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WK | LSN | TOPIC | SUB-TOPIC | OBJECTIVES | T/L ACTIVITIES | T/L AIDS | REFERENCE | REMARKS |
---|---|---|---|---|---|---|---|---|
1 | 1 |
Trigonometry
|
Pythagoras Theorem
|
By the end of the
lesson, the learner
should be able to:
Derive Pythagoras Theorem |
Deriving Pythagoras Theorem
|
Chalkboard Charts Illustrating derived theorem
|
KLB BK2 Pg 120 Discovering secondary pg 67
|
|
1 | 2 |
Trigonometry
|
Solutions of problems Using Pythagoras Theorem
Application to real life Situation Trigonometry Tangent, sine and cosines |
By the end of the
lesson, the learner
should be able to:
Solve problems using Pythagoras Theorem |
Solving problems using Pythagoras theorem
|
Charts illustrating Pythagoras theorem
Mathematical table Charts illustrating tangent, sine and cosine |
KLB BK2 Pg 121 Discovering secondary pg 67
|
|
1 | 3 |
Trigonometry
|
Trigonometric Table
Angles and sides of a right angled triangle |
By the end of the
lesson, the learner
should be able to:
Use trigonometric tables to find the sine, cosine and tangent |
Reading trigonometric tables of sines, cosines and tangent
|
Mathematical table
Mathematical table Charts Chalkboard |
KLB BK2 Pg 127, 138, 139 Discovering secondary pg 71
|
|
1 | 4 |
Trigonometry
|
Establishing Relationship of sine and cosine of complimentary angles
Sines and cosines of Complimentary angles Relationship between tangent, sine and cosine |
By the end of the
lesson, the learner
should be able to:
Establish the relationship of sine and cosine of complimentary angles |
Using established relationship to solve problems
|
Chalkboards
Chalkboard Charts illustrating the relationship of sines and cosines of complimentary angles Charts showing the three related trigonometric ratio |
KLB BK2 Pg 145
|
|
1 | 5 |
Trigonometry
|
Trigonometric ratios of special angles 30, 45, 60 and 90
Application of Trigonometric ratios in solving problems Logarithms of Sines |
By the end of the
lesson, the learner
should be able to:
Determine the trigonometric ratios of special angles without using tables |
Determining the trigonometric ratios of special angles 30,45,60 and 90 without using tables
|
Charts showing isosceles right angled triangle Charts illustrating Equilateral triangle
Chalkboard Chalkboard Mathematical tables |
KLB BK2 Pg 146-147
|
|
1 | 6 |
Trigonometry
|
Logarithms of cosines And tangents
Reading tables of logarithms of sines, cosines and tangents |
By the end of the
lesson, the learner
should be able to:
Read the logarithm of cosines and tangents from mathematical tables |
Reading logarithms of cosine and tangent from mathematical table
|
Chalkboard Mathematical table
|
KLB BK2 Pg 150-152
|
|
2 | 1 |
Trigonometry
|
Application of trigonometry to real life situations
Area of a triangle Area of a triangle given the base and height (A = ? bh) Area of a triangle using the formula (A = ? absin?) |
By the end of the
lesson, the learner
should be able to:
Solve problems in real life using trigonometry |
Solving problems using trigonometry in real life
|
Mathematical table
Chart illustrating worked problem Chalkboard Charts illustrating a triangle with two sides and an included angle Charts showing derived formula |
KLB BK2 Pg 153-154
|
|
2 | 2 |
Trigonometry
|
Area of a triangle using the formula A = ?s(s-a)(s-b)(s-c)
Area of Quadrilateral and Polygons Area of a square, rectangle, rhombus, parallelogram and trapezium Area of a kite |
By the end of the
lesson, the learner
should be able to:
Solve problems on the area of a triangle Given three sizes using the formula A = ?s(s-a)(s-b)(s-c) |
Solving problems on the area of triangle given three sides of a triangle
|
Charts illustrating a triangle with three sides Charts illustrating a worked example i.e. mathematical table
Charts illustrating formula used in calculating the areas of the quadrilateral Model of a kite |
KLB BK2 Pg 157-158
|
|
2 | 3 |
Trigonometry
|
Area of other polygons (regular polygon) e.g. Pentagon
Area of irregular Polygon |
By the end of the
lesson, the learner
should be able to:
Find the area of a regular polygon |
Calculating the area of a regular polygon
