# Singapore Mathematics Curriculum (Scope And Sequence) For 9th Grade and 10th Grade / Secondary 3 and Secondary 4 / GCE O LevelOur Singapore Math books for 9th Grade / Secondary 3 and Singapore Math books for 10th Grade / Secondary 4 are written in English and based on the Singapore Maths curriculum for 9th Grade and 10th Grade / Secondary 3 and Secondary 4 / GCE O Level, which covers the following topics. If your child uses our Singapore Math books for 9th Grade / Secondary 3 and Singapore Math books for 10th Grade / Secondary 4, he will be able to: (Some of the following symbols may not display properly.) ## Numbers - use natural numbers, integers (positive, negative and zero), prime numbers, common factors and common multiples, rational and irrational numbers, real numbers
- continue given number sequences, recognise patterns within and across different sequences and generalise to simple algebraic statements (including expressions for the nth term ) relating to such sequences
## Squares, square roots, cubes and cube roots - calculate squares, square roots, cubes and cube roots of numbers
## Vulgar and decimal fractions and percentages - use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts
- recognise equivalence and convert between these forms
## Ordering - order quantities by magnitude and demonstrate familiarity with the symbols =, (not equal)‚, >, <, (greater than or equal to symbol), (smaller than or equal to symbol)
## Standard form - use the standard form A x 10
^{n} where n is a positive or negative integer, and 1 (less than or equal to symbol) A <10 ## The four operations - use the four operations for calculations with whole numbers, decimal fractions and vulgar (and mixed) fractions, including correct ordering of operations and use of brackets
## Estimation - make estimates of numbers, quantities and lengths
- give approximations to specified numbers of significant figures and decimal places
- round off answers to reasonable accuracy in the context of a given problem
## Ratio, proportion, rate - demonstrate an understanding of the elementary ideas and notation of ratio, direct and inverse proportion and common measures of rate
- divide a quantity in a given ratio
- use scales in practical situations
- calculate average speed
- express direct and inverse variation in algebraic terms and use this form of expression to find unknown quantities
## Percentages - calculate a given percentage of a quantity
- express one quantity as a percentage of another
- calculate percentage increase or decrease
- carry out calculations involving reverse percentages, e.g. finding the cost price given the selling price and the percentage profit
## Use of a scientific calculator - use a scientific calculator efficiently
- apply appropriate checks of accuracy
## Everyday mathematics - use directed numbers in practical situations (e.g. temperature change, tide levels)
- use current units of mass, length, area, volume, capacity and time in practical situations (including expressing quantities in terms of larger or smaller units)
- calculate times in terms of the 12-hour and 24-hour clock (including reading of clocks, dials and timetables)
- solve problems involving money and convert from one currency to another
- use given data to solve problems on personal and household finance involving earnings, simple interest, compound interest (without the use of formula), discount, profit and loss
- extract data from tables and charts
## Graphs in practical situations - interpret and use graphs in practical situations including travel graphs and conversion graphs
- draw graphs from given data
- apply the idea of rate of change to easy kinematics involving distance-time and speed-time graphs, acceleration and retardation
- calculate distance travelled as area under a linear speed-time graph
## Graphs of functions - construct tables of values and draw graphs for functions of the form y = ax
^{n} where n = -2, -1, 0, 1, 2, 3, and simple sums of not more than three of these and for functions of the form y = kax where a is a positive integer - interpret graphs of linear, quadratic, reciprocal and exponential functions
- find the gradient of a straight line graph
- solve equations approximately by graphical methods
- estimate gradients of curves by drawing tangents
## Coordinate geometry - demonstrate familiarity with cartesian coordinates in two dimensions
- calculate the gradient of a straight line from the coordinates of two points on it
- interpret and obtain the equation of a straight line graph in the form y = mx + c
- calculate the length and the coordinates of the midpoint of a line segment from the coordinates of its end points
## Algebraic representation and formulae - use letters to express generalised numbers and express basic arithmetic processes algebraically
- substitute numbers for words and letters in formulae
- transform simple and more complicated formulae
- construct equations from given situations
## Algebraic manipulation - manipulate directed numbers
- use brackets and extract common factors
- expand products of algebraic expressions
- factorise expressions of the form ax + ay; ax + bx + kay + kby; a
^{2}x^{2} - b^{2}y^{2}; a^{2} + 2ab + b^{2}; ax^{2} + bx + c - manipulate simple algebraic fractions
## Indices - use and interpret positive, negative, zero and fractional indices
## Solutions of equations and inequalities - solve simple linear equations in one unknown
- solve fractional equations with numerical and linear algebraic denominators
- solve simultaneous linear equations in two unknowns
- solve quadratic equations by factorisation and either by use of the formula or by completing the square
- solve simple linear inequalities
## Geometrical terms and relationships - use and interpret the geometrical terms: point, line plane, parallel, perpendicular, right angle, acute, obtuse and reflex angles, interior and exterior angles, regular and irregular polygons, pentagons, hexagons, octagons, decagons
- use and interpret vocabulary of triangles, circles, special quadrilaterals
- solve problems (including problems leading to some notion of proof) involving similarity and congruence
- use and interpret vocabulary of simple solid figures: cube, cuboid, prism, cylinder, pyramid, cone, sphere
- use the relationships between areas of similar triangles, with corresponding results for similar figures and extension to volumes of similar solids
## Geometrical constructions - measure lines and angles
- construct simple geometrical figures from given data using protractors or set squares as necessary
- construct angle bisectors and perpendicular bisectors using straight edges and compasses only
- read and make scale drawings. (Where it is necessary to construct a triangle given the three sides, ruler and compasses only must be used.)
