An Introductory Example:

It frequently happens that a student does well at school, especially at high school maths, and then suddenly in a new term his/her marks start to drop.  iTutor - Calculator Used by Many StudentsThere as several reasons for this.  Success in sport may lead to extra practice time and a loss of study time. Clubs and the debating society may soak up study time.  Despite previous success new material to learn and use may find her/his core skills lacking.  For example in the previous term use of calculators may have masked a lack in numeracy.  When calculators have the capability shown at the right this is not surprising :-(  When new material requires that calculators are NOT used things can suddenly go very wrong.  For example a student is asked  to write down the third side of a right-angled triangle when two of the sides are 26 and 10 cm long.  This can be solved using Pythagoras' Theorem and that does require a calculator.  However, the instructions say that no calculator may be used.  Numeracy is required.  Number Sense shows that the sides are 2x13 and 2x5 cm.  The third side must be 2x12 cm  since the triangle is a 5-12-13 Pythagorean Triple with the sides doubled.              

Core Skills Entering Maths Grade 8

  •  Numeracy
  •  Elementary Reasoning Skills     

Core Skills Entering Maths Grade 9

  • Number Classification
    • Real, Rational, Integer, Counting etc
  • Patterns
    • Algebraic rules
  • Negative Numbers
    • Multiplication as successive addition and as scaling)
  • Algebra (Symbolic Arithmetic)
    • Simple algebra and applications, simplifying expressions, Algebraic (and numeric) Fractions.
  • Equations
    • First Order Linear Equations
  • Indices
    • Index Algebra, sums, products and quotients.  Rules
  • Graphs
    • Plotting Graphs, Discontinuous and Continuous Data, Scatter, Straight Lines and Curves
  • Geometry (Introduction)
    • Shapes, Angles, Parallel Lines, Triangles, Tessellation (tiling), platonic solids (regular polyhedra), Rotation, Reflection and Translation in the Cartesian Plane,  Circle Geometry, Pi, Chords, Sectors, Segments, Volumes of Solids.
  • Statistics (Introduction)
    • Mean, Median, Mode and Central Tendency
Core Skills Entering Maths Grade 10

  • Numbers, Numerical Operations, Relatedness
    • Numbers, Ratio, Proportion, Rate Exponents, Scientific Notation, Simple and Compound Interest
    • Distributive Law
    • Patterns, Functional Relatedness, Algebra
    • Sequences, Simplifying Algebraic Expressions, Factorizing, Algebraic Fractions, Equations, Graphs of Algebraic Relations, Gradient, Graphical Representation of Data, Product of Binomials,  Factorization, Difference of Two Squares  
  • Geometry
    • Quadrilaterals, Polygons, Similar and Congruent Triangles, Geometric Solids, Reflection, Rotation, Angles of Depression and Elevation.
  • Measurement
    • Distance, Speed, Time, Surface Area, Volume, Measurement (accuracy), Pythagoras (applications)
  • Statistics and Probability
    • Planning Data Collection, Organizing data, Graphing Data, Probability, Events, Experimental Probability.

 Core Skills Analysis

G10 -> G11

1 Identify as rational or irrational

  a)  5             b)            0.32           c)         Sqrt(0)          d)        π          e)         0.888rep

2  Write as fractions

a)   0.3rep     b)          0.23752375rec                            c)         1 +0.8rep

3  In the Pattern 10; 7; 4; 1; ...  determine

  The 5th term
  The formula for the nth term 
   Which term equals -11?         

4  Calculate without using a calculator   

  a)    21/2  - 51/8    +32/3            b)    sqrt(64 + 36)        c)   sqrt(64x36)

  d)   curt(64) - sqrt(64)  

5  Between which integers do the following values lie

  a)    2 x sqrt(21)                   b)     sqrt(62)                c)    curt(60)

6  Evaluate
          (2)^x.(4)(x-1)                                 (27)^(x-2)               (5)^1010 + 3x(5)^(1008)
a)      --------------                 b)             -------------  c)         -------------------------     


7   R12000  is invested for 7 years at 11% pa.

    What is the final value if simple interest is used?
     What is the final value if compound interest is used, compounded annually?


8   Factorize

    a)      2x^3   +   8x^2   +    16((x)^2)y              b)    16x^2  + 4y^2

9  Solve       
                            4(y+4)                           14
     10 - ------------------       =     ------    
                        y                                 y

       A wheel hub has a radius 'r'.  The wheel and tyre have a combined radius R 

    Find the area A of the side-wall of the tyre  in terms of r and R

    Make R the subject of the formula determined above.

