a-level-further-mathematics-roadmap-with-fs1_r.pptx

Roadmap

1. Holiday Holiday Holiday Holiday Holiday Holiday Holiday Algebra and Functions Series Further algebra and functions 1 Proof - induction Algorithms on Graphs I part one and Graph Theory Algorithms on Graphs II part one Critical Path Analysis Complex Numbers Matrices 1 – arithmetic and transformations Discrete Random Variables Poisson Distribution 1 Year 1 A Level Further Mathematics Roadmap Pure Decision Statistics Non-AS Pure Algorithms Linear Programming 1 Algebra and Functions Algorithms on Graphs I Year 2 Volumes of Revolution Vectors 1 – lines, angles, intro to planes Further Vectors ( planes,angles etc ) Matrices 2 - inverses Matrices 2 – sim eqns Algorithms and Graph Theory 2 Proof Algorithms on Graphs 1 part 2 NON AS Internal Exams Work Experience Vectors 1 Geometric Distribution Further Complex Numbers 1 – exponential Expansions, DMT and roots of unity Introduction to polar coordinates Further Vectors Matrices 2 - inverses Internal Exams Poisson Distribution 1 Poisson Distribution 2 Negative Binomial Distribution Hypothesis Testing

2. Holiday Holiday Holiday Holiday Holiday Holiday Holiday Polar Coordinates Not areas Year 2 A Level Further Mathematics Roadmap Pure Decision Statistics Further Calculus Differential Equations Differential Equations Algorithms on Graphs 2 part 2 Quality of Tests Linear Programming 2- Simplex Algorithm Year 1 Critical Path Analysis 2 Probability Generating Functions Polar Coordinates Areas Hyperbolic Functions 1 – definitions Further algebra and functions 2 Hyperbolic Functions 2 - integration Hyperbolic Functions 1 – inverses, differentiation Further Calculus Linear Programming 2 – Two stage simplex and Big M Internal Exams Further Volumes Chi-squared tests Central Limit Theorem

3. Year 12 Teacher One: Series Return to Routemap Specification content : Objectives : Notes: 4.3 Understand and use formulae for the sums of integers, squares and cubes and use these to sum other series * be able to use sigma notation; * understand and use formulae for the sums of integers, squares and cubes; * be able to use known formulae to sum more complex series.

4. Year 12 Teacher One: Further Algebra and Functions 1 Return to Routemap Specification content : Objectives: Notes: 4.4 Understand and use the method of differences for summation of series including use of partial fractions. * be able to use the method of differences to sum simple finite series. Do without partial fractions ( ie : set them up for them so they only have to do the method of differences) they will revisit this in Year 13 Maths

5. Specification content : Objectives : 1.1 The general ideas of algorithms and the implementation of an algorithm given by a flow chart or text. * understand what an algorithm is; * be able to trace an algorithm in the form of a flow chart; * be able to trace an algorithm given as instructions written in text; * know how to determine the output of an algorithm and how it links to the input; * be able to determine the order of a given algorithm and standard network problems. 1.2 Bin packing, bubble sort and quick sort. * know how to apply a bubble sort algorithm to a list of numbers or words; * know how to apply the quick sort algorithm to a list of numbers or words, clearly identifying the pivots used for each pass; * be able to identify the number of comparisons and swaps used in a given pass; * be able to identify size, efficiency and order of an algorithm and use them to make predictions; * know how to solve bin packing problems using full bin, first fit, and first fit decreasing algorithms, and understand their strengths and weaknesses. Year 12 Teacher One: Algorithms (part one) Return to Routemap

6. Year 12 Teacher One: Algebra and Functions Return to Routemap Specification content : Objectives : Notes: 4.1 Understand and use the relationship between roots and coefficients of polynomial equations up to the quartic equations * understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations 4.2 Form a polynomial equation whose roots are a linear transformation of the roots of a given polynomial equation (of at least cubic degree) * be able to form a polynomial equation whose roots are a linear transformation of the roots of a given polynomial equation (of at least cubic degree).

