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The toolkit and the mathematical exploration

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The toolkit and the mathematical exploration

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Investigative, problem-solving and modelling skills development leading to an individual exploration. The exploration is a piece of written work that involves investigating an area of mathematics.

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Number and Algebra
Log laws
Simplifying expressions rational exponents
Sum of Geometric Series
Complex numbers: sums, differences, products, quotients, by hand and
with technology. Powers of complex numbers, in Cartesian form, with technology. Complex Plane. Complex numbers as solutions to quadratic equations with real coefficients where b2−4ac<0.Cartesian form Polar form , Exponential form complex number and converting to different forms by hand and tech Calculate products, quotients and integer powers in polar or exponential forms
Adding sinusoidal functions with the same frequencies but different phase shift angles. Geometric interpretation of complex numbers
Matrices: Algebra, Multiplication, Identity, Zero, systems of linear equations, Solutions using Inverse matrix, Eigenvectors Eigenvalues, application to powers of 2X2
Functions
Composite, Inverse, transformations, composite transformations
Exponential models. Natural logarithmic models, Sinusoidal models, Logistic, Piecewise
Scaling very large or small numbers using logarithms.

Linearizing data using logarithms
Geometry and Trig
Radians, arc length, area of sector .
Definitions of sine x cosine x from a unit circle
Pythagorean identity
Definition of tanx = sinx/cosx .
Ambiguous case of Sine Rule
Geometric transformations of points in two dimensions using matrices
Geometric interpretation of the determinant of a transformation matrix.
Vectors, base unit vectors, column representation, position vectors. Rescaling and normalizing vectors.
Vector equation of a line in two and three dimensions
Vector applications to kinematics. Modelling linear motion with constant velocity in two and three dimensions. Motion with variable velocity in two dimensions.
Scalar product. Vector product.
Graph theory, simple directed subgraphs
Adjacency matrices. Walks. Number of k-length walks (or less thank-length walks) between two vertices.
Weighted adjacency tables. Construction of the transition matrix for a strongly-connected, undirected or directed graph.
Tree and cycle algorithms with undirected graphs
Eulerian trails and circuits. Hamiltonian paths and cycles. Minimum spanning tree (MST) graph algorithms: Kruskal’s and Prim’s algorithms for finding minimum spanning trees.
Chinese postman problem and algorithm
Travelling salesman problem to determine the Hamiltonian cycle
Nearest neighbour algorithm for determining an upper bound for the travelling salesman problem. Deleted vertex algorithm for determining a lower bound for the travelling salesman problem.
Statistics & Probability
Design of valid data collection methods. Selecting relevant variables from many variables. Choosing relevant and appropriate data to analyse.
Categorizing numerical data in a χ2 table and justifying the choice of categorisation. Choosing an appropriate number of degrees of freedom when estimating parameters from data when carrying out the χ2 goodness of fit test.
Definition of reliability and validity. Reliability tests. Validity tests.
Non linear regression

Least squares regression curves using technology

Sum of square residuals (SSres) as a measure of fit for a model
The coefficient of determination (R2) evaluating using tech
Linear transformation of a single random variable
Expected value and variance of linear combinations of n random variables
x bar as an unbiased estimate of μ. S2n−1 as an unbiased estimate of σ2.
A linear combination of n independent normal random variables is normally distributed
Confidence intervals for the mean of a normal population

Poisson distribution and sum of two independent Poisson distributions
Critical values and critical regions. Test for population mean for normal distribution. Test for proportion using binomial distribution. Test for population mean using Poisson distribution
Use of technology to test the hypothesis that the population PMCC (ρ) is 0 for bivariate normal distributions. Type I and II errors
Transition matrices. Powers of transition matrices, Markov chains,  Initial state probability matrices
steady state and long-term probabilities
Calculus
The derivatives of sin x, cos x, tan x, ex, ln x, xn where n∈ℚ. The chain rule, product rule and quotient rules. Related rates of change
Use of second derivative test to distinguish between a maximum and a minimum point.
Definite and indefinite integration of xn where n∈ℚ, including n= -1 , sin x, cos x,1/(cosx)2 and ex.
Integration by inspection, or substitution
Area of the region enclosed by a curve and the x or y-axes in a given interval.
Volumes of revolution about the x-ax is or y-axis
Kinematic problems involving displacement s, velocity v and acceleration a.
Setting up a model/differential equation from a context.
Solving by separation of variables
Slope fields and their diagrams
Euler’s method for finding the approximate solution to first order differential equations. Numerical solution of dydx=f(x,y).
Numerical solution of the coupled system
Phase portrait for the solutions of coupled differential equations of the form  dxdt=ax+by  dydt=cx+dy.

