Every lesson in the ladder
898 illustrated lessons in one ordered path, grouped by topic. Each topic page shows every lesson's first step, with free practice questions.
Syllabus coverage
- Singapore PSLE Mathematics
- Singapore O-Level Mathematics
- SAT Math
- GCSE Mathematics (Higher)
- Singapore A-Level H2 Mathematics
First Numbers
Making Ten
Groups and Shares
Bigger Numbers
Parts of a Whole
- Rounding Whole Numbers (2) flashcards
- Multiplying and Dividing in Your Head (6) flashcards
- Long Multiplication and Division (5) flashcards
- Factors and Multiples (13) flashcards
- Decimals (13) flashcards
- Area, Perimeter and Symmetry (4) flashcards
- Equivalent Fractions (6) flashcards
- Tables and Line Graphs (3) flashcards
- Time on a 24-Hour Clock (2) flashcards
Measure and Compare
Rates and Shapes
Letters for Numbers
- Negative Numbers and Linear Equations (8) flashcards
- Powers and Roots (15) flashcards
- Index Notation
- Laws of Indices
- The Power of a Power Rule
- Zero and Negative Indices
- Squares and Square Roots
- Cubes and Cube Roots
- Roots by Prime Factorization
- Powers in the Order of Operations
- Fractional Indices
- Standard Form
- Calculating in Standard Form
- Radicals
- Rationalizing the Denominator
- Adding and Multiplying Radicals
- Rational and Irrational Numbers
- Significant Figures and Estimation (5) flashcards
- Sequences (14) flashcards
- Term-to-Term Rules
- The nth Term
- Sequences Worth Knowing
- Quadratic Sequences
- Geometric Sequences
- Geometric Sequences with a Surd Ratio
- Sigma Notation
- The Sum of an Arithmetic Series
- The Sum of a Geometric Series
- The nth Term from the Sum
- The Sum to Infinity
- Recurrence Relations
- Proof by Induction: Summing 1 to n
- Proof by Induction: Divisibility
- Algebraic Expressions (14) flashcards
- Substitution
- Like Terms
- Expanding Brackets
- Factoring by Common Factor
- Factoring by Grouping
- Expanding Two Brackets
- Factoring Quadratics
- Factoring Quadratics with a Leading Coefficient
- Algebraic Identities
- The Sum and Difference of Two Cubes
- Finding Unknown Coefficients by Matching
- Algebraic Fractions
- Adding and Multiplying Algebraic Fractions
- Recognizing Equivalent Expressions
- Equations and Inequalities (26) flashcards
- Equations with Brackets
- Treating a Bracket as a Single Quantity
- Equations with the Unknown on Both Sides
- How Many Solutions an Equation Has
- Choosing a Coefficient to Fix the Solution Count
- Equations with Fractions
- Changing the Subject of a Formula
- Making a Twice-Appearing Letter the Subject
- Forming Equations
- Writing Formulas from Words
- Simultaneous by Substitution
- Simultaneous by Elimination
- Solving Simultaneous Equations by Scaling
- Writing a Pair of Equations from a Word Problem
- How Many Solutions a Pair of Equations Has
- Choosing a Coefficient in a Pair of Equations
- Solving for x + y Without Finding x and y
- Inequalities
- Two-Step Inequalities
- Integer Solutions of Inequalities
- Forming an Inequality in Two Variables
- Graphing an Inequality in Two Variables
- The Overlap of Two Inequality Regions
- Solving Absolute Value Equations
- Solving Absolute Value Inequalities
- Solution Sets of Inequalities
- Quadratic Equations (15) flashcards
- Solving by Factoring
- The Quadratic Formula
- The Sum and Product of the Roots
- The Discriminant and the Nature of the Roots
- Always Positive or Always Negative Quadratics
- Completing the Square
- Sketching from Vertex and Factored Form
- Deriving the Quadratic Formula
- Forming Quadratic Equations
- Solving a Quadratic Inequality
- A Linear Equation Paired with a Quadratic
- When a Line Meets, Touches or Misses a Curve
- Fractional Equations
- Solving Equations with a Square Root
- Solving a Rational Inequality
- Number Theory (9) flashcards
- Euclid’s Proof That Primes Never Run Out