7. Year 12 Teacher One: Proof Return to Routemap Specification content : Objectives : Notes: 1.1 Construct proofs using mathematical induction Contexts include sums of series, divisibility and powers of matrices * be able to obtain a proof for the summation of a series, using induction; * be able to use proof by induction to prove that an expression is divisible by a certain integer; * be able to use mathematical induction to prove general statements involving matrix multiplication.

8. Specification content : Objectives : 2.1 The minimum spanning tree (minimum connector) problem. Prim’s and Kruskal’s (greedy) algorithm. * understand the meaning of a minimum spanning tree; * be able to apply Kruskal’s algorithm to a network to find the minimum spanning tree; * be able to apply Prims algorithm to a network to find the minimum spanning tree; * be able to apply Prim’s algorithm to a distance matrix to find the minimum spanning tree. 2.2 Dijkstra’s algorithm for finding the shortest path. * be able to apply Dijkstra’s algorithm to find the shortest path between two vertices in a network; * be able to trace back through a network to be able to find the route corresponding to the shortest path; * be able to consider modifications to an original shortest path problem, for example by dealing with multiple start points or a different end point. 1.3 Use of the order of the nodes to determine whether a graph is Eulerian, semi-Eulerian or neither. * know the meaning of the vocabulary used in graph theory e.g. degree of a vertex, isomorphic graphs, walks, paths and cycles; * be familiar with different types of graph e.g. complete, planar, isomorphic, simple, connected; understand graphs represented in matrix form; * be familiar with k notation; * know the definition of a tree; * be able to determine if a graph is Eulerian, semi-Eulerian or neither, and find Eulerian cycles. Year 12 Teacher One: Algorithms on graphs I (part one) Return to Routemap

9. Specification content : Objectives: a) 3.1 Algorithm for finding the shortest route around a network, travelling along every edge at least once and ending at the start vertex (The Route Inspection Algorithm) * be able to determine whether a graph is traversable; * be able to apply an algorithm to solve the route inspection problem; * find a route by inspection; * understand the importance of the order of vertices of the graph in finding a route. Year 12 Teacher One: Algorithms on graphs II (part one) Return to Routemap

10. Specification content : Objectives : 4.1 Modelling of a project by an activity network, from a precedence table. *be able to model a project by an activity network from a precedence table; 4.2 Completion of the precedence table for a given activity network. *be able to complete a precedence table from a given network; *understand the use of dummies. 4.3 Algorithm for finding the critical path. Earliest and latest event times. Earliest and latest start and finish times for activities. Identification of critical activities and critical path(s). *know how to carry out a forward pass and backward pass using early and late event times; * be able to interpret and use dummies; * be able to identify critical activities and critical paths. Year 12 Teacher One: Critical path analysis (part one) page 1 of 2 Return to Routemap Continued on next page

11. Specification content : Objectives : 4.4 Calculation of the total float of an activity. Construction of Gantt (cascade) charts. * know how to determine the total float of activities; * be able to construct and interpret Gantt (cascade) charts. Year 12 Teacher One: Critical path analysis (part one) page 2 of 2 Return to Routemap Previous page

12. Specification content : Objectives : 5.1 Formulation of problems as linear programs. * know how to formulate a linear programming problem from a real-life problem (write inequalities from worded questions); * be able to form an appropriate objective function to maximise or minimise. 5.2 Graphical solution of two variable problems using objective line and vertex methods including cases where integer solutions are required. * know how to represent a linear programming problem graphically and identify the feasible region; * be able to solve linear programming problems to find a maximum or minimum; * be able to interpret solutions in the context of the original real life problem. Year 12 Teacher One: Linear programming (part one) Return to Routemap

13. Year 12 Teacher One: Volumes of revolution Return to Routemap Specification content: Objectives: Notes: 5.1 Derive formula for and calculate volumes of revolution. * be able to derive formulae for and calculate volumes of revolution about both the x and y -axes. Round x and y axes Only polynomials

