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Mathematics · Topics commonly taught around grade 3 · Fractions
Fractions, level 3
What a fraction names, fractions as numbers on a line, equivalent fractions, comparing fractions, whole numbers as fractions, the words numerator and denominator, fractions as decimals, and adding fractions with the same denominator.
Teaches: Add fractions with the same denominator, Compare fractions, Fractions as decimals, Equal fractions
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Mathematics · Topics commonly taught around grade 4
Mathematics level 4
Multiplicative comparison, factors and primes, numbers to a million, multi-digit multiplication and division with remainders, equivalent fractions and operations, decimals to hundredths, unit conversion, angles, lines and symmetry.
Teaches: Add and subtract like fractions, Compare any fractions, Equivalent fractions
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Mathematics · Topics commonly taught around grade 6
Mathematics level 6
Ratios, rates and percent, dividing fractions, negative numbers and the full coordinate plane, exponents and algebraic expressions, one-step equations and inequalities, area and volume, and statistics.
Teaches: Divide fractions
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Mathematics · Calculus BC
Calculus BC
Integration by parts and partial fractions; improper integrals at an infinite limit and at a singular endpoint; parametric and polar curves with their slopes, lengths and areas; the convergence tests and the hypotheses each one needs; power and Taylor series with the radius and the interval they converge on; the alternating and Lagrange error bounds; and Euler's method with the logistic model.
Teaches: Parts and partial fractions
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Mathematics · Advanced · Level 1
Calculus II
Integration by parts, reduction formulas, trigonometric integrals and substitutions, completing the square and partial fractions, with a strategy for choosing between them; area, volume by slicing and by shells, arc length, surface area, work and average value; improper integrals at an infinite limit and at a singular endpoint, with comparison; sequences, partial sums, and the divergence, integral, comparison, ratio, root and alternating series tests, with absolute and conditional convergence; power series, term-by-term calculus, Taylor series and the Lagrange remainder; and parametric and polar curves, with their slopes, lengths and areas.
Teaches: Partial fractions
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Mathematics · Advanced · Level 2
Differential equations
First-order equations — reading a slope field, separation, integrating factors, exact equations, existence and uniqueness, and the models they answer; linear second-order equations from the characteristic roots through the Wronskian, undetermined coefficients, variation of parameters and boundary value problems; the Laplace transform, what it does to an initial value problem, and how a switched forcing is handled; autonomous equations on a phase line, systems in matrix form and the classification of a phase portrait; series solutions at ordinary and at regular singular points; and the numerical methods a modeller reaches for, with the error and stability facts that decide the step size.
Teaches: Partial fractions and the inverse transform
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Mathematics · Advanced · Level 3
Number theory
Divisibility and the division algorithm, Euclid's algorithm and Bézout's identity, unique factorisation and the divisor functions; congruences, inverses, linear congruences and the Chinese remainder theorem, with the divisibility tests they explain; Fermat's little theorem, Euler's totient and theorem, Wilson's theorem, and the arithmetic public-key cryptography rests on; orders and primitive roots, quadratic residues, the Legendre symbol, Euler's criterion and quadratic reciprocity; linear Diophantine equations, Pythagorean triples, sums of two squares, continued fractions and Pell's equation; and the distribution of primes, from Euclid's proof and the sieve to the prime number theorem and Dirichlet's theorem on arithmetic progressions.
Teaches: Continued fractions and convergents