2.3.1

Identities & Proofs

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Proofs

To prove an 'identity', show one side is the same as the other. The identity symbol is ≡. E.g - Show that (n + 3)(n + 2) - 3n + 2 ≡ n2 + 2n + 8:

Proofs and Counter Examples

General numbers in proofs

General numbers in proofs

  • If we are proving something involving an even number, use 2n.
  • If we are proving something involving an odd number, use 2n + 1.
  • Consecutive numbers are shown by n, n + 1, n + 2, ... etc
Counter examples

Counter examples

  • A counter example is the simplest way to prove a statement is wrong.
  • You only need one counter example to show something is wrong.
  • To disprove the statement ‘The product of two primes is always odd':
    • Use the counter example of the primes 2 and 3 whose product is 6 which is even.
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1

Number

1.1

Using Numbers

1.1.1Equal Symbols1.1.2Negative Numbers1.1.3BIDMAS1.1.4Inverse Operations1.1.5End of Topic Test- Ordering1.1.6Factors1.1.7Multiples1.1.8Highest Common Factor1.1.9Lowest Common Multiple1.1.10Number of Ways to Do a Task1.1.11Number of Combinations1.1.12Number of Ways To Arrange Objects1.1.13End of Topic Test - Factors, Multiples & Combos1.1.14Standard Form1.1.15Ordinary Numbers to Standard Form1.1.16Addition in Standard Form1.1.17Subtraction in Standard Form1.1.18Multiplication in Standard Form1.1.19Division in Standard Form1.1.20Exam-Style Questions - Standard Form & Primes1.1.21Long Division1.1.22Prime Numbers1.1.23End of Topic Test - Standard Form & Long Division1.1.24Diagnostic Misconceptions - 1 NOT a Prime Number1.1.25Diagnostic Misconceptions - BIDMAS1.1.26Diagnostic Misconceptions - Adding Negatives
1.2

Fractions, Decimals & Percentages

1.2.1Adding Fractions1.2.2Subtracting Fractions1.2.3Multiplying Fractions1.2.4Dividing Fractions1.2.5Mixed Fractions1.2.6Calculating Percentages of Numbers1.2.7Calculating Fractions of Numbers1.2.8Multiply Decimals1.2.9Place Value1.2.10Recurring Decimals1.2.11Exam-Style Questions - Recurring Decimals1.2.12Convert Fractions & Terminating Decimals1.2.13Convert Recurring Decimals1.2.14Convert Decimals & Percentages1.2.15Compare Decimals1.2.16Compare Fractions1.2.17Exam-Style Questions - Percentages1.2.18End of Topic Test - Fractions & Decimals1.2.19Diagnostic Misconceptions - Adding Fractions
1.3

Powers & Roots

1.3.1Powers1.3.2Roots1.3.3Indices Advanced1.3.4Estimating Powers1.3.5Estimating Roots1.3.6Multiplying Powers1.3.7Dividing Powers1.3.8Power Raised to Power1.3.9Fractions Raised to a Power1.3.10Numbers Raised to Negative Powers1.3.11Roots of Numbers1.3.12Numbers to the Power of 1 & 01.3.13Surds1.3.14Simplify Surds1.3.15Rationalise Denominator1.3.16Exam-Style Questions - Indices1.3.17End of Topic Test - Powers & Roots1.3.18Diagnostic Misconceptions - Squaring1.3.19Diagnostic Misconceptions - Simplifying Surds
1.4

Accuracy

1.4.1Decimal Rounding1.4.2Significant Figure Rounding1.4.3Estimation1.4.4Truncation1.4.5Limits of Accuracy1.4.6Bounds Addition1.4.7Bounds Subtraction1.4.8Bounds Multiplication1.4.9Bounds Division1.4.10End of Topic Test - Rounding & Estimation1.4.11Diagnostic Misconceptions - Upper Bounds
2

