Singapore

S2.N Number and Algebra

• S2.N2 Ratio and proportion

  • S2.N2.4 map scales (distance and area)

    • Scale drawings: word problems (S2-H.12)

  • S2.N2.5 direct and inverse proportion

    • Identify proportional relationships by graphing (S2-I.3)
    • Write equations for proportional relationships from graphs (S2-I.5)
    • Identify proportional relationships (S2-I.6)
    • Graph proportional relationships (S2-I.7)

  • S2.N2.a Interpret the various scales used oh maps, floor plans and other scale drawings, and calculate the actual distance/length and area.

    • Scale drawings: word problems (S2-H.12)

  • S2.N2.b Work in groups to make a scale drawing of an existing or dream classroom/bedroom and explain the choices of the scale used.

  • S2.N2.c Discuss examples of direct and inverse proportion and explain the concepts using tables, equations and graphs.

    • Find the constant of proportionality from a table (S2-I.1)
    • Write equations for proportional relationships from tables (S2-I.2)
    • Graph proportional relationships (S2-I.7)
    • Interpret graphs of proportional relationships (S2-I.8)
    • Write direct variation equations (S3-S.4)
    • Write and solve direct variation equations (S3-S.5)

• S2.N5 Algebraic expressions and formulae

  • S2.N5.9 expansion of the product of algebraic expressions

    • Multiply using the distributive property (S2-W.12)
    • Simplify variable expressions using properties (S2-W.13)
    • Multiply monomials (S2-BB.6)
    • Multiply polynomials using algebra tiles (S2-BB.11)
    • Multiply binomials (S2-BB.12)
    • Multiply polynomials to find area (S2-BB.13)
    • Multiply trinomials (S2-BB.14)

  • S2.N5.10 changing the subject of a formula

  • S2.N5.11 finding the value of an unknown quantity in a given formula

    • Solve equations involving squares and square roots (S2-F.18)
    • Solve equations involving cubes and cube roots (S2-F.21)
    • Model and solve equations using algebra tiles (S2-X.3)
    • Write and solve equations that represent diagrams (S2-X.4)
    • Solve one-step equations (S2-X.6)
    • Solve two-step equations (S2-X.7)
    • Solve multi-step equations (S2-X.8)
    • Solve equations involving like terms (S2-X.9)
    • Solve equations with variables on both sides (S2-X.10)
    • Solve equations: mixed review (S2-X.11)
    • Solve equations: complete the solution (S2-X.12)
    • Solve equations: word problems (S2-X.13)

  • S2.N5.12 use of:

    • S2.N5.12.1 (a + b)² = a² + 2ab + b²

      • Factorise quadratics: special cases (S2-CC.3)
      • Multiply two binomials: special cases (S3-Z.9)

    • S2.N5.12.2 (a - b)² = a² - 2ab + b²

      • Factorise quadratics: special cases (S2-CC.3)
      • Multiply two binomials: special cases (S3-Z.9)

    • S2.N5.12.3 a² - b² = (a + b) (a- b)

      • Factorise quadratics: special cases (S2-CC.3)
      • Multiply two binomials: special cases (S3-Z.9)

  • S2.N5.13 factorisation of linear expressions of the form ax + bx + kay + by

    • Factorise by grouping (S3-AA.7)

  • S2.N5.14 factorisation of quadratic expressions ax² + bx + c

    • Factorise quadratics with leading coefficient 1 (S2-CC.1)
    • Factorise quadratics with other leading coefficients (S2-CC.2)

  • S2.N5.15 multiplication and division of simple algebraic fractions such as (3a/4b²) (5ab/3), 3a/4 ÷ 9a²/10

    • Multiply monomials (S2-BB.6)
    • Divide monomials (S2-BB.7)
    • Multiply and divide monomials (S2-BB.8)
    • Multiply and divide rational expressions (S2-DD.3)

  • S2.N5.16 addition and subtraction of algebraic fractions with linear or quadratic denominator such as 1/(x -2) + 2/(x-3), 1/(x² - 9) + 2/(x - 3), 1/(x - 3) + 2/((x -3)²)

    • Add and subtract rational expressions (S2-DD.4)

  • S2.N5.a Use algebra manipulatives, e.g. algebra discs, to explain the process of expanding the product of two linear expressions of the form px + q, where p and q are integers, to obtain a quadratic expression of the form ax² + bx + c.

