEARNING ACTIVITIES

VALUES AND POINTS TO NOTE

TEACHING Aids / CCTS/ VOCABULARY

3 WEEKS: CHAPTER 1 FUNCTION

1 Understand the concept of relations.

(i) Represent relations using:a) arrow diagrams,b) ordered pairs,c) graphs. (ii) Identify domain, object, image and range of a relation.(iii) Classify a relation shown on a mapped diagram as: one-to-one, many-to-one, one-to-many or many-to-many relation.

Use pictures, role-play and computer software to introduce the concept of relations.

Discuss the idea of set and introduce set notation.

functionrelationobjectimagerangedomaincodomainmapordered pairarrow diagram

2 Understand the concept of functions.

(i) Recognise functions as a special relation.

(ii) Express functions using function notation.

(iii) Determine domain, object, image and range of a function.

(iv) Determine the image of a function given the object and vice versa.

Use graphing calculators or computer software to explore the image of functions.

Represent functions using arrowdiagrams, ordered pairs orgraphs, e.g“ f : x→2x ” is read as“function f maps x to 2x”.“ f (x) = 2x ” is read as “2x is the image of x under the function f ”.Include examples of

notation

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functionsthat are not mathematicallybasedExamples of functions include algebraic (linear and quadratic), trigonometric and absolute value.Define and sketch absolute value functions.

3 Understand the concept of composite functions.

(i) Determine composition of two functions.

(ii) Determine the image of composite functions given the object and vice versa.

(iii) Determine one of the functions in a given composite function given the other related function.

Use arrow diagrams or algebraic method to determine composite functions.

Involve algebraic functions only.Images of composite functions include a range of values. (Limit to linear composite functions).

inversemappingcomposite function

4 Understand the concept of inverse functions.

(i) Find the object by inverse mapping given its image and function.

(ii) Determine inverse functions using algebra.(iii) Determine and state the condition for existence of an inverse function.

Use sketches of graphs to show the relationship between a function and its inverse.

Limit to algebraic functions.Exclude inverse of composite functions.Emphasise that inverse of afunction is not necessarily afunction

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2 WEEKS: CHAPTER 2 QUADRATIC EQUATIONS

1 Understand the concept of quadratic equations and their roots.

(i) Recognise a quadratic equation and express it in general form.

(ii) Determine whether a given value is the root of a quadratic equation by:a) substitution,b) inspection. (iii) Determine roots of quadratic equations by trial and improvement method.

Use graphing calculators orcomputer software such as theGeometer’s Sketchpad and spreadsheet to explore the concept of quadratic equations.

Questions for 1.2(b) are given in the form of; a and b are numerical values.

quadratic equationgeneral formrootsubstitutioninspectiontrial and improvementmethod

2 Understand the concept of quadratic equations.

(i) Determine the roots of a quadratic equation by:a) factorisation,b) completing the square,c) using the formula.

(ii) Form a quadratic equation from given roots.

Discuss when(x – p)(x – q) = 0, hence x – p = 0 or x – q = 0.Include cases when p = q.Derivation of formula for 2.1c is not required.If x = p and x = q are the roots, then the quadratic equation is(x – p)(x – q) = 0, that isx2 – (p + q)x + pq = 0.

factorisationcompleting the square

3 Understand and (i) Determine types of roots of b2 − 4ac > 0 discriminant3







use the conditions for quadratic equations to havea) two different roots;b) two equal roots;c) no roots.

quadratic equations from the value of b2 − 4ac.

(ii) Solve problems involvingb2 − 4ac in quadratic equations to:a) find an unknown value,b) derive a relation.

b2 − 4ac = 0b2 − 4ac < 0Explain that “no roots” means “no real roots”.

real roots

3 WEEKS: CHAPTER 3 QUADRATIC FUNCTIONS

1 Understand the concept of quadratic functions and their graphs.

(i) Recognise quadratic functions.(ii) Plot quadratic function graphs:a) based on given tabulated values,b) by tabulating values based on given functions.

(iii) Recognise shapes of graphs of quadratic functions.(iv) Relate the position of quadratic function graphs with types of roots forf (x) = 0 .

Use graphing calculators orcomputer software such asGeometer’s Sketchpad to explore the graphs of quadratic functions.

Use examples of everyday situations to introduce graphs of quadratic functions.

Discuss cases wherea > 0 and a < 0 forf(x) = ax2 + bx + c

quadratic functiontabulated valuesaxis of symmetryparabolamaximum pointminimum pointcompleting the squareaxis of symmetry

2 Find the maximum and minimum values of quadratic functions.

(i) Determine the maximum or minimum value of a quadratic function by completing the square.

Use graphing calculators ordynamic geometry softwaresuch as Geometer’s Sketchpad to explore the graphs of quadratic

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functions.