Qualitative analysis of future paths for distinct, real, complex and imaginary eigenvalues. Sketching trajectories and using phase portraits to identify key features such as equilibrium points, stable populations and saddle points

Solutions of d2xdt2=f(x,dxdt,t) by Euler’s method

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Number and Algebra
Permutations & combinations

Binomial theorem with negative and fractional indices

Partial Fractions

Complex numbers, the complex plane, polar and Euler form

Complex conjugate roots of polynomial equations

DeMoivre’s theorem

Power and roots of complex numbers

Proof by Induction, contraction, and counterexample

Systems of equations

Functions
Polynomial functions, including factor and remainder theorem

Sum and product of the roots of polynomial equations

Rational functions in the form  and

Odd and even functions

Inverse functions, including domain restrictions and self-inverses

Solving  both graphically and analytically

The graphs of

Solutions of absolute value equations and inequalities
Geometry & Trig
Reciprocal trig ratios

Pythagorean identities, double angle identities (including tan), compound angle identities

Inverse trig functions

Symmetry properties of trig graphs

Vectors

-       including definition, forms, operations, position, scalar product, vector product, angle between two vectors, perpendicular and parallel, application to kinematics

-       Vectors equation of a line in 2D and 3D

-      Scalar product:  Angle between two lines Vector Product

-       Coincident, parallel, intersecting and skew lines

-       Points of intersection

-       Vector & Cartesian equations of a plane

-       Normals

-       Intersections of a line with a plane, two planes, three planes

-       Angles between line and a place, two planes
Statistics and Probability
Bayes Theorem

Variance of discrete random variable

Mode and median of continuous random variables (using integration)

Mean, var, SD of both discrete and continuous random variables

The effect of linear transformations of X

Calculus
Informal understanding of continuity and differentiability

Understanding of limits (convergence and divergence)

Newton’s Quotient

Higher order derivatives

Limits in the form  using L’Hopital’s rule

Implicit differentiation

Related rates of change

Derivatives of tanx, secx, cscx, cotx arcsinx, arccosx, arctanx, a^x,
logb x
Indefinite integrals of tanx, secx, cscx,arcsinx, arccosx, arctanx

Partial fractions to rearrange the integrand

Integration by parts and repeated, Integration by substitution,

Area between  curve and the y-axis

Volumes of revolution about both the x and y axis

First order differential equations, including numerical solution using Euler’s method.

Separable DEs

Homogenous Des

Solving of DEs using the integrating factor

Maclaurin Series to obtain expansions of e^x, sinx, cosx, ln (1+x) (1+x)^p

Maclaurin series developed from DEs

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NUMBERS AND ALGEBRA

Approximation, decimal places, SFs

Upper and lower bounds of rounded numbers

Percentage errors

Estimation

Amortization and annuities using tech

Solving systems of linear equation (max 3 variables) using tech

Solving polynomial equations using tech

FUNCTIONS

Models

- Linear

- Quadratic

- Exponential growth and decay

- Direct and inverse variation

- Cubic

- Sinusoidal (with no phase shift)

Modelling skills

- Create, fit and use theoretical models

- Find the parameters of a model

- Test and reflection upon a model

- Justify choice of a model

- Use the model to interpret and make predictions

GEOMETRY AND TRIGONOMETRY

Arc length and area of sectors

Perpendicular bisectors

Voronoi diagrams

STATISTICS & PROBABILITY

Spearman’s rank correlation coefficient

Appropriateness and limitations to Pearson’s coefficient

The effect of outlier on both Spearman and Pearson’s coefficient

Formation of null and alternative hypotheses, p-values

Expected and observed frequencies

Chi squared  test

T-test

CALCULUS

Local extrema and critical points

Optimisation

Trapezoidal rule

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Number & Algebra

Simple deductive proofs

Exponent laws with rational exponents

Logarithm laws

Exponential equations (incl. logs)