    • Factorise quadratics using algebra tiles (S2-CC.4)

  • S2.N5.b Use the AlgeDisc application in AlgeTools to factorise a quadratic expression of the form ax² + bx + c into two linear factors where a, b and c are integers.

  • S2.N5.c Work in groups to identify and explain common errors in algebraic manipulations, e.g. (x + Y)² = x² + y.

• S2.N6 Functions and graphs

  • S2.N6.6 quadratic functions y = ax² + bx + c

  • S2.N6.7 graphs of quadratic functions and their properties:

    • S2.N6.7.1 positive or negative coefficient of x²

    • S2.N6.7.2 maximum and minimum points

      • Characteristics of quadratic functions (S2-CC.5)

    • S2.N6.7.3 symmetry

      • Characteristics of quadratic functions (S2-CC.5)

  • S2.N6.a Show graphically the relationship between the area of a square and the length of its side, and explain that the relation ship is a function, but not a linear function.

  • S2.N6.b Use spreadsheet or graphing software to study how the graph of y = ax2 + bx + c changes when either a, b or c varies.

• S2.N7 Equations and inequalities

  • S2.N7.6 graphs of linear equations in two variables (ax + by = c)

    • Complete a table and graph a linear function (S2-Z.8)
    • Graph a line using gradient (S2-Z.15)
    • Graph a line from an equation in y=mx+c form (S2-Z.16)
    • Graph a line from an equation in standard form (S2-Z.19)

  • S2.N7.7 solving simultaneous linear equations in two variables by:

    • S2.N7.7.1 substitution and elimination methods

      • Solve simultaneous equations using substitution (S2-AA.6)
      • Solve simultaneous equations using elimination (S2-AA.8)

    • S2.N7.7.2 graphical method

      • Solve simultaneous equations by graphing (S2-AA.2)

  • S2.N7.8 solving quadratic equations in one variable by factorisation

    • Solve a quadratic equation by factorising (S2-CC.8)

  • S2.N7.9 formulation a pair of linear equations in tow variables to solve problems

    • Solve simultaneous equations by graphing: word problems (S2-AA.3)
    • Solve simultaneous equations using substitution: word problems (S2-AA.7)
    • Solve simultaneous equations using elimination: word problems (S2-AA.9)

  • S2.N7.a Use Graphmatica, applets or other software to draw the graph of ax + by = c (a straight line), check that the coordinates of a point on the straight line satisfy the equation, and explain why the solution of a pair of simultaneous linear equations is the point of intersection of two straight lines.

  • S2.N7.b Draw the lines x = a and y = b, and describe the lines and their gradients.

    • Graph a horizontal or vertical line (S3-T.19)

• S2.N10 Problems in real-world context

  • S2.N10.1 solving problems based on real-world contexts:

    • S2.N10.1.1 in everyday life (including travel plans, transport schedules, sports and games, recipes, etc)

      • Multi-step word problems (S2-M.1)
      • Guess-and-check word problems (S2-M.2)
      • Use Venn diagrams to solve problems (S2-M.3)
      • Elapsed time word problems (S2-M.4)

    • S2.N10.1.2 involving personal and household finance (including simple interest, taxation, installments, utilities bills, money exchange, etc.)

      • Price lists (S2-K.1)
      • Unit prices (S2-K.2)
      • Unit prices: find the total price (S2-K.3)
      • Percent of a number: GST, discount and more (S2-K.4)
      • Find the percent: discount and mark-up (S2-K.5)
      • Sale prices: find the original price (S2-K.6)
      • Multi-step problems with percents (S2-K.7)
      • Estimate tips (S2-K.8)
      • Simple interest (S2-K.9)
      • Compound interest (S2-K.10)

  • S2.N10.2 interpreting and analysing data from tables and graphs, including distance-time and speed-time graphs

    • Constant rate of change (S2-Z.5)
    • Interpret tables (S2-EE.1)
    • Interpret bar graphs (S2-EE.2)
    • Interpret line graphs (S2-EE.4)
    • Interpret line plots (S2-EE.6)
    • Interpret stem-and-leaf plots (S2-EE.8)
    • Interpret histograms (S2-EE.10)
    • Interpret box-and-whisker plots (S2-EE.13)
    • Interpret pie charts (S2-EE.14)

  • S2.N10.3 interpreting the solution in the context of the problem

  • S2.N10.4 identifying assumptions made and the limitations of the solution

  • S2.N10.a Examine and make sense of data in a variety of contexts (including real data presented in graphs, tables and formulae/equations).

    • Interpret tables (S2-EE.1)
    • Interpret bar graphs (S2-EE.2)
    • Interpret line graphs (S2-EE.4)
    • Interpret line plots (S2-EE.6)
    • Interpret stem-and-leaf plots (S2-EE.8)
    • Interpret histograms (S2-EE.10)
    • Create frequency charts (S2-EE.12)
    • Interpret box-and-whisker plots (S2-EE.13)
    • Interpret pie charts (S2-EE.14)

  • S2.N10.b Work on tasks that incorporate some or all of the elements of the mathematical modelling process.

S2.G Geometry and Measurement

• S2.G2 Congruence and Similarity

  • S2.G2.1 congruent figures

    • Side lengths and angle measures of congruent figures (S2-P.9)
    • Congruence statements and corresponding parts (S2-P.10)
    • Congruent triangles: SSS, SAS and ASA (S2-P.11)
    • Similar and congruent figures (S2-Q.1)

  • S2.G2.2 similar figures

    • Similar and congruent figures (S2-Q.1)
    • Side lengths and angle measures of similar figures (S2-Q.5)

  • S2.G2.3 properties of similar triangles and polygons:

    • S2.G2.3.1 corresponding angles are equal

      • Side lengths and angle measures of similar figures (S2-Q.5)

    • S2.G2.3.2 corresponding sides are proportional

      • Side lengths and angle measures of similar figures (S2-Q.5)

  • S2.G2.4 enlargement and reduction of a plane figure

    • Dilations: graph the image (S2-Q.2)
    • Dilations: find the coordinates (S2-Q.3)
    • Dilations: scale factor and classification (S2-Q.4)

  • S2.G2.5 scale drawings

    • Scale drawings: word problems (S2-H.12)

  • S2.G2.6 solving simple problems involving congruence and similarity

    • Side lengths and angle measures of congruent figures (S2-P.9)
    • Congruence statements and corresponding parts (S2-P.10)
    • Congruent triangles: SSS, SAS and ASA (S2-P.11)
    • Similar and congruent figures (S2-Q.1)
    • Side lengths and angle measures of similar figures (S2-Q.5)

  • S2.G2.a Examine whether two figures are congruent, by checking if one figure can be mapped onto the other figure under translation, rotation, and reflection.

  • S2.G2.b identify similar triangles rectangles from cut-outs of triangles/rectangles and explain why they are similar.

  • S2.G2.c identify and suggest applications of congruency and similarity in real-world contexts, e.g. photocopying, tessellation patterns, etc.

• S2.G4 Pythagoras' theorem and trigonometry

  • S2.G4.1 use of Pythagoras' theorem

    • Pythagoras' theorem: find the length of the hypotenuse (S2-S.1)
    • Pythagoras' theorem: find the missing leg length (S2-S.2)
    • Pythagoras' theorem: find the perimeter (S2-S.3)
    • Pythagoras' theorem: word problems (S2-S.4)

  • S2.G4.2 determining whether a triangle is right-angled given the lengths of three sides

    • Converse of Pythagoras' theorem: is it a right triangle? (S2-S.5)

  • S2.G4.3 use of trigonometric ratios (sine, cosine and tangent) of acute angles to calculate unknown sides and angles in right-angles triangles

    • Trigonometric ratios: find a side length (S3-H.6)
    • Trigonometric ratios: find an angle measure (S3-H.7)
    • Solve a right triangle (S3-H.8)

  • S2.G4.a Either (i) use strings of 12 units (e.g. 1 unit = 10 cm) to form a right-angled triangle with sides of whole unit lengths (e.g. 3 units, 4 units and 5 units) and find out if there is a relationship involving the three sides; or (ii) use cut-out pieces of two squares with sides 3 units and 4 units respectively to form a square of sides 5 units.

