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Math AA SL

5 topics51 outcomesLocal syllabus version: 2021

Number and Algebra

9 outcomes

Open
  1. SL 1.1Operations with numbers in the form a × 10^k where 1 ≤ a < 10 and k is an integer (scientific notation); calculator notation is not acceptable
  2. SL 1.2Arithmetic sequences and series: formulae for the nth term and the sum of the first n terms; use of sigma notation; applications
  3. SL 1.3Geometric sequences and series: formulae for the nth term and the sum of the first n terms; use of sigma notation; applications
  4. SL 1.4Financial applications of geometric sequences and series: compound interest and annual depreciation; calculating the real value of an investment with interest and inflation
  5. SL 1.5Laws of exponents with integer exponents; introduction to logarithms with base 10 and e; awareness that a^x = b is equivalent to log_a(b) = x; numerical evaluation of logarithms using technology
  6. SL 1.6Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof; symbols and notation for equality and identity
  7. SL 1.7Laws of exponents with rational exponents; laws of logarithms (product, quotient, power rules); change of base of a logarithm; solving exponential equations including using logarithms
  8. SL 1.8Sum of infinite convergent geometric sequences; use of |r| < 1 and modulus notation
  9. SL 1.9The binomial theorem: expansion of (a + b)^n for n ∈ ℕ; use of Pascal's triangle and nCr; finding specific terms

Functions

11 outcomes

Open
  1. SL 2.1Different forms of the equation of a straight line: gradient-intercept (y = mx + c), general (ax + by + d = 0), point-gradient form; gradient; intercepts; parallel lines (m1 = m2) and perpendicular lines (m1 × m2 = −1)
  2. SL 2.2Concept of a function, domain, range and graph; function notation f(x), v(t), C(n); the concept of a function as a mathematical model; informal concept of inverse function; inverse notation f⁻¹(x); reflection in y = x
  3. SL 2.3The graph of a function y = f(x); creating a sketch from information given or a context; using technology to graph functions including sums and differences; all axes and key features labelled
  4. SL 2.4Key features of graphs: maximum and minimum values, intercepts, symmetry, vertex, zeros/roots, vertical and horizontal asymptotes using technology; finding the point of intersection of two curves or lines
  5. SL 2.5Composite functions (f ∘ g)(x) = f(g(x)); identity function; finding the inverse function f⁻¹(x); existence of an inverse for one-to-one functions
  6. SL 2.6The quadratic function f(x) = ax² + bx + c: graph, y-intercept, axis of symmetry; factored form f(x) = a(x−p)(x−q) with x-intercepts; vertex form f(x) = a(x−h)² + k with vertex (h, k)
  7. SL 2.7Solution of quadratic equations and inequalities using factorization, completing the square, and the quadratic formula; the discriminant Δ = b² − 4ac and the nature of the roots (two distinct real, two equal real, no real roots)
  8. SL 2.8The reciprocal function f(x) = 1/x and its self-inverse nature; rational functions f(x) = (ax + b)/(cx + d) and their graphs; equations of vertical and horizontal asymptotes
  9. SL 2.9Exponential functions f(x) = a^x, f(x) = e^x and their graphs; logarithmic functions f(x) = log_a x, f(x) = ln x and their graphs; exponential and logarithmic functions as inverses of each other
  10. SL 2.10Solving equations both graphically and analytically; use of technology to solve equations where there is no appropriate analytic approach; applications to real-life situations
  11. SL 2.11Transformations of graphs: translations y = f(x) + b and y = f(x − a); reflections in both axes y = −f(x) and y = f(−x); vertical stretch y = p f(x); horizontal stretch y = f(qx); composite transformations

Geometry and Trigonometry

8 outcomes

Open
  1. SL 3.1Distance between two points in three-dimensional space and their midpoint; volume and surface area of 3D solids including right pyramid, right cone, sphere, hemisphere and combinations; angle between two intersecting lines or between a line and a plane
  2. SL 3.2Use of sine, cosine and tangent ratios in right-angled triangles; the sine rule a/sinA = b/sinB = c/sinC; the cosine rule c² = a² + b² − 2ab cosC; area of a triangle as (1/2)ab sinC
  3. SL 3.3Applications of right and non-right-angled trigonometry including Pythagoras's theorem; angles of elevation and depression; construction of labelled diagrams from written statements; contexts may include bearings
  4. SL 3.4The circle: radian measure of angles; length of an arc; area of a sector
  5. SL 3.5Definition of cosθ and sinθ in terms of the unit circle; definition of tanθ as sinθ/cosθ; exact values of trig ratios of 0, π/6, π/4, π/3, π/2 and their multiples; extension of the sine rule to the ambiguous case
  6. SL 3.6The Pythagorean identity cos²θ + sin²θ = 1; double angle identities for sine and cosine; relationship between trigonometric ratios
  7. SL 3.7The circular functions sinx, cosx and tanx: amplitude, periodic nature and graphs; composite functions of the form f(x) = a sin(b(x + c)) + d; transformations; real-life contexts such as tide height or Ferris wheel motion
  8. SL 3.8Solving trigonometric equations in a finite interval both graphically and analytically; equations leading to quadratic equations in sinx, cosx or tanx