14. Specification content : Specification notes: Standard results and trig identities Integration by parts FAIDs Integration by substitution Year 12 Teachers One and Two: Integration: Preparing for Year 2 Return to Routemap

15. Specification content : Objectives : 2.1 Solve any quadratic equation with real coefficients. Solve cubic or quartic equations with real coefficients. * be able to solve any quadratic equation with real coefficients; 2.2 Add, subtract, multiply and divide complex numbers in the form with and real. Understand and use the terms ‘real part’ and ‘imaginary part’. * be able to add, subtract and multiply complex numbers in the form x + i y with x and y real; * understand and use the terms ‘real part’ and ‘imaginary part’. 2.3 Understand and use the complex conjugate. Know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs. Specification content : Objectives : 2.1 Solve any quadratic equation with real coefficients. Solve cubic or quartic equations with real coefficients. * be able to solve any quadratic equation with real coefficients; * be able to add, subtract and multiply complex numbers in the form x + i y with x and y real; * understand and use the terms ‘real part’ and ‘imaginary part’. 2.3 Understand and use the complex conjugate. Know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs. Year 12 Teacher Two: Complex Numbers (Slide 1 of 3) Return to Routemap

16. Specification content : Objectives : 2.4 Use and interpret Argand diagrams. * be able to use and interpret Argand diagrams 2.5 Convert between the Cartesian form and the modulus-argument form of a complex number. * be able to convert between the Cartesian form and the modulus-argument form of a complex number; 2.6 Multiply and divide complex numbers in modulus argument form. * be able to multiply and divide complex numbers in modulus-argument form. Year 12 Teacher Two: Complex Numbers (Slide 2 of 3) Return to Routemap Previous page

17. Specification content : Objectives : 2.7 Construct and interpret simple loci in the Argand diagram such as and arg  z  a    . * be able to construct and interpret simple loci in the Argand diagram such as and . Specification content : Objectives : Year 12 Teacher Two: Complex Numbers (Slide 3 of 3) Return to Routemap Previous page

18. Specification content : Objectives : 3.1 Add, subtract and multiply conformable matrices. Multiply a matrix by a scalar. * be able to find the dimension of a matrix; * be able to add and subtract matrices of the same dimension; * be able to multiply a matrix by a scalar; * be able to multiply conformable matrices. 3.2 Understand and use zero and identity matrices * understand and use zero and identity matrices; 3.3 Use matrices to represent linear transformations in 2D. Successive transformations. Single transformations in 3D. * be able to use matrices to represent 2D rotations, reflections, enlargements and translations; * be able to use matrix products to represent combinations of transformations; * be able to use matrices to represent linear transformations in three dimensions; * be able to use inverse matrices to reverse the effect of a linear transformation; * be able to use the determinant of a matrix to determine the area scale factor of a transformation; Year 12 Teacher Two: Matrices (Slide 1 of 2) Return to Routemap

19. Specification content : Objectives: 3.4 Find invariant points and lines for a linear transformation. * be able to find invariant points and lines for a linear transformation. Year 12 Teacher Two: Matrices (Slide 2 of 2) Return to Routemap Previous page

20. Specification content : Objectives: 3.5 Calculate determinants of 2×2 and 3×3 matrices and interpret as scale factors, including the effect on orientation. * be able to calculate determinants of 2×2 and 3×3 matrices; 3.6 Understand and use singular and non-singular matrices. Properties of inverse matrices. Calculate and use the inverse of non-singular 2×2 matrices and 3×3 matrices. * understand and use singular and non-singular matrices; * be able to know the properties of inverse matrices; * be able to calculate the inverse of non-singular 2×2 and 3×3 matrices. Year 12 Teacher Two: Matrices 2 - inverses Return to Routemap

21. Year 12 Teacher Two: Matrices 2 – simultaneous equations Return to Routemap Specification content : Objectives : Notes: 3.7 Solve three linear simultaneous equations in three variables by use of the inverse matrix * be able to use matrices and their inverses to solve linear simultaneous equations, including three linear simultaneous equations in three variables; 3.8 Interpret geometrically the solution and failure of solution of three simultaneous linear equations. * be able to interpret geometrically the solution and failure of solution of three simultaneous linear equations. This will be covered by teacher two in the Vectors 2 unit