  • S2.G4.b Discuss the use of trigonometric ratios in real life, e.g., find the height of a tree/a building by measuring the angle of elevation with a clinometer.

• S2.G5 Mensuration

  • S2.G5.6 volume and surface area of pyramid, cone and sphere

    • Volume of pyramids and cones (S2-U.12)
    • Surface area of pyramids and cones (S2-U.13)
    • Volume and surface area of spheres (S2-U.14)

  • S2.G5.a Visualize and make connection between the volumes of pyramid and cone, and volumes of pyramid/cone and the related prism/cylinder.

    • Volume of prisms and cylinders (S2-U.10)

  • S2.G5.b Make sense of the formulae for the volume and surface area of a sphere e.g. by relating to the formulae for the volume and curved surface area of the related cylinder.

• S2.G8 Problems in real-world contexts

  • S2.G8.1 solving problems in real-world contexts (including floor plans, surveying, navigation, etc) using geometry

    • Area and perimeter: word problems (S2-U.5)
    • Circles: word problems (S2-U.7)

  • S2.G8.2 interpreting the solution in the context of the problem

  • S2.G8.3 identifying the assumptions made and the limitation of the solution

  • S2.G8.a Work on tasks that incorporate some or all elements of the mathematical modelling process.

S2.S Statistics and Probability

• S2.S1 Data analysis

  • S2.S1.4 analysis and interpretation of:

    • S2.S1.4.1 dot diagrams

      • Interpret line plots (S2-EE.6)

    • S2.S1.4.2 histograms

      • Interpret histograms (S2-EE.10)

    • S2.S1.4.3 stem-and-leaf diagrams

      • Interpret stem-and-leaf plots (S2-EE.8)

  • S2.S1.5 purpose and uses, advantages and disadvantages of the different forms of statistical representations

    • Choose the best type of graph (S2-EE.16)

  • S2.S1.6 explaining why a given statistical diagram leads to misinterpretation of data

  • S2.S1.7 mean, mode and median as measures of central tendency for a set of data

    • Calculate mean, median, mode and range (S2-FF.1)
    • Interpret charts to find mean, median, mode and range (S2-FF.2)
    • Mean, median, mode and range: find the missing number (S2-FF.3)
    • Changes in mean, median, mode and range (S2-FF.4)

  • S2.S1.8 purposes and use of mean, mode and median

    • Interpret charts to find mean, median, mode and range (S2-FF.2)

  • S2.S1.9 calculation of the mean for grouped data

  • S2.S1.a Construct dot diagrams, histograms (including equal and unequal class intervals) and stem-and-leaf diagrams from given data.

    • Create line plots (S2-EE.7)
    • Create histograms (S2-EE.11)

  • S2.S1.b Predict, observe and explain how the different measures of central tendency are affected by extreme data values (or outliers).

  • S2.S1.c Discuss the appropriate use of the measures of central tendency in different contexts.

    • Interpret charts to find mean, median, mode and range (S2-FF.2)

• S2.S2 Probability

  • S2.S2.1 probability as a measure of chance

    • Probability of simple events (S2-GG.1)

  • S2.S2.2 probability of single events (including listing all the possible outcomes in a simple chance situation to calculate the probability)

    • Probability of simple events (S2-GG.1)
    • Probability of opposite, mutually exclusive and overlapping events (S2-GG.2)

  • S2.S2.a Discuss the concept of probability (or chance) using everyday events, including simple experiments such as tossing a coin.

    • Experimental probability (S2-GG.3)
    • Make predictions (S2-GG.4)

  • S2.S2.b Compare and discuss the experimental and theoretical values of probability using computer simulations.