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Standard Grade

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Past papers and their solutions can be found on the General page.

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Revision material posted below

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Quiz 1 - PowerPoint      Download            Answer Key

Course material - grouped by topic
click on links for more detail

 

 

Index:

Calculations and Mathematics in Society

Calculations, Calculators, Percentages
Standard Form
Fractions
Personal Finance
Distance, Speed & Time
Proportion

Probability, Information Handling & Statistics

Probability
Statistics

Geometry & Trigonometry

Similar Triangles and Similar Shapes
Pythagoras
Trigonometry
      SOHCAHTOA
      Sine and Cosine Rule
      Area of a triangle
Circle
Trigonometric Graphs and Equations
Areas and Volumes
      Cylinders & Prisms

Algebra

Positive and Negative Numbers
Removing Brackets, FOIL, Multiplication Tables
Inequalities
Simultaneous Equations
Factors and Factorising
      common factors
      pairs of brackets
      difference of two squares
Sequences
      Factors, odd and even numbers
Solving Quadratic Equations
Using and making formulae
Gradient and equation of a straight line
Functions - evaluating
Graphs - parabola, cubic, reciprocal (hyperbola)
The Quadratic Function and its properties
Simplifying algebraic Fractions
Solving algebraic fraction Equations
Indices
Surds

 

Calculations and Mathematics in Society

Calculations and the Calculator

    • Using the calculator

    • Standard form

    • Fractions

    • Percentage Calculations
      • Finding a percentage
      • Increasing and decreasing using a multiplier
      • Reversing the change - working backwards


Personal finance

    • Payslips, Wages & Salaries
      • Calculating Gross pay, Deductions, Net pay
      • Calculating Pension – superannuation
      • Using a table to find National Insurance (NI) deduction
      • Income Tax calculations - using a table

    • Simple interest calculations on savings
    • Compound Interest on savings

    • VAT - Value Added Tax

    • Telephone, Electric and Gas Bills

    • Hire Purchase

    • Holidays abroad - changing money into different currencies

    • Life Insurance

    • Calculating premiums

    • Appreciation and Depreciation


Distance, speed and time

    • DST triangle
    • Calculating distance, speed, time
    • Changing hours and minutes to decimal fractions of an hour
    • Interpreting graphs


Proportion in Practice

    • Tutorial on Proportion & Variation

    • Direct Proportion
      • Writing down a proportionality and then an equation using a constant.
      • Direct proportion calculations
      • Using squares and square roots

    • Inverse Proportion

    • Joint variation
      • Halving and doubling – including squares.
      • Variation with more than one variable changing.
        e.g. double one and treble another


Problem Solving

  • Mathematical Pictures
    • Matching graphs to pictures

 




Probability, Information Handling and Statistics


Probability

    • Calculating simple probability
      • Effect of non-replacement
      • Effect of changing probabilty by adding more items

    • Combining probabilities – two events
      • Mutually Exclusive Events
        • Table of outcomes or tree diagram
        • Adding probabilities when two events are mutually exclusive.
        • P(A or B) = P(A) + P(B)
          e.g. Probability of a 3 or a 4 when a dice is rolled = 1/6 + 1/6 = 2/6 = 1/3

      • Independent Events
        • Table of outcomes or tree diagram
        • Multiplying probabilities when two events are independent.
        • P(A and B) = P(A) x P(B)
          e.g. Probability of a 3 on the first roll and an 4 on the second roll of a dice
                =  1/6 x 1/6 = 1/36


    • Relative Frequency and Probability
    • Experimental Data
      • Observation, surveys, Relative frequency Table
      • Calculating the probability of an event from the relative frequency
    • Expectation = Number of events x P(outcome)

 

Statistics

    • Tutorial on Pie Charts

    • Graphs and Charts
      • Interpreting - bar graph, line graph, pie chart
      • Constructing a pie chart by calculating angles
      • Calculating values from a pie chart

    • Stem and Leaf Diagrams
    • Back to back stem and leaf

    • Dotplots – constructing
      • describing the distribution

    • Five Figure Summary
      • Box Plots
        • Construct a box plot from a five figure summary
        • Comparing distributions
          • typical value – Median
          • Spread of the marks - Interquartile range or Semi interquartile range

    • Calculating quartiles

    • Statistical Measures
      • Mean, median, mode, range
      • Interquartile range, semi-interquartile range

      • Standard Deviation
        • Calculating mean and standard deviation
        • Being able to use both formulae
        • Comparing two distributions

    • Frequency Tables
      • The meaning of a frequency table
      • Construction of  a frequency table using tally marks
      • Calculate mean of a frequency table

    • Frequency Tables with class intervals
      • Working with mid-value
      • Calculate mean

    • Cumulative Frequency Tables
      • Construction from a frequency table
      • Read information from a frequency table.

    • Cumulative Frequency Diagrams (graph)
      • Construction from a cumulative frequency table
      • Read information from a cumulativefrequency table.



Geometry


Similar Shapes


Pythagoras

    • Know Pythagoras' Theorem
    • Calculate hypotenuse or a shorter side in a right angled triangle.
    • Work with isosceles triangles
    • Converse of Pythagoras - Is it a right angle ?