22. Specification content : Objectives : * Calculation of the mean and variance of discrete probability distributions. Extension of expected value function to include E(g( X )). * be able to calculate the mean and variance of discrete probability distributions using and ; * know how to find the expectation of a function of a random variable. Specification content : Objectives : * Calculation of the mean and variance of discrete probability distributions. Extension of expected value function to include E(g( X )). Year 12 Teacher Two: Discrete Random Variables Return to Routemap

23. Year 12 Teacher Two: Poisson Distribution 1 Return to Routemap Specification content : Objectives: Notes: 2.1 The Poisson distribution. The additive property of Poisson distributions. * be able to use the Poisson distribution to model a real-world situation; * know the conditions for a Poisson distribution; * be able to comment critically on the appropriateness of using the Poisson distribution as a model; * know how to use calculators to calculate probabilities including cumulative probabilities; * know the additive property of Poisson distributions.

24. Year 12 Teacher Two: Poisson Distribution 2 Return to Routemap Specification content : Objectives: Notes: 2.2 The mean and variance of the binomial distribution and the Poisson distribution . * know how to find the mean and variance of the binomial distribution; * know how to find the mean and variance of the Poisson distribution; * be able to solve problems involving the mean and variance of the binomial and Poisson distributions. 2.3 The use of the Poisson distribution as an approximation to the binomial distribution. * be able to use the Poisson distribution as an approximation to the binomial distribution; * know when it is appropriate to use the Poisson distribution as an approximation to the binomial distribution.

25. Year 12 Teacher Two: Negative Binomial Distribution Return to Routemap Specification content : Objectives: Notes: 3.1 (Geometric) and negative binomial distributions. * be able to recognise when the negative binomial distribution is a suitable model and apply it to a problem in context; * know the necessary assumptions for the negative binomial distribution; * be able to explain the meaning of the parameters r and p; * be able to recognise the similarities and differences between the geometric and negative binomial distributions; Geometric distribution is covered in a separate unit 3.3 Mean and variance of negative binomial distribution with * be able to use the formulae for the mean and variance of the negative binomial distribution in a variety of problems. Specification content : Objectives: Notes: 3.1 (Geometric) and negative binomial distributions. * be able to recognise when the negative binomial distribution is a suitable model and apply it to a problem in context; * know the necessary assumptions for the negative binomial distribution; * be able to explain the meaning of the parameters r and p; * be able to recognise the similarities and differences between the geometric and negative binomial distributions; Geometric distribution is covered in a separate unit * be able to use the formulae for the mean and variance of the negative binomial distribution in a variety of problems.

26. Specification content : Objectives : 2.4 Extend ideas of hypothesis tests to test for the mean of a Poisson distribution. * understand and be able to apply the language of statistical hypothesis testing; * be able to carry out hypothesis tests to test for the mean of a Poisson distribution. 4.2 Extend hypothesis testing to test for the parameter p of a geometric distribution. * recognise from the context of a question when it is appropriate to use a geometric distribution; apply the method of hypothesis testing to carry out a test for the parameter p of a geometric distribution; * use the geometric distribution to calculate the p-value for their test; * state a conclusion to their test in context. Year 12 Teacher Two: Hypothesis Testing Return to Routemap

27. Year 12 Teacher Two: Vectors (Slide 1 of 2) Continued on next page Return to Routemap Specification content: Objectives: Notes: 6.1 Understand and use the vector and Cartesian forms of an equation of a straight line in 3D * know how to find the vector equation of a line in both two and three dimensions; * understand and use the Cartesian forms of an equation of a straight line in three dimensions; 6.2 Understand and use the vector and Cartesian forms of the equation of a plane * understand and use the vector and Cartesian forms of the equation of a plane. 6.3 Calculate the scalar product and use it to express the equation of a plane, and to calculate the angle between two lines, the angle between two planes and the angle between a line and a plane * be able to find the scalar product of two vectors; * be able to use the scalar product to express the equation of a plane; * be able to use the scalar product to calculate the angle between two lines; * be able to use the scalar product to calculate the angle between two planes; * be able to use the scalar product to calculate the angle between a line and a plane. Angle between two planes and angle between a line and a plane will be covered in Further Vectors