10  Solve

 a)                     27^(-x)                =       3x(9)^(x-1)        

b)                      x^2  - 5x   +  6    =       0     

c)                      2(x + 1)^2          =       (2x + 2)(2x + 3)


Core Skills Entering Maths Grade 11

  • Number Relationships and Patterns
    • Number Systems
    • Rational Number Inter-Conversions Decimal-Fraction
    • Rounding
    • Number Line
    • Number Patterns, Linear, Number Patterns by Exploration  and Conjecture
  • Exponents and Financial Maths 
    • Exponent Laws, Rational Exponent Laws, Operations with Variable Exponents    
  • Financial Maths
    • Simple Interest, Compound Interest, Hire Purchase Agreements
  • Algebra
    • Factorization of Trinomials,
    • Sum and Difference of Cubes    
    • Simplifying Algebraic Fractions
  • Equations
    • Linear
    • Quadratic - Solving
    • Changing the Subject of a Formula (Literal Equations)
    • Exponential
    • Simultaneous
    • Modelling.
  • Functions
    • Linear, Quadratic, Hyperbolic, Exponential
    • Finding Equations from Graphs
    • Domain, Range, Axes of Symmetry, Asymptotes
    • Continuity, increasing/decreasing
    • Average Gradient
    • Reflections and Translations
    • Trigonometric Functions
  • Trigonometry
    • Ratios
    • Use of Calculator
    • Trigonometric Equations
    • Trig and Pythagoras  
    • Special Angles
    • Angles of Elevation and Depression
  • Geometry
    • Triangles:  Similarity and Congruence
    • Quadrilaterals, Polygons
    • Midpoint Theorem 
  • Analytical Geometry
    •   Distance and Mid-Point Formulae
    • Average Gradient
  • Shapes
    • Solids, Prisms, pyramids, Spheres, Cones
    • Surface Area and volume
  • Statistics
    • Presentation of Data, Frequency Tables, Stem and Leaf Plots
    • Measures of Central Tendency, Mean, Median, Mode
    • Measures of Dispersion, Range, quartiles, Percentiles
    • Grouped Data
    • 5-Number Summary, Box and Whisker Plot
    • Sources of Bias and Error    
  • Probability
    • Events
    • Theoretical Probability, Experimental Probability
    • Venn Diagrams

Core Skills Entering Grade Maths 12    

      • Quadratic Equations
        • Completing the Square
        • Quadratic Formula
      • Quadratic Inequalities
      • Exponential Equations
  • Number Systems
  • Number Patterns
Financial Maths:
  • Simple Interest and Decay
  • Compound Interest and Decay
  • Graphs of Common Functions
  • Finding the equation from the Graph of a Function
  • Domain and Range
Exponents and Surds:
  • Exponents
  • Surd Laws

Linear Programming:

  • Feasible Region

Coordinate Geometry:

  • Analytical Geometry
  • Transformational Geometry
  • The Ratios
  • Identities
  • Special Angles
  • Reduction Formulae
  • Trigonometric Equations
  • Sine, Cosine and Area Rules. 
  Data Handling:
  • Statistics
    • Central Tendency
    • Cumulative Frequency
    • Bivariate Data
    • Grouped Data
      • Standard Deviation
  • Regular Solids

Force, Momentum, Impulse: 
  • Scalars and Vectors
  • Newtons Laws
  • Momentum
  • Conservation of Momentum
  • Impulse

Geometrical Optics:  

  • Real and Virtual images
  • Lenses, Parameters
  • Ray Diagrams  

Longitudinal Waves, Sound:

  • Transverse and Longitudinal Waves
  • The Wave Equation
  • Sound
  • Pitch and Loudness
  • The Human Ear
  • Music    

  • Electrostatic Force
  • Coulomb's Law
  • Electric Field:  Definition
  • Electrical Potential Energy and Potential
  • Potential Difference
  • Capacitance

  • Magnetic Fields and Current
  • Electromagnetic Induction
  • Lenz's Law

Electric Circuits:


Electronic Properties of Matter:
  • Electrical conduction
  • Semiconductors
  • p-n junction
  • Insulation

Chemical Bonding and Molecular Structure:
  • Lewis Theory
  • Electro-negativity
  • Bond polarity
  • Oxidation Number
  • Bond Energy and Length
  • Multiple bonds
  • Molecular Shape - VSE

Ideal Gasses:
  • Kinetic Theory
  • Heat and Temperature
  • Gas Pressure
  • Ideal Gas Law

Chemical Change and Stoichiometry:

  • Relative Atomic mass
  • Percentage Composition by Mass
  • Mole Concept
  • Empirical Formula, Molecular formula
  • Molar volume
  • Concentration
  • Stoichiometry from Chemical Equations

Energetic of Chemical Change:
  • Chemical reactions and chemical bonds
  • Exo and Endothermic reactions
  • Potential Energy diagrams
  • Activation Energy

Types of Reactions:
  • Acids and bases in solution
  • Protolytic Reactions
  • Acid-base reactions - neutralisation
  • Redox reactions
  • Oxidation and Reduction reactions, agents
  • Oxidation Numbers and Redox Reactions
  • Substitution, Addition and Elimination Reactions

The Atmosphere:

Industrial Chemistry: