H1 Mathematics
Lessons in h1 mathematics. This course is still being developed; independent practice and assessment coverage is incomplete.
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H1 Mathematics
- Functions and graphsSuggested next · 6 core lessonsConnect exponential and logarithmic models with graph features, numerical solutions and exact quadratic methods.
- DifferentiationSuggested next · 2 core lessonsConnect tangent gradients and chain rules to stationary-point tests, numerical methods and constrained optimisation.
- IntegrationSuggested next · 2 core lessonsReverse differentiation, evaluate exact or numerical integrals and choose the correct boundaries for non-negative regions.
- ProbabilitySuggested next · 1 core lessonChoose counting structures, update probability models and use a condition to identify the relevant sample space.
- Probability modelsSuggested next · 2 core lessonsChoose binomial and normal models, interpret their parameters and translate the event before calculating its probability.
- Sampling and inferenceSuggested next · 2 core lessonsConnect sampling design, mean uncertainty and unbiased variance estimates to planned population-mean tests and contextual conclusions.
- Correlation and regressionSuggested next · 1 core lessonUse scatter, correlation and directional least-squares fits to make contextual predictions and recognise their limits.
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H1 Mathematics 2024
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GCE A-Level H1 Mathematics 2027
Mixed practice
Practise a mix of topics from this course.
About 10 minutes
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Lessons in h1 mathematics. This course is still being developed; independent practice and assessment coverage is incomplete.
Learning goals
- Interpret a function as one output for each input
- Model and interpret exponential and logarithmic relationships
- Use logarithm laws with domain conditions
- Interpret and draw graphs of functions
- Use quadratic structure and discriminants to reason about roots
- Solve simultaneous equations by justified methods
- Solve quadratic equations by suitable methods
- Solve inequalities and represent their solution sets
- Formulate, use and interpret a mathematical model
- Interpret a derivative as gradient or rate of change
- Differentiate within the permitted function scope
- Differentiate power sums on the stated input interval
- Differentiate exponential compositions on the stated input interval
- Differentiate logarithmic compositions on the stated input interval
- Differentiate mixed function sums on the stated input interval
- Find a complete second derivative on the stated input interval
- Find the complete tangent to a polynomial at a stated input
- Find and classify every stationary point of a polynomial
- Find the greatest polynomial value and every attaining input on a closed interval
- Find the least polynomial value and every attaining input on a closed interval
- Apply the chain rule to composite functions
- Classify stationary points using justified derivative tests
- Find and interpret a tangent to a curve
- Model and solve an optimisation problem
- Find antiderivatives within the permitted scope
- Find complete power antiderivatives on the stated interval
- Find complete logarithmic antiderivatives on the stated interval
- Find complete exponential antiderivatives on the stated interval
- Find complete mixed antiderivatives on the stated interval
- Evaluate a polynomial definite integral with linked exact working
- Evaluate definite integrals
- Relate a definite integral to area under a curve
- Calculate event probabilities from an explicit sample space
- Distinguish independence from mutual exclusivity
- Calculate and interpret conditional probabilities
- Use counting principles, permutations and combinations
- Choose and use a binomial probability model
- Calculate an event probability in a specified binomial model
- Calculate a conditional probability in a specified binomial model
- Calculate the expectation of a specified binomial count
- Calculate the variance of a specified binomial count
- Recover the unique binomial parameters from exact moments
- Find the least trial count satisfying a binomial probability threshold
- Choose and use a normal probability model
- Calculate an event probability in a specified normal model
- Calculate a conditional probability in a specified normal model
- Recover a normal boundary from its specified tail probability
- Recover a normal mean from a probability and known variance
- Recover a normal variance from a probability and known mean
- Recover the unique normal mean and variance from two probabilities
- Calculate a linear combination's expectation from specified moments
- Calculate a linear combination's variance with justified dependence conditions
- Calculate a probability for a specified combination of normal variables
- Represent random variables, expectations and variances
- Reason about sampling and sampling distributions
- Calculate a population-mean estimate from a specified independent sample
- Calculate an unbiased population-variance estimate from a specified independent sample
- Calculate an estimated standard error from a specified independent sample
- Calculate a sample mean's expectation from specified population moments
- Calculate a sample mean's variance from specified population moments
- Calculate a sample mean's standard error from specified population moments
- Calculate a sample-mean probability under a specified exact or approximate normal model
- Calculate a minimum sample size for specified normal-model precision
- Formulate and interpret a justified hypothesis test
- Calculate a signed population-mean test statistic under a stipulated normal-test model
- Calculate a population-mean test p-value for a prespecified alternative
- Calculate all standardised critical boundaries for a planned population-mean test
- Calculate all sample-mean critical boundaries for a planned known-variance normal test
- Determine a population-mean test decision using the unrounded planned p-value and level
- Fit and interpret correlation and regression models
- Calculate product-moment correlation from specified paired data
- Calculate complete ordered intercept and gradient from specified paired data
- Calculate original-response model prediction from specified paired data
- Calculate signed residual on the fitted response scale from specified paired data
- Calculate sum of squared residuals on the fitted response scale from specified paired data
- Calculate original-input solution of a specified fitted calibration model from specified paired data
Syllabus statements covered
- Include: concept of function as a rule or relationship where for every input there is only one output; functions e^x and ln x and their graphs; exponential growth and decay; logarithmic growth; equivalence of y = e^x and x = ln y; laws of logarithms; use of a graphing calculator or graphing software to graph a given function; characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes (horizontal and vertical). Exclude: problems involving finding domain and range; use of notation f: x ↦ x² + 5; change of base of logarithms.