- Modular Arithmetic on a Clock Face
- Casting Out Nines to Check Arithmetic
- Why Every Fraction Terminates or Recurs
- Why 0.999… Equals Exactly One
- Turning Recurring Decimals into Fractions
- Egyptian Fractions and the Greedy Algorithm
- The Square Root of Two Is Irrational
- Continued Fractions
Balance and Proof
- Pythagoras and Similar Shapes (6) flashcards
- Graphs of Equations (25) flashcards
- The Coordinate Plane
- Simultaneous Equations
- y = mx + c
- Writing a Line as Ax + By = C
- Point-Slope Form of a Line
- Quadratic and Cubic Graphs
- The Average Rate of Change Over an Interval
- Estimating a Gradient with a Drawn Tangent
- Reciprocal Graphs
- The Graphs of y = axⁿ
- The Graph of y = |x|
- Taking the Modulus of a Whole Expression
- Replacing x with a Modulus
- The Graph of y = 1/f(x)
- Intersections of Graphs
- Solving Inequalities from Graphs
- The Equation of a Circle
- Center and Radius by Completing the Square
- The Tangent to a Circle at a Point
- Where a Line Meets a Circle
- The Equation of an Ellipse
- The Equation of a Hyperbola
- Distance-Time Graphs
- Speed-Time Graphs
- A Line in Graph, Table and Equation
- Lines and Angles (18) flashcards
- Angles at a Point
- Angles on a Line
- Complementary Angles
- Vertically Opposite Angles
- Parallel and Perpendicular Lines
- Drawing Parallel and Perpendicular Lines
- Angles in Parallel Lines
- Angles in a Triangle
- Angles in a Polygon
- The Interior Angle of a Regular Polygon
- Constructing a Perpendicular Bisector
- Constructing an Angle Bisector
- Constructing a Perpendicular from a Point
- Constructing a Triangle
- Loci
- Triangles on a Sphere Break the 180 Rule
- Why an Angle Cannot Be Trisected
- Angle Chases
- Polygons and Solids (12) flashcards
- Surface Area (5) flashcards
- Statistical Displays (13) flashcards
Curves and Angles
- Functions (26) flashcards
- Function Notation
- Domain and Range
- Piecewise Functions
- Graphing a Piecewise Function
- Making a Piecewise Function Continuous
- Composite Functions
- Inverse Functions
- Restricting the Domain for an Inverse
- When a Composite Function Exists
- Transforming Graphs
- Stretching a Graph Vertically
- Stretching a Graph Horizontally
- Combining Graph Transformations
- Exponentials and Logarithms
- Exponential Growth
- Exponential Decay
- Growth and Decay Problems
- Logistic Growth and Carrying Capacity
- The Laws of Logarithms
- The Change of Base Rule
- The Natural Logarithm
- Solving Exponential Equations
- Logarithmic Graphs
- Straightening Growth with Logarithms
- Straightening a Power Law
- Fitting a Logarithmic Model
- Rational Functions (6) flashcards
- Transformations (6) flashcards
- Congruence, Similarity and Circle Theorems (16) flashcards
- Congruence Tests
- Similarity Conditions
- The Midpoint Theorem
- Ratio of Areas
- Ratio of Volumes
- Angle in a Semicircle
- Chords, Arcs, Sectors and Segments
- Arcs and Sectors
- The Area of a Segment
- Angle at the Center
- Angles in the Same Segment
- Cyclic Quadrilaterals
- Tangent Properties
- The Alternate Segment Theorem
- Chords and the Center
- Circle Theorems with Algebra
- Measuring Data (19) flashcards
- Median and Mode
- Measures of Central Tendency
- Averages from a Frequency Table
- Mean of Grouped Data
- The Modal Class
- Range and IQR
- Measures of Spread
- Cumulative Frequency
- Quartiles from the Curve
- Percentiles
- Box and Whisker
- Judging Normality from a Box Plot
- Mean Absolute Deviation
- Standard Deviation
- Standard Deviation: Ungrouped
- Standard Deviation: Grouped
- Comparing Data Sets
- Simpson’s Paradox
- Finding a Frequency from the Mean
- Scatter Plots and Correlation (7) flashcards
- Simple and Compound Interest (4) flashcards
- Financial Mathematics (4) flashcards
- Quadratic Graphs and Coordinate Geometry (8) flashcards
Trigonometry
