Lion hunting

6002 /www/nrich/html/content/id/6002/ 1 2011-12-12T12:35:59 <mdoxml version="1.0"> A highly dangerous killer lion is on the loose. All we have to try and track down the lair of the lion are the sightings of the remains of its activity (use your imagination here!). By building up a series of sightings can you find the grid square in which the lion's lair is found? Each false guess costs you 10 points, whereas each new sighting costs you 1 point. What is the highest score you can get? Don't forget that you can drag the map around and zoom in/out as required. <mdo:flash height="400" width="550"><param value="/content/id/6002/YahooMap_Markers.swf" name="movie" ></param><param value="9" name="flashplayerversion" ></param><param value="true" name="allowfullscreen" ></param></mdo:flash> Now, these killer lions hunt according to a few clearly defined hunting patterns. By observing the behaviour of several different lions (click New hideout for a new lion) can you work out how many different sorts of hunting activity there are? Can you accurately describe them in terms of standard probability distributions? [Note: You might notice that killer lions can travel very long distances, and sometimes even hunt in the sea. That's why they are so dangerous.] </mdoxml> <mdoxml version="1.0"> One of our younger users, Ellie, noted that the place where there are the most sightings is the most likely location for the lair. Can you extend these ideas using standard statistics to hypothesise the location of the lair? </mdoxml> <mdoxml version="1.0"> <h3>Why do this problem?</h3> Spotting the patterns underlying apparently random data is one of the most important jobs of industrial mathematicians and statisticians. This problem allows students first to identify different sorts of patterns and then to try to refine their notions of these patterns. They will use a wide variety of mathematical and statistical problem solving skills in this activity. It can be approached on a variety of levels, from quite simplistic to very detailed. <h3>Possible approach</h3> <div>First approach the problem as a game. If the activity is being done individually, try to guess the location grid square in as few guesses /sightings as possible. If the activity is being done as a group, try taking turns to guess with a new sighting being given on each turn. The group will need to try to describe in words the square of interest if they do not have control of the whiteboard. This will help to focus their minds on the variable parameters in the problem.</div> <div>Next try to repeat for a few different lions. Can the group spot any recurring themes in the types of pattern exhibited? Once you have some suggestions for the patterns, make and test your hypothesis.</div> <div>Once the distributions have clearly been discovered, try to estimate the parameters involved.</div> <div>At each stage, try to be as presicse as possible with any statistical / probabilistic statements.</div> <h3>Key questions</h3> <ul> <li>In trying to maximise a score on the game, what features do we look for when making the decision to make a guess?</li> <li>How would you describe the configuration of sightings at each stage?</li> </ul> <h3>Possible extension</h3> Estimate the parameters used by the program to determine the behaviour of the lions by building up a set of data from several sets of sightings. <h3>Possible support</h3> <div>Playing the game several times is the best way into this activity. It will naturally raise statistical questions even if these are not formally explored. If several computers are available, see who can obtain the highest score in, say, a five minute period.</div> </mdoxml> <mdoxml version="1.0"> To help quantify your search, note that the behaviour of the lions does have a clear description in terms of standard probability distributions. The zoom in/out feature of the map can be very helpful in terms of seeing the structure of the sightings. Note that one of the possible behaviour patterns is rather tricky. Keep an open mind and, perhaps, develop a recording system for the location of the sightings. </mdoxml> <mdoxml version="1.0"><table border="1" cellpadding="0" cellspacing="0" classtable="" style="border-collapse:collapse;border:none"> <tbody> <tr> <td style="width:265.3pt;border:solid windowtext 