- Include: conditions for a quadratic equation to have two real roots, two equal roots or no real roots; conditions for ax² + bx + c to be always positive or always negative; solving simultaneous equations, one linear and one quadratic, by substitution; solving quadratic equations and inequalities in one unknown analytically; solving inequalities by graphical methods; formulating an equation or a system of linear equations from a problem situation; finding the approximate solution of an equation or a system of linear equations using a graphing calculator or graphing software.
- Include: derivative of f(x) as the gradient of the tangent to y = f(x) at a point; derivatives of x^n for any rational n, e^x and ln x, together with constant multiples, sums and differences; chain rule; graphical interpretation of f′(x) > 0, f′(x) = 0 and f′(x) < 0; first or second derivative tests for local maximum and minimum points and stationary points of inflexion in simple cases; locating maxima and minima and approximating a derivative at a point using a graphing calculator or graphing software; tangents to curves; local maxima and minima problems. Exclude: first-principles derivatives; products and quotients; dy/dx = 1/(dx/dy); implicit or parametric differentiation; non-stationary points of inflexion; connecting the graph of f to f′.
- Include: integration as the reverse of differentiation; integration of x^n for any rational n and e^x, with constant multiples, sums and differences; integration of (ax + b)^n for any rational n and e^(ax+b); definite integral as area under a curve; evaluation of definite integrals; areas bounded by a curve and lines parallel to coordinate axes, between a curve and a line, or between two curves; approximating a definite integral using a graphing calculator or graphing software. Exclude: definite integral as a limit of sum; trapezium rule; area below the x-axis.
- Include: addition and multiplication principles for counting; permutations nPr and combinations nCr; arrangements of distinct objects in a line with restrictions; addition and multiplication of probabilities; mutually exclusive and independent events; tables of outcomes, Venn diagrams, tree diagrams, permutations and combinations to calculate probabilities; conditional probabilities in simple cases; P(A′) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A|B) = P(A ∩ B)/P(B).
- Include: concept of discrete random variables; concept of binomial distribution B(n, p), its use as a probability model and conditions for suitability; mean and variance without proof. The concept of discrete random variables is for teaching and learning only.
- Include: concept of continuous random variables; concept of normal distribution as a continuous probability model with mean and variance; use of N(μ, σ²); standard normal distribution; finding P(X < x₁) or related probabilities given x₁, μ, σ, and finding a relationship between x₁, μ, σ given a probability; symmetry and properties of the normal curve; problems using E(aX + b) and Var(aX + b), and E(aX + bY) and Var(aX + bY) for independent X and Y. Exclude normal approximation to binomial distribution. The concept of continuous random variables is for teaching and learning only.
- Include: population and simple random sample; sample mean X̄ as a random variable with E(X̄) = μ and Var(X̄) = σ²/n; distribution of sample mean from a normal population; Central Limit Theorem for sufficiently large samples (e.g. n ≥ 30); unbiased estimates of population mean and variance from a sample, including summarised data Σx and Σx², or Σ(x − a) and Σ(x − a)².
- Include: null and alternative hypotheses H₀ and H₁, test statistic, critical region and value, significance level and p-value; formulating hypotheses and testing a population mean from a normal population of known variance or a large sample from any population; one- and two-tailed tests; contextual interpretation. Exclude the term Type I error, concept of Type II error and testing the difference between two population means.