## Bearings - interpret and use three-figure bearings measured clockwise from the north (i.e. 000
^{o}-360^{o}) ## Symmetry - recognise line and rotational symmetry (including order of rotational symmetry) in two dimensions, and properties of triangles, quadrilaterals and circles directly related to their symmetries
- recognise symmetry properties of the prism (including cylinder) and the pyramid (including cone)
- use the following symmetry properties of circles:
- equal chords are equidistant from the centre
- the perpendicular bisector of a chord passes through the centre
- tangents from an external point are equal in length
## Angle - calculate unknown angles and solve problems (including problems leading to some notion of proof) using the following geometrical properties:
- angles on a straight line
- angles at a point
- vertically opposite angles
- angles formed by parallel lines
- angle properties of triangles and quadrilaterals
- angle properties of polygons including angle sum
- angle in a semi-circle
- angle between tangent and radius of a circle
- angle at the centre of a circle is twice the angle at the circumference
- angles in the same segment are equal
- angles in opposite segments are supplementary
## Locus - use the following loci and the method of intersecting loci:
- set of points in two dimensions
- which are at a given distance from a given point
- which are at a given distance from a given straight line
- which are equidistant from two given points
- sets of points in two dimensions which are equidistant from two given intersecting straight lines
## Mensuration - solve problems involving:
- the perimeter and area of a rectangle and a triangle
- the circumference and area of a circle
- the area of a parallelogram and a trapezium
- the surface area and volume of a cuboid, cylinder, prism, sphere, pyramid and cone. (Formulae will be given for the sphere, pyramid and cone.)
- arc length and sector area as fractions of the circumference and area of a circle
## Trigonometry - apply Pythagoras theorem and the sine, cosine and tangent ratios for acute angles to the calculation of a side or of an angle of a right-angled triangle (angles will be quoted in, and answers required in, degrees and decimals of a degree to one decimal place)
- solve trigonometrical problems in two dimensions including those involving angles of elevation and depression and bearings
- extend sine and cosine functions to angles between 90
^{o} and 180^{o} - solve problems using the sine and cosine rules for any triangle and the formula 1/2 ab sinC for the area of a triangle
- solve simple trigonometrical problems in three dimensions. (Calculations of the angle between two planes or of the angle between a straight line and a plane will not be required.)
## Statistics - collect, classify and tabulate statistical data
- read, interpret and draw simple inferences from tables and statistical diagrams
- construct and use bar charts, pie charts, pictograms, dot diagrams, stem-and-leaf diagrams, simple frequency distributions and frequency polygons
- use frequency density to construct and read histograms with equal and unequal intervals
- calculate the mean, median and mode for individual data and distinguish between the purposes for which they are used
- construct and use cumulative frequency diagrams
- estimate the median, percentiles, quartiles and interquartile range from the cumulative frequency diagrams
- calculate the mean for grouped data
- identify the modal class from a grouped frequency distribution
## Probability - calculate the probability of a single event as either a fraction or a decimal (not a ratio)
- calculate the probability of simple combined events, using possibility diagrams and tree diagrams where appropriate (in possibility diagrams outcomes will be represented by points on a grid and in tree diagrams outcomes will be written at the end of branches and probabilities by the side of the branches)
## Transformations - use the following transformations of the plane: reflection (M), rotation (R), translation (T), enlargement (E), shear (H), stretch (S) and their combinations (if M(a)=b and R(b)=c the notation RM(a)=c will be used; invariants under these transformations may be assumed)
- identify and give precise descriptions of transformations connecting given figures
## Vectors in two dimensions - describe a translation by using a vector represented by (
^{x}_{y}), AB or a; . add vectors and multiply a vector by a scalar - calculate the magnitude of a vector (
^{x}_{y}) as the square root of (x^{2}+y^{2}). (Vectors will be printed as AB or a and their magnitudes denoted by modulus signs, e.g. |AB| or |a|. In their answers to questions candidates are expected to indicate a in some definite way, e.g. by an arrow or by underlining, thus AB or a) - represent vectors by directed line segments
- use the sum and difference of two vectors to express given vectors in terms of two coplanar vectors
- use position vectors
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