28. Year 12 Teacher Two: Vectors (Slide 2 of 2) Previous page Return to Routemap Specification content: Objectives: Notes: 6.4 Check whether vectors are perpendicular by using the scalar product *be able to check whether vectors are perpendicular by using the scalar product;

29. Year 12 Teacher Two: Further Vectors Return to Routemap Specification content: Objectives: Notes: 6.5 Find the intersection between a line and a plane Calculate the perpendicular distance between two lines, from a point to a line and from a point to a plane 3.8 Interpret geometrically the solution and failure of solution of three simultaneous linear equations. * be able to interpret geometrically the solution and failure of solution of three simultaneous linear equations. Students learned how to use the inverse matrix to solve 3 linear simultaneous equations in the Matrices unit.

30. Year 12 Teacher Two: Hyperbolic Functions (Slide 1 of 2) Return to Routemap Specification content : Objectives : Notes: 8.1 Understand the definitions of hyperbolic functions sinh   x , cosh  x and tanh   x , including their domains and ranges, and be able to sketch their graphs. * know the definitions of sinh   x , cosh  x and tanh   x including their domains and ranges; * be able to sketch graphs of the hyperbolic functions; * be able to differentiate and integrate the hyperbolic functions and know the standard results; 8.2 Differentiate and integrate hyperbolic functions. 8.3 Understand and be able to use the definitions of the inverse hyperbolic functions and their domains and ranges. * understand and be able to use the inverse hyperbolic functions including domains and ranges.

31. Year 12 Teacher Two: Hyperbolic Functions (Slide 2 of 2) Return to Routemap Previous page Specification content : Objectives : Notes: 8.4 Derive and use the logarithmic forms of the inverse hyperbolic functions.

32. Year 13 Teacher Two: Hyperbolic Functions 2 - Integration Return to Routemap Previous page Specification content : Objectives : Notes: 8.5 Integrate functions of the form and and be able to choose substitutions to integrate associated functions. Continuation of Further Calculus Specification content : Objectives : Notes: Continuation of Further Calculus

33. Specification content : Specification notes: a) Chain, product and quotient rule; including e, ln and trig Year 12 Teacher Two: Differentiation: Preparing for Year 2 Return to Routemap

34. Year 12 Teacher Two: Further Algebra and Functions 2 Return to Routemap Specification content : Objectives : Notes: 4.5 Find the Maclaurin series of a function including the general term. * be able to find and use higher derivatives of functions; * know how to express functions as an infinite series in ascending powers using Maclaurin’s expansion; * be able to find the series expansion of composite functions. 4.6 Recognise and use the Maclaurin series for e x , ln(1 + x ), sin  x , cos  x and (1 + x ) n , and be aware of the range of values of x for which they are valid (proof not required).

35. Year 12 Decision Teacher: Polar Coordinates Return to Routemap Specification content: Objectives: Notes: 7.1 Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates. * understand and be able to use polar coordinates and be able to convert between polar and Cartesian coordinates; Prove the identities for and using De Moivre’s Theorem if they have not done double angle formulae in Maths yet 7.2 Sketch curves with r given as a function of θ , including use of trigonometric functions. * know how to sketch standard polar curves. Specification content: Objectives: Notes: 7.1 Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates. * understand and be able to use polar coordinates and be able to convert between polar and Cartesian coordinates; 7.2 Sketch curves with r given as a function of θ , including use of trigonometric functions. * know how to sketch standard polar curves.