3 Sketch graphs of quadratic functions.

(i) Sketch quadratic function graphs by determining the maximum or minimum point and two other points.

Use graphing calculators ordynamic geometry softwaresuch as the Geometer’sSketchpad to reinforce the understanding of graphs of quadratic functions.

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 byfinding the intersection with the x-axis (if it exists).

sketchintersectionvertical linequadratic inequalityrangenumber line

4 Understand and use the concept of quadratic inequalities.

(i) Determine the ranges of values of x that satisfies quadratic inequalities.

Use graphing calculators ordynamic geometry softwaresuch as the Geometer’sSketchpad to explore the concept of quadratic inequalities.

Emphasise on sketching graphs and use of number lines when necessary.

1ST MID-TERM EXAMINATION

1ST MID-TERM HOLIDAY

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1 WEEK: CHAPTER 4 SIMULTANEOUS EQUATIONS

1 Solve simultaneous equations in two unknowns: one linear equation and one non-linear equation.

(i) Solve simultaneous equations using the substitution method. (ii) Solve simultaneous equations involving real-life situations.

Use graphing calculators ordynamic geometry softwaresuch as the Geometer’sSketchpad to explore the concept of simultaneous equations.Use examples in real-life situations such as area, perimeter and others.

Limit non-linear equations up to second degree only.

simultaneousequationsintersectionsubstitution method

3 WEEKS: CHAPTER 5 INDICES AND LOGARITHMS

1 Understand and use the concept of indices and laws of indices to solve problems.

(i) Find the values of numbers given in the form of:a) integer indices,b) fractional indices. (ii) Use laws of indices to find the values of numbers in index form that are multiplied, divided or raised to a power. (iii) Use laws of indices to simplify algebraic expressions.

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.

Discuss zero index and negative indices.

baseinteger indicesfractional indicesindex formraised to a powerlaw of indices

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2 Understand and use the concept of logarithms and laws of logarithms to solve problems.

(i) Express equation in index form to logarithm form and vice versa. (ii) Find logarithm of a number.

(iii) Find logarithm of numbers by using laws of logarithms.

(iv) Simplify logarithmic expressions to the simplest form.

Use scientific calculators to enhance the understanding of the concept of logarithms.

Explain definition of logarithm.N = ax; loga N = x with a > 0, a ≠ 1.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.

Discuss laws of logarithms.

index formlogarithm formlogarithmundefined

3 Understand and use the change of base of logarithms to solve problems.

(i) Find the logarithm of a number by changing the base of the logarithm to a suitable base.

(ii) Solve problems involving the

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change of base and laws of logarithms.

4 Solve equations involving indices and logarithms.

(i) Solve equations involving indices. (ii) Solve equations involving logarithms.

Equations that involve indices and logarithms are limited to equations with single solution only.Solve equations involvingindices by:a) comparison of indices and bases,b) using logarithms.

4 WEEKS: CHAPTER 6 COORDINATE GEOMETRY

1 Find distance between two points.

(i) Find the distance between two points using formula.

Use examples of real-life situations to find the distance between two points.

Use the Pythagoras’ Theorem to find the formula for distance between two points.

distancemidpoint

2 Understand the concept of division of line segments.

(i) Find the midpoint of two given points.

(ii) Find the coordinates of a point that divides a line according to a given ratiom : n.

Limit to cases where m and n are positive.

coordinateratio

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3 Find areas of polygons.

(i) Find the area of a triangle based on the area of specific geometrical shapes.

(ii) Find the area of a triangle by using formula.

(iii) Find the area of a quadrilateral by using formula.

Use dynamic geometrysoftware such as theGeometer’s Sketchpad to explore the concept of areas of polygons.

Limit to numerical values.Emphasise the relationshipbetween the sign of the value forarea obtained with the order of the vertices used.

Emphasise that when the area of polygon is zero, the given points are collinear.

areapolygongeometrical shapequadrilateralvertexverticesclockwiseanticlockwisemoduluscollinear

4 Understand and use the concept of equation of a straight line.

(i) Determine the x-intercept and the y-intercept of a line. (ii) Find the gradient of a straight line that passes through two points.(iii) Find the gradient of a straight line using the x-intercept and y-intercept.(iv) Find the equation of astraight line given:a) gradient and one point,b) points,c) x-intercept and y-intercept. (v) Find the gradient and the intercepts of a straight line given

Use dynamic geometrysoftware such as theGeometer’s Sketchpad to explore the concept of equation of a straight line.

Answers for learning outcomes4.4(a) and 4.4(b) must be stated in the simplest form.Involve changing the equation into gradient and intercept form.

x-intercepty-interceptgradientstraight linegeneral formintersectiongradient formintercept form

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the equation.(vi) Change the equation of a straight line to the general form.(vii) Find the point of intersection of two lines.

