# HL analysis and approaches (AA) Homepage

## Algebra

## Series and Expansions

## Vectors

## Differentiation

## Probability

## Functions

## Trigonometry

## Statistics

## Integration

## Revision

## Number and Logs

## Complex Numbers

## Reasoning and Proof

## Differential Equations

##### Algebra

Find equations of horizontal and vertical straight lines |

Understand the the terms gradient and y-intercept. |

Find equations of diagonal straight lines. |

Factorise simple quadratic equations such as x2+bx+c |

Factorise simple quadratic equations such as ax2+bx+c where a>1. |

Solve simple quadratic equations such as x2+bx+c=0 |

Solve simple quadratic equations such as ax2+bx+c=0 where a>1. |

Solve quadratic equations by use of a GDC. |

Understand the term discriminant and how it is used to determine the number of roots of a quadratic. |

To be able to complete the square for a quadratic. |

To solve an equation from the completed the square form. |

Find minimum points and lines of symmetry from graphs of quadratics. |

Show simple numeric and algebraic proofs. |

Use of the notation for equality and identity. |

Write an algebraic fraction in linear terms. |

Solve simultaneous equations with 3 unknowns using a GDC. |

Solve simultaneous equations with 3 unknowns using row reduction for a unique solution. |

Solve simultaneous equations with 3 unknowns using row reduction, showing infinite and no solutions. |

##### Functions

Understanding the concept of a function. |

Substitute and solve equations using algebra. |

Understand the terms domain and range. |

Find the inverse of a function |

Understand the terms one-to-one; many-to-one; one-to-many and their applications to functions and inverses. |

Graph functions with and without a GDC. Note key points on graphs (intersection with axes, maximum and minimum). |

Understand and use composite functions. |

Substitute and solve equations using graphs, including composite functions. |

Using reciprocal graphs, including finding asymptotes from functions with and without sketches. |

Solve intersection of functions using a GDC. |

Transformation of functions: reflections in x and y axes; reflection in y=x line; translations; vertical and horizontal stretches. |

Using the inverse function and the reflection to solve equations. |

Odd and even functions |

Sum and roots of a quadratic equation (alpha and beta values). |

Sum and roots of all polynomial functions, where order is greater than 2. |

Inequalities of functions and their graphs. |

The reciprocal of a function and its graph. |

The modulus of a function and its graph. |

Factor theorem of polynomials. |

Remainder theorem of polynomials. |

##### Number and Logarithms

Write numbers in a x 10k (standard form). |

Calculate in standard form. |

Understand basic rules of exponents such as (multiplying, division, powers, fractional powers, negative powers, etc). |

Find exponents of integers without a calculator. |

Convert between exponent and logarithm form. |

Use a calculator to solve logarithms. |

Understand the basic rules of logarithms and relationship with exponent forms. |

Understand and use the change of base rule. |

Use the natural logarithm (ln) and interchange with its base (e). |

Graphing logarithms and use with transformations and function rules. |

Modelling with logarithms, such as population models. |

##### Series and Expansions

Understand the term arithmetic sequence. |

Understand the term geometric sequence. |

Find the nth term of an arithmetic sequence. |

Find the sum of n terms of an arithmetic series. |

Find common differences and problem solve with arithmetic sequence and series. |

Find the nth term of a geometric sequence. |

Find the sum of n terms of a geometric series. |

Find common differences and problem solve with geometric sequence and series. |

Find the infinite sum of a geometric series. |

Use the sigma sign for sums of both arithmetic and geometric series. |

Use series to apply to compound interest problems. |

Understand, use and generate Pascal’s triangle with and without a GDC. |

Expanding a bracket such as (ax+b)n, where n>2. |

Problem solving with binomial expansions, including finding terms and constants (independent) terms. |

Extending binomial expansions for fractional and negative powers. |

Counting permutations and combinations. |

##### Trigonometry

Trigonometric ratios in a right angled triangle. SOHCAHTOA. |

Sine, cosine rules in non-right angled triangles; ambiguous case of the sine rule. |

Area of a triangle ½absinC. |

Using SOHCAHTOA with 3d shapes, including pyramids, cones, spheres, and combinations of these. |

Using trigonometry in problem solving (Pythagoras’, elevation and depression). |

Radian and degree measure conversion. |

Using radian measure, length of an arc and area of a sector. |

Unit circle and Pythagorean identities, sin2x+cos2x=1, tanx=sinx/cosx. |

Exact trig ratios for 2pi, pi, pi/2, pi/3, pi/4 and pi/6 and multiples of these. |

Double angle identities for sine and cosine. |

Circular functions and their periodic nature. |

Trig graphs and the transformation of trig graphs. |

Real life trigonometric graphs and modelling. |

Solving trigonometric equations with and without a GDC. |

Quadratic equations with trigonometric equations. |

Reciprocal trigonometric ratios, secx, cosecx, and cotx, and their Pythagorean identities. |

Inverse trigonometric functions, e.g. arcsinx. |

Compound angle identities. |

Double angle identity for tan. |

Symmetry of trigonometric identities |

##### Complex Numbers

Understand the concept of the imaginary number, i, and the powers of i. |

Calculations with complex numbers in the form a+bi (cartesian form), including addition/substraction, multiplication/division. |