- Compass Directions and Bearings (3) flashcards
- Triangle Trigonometry (25) flashcards
- Sine, Cosine and Tangent
- Exact Sine, Cosine and Tangent at 30, 45 and 60°
- Sine and Cosine of Complementary Angles
- Solving Right Triangles
- Finding an Angle from a Ratio
- Elevation and Depression
- Sine and Cosine of Angles Past 90 Degrees
- The Graphs of Sine and Cosine
- The Graph of Tangent
- Amplitude and the Principal Axis
- The Period and Phase Shift of a Wave
- Solving Simple Trigonometric Equations
- The Principal Values of the Inverse Functions
- Quadratic Equations in Sine, Cosine or Tangent
- Trigonometric Equations with a Multiple Angle
- Modeling Tides and Daylight with a Sine Function
- Area with Sine
- The Sine Rule
- The Cosine Rule
- Finding Angles with the Sine Rule
- The Ambiguous Case of the Sine Rule
- Finding Angles with the Cosine Rule
- Trigonometry in Three Dimensions
- Exact Trigonometric Ratios with Surds
- Bearings with the Sine and Cosine Rules
Vectors
- Vectors in the Plane (17) flashcards
- Vectors
- Column Vectors
- Adding Vectors
- Subtracting Vectors
- Scalar Multiplication
- Magnitude of a Vector
- Unit Vectors
- The Unit Vectors i and j
- Position Vectors
- Vector Proofs with Midpoints
- The Ratio Theorem
- Collinear Vectors
- The Dot Product
- The Angle Between Vectors
- Vectors in Three Dimensions
- Distance and Midpoint in Three Dimensions
- The Vector Equation of a Line
- Planes and the Vector Product (13) flashcards
- The Vector Product
- Expanding Products of Vectors
- Parallelogram Area from the Vector Product
- The Vector Equation of a Plane
- The Cartesian Equation of a Plane
- The Angle Between a Line and a Plane
- The Angle Between Two Planes
- The Intersection of a Line and a Plane
- The Distance from a Point to a Plane
- The Distance from a Point to a Line
- The Cartesian Form of a Line
- Intersecting and Skew Lines
- The Intersection of Two Planes
Matrices
Probability with Several Events
- Combined Events (16) flashcards
- Naming the Regions of a Venn Diagram
- Possibility Diagrams
- Chance from Areas
- Mutually Exclusive Events
- The Addition Rule for Overlapping Events
- Independent Events
- Tree Diagrams
- The Chance of At Least One
- Without Replacement
- Multiplying Along a Tree Without Replacement
- Conditional Probability
- Conditional Probability Notation
- The Monty Hall Problem
- The Birthday Paradox
- Why a Positive Test Can Still Mean Healthy
- Reversing a Tree: the Base-Rate Answer
Introduction to Calculus
- Calculus (9) flashcards
- Radians and Trigonometric Identities (12) flashcards
- Radians
- Exact Trigonometric Values in Radians
- Arc Length and Sector Area
- Trigonometric Identities
- Proving a Trigonometric Identity
- The Compound Angle Formulas
- The Double Angle Formulas
- Secant, Cosecant and Cotangent
- The Identity 1 + tan²θ = sec²θ
- The Harmonic Form R sin(x + α)
- Solving a sin x + b cos x = c with the R Form
- Every Solution of a Trigonometric Equation
- Polynomials and the Binomial Theorem (8) flashcards
- Solving Equations Numerically (3) flashcards
Probability Distributions and Inference
- Sets and Counting (14) flashcards
- Distributions (16) flashcards
- Random Variables
- Probability Distributions
- Expected Value
- Variance
- Transforming a Random Variable
- Bernoulli Trials
- The Binomial Distribution
- Binomial Mean and Variance
- Modeling with a Binomial
- Continuous Variables
- The Normal Distribution
- Standardizing
- Reading the Z-Table
- Reading Normal Probabilities
- Inverse Normal
- Combining Normals
- Sampling (11) flashcards
- Inference (18) flashcards
- Point Estimates
- Unbiased Sample Variance
- Estimates from Summarized Data
- Null and Alternative Hypotheses
- One-Tailed and Two-Tailed
- The Z-Test
- Critical Regions
- P-Values
- Testing a Proportion with the Binomial
- Type One and Type Two Errors