Algebra

2.1

Introduction to Algebra

2.1.1Algebra Basics2.1.2Substituting Values2.1.3Algebraic Vocabulary2.1.4Collecting Terms2.1.5End of Topic Test - Algebra Basics & Power Rules2.1.6Expanding Brackets I2.1.7Common Factors2.1.8Expanding Brackets II2.1.9Rearrange For x2.1.10Rearrange with Powers2.1.11Exam-Style Questions - Basics of Algebra2.1.12End of Topic Test - Expanding & Rearranging
2.2

Manipulating Algebra

2.2.1How to Factorise2.2.2Factorising Quadratics2.2.3Algebraic Fractions - Addition & Subtraction2.2.4Algebraic Fractions - Multiplication & Division2.2.5Diagnostic Misconceptions - Common Denominators2.2.6Factorising Quadratics (tt)2.2.7Simplify algebraic fractions - cancelling factors2.2.8Simplify algebraic fractions - factorisation
2.3

Proofs & Functions

2.3.1Identities & Proofs2.3.2Exam-Style Questions - Algebraic Proof2.3.3End of Topic Test - Algebra & Proofs2.3.4Functions2.3.5Combine Functions2.3.6Composite Functions2.3.7Inverse Functions2.3.8Functions (tt)
2.4

Straight Line Graphs

2.4.1Straight Line Graphs2.4.2y = mx + c2.4.3Grad + YIntercept2.4.4Exam-Style Questions - Straight Line Graphs2.4.5End of Topic Test - Functions & Straight Lines2.4.6Diagnostic Misconceptions - The x Coefficient
2.5

Common Graphs

2.5.1Shape of Quadratic Graphs2.5.2Roots of a Quadratic Graph2.5.3Completing Squares2.5.4Turning Points2.5.5Common Functions 12.5.6Common Functions 22.5.7Trigonometric functions2.5.8End of Topic Test - Common Graphs
2.6

Transformations & Tangents

2.6.1Transformations2.6.2Tangent on a Circle
2.7

Properties of Graphs

2.7.1Estimate Gradient2.7.2Estimate Area2.7.3Represent Gradient & Area2.7.4Distance-Time Graph2.7.5Velocity-Time Graph2.7.6End of Topic Test - Transformations & Graphs
2.8

Solving Equations

2.8.1Solve Linear Equations2.8.2Linear Equations with Fractions2.8.3Solve Linear Equations Graphically2.8.4Quadratic ±2.8.5Factorise Quadratics2.8.6Quadratic Formula2.8.7Complete the Square2.8.8Solve Quadratics Graphically2.8.9End of Topic Test - Solving Equations2.8.10Simultaneous Equations2.8.11Quadratic Simultaneous Equations2.8.12Simultaneous Equations - Graphs2.8.13Simultaneous Equations - Word Problems2.8.14Approximate Solutions - Number of Iterations2.8.15Approximate Solutions - Iterations to 1 DP2.8.16Approximate Solutions - Iteration Formula2.8.17Exam-Style Questions - Simultaneous Equations2.8.18End of Topic Test - Simultaneous Equations2.8.19Diagnostic Misconceptions - Solving Quadratics
2.9

Inequalities

2.9.1Solving Inequalities2.9.2Solving Double Inequalities2.9.3Solving Quadratic Inequalities2.9.4Representing Inequalities2.9.5Exam-Style Questions - Inequalities2.9.6End of Topic Test - Inequalities
2.10

Sequences

2.10.1Nth Term Rules2.10.2Recurrence Relationships2.10.3Find Nth Term in Linear Sequence2.10.4Check If a Number is In a Sequence2.10.5Exam-Style Questions - Sequences2.10.6Arithmetic & Geometric Progressions2.10.7Key Sequences2.10.8Quadratic Sequences2.10.9Find Nth Term in Quadratic Sequence2.10.10Exam-Style Questions - Geometric Series2.10.11End of Topic Test - Sequences2.10.12Nth Term Rules (tt)
3

Ratio

3.1

Ratios in Practice

3.1.1Converting Between Units3.1.2Formula Triangles3.1.3Scale Factors3.1.4Ratios in Shapes3.1.5Diagnostic Misconceptions - Writing Ratios
3.2