36. Year 13 Teacher One: Polar Coordinates Return to Routemap Specification content: Objectives: Notes: 7.3 Find tangents and normals to polar curves. * be able to find tangents parallel and perpendicular to the initial line;

37. Year 13 Teacher One: Polar Coordinates Return to Routemap Specification content: Objectives: Notes: 7.3 Find the area enclosed by a polar curve. * be able to find (compound) areas under polar graphs using the formula Specification content: Objectives: Notes: 7.3 Find the area enclosed by a polar curve.

38. Year 13 Teacher One: Further Complex Numbers Return to Routemap Specification content : Objectives : Notes: 2.8 Understand de Moivre’s theorem and use it to find multiple angle formulae and sums of series. * understand, remember and be able to use de Moivre’s theorem: z n = r n e i nθ = r n (sin  nθ + i  cos  nθ ); * be able to derive multiple angle formulae/expressions e.g. cos 3 θ in terms of powers of cos  θ , and sin 3   θ in terms of multiple angles of sin  θ ; * be able to apply de Moivre’s theorem to sum a geometric series 2.9 Know and use the definition e i θ = cos  θ + i  sin θ and the form z = r e i θ . * be able to multiply and divide complex numbers in modulus-argument and exponential form; * know and use cosine and sine in terms of the exponential form. 2.10 Find the n distinct n th roots of r e i θ for r ≠ 0 and know that they form the vertices of a regular n - gon in the Argand diagram. *know how to solve completely equations of the form z n – a – i b = 0, giving special attention to cases where a = 1, b = 0 2.11 Use complex roots of unity to solve geometric problems.

39. Year 13 Teacher One: Further Calculus (Slide 1 of 2) Continued on next page Return to Routemap Specification content: Objectives: Notes: 5.2 Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity. * know how to deal with infinity as a limit of a definite integral; * be able to integrate functions across limits which include values when the function is undefined i.e. deal with discontinuous integrands. 5.3 Understand and evaluate the mean value of a function. * understand and be able to evaluate the mean value of a function. 5.4 Integrate using partial fractions. * be able to integrate functions which can be split into partial fractions up to denominators with quadratic factors. Could have non-constant terms on the numerator

40. Year 13 Teacher One: Further Calculus (Slide 2 of 2) Previous page Return to Routemap Specification content: Objectives: Notes: 5.5 Differentiate inverse trigonometric functions. be able to differentiate inverse trigonometric functions such as   arctan   x 2 ; Worth revisiting the Maclaurin Expansions as well 5.6 Integrate functions of the form and and be able to choose trigonometric substitutions to integrate associated functions. know how to integrate functions of the form and and be able to choose trigonometric substitutions to integrate associated functions. Links with hyperbolic functions Specification content: Objectives: Notes: 5.5 Differentiate inverse trigonometric functions. Worth revisiting the Maclaurin Expansions as well Links with hyperbolic functions

41. Year 13 Teacher One: Further Volumes Return to Routemap Specification content: Objectives: Notes: Calculate volumes of revolution around either axis for more complicated functions They will have done this in Year 12 with polynomials. You will also need to do this with parametric equations.

42. Year 13 Teacher One: Differential Equations (Slide 1 of 3) Continued on next page Return to Routemap Specification content: Objectives: Notes: 9.1 Find and use an integrating factor to solve differential equations of form + P ( x ) y = Q ( x ) and recognise when it is appropriate to do so. * be able to identify the form of first order differential equations that can be solved by an integrating factor and carry out the solution; 9.2 Find both general and particular solutions to differential equations. * be able to find general and particular solutions of differential equations of this form. 9.3 Use differential equations in modelling in kinematics and in other contexts. Specification content: Objectives: Notes: * be able to identify the form of first order differential equations that can be solved by an integrating factor and carry out the solution; 9.2 Find both general and particular solutions to differential equations. * be able to find general and particular solutions of differential equations of this form. 9.3 Use differential equations in modelling in kinematics and in other contexts.