- Include: scatter diagrams and plausibility of a linear relationship; correlation coefficient as a measure of fit; interpretation of product moment correlation near −1, 0 and 1; linear regression and least squares; interpolation and extrapolation; appropriate regression line for prediction or estimation in practical situations, including how well it models the situation. Exclude derivation of formulae, r² = b₁b₂ for regression coefficients, hypothesis tests and transformations to achieve linearity.
- Include: concept of function as a rule or relationship where for every input there is only one output; functions e^x and ln x and their graphs; exponential growth and decay; logarithmic growth; equivalence of y = e^x and x = ln y; laws of logarithms; use of a graphing calculator or graphing software to graph a given function; characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes (horizontal and vertical). Exclude: problems involving finding domain and range; use of notation f: x ↦ x² + 5; change of base of logarithms.
- Include: conditions for a quadratic equation to have two real roots, two equal roots or no real roots; conditions for ax² + bx + c to be always positive or always negative; solving simultaneous equations, one linear and one quadratic, by substitution; solving quadratic equations and inequalities in one unknown analytically; solving inequalities by graphical methods; formulating an equation or a system of linear equations from a problem situation; finding the approximate solution of an equation or a system of linear equations using a graphing calculator or graphing software.
- Include: derivative of f(x) as the gradient of the tangent to y = f(x) at a point; derivatives of x^n for any rational n, e^x and ln x, together with constant multiples, sums and differences; chain rule; graphical interpretation of f′(x) > 0, f′(x) = 0 and f′(x) < 0; first or second derivative tests for local maximum and minimum points and stationary points of inflexion in simple cases; locating maxima and minima and approximating a derivative at a point using a graphing calculator or graphing software; tangents to curves; local maxima and minima problems. Exclude: first-principles derivatives; products and quotients; dy/dx = 1/(dx/dy); implicit or parametric differentiation; non-stationary points of inflexion; connecting the graph of f to f′.
- Include: integration as the reverse of differentiation; integration of x^n for any rational n and e^x, with constant multiples, sums and differences; integration of (ax + b)^n for any rational n and e^(ax+b); definite integral as area under a curve; evaluation of definite integrals; areas bounded by a curve and lines parallel to coordinate axes, between a curve and a line, or between two curves; approximating a definite integral using a graphing calculator or graphing software. Exclude: definite integral as a limit of sum; trapezium rule; area below the x-axis.
- Include: addition and multiplication principles for counting; permutations nPr and combinations nCr; arrangements of distinct objects in a line with restrictions; addition and multiplication of probabilities; mutually exclusive and independent events; tables of outcomes, Venn diagrams, tree diagrams, permutations and combinations to calculate probabilities; conditional probabilities in simple cases; P(A′) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A|B) = P(A ∩ B)/P(B).
- Include: concept of discrete random variables; concept of binomial distribution B(n, p), its use as a probability model and conditions for suitability; mean and variance without proof. The concept of discrete random variables is for teaching and learning only.
- Include: concept of continuous random variables; concept of normal distribution as a continuous probability model with mean and variance; use of N(μ, σ²); standard normal distribution; finding P(X < x₁) or related probabilities given x₁, μ, σ, and finding a relationship between x₁, μ, σ given a probability; symmetry and properties of the normal curve; problems using E(aX + b) and Var(aX + b), and E(aX + bY) and Var(aX + bY) for independent X and Y. Exclude normal approximation to binomial distribution. The concept of continuous random variables is for teaching and learning only.
- Include: population and simple random sample; sample mean X̄ as a random variable with E(X̄) = μ and Var(X̄) = σ²/n; distribution of sample mean from a normal population; Central Limit Theorem for sufficiently large samples (e.g. n ≥ 30); unbiased estimates of population mean and variance from a sample, including summarised data Σx and Σx², or Σ(x − a) and Σ(x − a)².
- Include: null and alternative hypotheses H₀ and H₁, test statistic, critical region and value, significance level and p-value; formulating hypotheses and testing a population mean from a normal population of known variance or a large sample from any population; one- and two-tailed tests; contextual interpretation. Exclude the term Type I error, concept of Type II error and testing the difference between two population means.
- Include: scatter diagrams and plausibility of a linear relationship; correlation coefficient as a measure of fit; interpretation of product moment correlation near −1, 0 and 1; linear regression and least squares; interpolation and extrapolation; appropriate regression line for prediction or estimation in practical situations, including how well it models the situation. Exclude derivation of formulae, r² = b₁b₂ for regression coefficients, hypothesis tests and transformations to achieve linearity.