- Confidence Intervals
- Expected Frequencies
- The Chi-Squared Test for Independence
- The Chi-Squared Goodness of Fit Test
- The One-Sample T-Test
- The Two-Sample T-Test
- Testing a Variance with Chi-Squared
- Comparing Two Variances with the F-Test
- Correlation and Regression (9) flashcards
Complex Numbers
- Complex Arithmetic (9) flashcards
- The Complex Plane (17) flashcards
- The Argand Diagram
- Moving Points on the Argand Diagram
- The Modulus of a Complex Number
- The Argument of a Complex Number
- Polar Form
- The Form r(cos θ + i sin θ)
- Dividing in Polar Form
- De Moivre’s Theorem
- The Roots of Unity
- Regular Polygons from the Roots of Unity
- Multiple Angles by De Moivre
- The Identity z + 1/z = 2 cos θ
- Summing a Series with De Moivre
- The Exponential Form of a Complex Number
- The nth Roots of a Complex Number
- Circles and Bisectors on the Argand Diagram
- Rays and Regions on the Argand Diagram
Limits and Continuity
Differentiation
- Rules of Differentiation (19) flashcards
- The Derivative As a Limit
- Derivative Notation
- Differentiating From First Principles
- Where a Derivative Fails to Exist
- Differentiating Linear Functions
- The Power Rule
- The Constant Multiple and Sum Rules
- The Product Rule
- The Quotient Rule
- The Chain Rule
- Differentiating Trigonometric Functions
- Differentiating tan x and the Reciprocal Ratios
- The Number E
- Differentiating Exponentials and Logarithms
- Differentiating aˣ and logₐ x
- Implicit Differentiation
- The Derivative of an Inverse Function
- Differentiating sin⁻¹ x and tan⁻¹ x
- Higher Derivatives
- Applications of Differentiation (20) flashcards
- Tangents and Normals to Curves
- The Tangent-Line Approximation
- The Mean Value Theorem
- Increasing and Decreasing Functions
- Finding Stationary Points
- Critical Points and Local Extrema
- The Second Derivative Test
- The First Derivative Test
- Absolute Extrema on a Closed Interval
- Points of Inflection
- Curve Sketching with Derivatives
- Optimization Problems
- Motion in a Straight Line
- Related Rates
- Parametric Equations
- Parametric Differentiation
- Maclaurin Series
- Where a Series Is Valid
- L'Hôpital's Rule
- From the Derivative Back to the Function
- Motion in Two Dimensions (5) flashcards
- Series and Convergence (11) flashcards
- Hyperbolic Functions (5) flashcards
Integration
- Techniques of Integration (23) flashcards
- The Antiderivative
- Integrating Powers
- Finding the Constant of Integration
- Reading the Integral Sign
- Differentials
- The Definite Integral
- Left, Right and Midpoint Riemann Sums
- The Riemann Sum in Sigma Notation
- The Fundamental Theorem of Calculus
- Area Below the Axis
- Properties of the Definite Integral
- Integration by Substitution
- Integrating f(ax + b)
- Integration by Parts
- Integrating Trigonometric Functions
- Integrating sin² and cos²
- Integrating Exponentials and Logarithms
- Integration by Partial Fractions
- Integrals of f′ over f
- Integrating sec²x, sec x tan x and tan x
- Integrals That Give tan⁻¹ and sin⁻¹
- Choosing a Trigonometric Substitution
- Integrals That Give arsinh and arcosh
- Applications of Integration (21) flashcards
- Velocity and Displacement by Integration
- Total Distance Traveled
- Area Between Two Curves
- Volumes of Revolution
- Volumes by Washers
- Volumes with Known Cross Sections
- Arc Length
- The Mean Value of a Function
- Improper Integrals
- Numerical Integration
- Simpson's Rule
- Separable Differential Equations
- Solving a Differential Equation by Substitution
- Forming Differential Equations
- Slope Fields
- Euler’s Method
- The Integrating Factor
- The Auxiliary Equation
- Complementary Function and Particular Integral
- Simple Harmonic Motion
- Damped Oscillations
- Polar Coordinates and Curves (4) flashcards
- Coupled Differential Equations (4) flashcards