|
Mathematical table Charts illustrating Polygons
Charts illustrating various irregular polygons Polygonal shapes |
KLB BK2 Pg 164
|
|
2 | 4 |
Trigonometry
|
Area of part of a circle Area of a sector (minor sector and a major sector)
Defining a segment of a circle Finding the area of a segment of a circle Area of a common region between two circles given the angles and the radii |
By the end of the
lesson, the learner
should be able to:
- Find the area of a sector given the angle and the radius of a minor sector Calculate the area of a major sector of a circle |
Finding the area of a minor and a major sector of a circle
|
Charts illustrating sectors
Chart illustrating a Segment Charts illustrating common region between the circles Use of a mathematical table during calculation |
KLB BK 2 Pg 167
|
|
2 | 5 |
Trigonometry
|
Area of a common region between two circles given only the radii of the two circles and a common chord
Surface area of solids Surface area of prisms Cylinder (ii) Triangular prism (iii) Hexagonal prism Area of a square based Pyramid |
By the end of the
lesson, the learner
should be able to:
Calculate the area of common region between two circle given the radii of the two intersecting circles and the length of a common chord of the two circles |
Finding the area of a common region between two intersecting
|
Charts illustrating common region between two intersecting circles
Models of cylinder, triangular and hexagonal prisms Models of a square based pyramid |
KLB BK 2 Pg 176
|
|
2 | 6 |
Trigonometry
|
Surface area of a Rectangular based Pyramid
Surface area of a cone using the formula A = ?r2 + ?rl |
By the end of the
lesson, the learner
should be able to:
Find the surface area of a rectangular based pyramid |
Finding the surface area of a rectangular based pyramid
|
Models of a Rectangular based pyramid
Models of a cone |
KLB BK 2 Pg 179-180
|
|
3 | 1 |
Trigonometry
|
Surface area of a frustrum of a cone and a pyramid
Finding the surface area of a sphere Surface area of a Hemispheres |
By the end of the
lesson, the learner
should be able to:
Find the surface area of a frustrum of a cone and pyramid |
Finding the surface area of a frustrum of a cone and a pyramid
|
Models of frustrum of a cone and a pyramid
Models of a sphere Charts illustrating formula for finding the surface area of a sphere Models of a hemisphere |
KLB BK 2 Pg 182
|
|
3 | 2 |
Trigonometry
|
Volume of Solids Volume of prism (triangular based prism)
Volume of prism (hexagonal based prism) given the sides and angle Volume of a pyramid (square based and rectangular based) |
By the end of the
lesson, the learner
should be able to:
Find the volume of a triangular based prism |
Finding the volume of a triangular based prism
|
Models of a triangular based prism
Models of hexagonal based prism Models of square and Rectangular based Pyramids |
KLB BK 2 Pg 186
|
|
3 | 3 |
Trigonometry
|
Volume of a cone
Volume of a frustrum of a cone |
By the end of the
lesson, the learner
should be able to:
Find the volume of a cone |
Finding the volume of a cone
|
Model of a cone
Models of a frustrum of a cone |
KLB BK 2 Pg 191
|
|
3 | 4 |
Trigonometry
|
Volume of a frustrum of a pyramid
Volume of a sphere (v = 4/3?r3) Volume of a Hemisphere {(v = ? (4/3?r3)} |
By the end of the
lesson, the learner
should be able to:
Find the volume of a frustrum of a Pyramid |
Finding volume of a full pyramid Finding volume of cutoff pyramid Find volume of the remaining fig (frustrum) by subtracting i.e. Vf = (V ? v)
|
Models of frustrum of a pyramid
Model of a sphere Mathematical table Models of hemisphere |
KLB BK 2 Pg 194
|
|
3 | 5 |
Trigonometry
Trigonometric Ratios Trigonometric Ratios |
Application of area of triangles to real life
Tangent of an angle Tangent of an angle |
By the end of the
lesson, the learner
should be able to:
Use the knowledge of the area of triangles in solving problems in real life situation |
Solving problems in real life using the knowledge of the area of triangle
|
Mathematical table Chart illustrating formula used
Protractor Ruler Right corners Mathematical tables |
KLB BK 2 Pg 159
|
|
3 | 6 |
Trigonometric Ratios
|
Using tangents in calculations
Application of tangents |
By the end of the
lesson, the learner
should be able to:
calculate the size of an angle given two sides and an angle from tables |
Measuring lengths/angles
Dividing numbers Drawing right angles Reading mathematical tables |
Protractor
Ruler Right corners Mathematical tables |
KLB Maths Bk2 Pg. 119-122
|
|
4 | 1 |
Trigonometric Ratios
|
The sine of an angle
The cosine of an angle Application of sine and cosine |
By the end of the
lesson, the learner
should be able to:
find the sine of an angle by calculations and through tables |
Measuring lengths/angles
Dividing numbers Drawing right angles Reading mathematical tables |
Protractor
Ruler Right corners Mathematical tables |
KLB Maths Bk2 Pg. 119-122
|
|
4 | 2 |
Trigonometric Ratios
|
Complementary angles
Special angles Application of Special angles |
By the end of the
lesson, the learner
should be able to:
define complementary angles. Work out sines of an angle given the cosine of its complimentary and vice versa |
Measuring lengths/angles
Dividing numbers Drawing right angles Reading mathematical tables |
Protractor
Ruler Right corners Mathematical tables |
KLB Maths Bk2 Pg. 119-122
|
|
4 | 3 |
Trigonometric Ratios
|
Logarithms of sines, cosines and tangents
Relationship between sin, cos and tan |
By the end of the
lesson, the learner
should be able to:
solve problems using logarithms of sines cosines and tangents |
Measuring lengths/angles
Dividing numbers Drawing right angles Reading mathematical tables |
Protractor
Ruler Right corners Mathematical tables |
KLB Maths Bk2 Pg. 119-122
|
|
4 | 4 |
Trigonometric Ratios
Area of A Triangle |
Application to real life situation
Problem solving Area = |
By the end of the
lesson, the learner
should be able to:
apply the knowledge of trigonometry to real life situations |
Measuring lengths/angles
Dividing numbers Drawing right angles Reading mathematical tables |
Protractor
Ruler Right corners Mathematical tables |
KLB Maths Bk2 Pg. 119-122
|
|
4 | 5 |
Area of A Triangle
|
Solve problems involving =
A =?s(s-a) (s-b) (s-c) Problem solving |
By the end of the
lesson, the learner
should be able to:
solve problems involving area of triangles using the formula Area = |
Discussions
Drawing triangles Measuring lengths/angles Calculating area |
Protractor
Ruler Right corners Mathematical tables |
KLB Maths Bk2 Pg. 155-157
|
|
4 | 6 |
Area of Quadrilaterals
|
Area of parallelogram
Area of Rhombus |
By the end of the
lesson, the learner
should be able to:
find the area of quadrilaterals like trapeziums, parallelogram etc. by dividing the shape of triangles |
Drawing trapeziums/polygons
Measuring lengths/angles Reading mathematical tables Discussions |
Parallelograms
Trapeziums Polygons Squares/rectangles Mathematical tables |
KLB Maths Bk2 Pg. 160
|
|
5 | 1 |
Area of Quadrilaterals
|
Area of trapezium and kite
Area of regular polygons Problem solving |
By the end of the
lesson, the learner
should be able to:
solve problems on the area of a regular polygon |
Drawing trapeziums/polygons
Measuring lengths/angles Reading mathematical tables Discussions |
Parallelograms
Trapeziums Polygons Squares/rectangles Mathematical tables Mathematical tables Chalkboard illustrations |
KLB Maths Bk2 Pg. 162-163
|
|
5 | 2 |
Area of Part of a Circle
|
Area of a sector
Area of a segment Common region between two circles |
By the end of the
lesson, the learner
should be able to:
find area of a sector |
Drawing circles
Measuring radii/diameters Measuring angles Calculating the area of a circle Discussions |
Circles
Chart illustrating the area of a sector Chart illustrating the area of a minor segment |
KLB Maths Bk2 Pg. 167-169
|
|
5 | 3 |
Area of Part of a Circle
|
Common region between two circles
Problem solving |
By the end of the
lesson, the learner
should be able to:
find the area of the common region between two circles and solve problems related to that |
Drawing circles
Measuring radii/diameters Measuring angles Calculating the area of a circle Discussions |
Circles
Chart illustrating the area of a minor segment Chart illustrating the area of a minor segment Chalkboard illustrations |
