# Teaching Units - content details

Content Details for Maths HL and Maths SL Teaching Units

#### Maths HL teaching units

0. Fundamentals HL
the real numbers (and subsets); set notation; sets relations & operations; inequalities & inequality properties; absolute value (modulus); properties of real numbers; roots and radicals (surds); exponents (indices); integer exponents; rational exponents; scientific notation (standard form); polynomials; expanding & factorizing polynomials; algebraic fractions; equations of lines; distance formula; midpoint formula; systems of linear equations

1. Functions – Basics HL
concept of function; domain; range; one-to-one & many-to-one functions; composite functions; identity function; inverse function (including domain restriction); function graphing skills and use of GDC; x- and y-intercepts; vertical & horizontal asymptotes; transformations of graphs; composite transformations

2. Functions, Equations & Inequalities HL
polynomial functions; odd & even functions; quadratic functions; completing the square; sum & product of the roots of a quadratic equation; quadratic formula and use of discriminant; polynomial division; synthetic division [optional]; remainder & factor theorems; sum & product of the roots of any polynomial equation; rational functions; solving equations involving radicals, absolute value and in quadratic form; solving inequalities; partial fractions [optional]

3. Sequences & Series HL
explicitly and recursively defined sequences; arithmetic sequences & series; sum of finite arithmetic sequences; geometric sequences & series; sum of finite & infinite geometric sequences; sigma notation; problems involving compound interest & other examples of exponential growth & decay

4. Counting Principles; Binomial Theorem; Induction HL
counting principles including permutations & combinations; factorial notation; expansion of binomial expressions; binomial theorem; binomial coefficients; Pascal's triangle; proof by mathematical induction

5. Exponential & Logarithmic Functions HL
exponential functions; laws of exponents; graphs of exponential functions; exponential growth & decay; compound interest; the number e; logarithmic functions; properties of logarithms; change of base; solving exponential and logarithmic equations

6. Trigonometric Functions & Equations HL
circles; radian measure of angles; arc length; area of a sector; the unit circle; sine, cosine, tangent ratios and reciprocal ratios of cosecant, secant and cotangent; exact values of trig ratios for special angles; trigonometric identities; odd & even functions; characteristics of trigonometric functions & their graphs; graphs of composite trig functions; the inverse trig functions of arcsine, arccosine & arctangent; solving trigonometric equations analytically & graphically

7. Triangle Trigonometry HL
right triangle definition of sine, cosine and tangent ratios; solution of triangles; angles of depression & elevation; area of a triangle; the sine rule (including ambiguous case); cosine rule; further applications of triangle trigonometry (including three dimensional problems)

8. Vectors HL
vectors as displacements in the plane; component form & column form of a vector; addition & subtraction of vectors; scalar multiplication of a vector; parallel vectors; magnitude of a vector; unit vectors; direction vectors; base vectors; position vectors; scalar (dot) product; perpendicular vectors; angle between vectors; vector equation of a line in 2 and 3 dimensions; Cartesian & parametric equations of a line in 3 dimensions; angle between lines; intersection of two lines; application of lines to motion; vector (cross) product; properties & geometric interpretation of vector product; vector, Cartesian & parametric equations of a plane; distance between a point and a plane; intersection and angle between line and a plane, and between two planes

9. Complex Numbers HL
the imaginary number i; characteristics of complex numbers; modulus and argument of a complex number; conjugate roots of polynomial equations; the complex plane; Cartesian form; sums, products & quotients of complex numbers; modulus-argument (polar) form; powers & roots of complex numbers; de Moivre’s theorem; exponential form for complex numbers; Euler’s formula

10. Differential Calculus I HL
limits of functions and convergence; limit definition of derivative (first principles); derivative as gradient function and rate of change; basic differentiation rules; maximum and minimum points of functions; points of inflection; first & second derivative tests; equations of tangents & normals

11. Differential Calculus II HL
problems involving displacement, velocity & acceleration (kinematics); derivatives of trigonometric, exponential & logarithmic functions; derivative of composite functions (chain rule); product & quotient rules; implicit differentiation; derivative of inverse trigonometric functions; related rates; optimization problems

12. Integral Calculus HL
indefinite integration as anti-differentiation; integration of power, reciprocal. trigonometric & exponential functions; integration by substitution; integration by parts; integration involving trigonometric substitutions; integration involving partial fractions [optional]; definite integrals; areas under curves; areas between curves; volumes of revolution about the x-axis or the y-axis; problems involving displacement, velocity & acceleration; displacement & total distance travelled

13. Statistics HL
concepts of population, sample, random sample, discrete & continuous data; frequency tables; frequency histograms; box-and-whisker plots; outliers; displaying grouped data; cumulative frequency tables & graphs; mean; median; mode; percentiles; quartiles; range; interquartile range; variance; standard deviation