Manipulating Ratios

3.2.1Simplifying Ratios3.2.2Turn Ratios into Equations3.2.3Ratio Notation & Reduction3.2.4End of Topic Test - Units & Ratios3.2.5Part : Part Ratio3.2.6Part : Whole Ratio3.2.7Two Ratios3.2.8Differences in Ratios3.2.9Fractions in Ratios3.2.10Numbers as Fractions3.2.11Ratios to Fractions3.2.12Exam-Style Questions - Ratios3.2.13End of Topic Test - Fractions & Ratios3.2.14Diagnostic Misconceptions - Combining Ratios
3.3

Percentage & Interest

3.3.1Convert Percentages to Decimals3.3.2Convert Percentages to Fractions3.3.3Find the Percentage3.3.4More/Less Percentage3.3.5Price from Percentage Change3.3.6Calculate Percentage Change from Price3.3.7End of Topic Test - Percentages3.3.8Profit & Interest & Discount3.3.9Original Price after Discounts3.3.10Simple Interest3.3.11Compound Growth & Decay3.3.12Compound Interest3.3.13Compound Interest - After How Many Days?3.3.14End of Topic Test - Percentages in Finance
3.4

Proportion

3.4.1Proportion3.4.2Direct Proportion3.4.3Inverse Proportion3.4.4Exam-Style Questions - Proportionality
3.5

Gradient

3.5.1Gradient on a Straight Line3.5.2Direct & Inverse Proportions as Graphs3.5.3Gradient of a Curve3.5.4Positive & Negative Gradients3.5.5End of Topic Test - Proportion & Gradient
4

Geometry

4.1

Introduction to Geometry

4.1.1Geometry Basics4.1.2Geometry Basics 24.1.3Regular Polygons4.1.4Triangles & Angles4.1.5Midpoint4.1.6Bisectors & Perpendiculars4.1.7Locus4.1.8Angle rules4.1.9Parallel Angle Rules4.1.10Angles in a Polygon4.1.11Angles in a Triangle4.1.12Exam-Style Questions - Angles in Shapes4.1.13End of Topic Test - Geometry
4.2

Triangles & Quadrilaterals

4.2.1Types of Triangle4.2.2Angles & Triangles4.2.3Congruence4.2.4Similarity4.2.5Quadrilaterals
4.3

Transformations

4.3.1Types of Transformations4.3.2Invariant Points4.3.3Effects of Transformations4.3.4Exam-Style Questions - Reflection
4.4

Circle Basics

4.4.1Properties of a Circle 14.4.2Chords4.4.3Properties of a Circle 24.4.4Area of a Circle4.4.5Circumference of a Circle4.4.6Fractions of Circles4.4.7Exam-Style Questions - Circle Formulae4.4.8End of Topic Test - Triangles & Circles
4.5

Circle Theorems

4.5.1Perpendicular Bisector of a Chord4.5.2Angle at the Centre4.5.3Angle of a Semicircle4.5.4Opposite Angles in Cyclic Quadrilaterals4.5.5Tangent Lengths4.5.6Angle in the Same Segment4.5.7Alternate Segment Theorem4.5.8Exam-Style Questions - Circle Theorems4.5.9Diagnostic Misconceptions - Circumference & Radius
4.6

Measurements & Units

4.6.1Length4.6.2Area4.6.3Volume4.6.4Capacity4.6.5Time4.6.6Money4.6.7Mass4.6.8Bearings4.6.9Scale Factor & Enlargement4.6.10Exam-Style Questions - Enlargements4.6.11Diagnostic Misconceptions - Hours
4.7

Calculating Area

4.7.1Area of 2D Shapes4.7.2Exam-Style Questions - Area4.7.3End of Topic Test - Circle Theorems, Units & Area
4.8