Statistics and Probability

12 outcomes

Open
  1. SL 4.1Concepts of population, sample, random sample, discrete and continuous data; reliability of data sources and bias in sampling; interpretation of outliers (data more than 1.5 × IQR from nearest quartile); sampling techniques: simple random, convenience, systematic, quota, stratified
  2. SL 4.2Presentation of data: frequency distributions and tables; histograms; cumulative frequency graphs; use of cumulative frequency to find median, quartiles, percentiles, range and IQR; box-and-whisker diagrams; comparing two distributions
  3. SL 4.3Measures of central tendency: mean, median and mode; estimation of mean from grouped data; modal class; measures of dispersion: IQR, standard deviation and variance; effect of constant changes on original data
  4. SL 4.4Pearson's product-moment correlation coefficient r; linear correlation of bivariate data; scatter diagrams; positive, zero, negative; strong, weak, no correlation; equation of the regression line of y on x; use for prediction; distinguish correlation from causation
  5. SL 4.5Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space and event; probability of an event P(A) = n(A)/n(U); complementary events A and A′; expected number of occurrences
  6. SL 4.6Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes; combined events P(A ∪ B) = P(A) + P(B) − P(A ∩ B); mutually exclusive events; conditional probability P(A|B) = P(A ∩ B)/P(B); independent events P(A ∩ B) = P(A)P(B)
  7. SL 4.7Concept of discrete random variables and their probability distributions; expected value (mean) E(X) for discrete data; applications including fair games
  8. SL 4.8Binomial distribution B(n, p): situations where it is an appropriate model; mean and variance of the binomial distribution; probabilities found using technology
  9. SL 4.9The normal distribution and curve; properties of the normal distribution; diagrammatic representation; normal probability calculations using technology; inverse normal calculations
  10. SL 4.10Equation of the regression line of x on y; use for prediction; awareness that a y-on-x line cannot always be used to predict x from a given y value
  11. SL 4.11Formal definition and use of the conditional probability formula P(A|B) = P(A ∩ B)/P(B); testing for independence: P(A|B) = P(A) = P(A|B′) for independent events
  12. SL 4.12Standardization of normal variables (z-values): z = (x − μ)/σ; the standardized value gives the number of standard deviations from the mean; inverse normal calculations where mean and standard deviation are unknown using z-values

Calculus

11 outcomes

Open
  1. SL 5.1Introduction to the concept of a limit; derivative interpreted as gradient function and as rate of change; informal understanding of the gradient of a curve as a limit; derivative notation dy/dx, f′(x)
  2. SL 5.2Increasing and decreasing functions; graphical interpretation of f′(x) > 0 (increasing), f′(x) = 0 (stationary), f′(x) < 0 (decreasing)
  3. SL 5.3Derivative of f(x) = ax^n is f′(x) = anx^{n−1} for n ∈ ℤ; derivative of functions of the form f(x) = ax^n + bx^{n−1} + … with integer exponents
  4. SL 5.4Tangents and normals at a given point and their equations; use of both analytic approaches and technology
  5. SL 5.5Introduction to integration as anti-differentiation of f(x) = ax^n + bx^{n−1} + … for n ∈ ℤ, n ≠ −1; anti-differentiation with boundary condition to find the constant term; definite integrals using technology; area under a curve y = f(x) where f(x) > 0
  6. SL 5.6Derivative of x^n (n ∈ ℚ), sinx, cosx, e^x and lnx; differentiation of a sum and a multiple of these functions; the chain rule for composite functions; the product and quotient rules
  7. SL 5.7The second derivative f″(x) / d²y/dx²; graphical behaviour of functions including the relationship between the graphs of f, f′ and f″
  8. SL 5.8Local maximum and minimum points; testing using change of sign of f′ or sign of f″; optimization; points of inflexion with zero and non-zero gradients; concave-up (f″ > 0) and concave-down (f″ < 0)
  9. SL 5.9Kinematic problems involving displacement s, velocity v = ds/dt, acceleration a = dv/dt = d²s/dt²; total distance travelled; displacement and distance as definite integrals of velocity
  10. SL 5.10Indefinite integrals of x^n (n ∈ ℚ), sinx, cosx, 1/x and e^x; composites of any of these with linear function ax + b; integration by inspection (reverse chain rule) or by substitution for ∫k g′(x) f(g(x)) dx
  11. SL 5.11Definite integrals using analytical approach ∫_a^b g′(x) dx = g(b) − g(a); areas of a region enclosed by y = f(x) and the x-axis where f(x) can be positive or negative; areas between curves