Sum of infinite geometric series

Binomial theorem (excluding negative and fractional indices)

Use of Pascal’s Triangle and nCr

Functions

Composite function

Identity functions

Quadratic functions including discriminant

Solutions quadratic equations and inequalities

Reciprocal functions & Rational functions in the form of y=linear/linear

Exponential & logarithmic functions and their graphs

Solving equations both graphically and analytically

Transformations of graphs (translations, reflections, stretches) - composite

Geometry & Trig

Radian measure, length of arc, area of sector

Definition of sine, cosine, tangent and the unit circle

Sine law – ambiguous case

The Identities (Pythagorean and double angle identities)

Sinusoidal functions, including transformation and real-life problems

Solving trigonometric equations in a finite interval both graphically and analytically, including quadratic equations

Probability & Statistics

Standardisation of normal variables (z-scores) & inverse normal to calculate mean and standard deviation

Linear regression of x on y (interpolation & extrapolation)

Conditional probability and Independent events

Calculus

Derivatives of x n (where n can be rational) sine, cosine, natural exponential and logarithms

Chain, product and quotient

Second derivatives and graphical behaviour of f(x) and the first and second derivative

Local max and min points, and testing for both

Optimization

Points of infection - both stationary & non-stationary

Kinematics

Indefinite integrals of x n (where n is rational), sine, cosine, natural exponential and reciprocal as well as composites of any of the above with linear functions

Integration by inspection or substitution

Definite integrals (including analytically)

Area between y = f(x) and the x-axis, with and without technology

Area between curves

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Mathematics HL Applications and Interpretation

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Mathematics HL Applications and Interpretation

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Mathematics SL Applications and Interpretation

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Mathematics SL Applications and Interpretation

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NUMBERS AND ALGEBRA

Operations with numbers in the form 𝑎𝑥104

Arithmetic sequences and series

Geometric sequence and series, including financial applications (excluding

sum of infinite series)

Exponent laws

FUNCTIONS

Linear functions including parallel and perpendicular

Domain, range, and functional notation

Inverse functions (as a reflection in 𝑦 = 𝑥)

The graph of a function (sketching and using tech to find sums and

differences)

Key features of graphs

Points of intersection using tech

GEOMETRY AND TRIGONOMETRY

Distance between, and midpoint of, two points

Volume and surface area of 3D solids

Angles between intersecting lines or a line and a plane

Trig ratios (sin, cos, tan)

Sine law (excluding ambiguous case) Cosine law

Area of triangle - 𝐴 = 1/2 𝑎𝑏𝑠𝑖𝑛𝐶

Applications of right and non-right angled trig (including angles of elevation

and depression)

STATISTICS & PROBABILITY

Concepts of population, sample, random sample, discrete and continuous data

Reliability of data sources and bias in sampling
Outliers

Sampling techniques and their effectiveness

Frequency tables & histograms

Cumulative frequency graphs to find median, quartiles, percentiles, range, and IQR

Box and whisker plots

Measures of central tendency (mean, median, mode)

Measures of dispersion (IQR, SD, Var, quartiles)

Effect of constant changes on the original data

Linear correlation & Pearson’s Coefficient

Scatterplots & line of best fit

Linear regression equations (𝑦 on 𝑥 only)

Concepts of trial, outcome, sample space…

Probability formula

Complimentary events

Expected value

Venn diagrams, tree diagrams,…

𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)

Mutually exclusive events

Conditional probability, 𝑃(𝐴|𝐵) = H(I∩J/)H(J) only

Independent events

Probability distributions & expected value of discrete random variables

Binomial distribution, including mean and variance

Normal distribution, including inverse normal (excluding Z-scores)

CALCULUS

Concept of limit

Derivatives as gradients or rates of change

Increasing and decreasing functions, including graphical interpretation of

𝑓L(𝑥) > 0, 𝑓L(𝑥) = 0, 𝑓L(𝑥) < 0

Sum/difference & power rule for polynomial derivatives where all exponents are integers

Tangents and normal

Integration including finding C

Definite integrals using tech

Area between 𝑦 = 𝑓(𝑥) > 0 and the x-axis