Finding solutions to quadratic and polynomial equations where the roots are not real. |

Drawing number in the complex plane, the argand diagram. |

Writing complex numbers in modulus-argument form. |

Representing complex numbers in Euler form. |

Converting between polar-argument and cartesian form with and without technology. |

Calculations (addition/substraction, multiplication/division) in modulus argument form. |

Using and proof of de. Moivre’s theorem and using Powers in modulus-argument form. |

##### Vectors

Concept of a vector in 2d and 3d. |

Use of vector geometry to show movement, including displacement and position vectors. |

Ading, subtracting, scalar multiples of vectors. |

The magnitude of 2d and 3d vectors. |

Scalar product of two vectors. |

Finding angles between vectors, including perpendicular and parallel vectors. |

Vector equations of lines in 3d. |

Angles between lines. |

Kinetic applications of vectors. |

Intersections of vectors lines, angles, skew and coincident. |

The vector product rule, and the normal vector. |

Vector and cartesian equation of a plane. |

Intersecting planes and link to linear algebra (3×3 solutions). |

Angles between planes and planes or lines. |

Intersection points between planes and lines. |

##### Statistics

Types of data: continuous and discrete. |

Sampling: random samples, bias in samples and outliers. |

Understand how to take a random sample, stratified data, systematic data. |

Central tendency measure: mean, mode, median, range and quartiles, including grouped data. |

Understand statistical diagrams such as histograms with even bar width. |

Reading and constructing cumulative frequency and box and whisker plots. |

Variance and standard deviation, and effect of constant changes on data. |

Correlation and regression values, Pearsons’ product moment correlation coefficient. |

Scatter diagrams, lines of best fit and predicting data from lines. |

##### Reasoning and Proof

Proof by mathematical induction. |

Use of proof by contradiction. |

Use counter-examples to show untrue statements. |

##### Differentiation

Understand the concept of a limit from a table or graph. |

Understand that the derivative is a gradient or rate of change function, and its notation. |

Understand when functions are increasing or decreasing. |

Simple derivatives by reducing the powers, e.g axn |

Find the equations of tangents and normals to functions. |

Derivatives of sinx, cosx, lnx, and ex. |

Use and understand the chain rule. |

Use and understand the product and quotient rules. |

Understand the concept of maximum and minimum points. |

Find the second derivative, and use this distinguishing between maximum and minimum points |

Finding points of inflexion. |

Using differentiation for solving optimization problems. |

Understand informally continuity and differentiability of a function at a point. |

Differentiate from first principles using the concept of a limit. |

Use differentiation for higher orders (greater than the second derivative). |

Evaluation of a limit using Maclaurin’s Series or l’Hopital’s rule, and the repeated use of l’Hopital’s rule. |

Use implicit differentiation for related rates of change and optimization problems. |

##### Integration

Integrate by using anti-differentiation techniques. |

Integration and finding the value of the constant, C. |

Using definite integration in order to find areas under curves. |

Integration techniques to find integrals of sin, cos, ln and x. |

Integration to find composite functions such as integral of cos(3x+1). |

Using reverse chain rule to integrate functions. |

Using technology to find definite integrals. |

Finding regions enclosed by two functions. |

Using integration to solve kinetic problems: displacement, velocity, acceleration. |

Derivatives of tanx, cosecx, secx, cotx, powers of x, logs, arcsinx, arccosx, arctanx. |

Use of partial fractions to integrate. |

Use the substitution rule for integration. |

Use integration by parts, including repeated integration by parts. |

Find volumes of revolution for volumes bounded by x and y axes. |

##### Differential Equations

Understand and use first order differential equations. |

Numerical solutions to differential equations using Euler’s method. |

Separating variables to solve differential equations. |

Homogeneous solutions to differential equations using substitution such as y=vx. |

Solution of y’+P(x)y=Q(x) using an integrating factor. |

Applying the MacLaurin’s series for differential equations |

##### Probability

Understand set notation for number, union, intersection, contained in, element, inverse, null and universal. |

Be able to draw and shade Venn diagrams with the use of set notation. |

Solve problems by use of sets and Venn diagrams. |

Understand the the terms mutually exclusive and independent. |

Find simple probabilities involving one or two events, using the terms AND and OR. |

Draw tree diagrams to represent probabilities. |

Use the independence formulae to solve problems of independent probability. |

Find conditional, or ‘given that’ probabilities. |

Use Venn diagrams to solve probability problems. |

Draw and interpret tree diagrams, using them for conditional and independent events. |

Discrete random variables and their probability distributions. |

Expected value (mean) and variance for discrete standard variables. |

Binomial distributions, including mean and variance of these distributions. |

Normal distribution diagrammatic representation and using this for non-GDC calculations. |

Using a GDC to find probabilities based on knowing the mean and standard deviation. |

Finding z-values with normal distributions, and using for inverse calculations. |

Continuous random variables and their density functions. |

Mean, mode, median of continuous random variables. |

Variance of continuous random variables. |

Effect of linear transformations on continuous random variables. |

Using and applying Bayes Theorem. |