1.0pt; background:#33CCCC;padding:0cm 5.4pt 0cm 5.4pt" valign="top" width="354"> AS Core Content </td> <td style="width:265.35pt;border:solid windowtext 1.0pt; border-left:none;background:#33CCCC;padding:0cm 5.4pt 0cm 5.4pt" valign="top" width="354"> A2 Core Content </td> <td style="width:265.35pt;border:solid windowtext 1.0pt; border-left:none;background:#33CCCC;padding:0cm 5.4pt 0cm 5.4pt" valign="top" width="354"> Further Pure Content </td> </tr> <tr style="height:18.45pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:18.45pt" valign="top" width="1061"> Indices and Surds </td> </tr> <tr> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt" valign="top" width="354"> Rational indices (positive, negative and zero) Laws of indices <a href="http://nrich.maths.org/7039">Power Stack</a> W Equivalence of Surd and Index notation Properties of Surds; rationalising denominators. <a href="http://nrich.maths.org/901">The Root of the Problem</a> A <a href="http://nrich.maths.org/324">Climbing Powers</a> B <a href="http://nrich.maths.org/7448">Irrational Arithmagons</a> B <a href="http://nrich.maths.org/7040">Quick Sum</a> W </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt" valign="top" width="354"> </td> </tr> <tr style="height:1.5pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.5pt" valign="top" width="1061"> Polynomials </td> </tr> <tr style="height:53.05pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:53.05pt" valign="top" width="354"> Addition, subtraction, multiplication of polynomials; collecting like terms, expansion of brackets, simplifying. <a href="http://nrich.maths.org/7042">Common Divisor</a> W Completing the square; using this to find the vertex. The discriminant of a quadratic polynomial; using the discriminant to determine the number of real roots. <a href="http://nrich.maths.org/6406">Implicitly</a> B Solution of quadratic equations, and linear and quadratic inequalities in one unknown. <a href="http://nrich.maths.org/7105">Inner Equality</a> W <a href="http://nrich.maths.org/7051">Unit Interval</a> W <a href="http://nrich.maths.org/7041">Quad Solve</a> W Solution of simultaneous equations, one linear and one quadratic. <a href="http://nrich.maths.org/543">System Speak</a> A Solutions of equations in x which are quadratic in some function of x. <a href="http://nrich.maths.org/6333">Direct Logic</a> A </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:53.05pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:53.05pt" valign="top" width="354"> Using relationships between the roots of a quadratic/cubic and the coefficients. Using substitution to get equations with roots simply related to the roots of an original equation. </td> </tr> <tr style="height:1.0pt"> <td colspan="2" style="width:530.65pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="708"> Coordinate Geometry and Graphs </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Polar Coordinates </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Finding length, gradient and midpoint of a line segment given its endpoints Equations of straight lines (y=mx+c, y-y1=m(x-x1), ax+by+c=0 Gradients of parallel or perpendicular lines <a href="http://nrich.maths.org/785">Parabella</a> A Equation of a circle with centre (a,b) and radius r: (x-a)2+(y-b)2=r2 Circle geometry: equation of a circle in expanded form x2+y2+2gx+2fy+c=0, angle in a semicircle is a right angle, perpendicular from centre to chord bisects the chord, radius is perpendicular to tangent. Solving equations using intersections of graphs, interpreting geometrically the algebraic solution of equations. <a href="http://nrich.maths.org/5438">Intersections</a> B Curve sketching: y=kxn, where n is an integer and k is a constant y=k?x where k is a constant y=ax2+bx+c where a, b and c are constants y=f(x), where f(x) is the product of at most 3 linear factors, not necessarily distinct <a href="http://nrich.maths.org/6493">Curve Match</a> B Transformations of graphs: Relationship between y=f(x) and y=af(x), y=f(x) + a, y=f(x+a), y=f(ax) where a is constant. <a href="http://nrich.maths.org/6842">Erratic Quadratic</a> B <a href="http://nrich.maths.org/6500">Whose Line Graph Is It Anyway?