Triangle Formulae

4.8.1Pythagoras' Theorem4.8.2Pythagoras in 3D4.8.3Trigonometry4.8.4Exact Values4.8.5Sine Rule4.8.6Cosine Rule4.8.7Area of a Triangle4.8.8Exam-Style Questions - Trigonometry & Area4.8.9Diagnostic Misconceptions - Bearings from Angles4.8.10Diagnostic Misconceptions - Pythagoras' Theorem
4.9

3D Shapes

4.9.1Types of 3D Shape4.9.2Plans & Elevations4.9.3Nets4.9.4Volume of 3D Shapes4.9.5Volume of a Cuboid4.9.6Volume of a Cylinder4.9.7Volume of a Sphere4.9.8Volume of a Hemisphere4.9.9Volume of a Cone4.9.10Volume of a Pyramid4.9.11Surface Area & Composite Shapes4.9.12Surface Area of a Cuboid4.9.13Surface Area of a Cylinder4.9.14Surface Area of a Sphere/Hemisphere4.9.15Surface Area of a Cone4.9.16Exam-Style Questions - Volume
4.10

Vectors

4.10.1Vectors Introduction4.10.2Vectors4.10.3Translation4.10.4Scalar4.10.5Exam-Style Questions - Vectors4.10.6End of Topic Test - Trigonometry, Shapes & Vectors4.10.7Diagnostic Misconceptions - Column Vectors
5

Probability

5.1

Introduction to Probability

5.1.1Randomness & Fairness5.1.2Sum of Probabilities5.1.3Relative & Expected Frequency5.1.4Frequency Tables5.1.5Using Frequency Trees5.1.6Complete Frequency Trees
5.2

Representing Probabilities

5.2.1Probability5.2.2Probability Scale5.2.3Sample Space Diagrams5.2.4Sets5.2.5Venn Diagrams5.2.6Probabilities in Venn Diagrams5.2.7Exam-Style Questions - Venn Diagrams
5.3

Probability Calculations

5.3.1Theoretical Probabilities5.3.2Multiple Events5.3.3Probability Calculations5.3.4Dependent Events5.3.5Independent Events5.3.6Probability Tree Diagram5.3.7Complex Tree Diagrams5.3.8Conditional Probability5.3.9Conditional Probability From a Two Way Table5.3.10Conditional Probability Venn Diagrams5.3.11Conditional Probability Tree Diagrams5.3.12Exam-Style Questions - Probability5.3.13End of Topic Test - Probability5.3.14Diagnostic Misconceptions - Denominator
6

Statistics

6.1

Introduction to Statistics

6.1.1Statistics Basics6.1.2Statistics Key Terms6.1.3Frequency Tables6.1.4Exam-Style Questions - Statistics & Probability6.1.5Diagnostic Misconceptions - Median6.1.6Diagnostic Misconceptions - Mode6.1.7Diagnostic Misconceptions - Mean
6.2

Charts & Diagrams

6.2.1Bar Charts6.2.2Pie Charts6.2.3Stem & Leaf Diagrams6.2.4Back to Back Stem & Leaf Diagrams6.2.5End of Topic Test - Statistics6.2.6Pictograms6.2.7Time Series Graphs6.2.8Histograms6.2.9Histograms 26.2.10Frequency Polygons6.2.11Cumulative Frequency6.2.12Cumulative Frequency 26.2.13Scatter Graphs6.2.14Scatter Graphs - Predictions6.2.15Correlation & Causation6.2.16Exam-Style Questions - Cumulative Frequency
6.3

Comparing Datasets

6.3.1Comparing Datasets6.3.2End of Topic - Datasets

Practice questions on Identities & Proofs

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

  1. 1
    What is the general expression for even numbers?Multiple choice
  2. 2
    What is the general expression for odd numbers?Multiple choice
  3. 3
    What is the general expression for consecutive numbers?Multiple choice
  4. 4
    Give a counter example to disprove the statement: <br> x&sup2; is always more than xFlashcard
  5. 5
    Give a counter example to disprove the statement: <br> Any number larger than 11 is divisible by 3Flashcard
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Simplify algebraic fractions - factorisation

Exam-Style Questions - Algebraic Proof