43. Year 13 Teacher One: Differential Equations (Slide 2 of 3) Continued on next page Previous page Return to Routemap Specification content: Objectives: Notes: 9.4 Solve differential equations of form y ''+ ay '+ by = 0, where a and b are constants, by using the auxiliary equation. * be able to solve second order differential equations of the form y ′′ + ay ′ + by = f( x ) where f( x ) is a polynomial, exponential or trigonometric function; * be able to find general and particular solutions of second order differential equations of this form. 9.5 Solve differential equations of form y ''+ ay '+ by = f( x ), where a and b are constants, by solving the homogeneous case and adding a particular integral to the complementary function (in cases where f( x ) is a polynomial, exponential or trigonometric function). 9.6 Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the differential equation.

44. Year 13 Teacher One: Differential Equations (Slide 3 of 3) Previous page Return to Routemap Specification content: Objectives: Notes: 9.7 Solve the equation for simple harmonic motion and relate the solution to the motion. * be able to use differential equations in modelling in kinematics and in other contexts; * be able to solve the equation for simple harmonic motion and relate the solution to the motion; 9.8 Model damped oscillations using second order differential equations and interpret their solutions. * be able to model damped oscillations using second order differential equations and interpret their solutions. 9.9 Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled first order simultaneous equations and be able to solve them, for example predator-prey models. Specification content: Objectives: Notes: 9.8 Model damped oscillations using second order differential equations and interpret their solutions. * be able to model damped oscillations using second order differential equations and interpret their solutions. 9.9 Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled first order simultaneous equations and be able to solve them, for example predator-prey models.

45. Specification content: Objectives: 1.4 The planarity algorithm for planar graphs. * be able to apply the planarity algorithm for planar graphs; * be able to determine if a graph contains a Hamiltonian cycle. Year 12 Teacher One: Algorithms and graph theory (part two) Return to Routemap

46. Specification content: Objectives: 2.2 Floyd’s algorithm for finding the shortest path. * be able to find all the shortest paths between all the pairs of vertices using Floyd’s algorithm. Year 12 Teacher One: Algorithms on graphs I part two Return to Routemap

47. Specification content: Objectives: 3.2 The practical and classical Travelling Salesman problems. The classical problem for complete graphs satisfying the triangle inequality. Determination of upper and lower bounds using minimum spanning tree methods. The nearest neighbour algorithm. * understand the travelling salesman problem and that there is no simple algorithm to solve it for complex networks; * be able to use the nearest neighbour algorithm to find upper bounds for the problem; * be able to find lower bounds for a problem; * understand that not all upper and lower bounds give a solution to the problem; * know how to identify the best upper and lower bounds; * be able to solve the travelling salesman problem and interpret this solution in the context of the problem. Note: It is unlikely that the students will be asked to solve a TSP unless one of the bounds turns out to be optimal, in which case they may be asked to explain why this is the case. Year 13 Teacher One: Algorithms on graphs II part two Return to Routemap

48. Specification content: Objectives: 5.3 The Simplex algorithm and tableau for maximising and minimising problems with ≤ constraints. * understand and use slack, surplus and artificial variables * be able to use slack variables to write inequality constraints as equations; * know how to rewrite LP problems so that each equation contains all the variables x, y, s, and t ; * be able to put the information in an initial tableau; * be able to find the pivot and use it to form a new tableau; * be able to identify if a tableau satisfies the optimality condition. Year 13 Teacher One: Linear programming 2 Return to Routemap

49. Specification content: Objectives: 5.4 The two-stage Simplex and big-M methods for maximising and minimising problems which may include both ≤ and ≥ constraints. * know how to use slack and surplus variables; * understand and be able to use artificial variables; * be able to use the two-stage simplex algorithm; * be able to use the big-M method; * be able to relate the solution to the original problem. Year 13 Teacher One: Linear programming 2 Return to Routemap

50. Specification content: Objectives: 4.5 Construct resource histograms (including resource levelling) based on the number of workers required to complete each activity. * be able to draw and interpret resource histograms; * be able to level resource histograms. 4.6 Scheduling the activities using the least number of workers required to complete the project. * be able to construct a scheduling diagram; * be able to interpret and modify schedules to meet requirements. Year 13 Teacher One: Critical path analysis 2 Return to Routemap

