Use Pictures, Role-Play and Computer Software to Introduce the Concept of Relations

WEEK/
DATE / LEARNING OBJECTIVES / SUGGESTED TEACHING AND LEARNING ACTIVITIES / LEARNING OUTCOMES
Students will be taught to: / Students will be able to:
3 / FUNCTIONS
1.Understand the concept of relations. /

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

/ 1.1Represent relations using
a) arrow diagrams
b) ordered pairs
c) graphs
1.2Identify domain, codomain, object, image and range of a relation.
1.3Classify a relation shown on a mapped diagram as: one to one, many to one, one to many or many to many relation.
2.Understand the concept of functions. /

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

/ 2.1Recognize functions as a special relation.
2.2Express functions using function notation.
2.3Determine domain, object, image and range of a function.
2.4Determine the image of a function given the object and vice versa.
3.Understand the concept of composite functions. /

• Use arrow diagrams or algebraic method to determine composite functions.

/ 3.1Determine composition of two functions.
3.2Determine 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.
4.Understand the concept of inverse functions. /

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

/ 4.1Find the object by inverse mapping given its image and function.
4.2Determine inverse functions using algebra.

• 4.3 Determine and state the condition for existence of an inverse function.

4 / QUADRATIC EQUATIONS
1. Understand the concept of quadratic equation and its roots. /

• Use graphing calculators or computer software such as the Geometer’s Sketchpad and spreadsheet to explore the concept of quadratic equations.

/ 1.1Recognise a quadratic equation and express it in general form.
1.2Determine whether a given value is the root of a quadratic equation by
a) substitution;
b) inspection.
1.3Determine roots of quadratic equations by trial and improvement method.
2.Understand the concept of quadratic equations. / 2.1Determine the roots of a quadratic equation by
a) factorisation;
b) completing the square
c) using the formula.
2.2Form a quadratic equation from given roots.
3. Understand and use the conditions for quadratic equations to have
a)two different roots;
b)two equal roots;
c)no roots. / 3.1Determine types of roots of quadratic equations from the value of b2  4ac.
3.2Solve problems involving
b2  4ac in quadratic equations to:
a)find an unknown value;
b)derive a relation.
4 / QUADRATIC FUNCTIONS
1. Understand the concept of quadratic functions and their graphs. /

• Use graphing calculators or computer software such as Geometer’s Sketchpad to explore the graphs of quadratic functions.

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

/ 1.1Recognise quadratic functions.
1.2Plot quadratic function graphs
a) based on given tabulated values;
b) by tabulating values based on given functions.
1.3Recognise shapes of graphs of quadratic functions.
1.4Relate the position of quadratic function graphs with types of roots for f (x)  0.
2.Find the maximum and minimum values of quadratic functions. /

• Use graphing calculators or dynamic geometry software such as the Geometer’s Sketchpad to explore the graphs of quadratic functions.

/ 2.1Determine the maximum or minimum value of a quadratic function by completing the square.
3.Sketch graphs of quadratic functions. /

• Use graphing calculators or dynamic geometry software such as the Geometer’s Sketchpad to reinforce the understanding of graphs of quadratic functions.

/ 3.1Sketch quadratic function graphs by determining the maximum or minimum point and two other points.
4.Understand and use the concept of quadratic inequalities. /

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

/ 4.1Determine the ranges of values of x that satisfies quadratic inequalities.
1 / SIMULTANEOUS EQUATIONS
1.Solve simultaneous equations in two unknowns: one linear equation and one non-linear equation. /

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

• Use examples in real-life situations such as area, perimeter and others.

/ 1.1Solve simultaneous equations using the substitution method.
1.2Solve simultaneous equations involving real-life situations.
4 / INDICES AND LOGARITHMS
1.Understand and use the concept of indices and laws of indices to solve problems. /

• 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.

/ 1.1Find the value of numbers given in the form of:
a)integer indices.
b)fractional indices.
1.2Use laws of indices to find the value of numbers in index form that are multiplied, divided or raised to a power.
1.3Use laws of indices to simplify algebraic expressions.
2.Understand and use the concept of logarithms and laws of logarithms to solve problems. /

• Use scientific calculators to enhance the understanding of the concept of logarithm.

/ 2.1Express equation in index form to logarithm form and vice versa.
2.2 Find logarithm of a number.
2.3Find logarithm of numbers by using laws of logarithms.
2.4Simplify logarithmic expressions to the simplest form.
3.Understand and use the change of base of logarithms to solve problems. / 3.1Find the logarithm of a number by changing the base of the logarithm to a suitable base.
3.2Solve problems involving the change of base and laws of logarithms.
4.Solve equations involving indices and logarithms. / 4.1 Solve equations involving indices.
4.2 Solve equations involving logarithms.
4 / COORDINATE GEOMETRY
1.Find distance between two points. /

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

/ 1.1Find the distance between two points using formula.
2.Understand the concept of division of a line segment. / 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.
3.Find areas of polygons. /

• Use dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of area of polygons.

• Use for

substitution of coordinates into the formula. / 3.1Find the area of a triangle based on the area of specific geometrical shapes.
3.2 Find the area of a triangle by using formula.
3.3Find the area of a quadrilateral using formula.
4.Understand and use the concept of equation of a straight line. /

• Use dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of equation of a straight line.

/ 4.1 Determine the x-intercept and the y-intercept of a line.
4.2Find the gradient of a straight line that passes through two points.
4.3Find the gradient of a straight line using the x-intercept and y-intercept.
4.4Find the equation of a straight line given:
a)gradient and one point;
b)two points;
c) x-intercept and y-intercept.
4.5 Find 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.
5. Understand and use the concept of parallel and perpendicular 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.

/ 5.1Determine whether two straight lines are parallel when the gradients of both lines are known and vice versa.
5.2Find the equation of a straight line that passes through a fixed point and parallel to a given line.
5.3Determine whether two straight lines are perpendicular when the gradients of both lines are known and vice versa.
5.4Determine the equation of a straight line that passes through a fixed point and perpendicular to a given line.
5.5Solve problems involving equations of straight lines.
6. Understand and use the concept of equation of locus involving distance between two points. /

• 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.

/ 6.1Find 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.
6.2Solve problems involving loci.
4 / STATISTICS
1. Understand and use the concept of measures of central tendency to solve problems. /

• 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.

/ 1.1Calculate the mean of ungrouped data.
1.2Determine the mode of ungrouped data.
1.3Determine the median of ungrouped data.
1.4Determine the modal class of grouped data from frequency distribution tables.
1.5Find the mode from histograms.
1.6Calculate the mean of grouped data.
1.7 Calculate the median of grouped data from cumulative frequency distribution tables.
1.8Estimate the median of grouped data from an ogive.
1.9Determine 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.
1.10Determine the most suitable measure of central tendency for given data.
2.Understand and use the concept of measures of dispersion to solve problems. / 2.1Find the range of ungrouped data.
2.2Find the interquartile range of ungrouped data.
2.3 Find the range of grouped data.
2.4Find the interquartile range of grouped data from the cumulative frequency table.
2.5Determine the interquartile range of grouped data from an ogive.
2.6Determine the variance of
a) ungrouped data;
b) grouped data.
2.7Determine the standard deviation of:
a) ungrouped data
b) grouped data.
2.8Determine the effects on range, interquartile range, variance and standard deviation for a set of data when:
a) each data is changed uniformly;
b) extreme values exist;
c) certain data is added or removed.
2.9Compare measures of central tendency and dispersion between two sets of data.
2 / CIRCULAR MEASURES
1.Understand the concept of radian. /

• Use dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of circular measure.

/ 1.1Convert measurements in radians to degrees and vice versa.
2. Understand and use the concept of length of arc of a circle to solve problems. /

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

/ 2.1Determine:
a)length of arc;
b)radius; and
c)angle subtended at the centre of a circle
based on given information.
2.2Find perimeter of segments of circles.
2.3Solve problems involving lengths of arcs.
3. Understand and use the concept of area of sector of a circle to solve problems. / 3.1Determine the:
a)area of sector;
b)radius; and
c)angle subtended at the centre of a circle
based on given information.
3.2Find the area of segments of circles.
3.3 Solve problems involving areas of sectors.
5 / DIFFERNTATIONS
1.Understand and use the concept of gradients of curve and differentiation. /

• Use graphing calculators or dynamic geometry software such as Geometer’s Sketchpad to explore the concept of 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.
1.3 Find the first derivative of a function y = f(x), as the gradient of tangent to its graph.
1.4Find the first derivative of polynomials using the first principles.
1.5Deduce the formula for first derivative of the function y = f(x) by induction.
2.Understand and use the concept of first derivative of polynomial functions to solve problems. / 2.1Determine the first derivative of the function y = axn using formula.
2.2Determine value of the first derivative of the function y = axn for a given value of x.
2.3Determine first derivative of a function involving:
a)addition, or
b)subtraction
of algebraic terms.
2.4Determine the first derivative of a product of two polynomials.
2.5Determine 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.
2.9Determine the equation of normal at a point on a curve.
3.Understand and use the concept of maximum and minimum values to solve problems. /

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

/ 3.1Determine coordinates of turning points of a curve.
3.2Determine whether a turning point is a maximum or a minimum point.
3.3Solve problems involving maximum or minimum values.
4.Understand and use the concept of rates of change to solve problems. /

• Use graphing calculators with computer base ranger to explore the concept of rates of change.

/ 4.1Determine rates of change for related quantities.
5.Understand and use the concept of small changes and approximations to solve problems. / 5.1Determine small changes in quantities.
5.2Determine approximate values using differentiation.
6.Understand and use the concept of second derivative to solve problems. / 6.1Determine the second derivative of function y = f (x0
6.2 Determine whether a turning
point is maximum or minimum
point of a curve using the second
derivative.
2 / SOLUTION OF TRIANGLES
1.Understand and use the concept of sine rule to solve problems. /

• 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.

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

• 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.
. / 2.1Verify cosine rule.
2.2Use cosine rule to find unknown sides or angles of a triangle.
2.3Solve problems involving cosine rule.
2.4Solve problems involving sine and cosine rules.
3.Understand and use the formula for areas of triangles to solve problems. /

• Use dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of areas of triangles.

• Use examples of real-life situations to explore area of triangles.

/ 3.1Find the area of triangles using the formula ab sin C or its equivalent.
3.2Solve problems involving three-dimensional objects.
2 / INDEX NUMBER
1.Understand and use the concept of index number to solve problems. /

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

/ 1.1Calculate index number.
1.2Calculate price index.
1.3 Find Q0 or Q 1 given relevant information.
2. Understand and use the concept of composite index to solve problems /

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

/ 2.1Calculate composite index.
2.2Find index number or weightage given relevant information.
2.3 Solve problems involving index number and composite index.
PROJECT WORK
1. Carry out project work. /

• 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.

/ 1.1Define the problem/situation to be studied.
1.2 State relevant conjectures.
1.3 Use problem solving strategies to solve problems.
1.4 Interpret and discuss results.
1.5 Draw conclusions and/or generalisations based on critical evaluation of results.
1.6 Present systematic and comprehensive written reports.