KLB Maths Bk2 Pg. 167-169
|
|
5 | 4 |
Surface Area of Solids
|
Surface area of prisms
Surface area of pyramid Surface area of a cone |
By the end of the
lesson, the learner
should be able to:
find the surface area of a prism. |
Drawing prisms
Measuring lengths Opening prisms to form nets Discussions Calculating area |
Prism Chalkboard illustrations
Pyramids with square base, rectangular base, triangular base Cone |
KLB Maths Bk2 Pg. 177
|
|
5 | 5 |
Surface Area of Solids
|
Surface area of frustrum with circular base
Surface area of frustrum with square base Surface area of frustrum with rectangular base |
By the end of the
lesson, the learner
should be able to:
find the surface area of frustrum with circular base |
Drawing cones/frustums
Making cones/frustums Measuring lengths/ angles Discussions |
Chart illustrating the surface area of a frustrum
Chart illustrating frustrum with a square base Chart illustrating frustrum with a rectangular base |
KLB Maths Bk2 Pg. 181-283
KLBMathematics Bk2 Discovering Secondary Mathematics Bk2 |
|
5 | 6 |
Surface Area of Solids
Volume of Solids |
Surface area of spheres
Problem solving Volume of prism |
By the end of the
lesson, the learner
should be able to:
find the surface area of a sphere |
Sketching spheres
Making spheres Measuring diameters/ radii of spheres Discussions |
Chalkboard illustrations
Past paper questions Prism |
KLB Maths Bk2 Pg. 183
|
|
6 | 1 |
Volume of Solids
|
Volume of pyramid
Volume of a cone |
By the end of the
lesson, the learner
should be able to:
find the volume of a pyramid |
Drawing pyramids
Making pyramids Opening pyramids to form nets Discussions |
Pyramid
Cone |
KLB Maths Bk2 Pg. 189-190
|
|
6 | 2 |
Volume of Solids
|
Volume of a sphere
Volume of frustrum Volume of frustrum with a square base |
By the end of the
lesson, the learner
should be able to:
find the volume of a sphere |
Identifying spheres
Sketching spheres Measuring radii/ diameters Discussions |
Sphere
Frustrum with circular base Frustrum with square base |
KLB Maths Bk2 Pg. 195
|
|
6 | 3 |
Volume of Solids
|
Volume of frustrum with a rectangular base
Application to real life situation Problem solving |
By the end of the
lesson, the learner
should be able to:
find the volume of a frustrum with a rectangular base |
Making cones/frustums
Opening cones/frustums to form nets |
Frustrum with rectangular base
Models of pyramids, prism, cones and spheres Past paper questions |
KLB Maths Bk2 Pg. 192-193
|
|
6 | 4 |
Quadratic Expressions and Equations
|
Expansion of Algebraic Expressions
Quadratic identities |
By the end of the
lesson, the learner
should be able to:
expand algebraic expressions |
Discussions
Multiplying numbers Dividing numbers Adding numbers Subtracting numbers Exercises |
Real-life experiences
Worked out expressions |
KLB Maths Bk2 Pg. 203
|
|
6 | 5 |
Quadratic Expressions and Equations
|
Application of identities
Factorise the Identities Factorise other quadratic expressions |
By the end of the
lesson, the learner
should be able to:
identify and use the three Algebraic identities |
Discussions
Multiplying numbers Dividing numbers Adding numbers Subtracting numbers Exercises |
Real-life experiences
Worked out expressions Chart illustrating factorization of a quadratic expression |
KLB Maths Bk2 Pg. 204-205
|
|
6 | 6 |
Quadratic Expressions and Equations
|
Factorisation of expressions of the form k2-9y2
Simplification of an expression by factorisation Solving quadratic equations |
By the end of the
lesson, the learner
should be able to:
factorise a difference of two squares |
Discussions
Multiplying numbers Dividing numbers Adding numbers Subtracting numbers Exercises |
Real-life experiences
Worked out expressions |
KLB Maths Bk2 Pg. 205-208
|
|
7 | 1 |
Quadratic Expressions and Equations
|
The formation of quadratic equations
Formation and solving of quadratic equations from word problems |
By the end of the
lesson, the learner
should be able to:
form quadratic equations from information |
Discussions
Multiplying numbers Dividing numbers Adding numbers Subtracting numbers Exercises |
Real-life experiences