14. Probability HL
concepts of trial, outcome, equally likely outcomes, sample space & event; definition of probability of event; complementary events; independent events; Venn diagrams; tree diagrams; probability rules for: combined events, mutually exclusive events & independent events; probabilities with and without replacement; conditional probability; Bayes’ theorem

15. Probability Distributions HL
concept of a random variable; discrete random variables and their probability distributions; continuous random variables and their probability distributions; expected value (mean), variance & standard deviation for discrete & continuous data; properties of expected value; binomial distribution; mean, variance & standard deviation for binomial distribution; Poisson distribution; mean, variance & standard deviation for Poisson distribution; continuous random variables & probability density functions; normal distribution; standardization of normal variables; inverse normal distribution

#### Maths SL teaching units

0. Fundamentals SL
the real numbers (and subsets); sets & intervals; absolute value (modulus); properties of real numbers; roots and radicals (surds); exponents (indices); integer exponents; rational exponents; scientific notation (standard form); polynomials; expanding & factorizing polynomials; algebraic fractions; equations of lines; distance formula; midpoint formula; systems of linear equations

1. Functions & Equations SL
concept of function; domain; range; composite functions; identity function; inverse function; function graphing skills and use of GDC; x- and y-intercepts; vertical & horizontal asymptotes; transformations of graphs; composite transformations; quadratic functions; reciprocal function; rational functions; graphical & analytical solutions of equations; quadratic formula and use of discriminant

2. Sequences & Series; Binomial Theorem SL
arithmetic sequences & series; geometric sequences & series; sigma notation; compound interest; expansion of binomial expressions; binomial theorem; binomial coefficients; Pascal's triangle

3. Exponential & Logarithmic Functions SL
exponential functions; laws of exponents; graphs of exponential functions; exponential growth & decay; compound interest; the number e; logarithmic functions; properties of logarithms; change of base; solving exponential and logarithmic equations

4. Trigonometric Functions & Equations SL
circles; radian measure of angles; arc length; area of a sector; the unit circle; sine, cosine and tangent ratios; exact values of trig ratios for special angles; trigonometric identities; characteristics of trigonometric functions & their graphs; graphs of composite trig functions; solving trigonometric equations

5. Triangle Trigonometry SL
right triangle definition of sine, cosine and tangent ratios; solution of triangles; angles of depression & elevation; area of a triangle; the sine rule (including ambiguous case); cosine rule; further applications of triangle trigonometry (including three dimensional problems)

6. Vectors SL
vectors as displacements in the plane; component form & column form of a vector; addition & subtraction of vectors; scalar multiplication of a vector; parallel vectors; magnitude of a vector; unit vectors; direction vectors; base vectors; position vectors; scalar (dot) product of two vectors; perpendicular vectors; angle between two vectors; vector equation of a line in two and three dimensions; angle between two lines; point of intersection of two lines; application of lines to motion

7. Differential Calculus I SL
limits of functions and convergence; limit definition of derivative (first principles); derivative as gradient function and rate of change; basic differentiation rules; maximum and minimum points of functions; points of inflection; first & second derivative tests; equations of tangents & normals

8. Differential Calculus II SL
problems involving displacement, velocity & acceleration (kinematics); derivatives of trigonometric, exponential & logarithmic functions; derivative of composite functions (chain rule); product & quotient rules; optimization problems

9. Integral Calculus SL
indefinite integration as anti-differentiation; integration of power, reciprocal. trigonometric & exponential functions; integration by inspection or substitution; definite integrals; areas under curves; areas between curves; volumes of revolution about the x-axis; problems involving displacement, velocity & acceleration; displacement & total distance travelled

10. Statistics SL
concepts of population, sample, random sample, discrete & continuous data; frequency tables; frequency histograms; box-and-whisker plots; outliers; displaying grouped data; cumulative frequency tables & graphs; mean; median; mode; percentiles; quartiles; range; interquartile range; variance; standard deviation; linear correlation of bivariate data; Pearson's product-moment correlation coefficient r; scatter diagrams; line of best fit; equation of regression line of y on x

11. Probability SL
concepts of trial, outcome, equally likely outcomes, sample space & event; definition of probability of event; complementary events; independent events; Venn diagrams; tree diagrams; probability rules for: combined events, mutually exclusive events & independent events; probabilities with and without replacement; conditional probability

12. Probability Distributions SL
concept of a random variable; discrete random variables and their probability distributions; expected value (mean), variance & standard deviation for discrete data; properties of expected value; binomial distribution; mean, variance & standard deviation of binomial distribution; continuous random variables & probability density functions; normal distribution; standardization of normal variables; inverse normal distribution

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