# HL applications and interpretations (AI) Homepage

## Number and Algebra

## Number and Logs

## Vectors

## Hypothesis

## Differential Equations

## Probability

## Functions

## Series

## Complex Numbers

## Differentiation

## Matrices

## Revision

## Voronoi Diagrams

## Trigonometry

## Statistics

## Integration

## Graph Theory

##### Complete Learning Outcomes

##### Algebra

Find equations of horizontal and vertical straight lines |

Understand the the terms gradient and y-intercept. |

Find equations of diagonal straight lines. |

Use straight lines to solve 2×2 simultaneous equations. |

Use a GDC to solve 2×2 simultaneous equations. |

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 quadratic graph, including roots, lines of symmetry and minimum point. |

##### 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, including the reflection of a function and its inverse. |

Graph functions with a GDC. Note key points on graphs (intersection with axes, maximum and minimum). |

Graphing linear, quadratics and cubic functions. |

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

Solve intersection of functions using a GDC. |

Apply knowledge of functions to create and understand models. |

Understand direct and inverse proportion. |

Recognise appropriate models and their parameters. |

Understand piecewise functions. |

Understand and use composite functions. |

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

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

##### Voronoi Diagrams

Finding the coordinate of a bisector of a line segment, or between two points. |

Finding perpendicular line equations. |

Understanding language of Voronoi diagrams: sites, vertices, edges, cells. |

Nearest neighbour interpolation. |

Solving Voronoi problems in context. |

##### 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. |

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

Modelling with logarithms, such as population models. |

Understanding and calculating simple and compound interest. |

Using a GDC and other technology to find amortization and annuities. |

Approximation to decimal places and significant figures. |

Understanding and finding upper and lower bounds of rounded number. |

Finding percentage error of rounded number. |

Estimation of number. |

Convert between exponent and logarithm form. |

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). |

Using logarithm models with large and small numbers. |

Interpretation of log-log and semi-log graphs. |

##### Series

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 problem solving in modelling. |

##### 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. |

Circular functions and their periodic nature. |

Trig graphs and the transformation of trig graphs. |

Real life trigonometric graphs and modelling. |

##### 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 equations where the roots are not real. |

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

Writing complex numbers in modulus-argument form. |

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

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

Powers in modulus-argument form. |

Understand sinusoidal functions with the same frequencies, but different shift-phases. |

##### 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. |

Vector applications to motion in 2d and 3d, including kinetics. |

Geometric interpretation of vectors, such as magnitude of the cross product. |

Components and direction of vectors. |

##### 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. |

Biased and unbiased estimators |

##### Hypothesis Testing

Calculate Spearman’s Rank coefficient and understand limitations. |

Analyse outliers and use of Spearman’s rank and Pearson’s correlation on data. |

Understand the null and alternative hypothesis for testing, the p-value, and significance levels. |

Conduct and analyse a Chi Squared test for goodness of fit. |

Conduct and analyse a Chi Squared test for independence. |

Conduct a t-test to analyse the means of two populations, using one and two tailed data. |

Design appropriate data collection techniques. |

Choosing appropriate data to analyse and fitting to relevant data types. |

Understanding reliability, testing and results with statistics |

Analysing non-linear regression, and evaluation of least squares regression on data. |

Sum of square residuals to fit a model. |

Using and finding the coefficient of determination (R2). |

##### 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 and related rates of change. |

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 |

Understand the terms concave up and concave down. |

Using differentiation for solving 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. |

Using technology to find definite integrals. |

Using the trapezoid rule to estimate areas. |

Areas enclosed between the x and y axes. |

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

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. |

Finding volumes of revolution about the 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. |

Numerical solutions with coupled systems. |

Phase portrait for the solutions of coupled differential equations. |

Qualitative analysis of future paths for distinct, real and imaginary eigenvalues. |

Sketching trajectories and using phase portraits to identify special points. |

Solutions of the second differential by use of Euler’s method. |

##### 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) 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. |

Poisson distributions, mean, variance, and sums of poisson distributions. |

Confidence intervals for normal distributions with and without a known standard deviation. |

Test for critical values and regions with normal distributions. |

Calculating Type I and Type II errors, includes link with Statistics module. |

##### Matrices

Understand definition and order of matrices. |

Simple operations with matrice: addition, subtraction, scalar multiplying. |

Multiplication of matrices. |

Identity and zero matrices and determinants of matrices using technology and by hand. |

Solving simultaneous equations with matrices and using inverse matrices for solutions. |

Understanding eigenvalues and eigenvectors, including diagonalization. |

Powers of 2×2 matrices. |

Using matrices for transformations. |

Determinant of a transformation matrix. |

Transition matrices and powers of transition matrices. |

Markov chains and initial state probability matrices. |

Steady and long term probabilities by continued multiplications. |

##### Graph Theory

Definitions of graphs: vertices, edges, degree, adjacency. |

Draw and understand simple graphs, weighted graphs and connected graphs. |

Draw and understand trees, subgraphs, directed graphs, degree in and degree out graphs. |

Understand walks, trials, paths and circuits. |

Understand Eulerian trails and circuits. |

Understand Hamiltonian paths and cycles. |

Use and understand the minimum spanning tree, including Kruskal’s and Prim’s algorithms. |

Solve Chinese Postman Problems (Eulerian). |

Using the nearest neighbour algorithm. |

Using vertex deletion to find lower bounds for the travelling salesman problem (Hamiltonian). |

Adjacency matrices for walks and graphs. |

Using weighted adjacency tables for construction of transition matrices. |