Worked out expressions |
KLB Maths Bk2 Pg. 208
|
|
7 | 2 |
Quadratic Expressions and Equations
Linear Inequalities |
Solving on quadratic equations
Forming quadratic equations from the roots Inequalities symbols |
By the end of the
lesson, the learner
should be able to:
solve problems on quadratic equations |
Discussions
Multiplying numbers Dividing numbers Adding numbers Subtracting numbers Exercises |
Real-life experiences
Worked out expressions Number lines Graph papers Square boards Negative and positive numbers |
KLB Maths Bk2 Pg. 208-210
|
|
7 | 3 |
Linear Inequalities
|
Number line
Inequalities in one unknown Graphical representation |
By the end of the
lesson, the learner
should be able to:
illustrate inequalities on a number line |
Drawing graphs of
inequalities Determining the scale of a graph Shading unwanted regions Discussions |
Number lines
Graph papers Square boards Negative and positive numbers Number lines Graph papers |
KLB Maths Bk2 Pg. 213-224
|
|
7 | 4 |
Linear Inequalities
|
Graphical solutions of simultaneous linear inequalities
|
By the end of the
lesson, the learner
should be able to:
solve the linear inequalities in two unknowns graphically |
Drawing graphs of
inequalities Determining the scale of a graph Shading unwanted regions Discussions |
Number lines
Graph papers Square boards Negative and positive numbers |
KLB Maths Bk2 Pg. 213-224
|
|
7 | 5 |
Linear Inequalities
|
Area of the wanted region
Inequalities from inequality graphs Problem solving. |
By the end of the
lesson, the learner
should be able to:
calculate the area of the wanted region |
Drawing graphs of
inequalities Determining the scale of a graph Shading unwanted regions Discussions |
Number lines
Graph papers Square boards Negative and positive numbers |
KLB Maths Bk2 Pg. 213-224
|
|
7 | 6 |
Linear Motion
|
Displacement, velocity, speed and acceleration
Distinguishing terms Distinguishing velocity and acceleration |
By the end of the
lesson, the learner
should be able to:
Define displacement, speed velocity and acceleration |
Teacher/pupil discussion
Plotting graphs Drawing graphs |
Graph papers
Stones Pieces of paper |
KLB Maths Bk2 Pg. 228-238
|
|
8 |
Mid term exams |
|||||||
9 | 1 |
Linear Motion
|
Distance time graphs
Interpret the velocity time graph |
By the end of the
lesson, the learner
should be able to:
plot and draw the distance time graphs |
Plotting graphs
Drawing graphs |
Graph papers
Stones Pieces of paper Drawn graphs |
KLB Maths Bk2 Pg. 228-238
|
|
9 | 2 |
Linear Motion
|
Interpreting graphs
Relative speed (objects moving in the same direction) Problem solving |
By the end of the
lesson, the learner
should be able to:
interpret graphs of linear motion |
Learners interpret graphs
|
Drawn graphs
Real life situation Chalkboard illustrations Past paper questions |
KLB
Maths Bk2 Pg.334 |
|
9 |
Half term break |
|||||||
10 | 1 |
Statistics
|
Definition
Collection and organization of data Frequency tables |
By the end of the
lesson, the learner
should be able to:
define statistics |
Collecting data
Measuring length/mass/age Drawing graphs Drawing tables Using symbols to represent data Discussion |
Weighing balance
Ruler Tape measure Pieces of stick Arm length Foot length Graph papers |
KLB Maths Bk2 Pg. 241-252
|
|
10 | 2 |
Statistics
|
Grouped data
Mean of ungrouped data |
By the end of the
lesson, the learner
should be able to:
group data into reasonable classes |
Collecting data
Measuring length/mass/age Drawing graphs Drawing tables Using symbols to represent data Discussion |
Weighing balance
Ruler Tape measure Pieces of stick Arm length Foot length Graph papers |
KLB Maths Bk2 Pg. 241-252
|
|
10 | 3 |
Statistics
|
Median of ungrouped data
Mean of ungrouped data Median of a grouped data modal class |
By the end of the
lesson, the learner
should be able to:
calculate the median of ungrouped data and state the mode |
Collecting data
Measuring length/mass/age Drawing graphs Drawing tables Using symbols to represent data Discussion |
Weighing balance
Ruler Tape measure Pieces of stick Arm length Foot length Graph papers |
KLB Maths Bk2 Pg. 241-252
|
|
10 | 4 |
Statistics
|
Data
Representation.