Trigonometry

    • SOH-CAH-TOA – in a right angled triangle.
      • Knowing ratios for sine, cosine, tangent.
      • Finding a short side
      • Finding the hypotenuse
      • Finding an angle
      • Applications of trigonometry to isosceles triangles

    • The Sine Rule
      • Finding a side, Finding an angle, Finding the altitude of a triangle

    • The Cosine rule – with cyclic permutation
      • Finding a side  SAS, Finding an angle  SSS

    • Area of a triangle  
      • Area =     (SAS)

    • Selecting a strategy
      • If a right angled triangle - use SOHCAHTOA
      • If not:
        • SAS, SSS use cosine rule
        • All other cases use sine rule
      • Applications


Symmetry in the Circle

    • Angles in circles
      • Isosceles triangles
      • calculating base angles and centre angle

    • Angles in a semi-circle
      • are right angles

    • Tangents to a circle
      • are at right angles to the radius at the point of contact

    • Symmetry about a diameter
      • Chords
      • Isosceles triangles
      • Applications
        • Using Pythagoras
          • Milk and oil tanker
          • Bridges
          • Tunnels
          • Shelters
          • Widths and heights
        • Using Trigonometry – SOH-CAH-TOA

    • Fractions of a circle
      • Lengths of Arcs
      • Areas of Sectors


Trigonometry – Calculations, Graphs and Equations


Areas and Volumes

    • Rectangle, Triangle and Circle
      • Areas, Perimeters, Using   on the calculator

    • Composite shapes
      • Area calculations
      • Perimeter calculations

    • Areas of quadrilaterals
      • Rhombus and kite
      • Parallelogram
      • Trapezium

    • Using Pythagoras

    • Using Trigonometry - SOH-CAH-TOA

    • Prisms
      • Identifying a prism – uniform cross-section
      • Volume = Area of cross section x height

    • Cylinders
      • volume 
      • Working with cubic centimetres and litres
      • Calculating the height  or radius
      • Area - curved surface area:   
      • Open and closed cylinders - Area of lid or base =

 


 

Algebra

Positive and negative numbers

    • Add and subtract positive and negative numbers
    • Multiply and divide positive and negative numbers
    • Simplifying expressions
    • Solving simple equations


Brackets and Equations

    • Examples
    • Solutions

    • Removing a single bracket
    • Solving equations with single brackets
    • Using FOIL - to multiply a pair of brackets
    • Using a multiplication table
    • Breaking brackets when squared    e.g.    


In
equalities

  • Solving simple inequalities – with and without brackets
    • Know that multiplication or division by a negative number changes direction of inequality



Simultaneous Equations

    • Tutorial

    • Forming simultaneous equations to describe a problem

    • Solving simultaneous equations
      • using graphs
      • by substitution
      • by elimination

    • Applications to problem solving

 
Factors


Quadratic Equations

    • Tutorial

    • Solving quadratic equations
      • Using graphs
      • Factorising
        • Using common factor   to obtain solutions
        • Difference of two squares
        • Putting a trinomial into two brackets
      • Solving using the formula   
    • Quadratic equations as mathematical models - Problem solving


Formulae

    • Substituting numbers into formulae

    • Making up your own formulae when reading a question

    •  Changing the subject of the formula 

    • Being able to reason the result if variables are halved or doubled



Gradient and Equation of a Straight Line

    • Tutorial
    • Calculating a gradient:    

    • Equation of a straight line
      • y = mx + c      m = gradient    c = y-intercept
        • Write down gradient and y-intercept from the basic equation
        • Sketching a line from the equation
        • A point lying on a line satisfies the equation of the line
        • Finding equation of a line with gradient and a point
        • Finding equation of a line through two points
        • Rearranging an equation to find the gradient and y-intercept.

    • Straight Lines as mathematical models
      • Writing down the equation from the graph
      • Problem solving applications

    • Best-fitting straight line
      • Plotting experimental data, including the mean
      • Finding the best fitting straight line
      • Find the equation of the line
      • Use the equation to predict a value



Functions and Graphs

    • Tutorial

    • Evaluating a function   
      • e.g. If  f(x) =3x - 7, evaluate when x = -1

    • Work back to the input  
      • e.g. If f(x) = 7x+3, find a, if f(a) = 10

    • Graphs of functions
      • Recognise:  Linear, Quadratic, Cubic, Reciprocal functions

    • Knowing and understanding properties of the quadratic function

        • The parabola   is a quadratic function

        • Minimum value
          • of  is 0, when x = 0

        • Axis of symmetry
          • goes through the minimum and is the line x = 0

        • Graphs of parabolas

          • Identify orientation
            • (which way up it is)    u  or n

          • Identify maximum or minimum
            • value of x for which max or min occurs

          • identify axis of symmetry
            • x = ........ (value of x on which the turning point lies)

          • Identify roots of the equation
            • where f(x) = 0  ie. where graph cuts the x-axis



Fractions and Equations

    • Tutorial

    • Simplifying fractions
      • cancelling down
        • spotting and using common factors
        • spotting and using difference of two squares

    • Multiplication and division
      • Multiply - multiply tops, multiply denominators
      • Divide - invert (flip) second fraction and change to multiply

    • Addition and subtraction
      • common denominator

    • Solving equations with fractions
      • Remove fractions first
      • Remove brackets



Indices and Surds

    • Tutorial on Indices

    • Rules of indices







      • Simplifying expressions

    • Zero and negative indices





    • Dealing with multipliers

      • ,   



    • Fractional indices





    • Surds

    • Applications

      • Simplifying
        • by forming a product with the largest perfect square factor

      • Adding and subtracting surds

      • Simplifying expressions by combining them

      • Rationalising the denominator
        • multiply top and bottom by appropriate surd

          e.g.   since 

 



      © CPD Publications 2006

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