5 Understand and use the concept of parallel and perpendicular lines.

(i) Determine whether two straight lines are parallel when the gradients of both lines are known and vice versa. (ii) Find the equation of a straight line that passes through a fixed point and parallel to a given line. (iii) Determine whether two straight lines are perpendicular when the gradients of both lines are known and vice versa.(iv) Determine the equation of a straight line that passes through a fixed point and perpendicular to a given line.(v) Solve problems involving equations of straight lines.

Use examples of real-life situations to explore parallel and perpendicular lines.

Use graphic calculator and dynamic geometry software such as Geometer’s Sketchpad to explore the concept of parallel and perpendicular lines.

Emphasise that for parallel lines:m1 = m2.

parallelperpendicular

6 Understand and use the concept of equation of locus

(i) Find the equation of locus that satisfies the condition if:a) the distance of a moving point

Use examples of real-lifesituations to explore equation of locus involving

equation of locusmoving point

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involving distance between two points.

from afixed point is constant,b) the ratio of the distances of a moving point from two fixed points is constant. (ii) Solve problems involving loci.

distance between two points.Use graphing calculators anddynamic geometry softwaresuch as the Geometer’sSketchpad to explore the concept of parallel and perpendicular lines.

loci

MID-YEAR EXAMINATION

MID-YEAR HOLIDAY

2 WEEKS: CHAPTER 7 STATISTICS

1 Understand and use the concept of measures of central tendency to solve problems.

(i) Calculate the mean of ungrouped data. (ii) Determine the mode of ungrouped data. (iii) Determine the median of ungrouped data.(iv) Determine the modal class of grouped data from frequency distribution tables.

Use scientific calculators, graphing calculators and spreadsheets to explore measures of central tendency.

Pupils collect data from real life situations to investigate measures of central tendency.

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.

measure of centraltendencymeanmodemedianungrouped datafrequencydistribution tablemodal classuniform class

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(v) Find the mode from histograms.(vi) Calculate the mean of grouped data. (vii) Calculate the median of grouped data from cumulative frequency distribution tables. (viii) Estimate the median of grouped data from an ogive.

(ix) Determine the effects on mode, median and mean for a set of data when:a) each data is changed uniformly,b) extreme values exist,c) certain data is added or removed. (x) Determine the most suitable measure of central tendency for given data.

Involve grouped and ungrouped data.

intervalhistogrammidpointcumulative frequencydistribution tableogiverangeinterquartilemeasures ofdispersionextreme valuelower boundary

2 Understand and use the concept of measures of dispersion to solve

(i) Find the range of ungrouped data.

(ii) Find the interquartile range of ungrouped data.(iii) Find the range of grouped data.(iv) Find the interquartile range of grouped data from the cumulative

Determine the upper and lower quartiles by using the first principle.

Emphasise that comparison between two sets of data using only measures of central tendency is not

standard deviationclass intervalupper quartilelower quartilevariance

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frequency table.

(v) Determine the interquartile range of grouped data from an ogive.(vi) Determine the variance of:a) ungrouped data,b) grouped data. (vii) Determine the standard deviation of:a) ungrouped data,b) grouped data.(viii) Determine the effects on range, interquartile range, variance and standard deviation for a set of datawhen:a) each data is changed uniformly,b) extreme values exist,c) certain data is added or removed.(ix) Compare measures of central tendency and dispersion between two sets of data.

sufficient.

2 WEEKS: CIRCULAR MEASURES

1 Understand the (i) Convert measurements in Use dynamic geometry Discuss the definition of radian13







concept of radian. radians to degrees and vice versa. software such as theGeometer’s Sketchpad to explore the concept of circular measure.

one radian.“rad” is the abbreviation of radian.Include measurements in radians expressed in terms of π.

degree

2 Understand and use the concept of length of arc of a circle to solve problems.

(i) Determine:a) length of arc,b) radius,c) angle subtended at the centre of a circle based on given information.(ii) Find perimeter of segments of circles. (iii) Solve problems involving lengths of arcs.

Use examples of real-life situations to explore circular measure.

length of arcangle subtendedcircleperimetersegment

3 Understand and use the concept of area of sector of a circle to solve problems.

(i) Determine the:a) area of sector,b) radius,c) angle subtended at the centre of a circle based on given information.

(ii) Find the area of segments of circles.(iii) Solve problems involving areas of sectors.

areasector

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2 WEEKS: CHAPTER 10 SOLUTION OF TRIANGLES

1 Understand and use the concept of sine rule to solve problems.

(i) Verify sine rule.

(ii) Use sine rule to find unknown sides or angles of a triangle. (iii) Find the unknown sides and angles of a triangle involving ambiguous case.(iv) Solve problems involving the sine rule.

Use dynamic geometrysoftware such as theGeometer’s Sketchpad to explore the sine rule.

Use examples of real-life situations to explore the sine rule.

Include obtuse-angled triangles.

sine ruleacute-angled triangleobtuse-angled triangleambiguous

2 Understand and use the concept of cosine rule to solve problems.

(i) Verify cosine rule. (ii) Use cosine rule to find unknown sides or angles of a triangle.(iii) Solve problems involving cosine rule.(iv) Solve problems involving sine and cosine rules.

Use dynamic geometrysoftware such as theGeometer’s Sketchpad to explore the cosine rule.

Use examples of real-life situations to explore the cosine rule.

Include obtuse-angled triangles

cosine rule

3 Understand and use the formula for

(i) Find the areas of triangles using the formula ½ ab sin C or its

Use dynamic geometrysoftware such as the

three-dimensional

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areas of triangles to solve problems.

equivalent. (ii) Solve problems involving three-dimensional objects.

Geometer’s Sketchpad to explore the concept of areas of triangles.Use examples of real-life situations to explore areas of triangles.

object

2 WEEKS: CHAPTER 11 INDEX NUMBER

1 Understand and use the concept of index number to solve problems.

(i) Calculate index number. (ii) Calculate price index.(iii) Find Q0 or Q 1 given relevant information.

Use examples of real-life situations to explore index numbers.

Explain index number. Q0 = Quantity at base time.Q1 = Quantity at specific time.

index numberprice indexquantity at base timequantity at specifictime

2 Understand and use the concept of composite index to solve problems.

(i) Calculate composite index.(ii) Find index number or weightage given relevant information.(iii) Solve problems involving index number and composite index.

Use examples of real-life situations to explore composite index.

Explain weightage and composite index.

2ND MID-TERM EXAMINATION

2ND MID-TERM HOLIDAY

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5 WEEKS: CHAPTER 9 DIFFERENTIATION

1 Understand and use the concept of gradients of curve and differentiation.

(i) Determine the value of a function when its variable approaches a certain value.(ii) Find the gradient of a chord joining two points on a curve. (iii) Find the first derivative of a function y = f (x), as the gradient of tangent to its graph. (iv) Find the first derivative of polynomials using the first principle. (v) Deduce the formula for first derivative of the function y = f (x) by induction.

Use graphing calculators or dynamic geometry software such as Geometer’s Sketchpad to explore the concept of differentiation.

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 and can be illustrated using graphs.

Limit to y = axn ; a, n are constants, n = 1, 2, 3…

Notation of f '(x) is equivalent to dx/dy when y = f (x), f '(x) read as “f prime of x”.

limittangentfirst derivativegradientinductioncurvefixed point

2 Understand and use the concept of first derivative of polynomial functions to solve problems.

(i) Determine the first derivative of the function y = axn using formula.

(ii) Determine value of the first derivative of the function y = axn

for a given value of x.(iii) Determine first derivative of a

Limit cases in LearningOutcomes 2.7 through 2.9 to rules introduced in 2.4 through 2.6.

productquotientcompositefunctionchain rulenormal

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function involving:a) addition, orb) subtractionof algebraic terms.(iv) Determine the first derivative of a product of two polynomials. (v) Determine the first derivative of a quotient of two polynomials. (vi) Determine the first derivative of composite function using chain rule.(vii) Determine the gradient of tangent at a point on a curve. (viii) Determine the equation of tangent at a point on a curve.

(ix) Determine the equation of normal at a point on a curve.

3 Understand and use the concept of maximum and minimum values to solve problems.

(i) Determine coordinates of turning points of a curve. (ii) Determine whether a turning point is a maximum or a minimum point.

(iii) Solve problems involving maximum or minimum values.

Use graphing calculators or dynamic geometry software to explore the concept of maximum and minimum values.

Emphasise the use of first derivative to determine the turning points.

Exclude points of inflexion.

Limit problems to two

turning pointminimum pointmaximum point

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variables only.

4 Understand and use the concept of rates of change to solve problems.

(i) Determine rates of change for related quantities.

(i) Determine rates of change for related quantities.

Limit problems to 3 variables only.

rates of change

5 Understand and use the concept of small changes and approximations to solve problems.

(i) Determine small changes in quantities.

(ii) Determine approximate values using differentiation.

Exclude cases involving percentage change.

approximation

6 Understand and use the concept of second derivative to solve problems.

(i) Determine the second derivative of y = f (x). (ii) Determine whether a turning point is maximum or minimum point of a curve using the second derivative.

second derivative

REVISON

YEAR END EXAMINATION

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yearly teaching plan for additional web viewsubstitution method. 3 weeks: ... find the gradient of a...

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