Line graphs
Bar graphs Pictogram |
By the end of the
lesson, the learner
should be able to:
represent data in form of a line graph |
Collecting data
Measuring length/mass/age Drawing graphs Drawing tables Using symbols to represent data Discussion |
Weighing balance
Ruler Tape measure Pieces of stick Arm length Foot length Graph papers Pictures which are whole, half, quarter |
KLB Maths Bk2 Pg. 241-252
|
|
10 | 5 |
Statistics
|
Histograms
Frequency polygons |
By the end of the
lesson, the learner
should be able to:
represent data in form of histograms |
Collecting data
Measuring length/mass/age Drawing graphs Drawing tables Using symbols to represent data Discussion |
Weighing balance
Ruler Tape measure Pieces of stick Arm length Foot length Graph papers Histograms drawn. Data |
KLB Maths Bk2 Pg. 241-252
|
|
10 | 6 |
Statistics
|
Histograms with uneven distribution
Interpretation of data Problem solving |
By the end of the
lesson, the learner
should be able to:
draw histograms with uneven distribution |
Collecting data
Measuring length/mass/age Drawing graphs Drawing tables Using symbols to represent data Discussion |
Data with uneven classes
Real life situations Past paper questions |
KLB Maths Bk2 Pg. 241-252
|
|
11 | 1 |
Angle Properties of a Circle
|
Arc chord segment
Angles subtended by the same arc in the same segment Angle at the centre and at the circumference |
By the end of the
lesson, the learner
should be able to:
identify an arc, chord and segment |
Discussions
Drawing circles Measuring radii/ diameters/angles Identifying the parts of a circle |
Chart illustrating arc chord and segment
Chart illustrating Angles subtended by the same arc in same segment are equal Chart illustrating Angles subtended at the centre by an arc and one subtended at the circumference |
KLB Maths Bk2 Pg. 264-278
|
|
11 | 2 |
Angle Properties of a Circle
|
Angles subtended by the diameter at the circumference
Cyclic quadrilateral |
By the end of the
lesson, the learner
should be able to:
state the angle in the semi-circle |
Discussions
Drawing circles Measuring radii/diameters/angles Identifying the parts of a circle |
Circles showing the
different parts |
KLB Maths Bk2 Pg. 264-278
|
|
11 | 3 |
Angle Properties of a Circle
|
Cyclic quadrilateral
Exterior angle property Problem solving |
By the end of the
lesson, the learner
should be able to:
find and compute angles of a cyclic quadrilateral |
Discussions
Drawing circles Measuring radii/diameters/angles Identifying the parts of a circle |
Circles showing the
different parts different parts Past paper questions |
KLB Maths Bk2 Pg. 264-278
|
|
11 | 4 |
Angle Properties of a Circle
Vectors Vectors |
Problem solving
Definition and Representation of vectors Equivalent vectors |
By the end of the
lesson, the learner
should be able to:
state all the properties and use them selectively to solve missing angles. |
Discussions
Drawing circles Measuring radii/diameters/angles Identifying the parts of a circle |
Circles showing the
different parts Past paper questions 1x2 matrices Graph papers Square boards Ruler |
KLB Maths Bk2 Pg. 264-278
|
|
11 | 5 |
Vectors
|
Addition of vectors
Multiplication of vectors |
By the end of the
lesson, the learner
should be able to:
add vectors |
Writing position vectors
Adding/subtracting numbers Squaring and getting the square root of numbers |
1x2 matrices
Graph papers Square boards Ruler |
KLB Maths Bk2 Pg. 286-289
|
|
11 | 6 |
Vectors
|
Position vectors
Column vector Magnitude of a vector Mid - point Translation vector |
By the end of the
lesson, the learner
should be able to:
define a position vector illustrate position vectors on a Cartesian plane |
Writing position vectors
Adding/subtracting numbers Squaring and getting the square root of numbers |
1x2 matrices
Graph papers Square boards Ruler |
KLB Maths Bk2 Pg.298
|
|
12 |
End term exams |
|||||||
13 |
Marking and closing |
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