51. Year 13 Teacher Two: Probability Generating Functions Return to Routemap Specification content: Objectives: Notes: 7.1 Definitions, derivations and applications. Use of the probability generating function for the negative binomial, geometric, binomial and Poisson distributions. * be able to use the definition of a probability generating function ( pgf ) to find the pgf of a given probability distribution; * be able to check the validity of a given pgf ; * know how to use the power series form of a pgf to find individual probabilities; * be able to use the pgf of a discrete random variable to find the pgf of a closely related random variable. * be able to use the pgfs given in the formulae book for each of the negative binomial, geometric, binomial and Poisson distributions to find the mean and variance or probabilities from these distributions. 7.2 Use to find the mean and variance. * know how to use a given pgf to find the mean and variance of the probability distribution; 7.3 Probability generating function of the sum of independent random variables. * know how to use the result given in the formulae book to find the pgfs of the sum of independent random variables; be able to use the result in context to solve simple problems. Specification content: Objectives: Notes: 7.1 Definitions, derivations and applications. Use of the probability generating function for the negative binomial, geometric, binomial and Poisson distributions. 7.2 Use to find the mean and variance. * know how to use a given pgf to find the mean and variance of the probability distribution; 7.3 Probability generating function of the sum of independent random variables. * know how to use the result given in the formulae book to find the pgfs of the sum of independent random variables; be able to use the result in context to solve simple problems.

52. Year 13 Teacher Two: Chi Squared Tests Return to Routemap Specification content: Objectives: Notes: 3.1 Goodness of fit tests and Contingency Tables. The null and alternative hypothesis. The use of as an approximate  2 statistic. Degrees of freedom. * know and be able to use the process of a goodness of fit test; * know how to find the number of degrees of freedom of the expected values, including when one or more parameters are estimated from the data; * be able to use as an approximate  2 statistic; * be able to apply goodness of fit tests to include the discrete uniform, binomial and Poisson distributions; * be able to show frequencies by means of a contingency table; * know how to obtain p-values from calculators; * be able to use tables to find critical values. * be able to apply the Chi squared test to determine whether or not a set of data comes from a geometric distribution; * know how to determine the degrees of freedom for the test depending on whether or not the parameter of the geometric distribution is given; * be able to state appropriate hypotheses and conclusions. Specification content: Objectives: Notes:

53. Year 13 Teacher Two: Quality of Tests Return to Routemap Specification content: Objectives: Notes: 8.1 Type I and Type II errors. Size and power of test. The power function. * be able to state a definition of a Type I error and explain in the context of the question; * be able to state a definition of a Type II error and explain in the context of the question; * be able to use conditional probability with standard probability distributions to calculate the value of a Type II error. * know the definition of the size of a test; * know the definition of the power of a test; be able to explain how the power of a test relates to a Type II error; * be able to use standard probability distributions to calculate the power of a test. * be able to find and/or use a power function for a particular test; * be able to comment on the test using the results of a power function for different values of the parameter.

54. Year 13 Teacher Two: Central Limit Theorem Return to Routemap Specification content: Objectives: Notes: 5.1 Applications of the Central Limit Theorem to other distributions. * be able to quote the Central Limit Theorem; know the conditions for the use of the Central Limit Theorem; * recognise, from problems in context, where the Central Limit Theorem is required.

55. Year 12 Teacher Two: Geometric Distribution Return to Routemap Specification content: Objectives: Notes: 3.1 Geometric and (negative binomial) distributions. 3.2 Mean and variance of a geometric distribution with parameter p . * know when the geometric distribution is a suitable model and be able to apply it to a problem in context; * know the necessary assumptions for the geometric distribution; * be able to explain the meaning of the parameter p ; * be able to use the formulae for the mean and variance of the geometric distribution in a variety of problems. Negative binomial covered in a separate unit