</a> B </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Composition of transformations of graphs ? relationship between y=f(x) and y=af(x+b) The modulus function, the relationship between the graphs y=f(x) and y=|f(x)| Parametric equations of curves; converting between parametric and cartesian forms </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Converting equations between Cartesian and polar form. Sketching simple polar curves. <a href="http://nrich.maths.org/2679">Polar Flower</a> A Finding the area of a sector using integration. </td> </tr> <tr style="height:1.0pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="1061"> Differentiation and Integration </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Gradient of a curve as the limit of gradients of a sequence of chords. <a href="http://nrich.maths.org/7060">Gradient Match</a> W Derivative and second derivative; notation f'(x) and f''(x), dy/dx, d2y/dx2 The derivative of xn where n is rational, together with constant multiples, sums, differences. Gradients, tangents, normals, rates of change, increasing/decreasing functions, stationary points, classifying stationary points. <a href="http://nrich.maths.org/7087">Calculus Analogies</a> C <a href="http://nrich.maths.org/7197">Patterns of Inflection</a> C <a href="http://nrich.maths.org/7084">Turning to Calculus</a> C <a href="http://nrich.maths.org/7085">Curvy Catalogue</a> C <a href="http://nrich.maths.org/7091">The Sign of the Times</a> W Indefinite integration as the reverse process of differentiation. <a href="http://nrich.maths.org/6412">Integration Matcher</a> C Integrating xn for rational n (n?-1) together with constant multiples, sums and differences. Definite integrals, constants of integration. Using integration to find the area of a region bounded by curves and lines. Estimating areas under curves using the Trapezium Rule. </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Derivative of ex and ln x, together with constant multiples, sums and differences. Chain rule, product rule, quotient rule. <a href="http://nrich.maths.org/6552">Calculus Countdown</a> B dx/dy as 1 ÷ dy/dx <a href="http://nrich.maths.org/6406">Implicitly</a> B Integral of ex and 1/x together with constant multiples, sums and differences Integrating expressions involving a linear substitution. Volumes of revolution <a href="http://nrich.maths.org/6426">Brimful</a> A <a href="http://nrich.maths.org/6430">Brimful 2</a> A <a href="http://nrich.maths.org/6495">The Right Volume</a> W Derivative of sin x, cos x and tan x together with constant multiples, sums and differences. <a href="http://nrich.maths.org/7059">Trig Trig Trig</a> W Derivatives of functions defined parametrically. Integration of trigonometric functions (through the notion of "reverse differentiation) <a href="http://nrich.maths.org/6382">Mind Your Ps and Qs</a> B Integration of rational functions Integration of functions of the form y=kf'(x)/f(x) Integration by parts </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Derivatives of inverse trig functions, hyperbolic functions, inverse hyperbolic functions. Derivation of first few terms of Maclaurin series of simple functions. <a href="http://nrich.maths.org/4750">Towards Maclaurin</a> B Integrals such as 1/?(a2-x2), 1/?(x2-a2), 1/( a2+x2), 1/?(x2+a2), using appropriate trigonometric or hyperbolic substitutions. Reduction formulae to evaluate definite integrals Using areas of rectangles to estimate or bound the area under a curve or to derive inequalities concerning sums. </td> </tr> <tr style="height:1.0pt"> <td colspan="2" style="width:530.65pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="708"> Trigonometry </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Hyperbolic Functions </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Sine and Cosine rules. Area formula for triangles A=½ab sinC Relationship between degrees and radians Arc length s=r?, Area of a sector A = ½r2? <a href="http://nrich.maths.org/7475">Stand Up Arcs</a> W <a href="http://nrich.maths.org/6328">Curved Square</a> B Graphs, periodicity and symmetry for sine, cosine and tangent functions <a href="http://nrich.maths.org/7063">Trigger</a> W Identities tan ? = sin ?/cos ?, cos2? + sin2?=1 <a href="http://nrich.maths.org/7052">Geometric Trig</a> W Exact values of sine, cosine and tangent of 30° , 45° , 60° <a href="http://nrich.maths.org/5869">Impossible Square?</a> B <a href="http://nrich.maths.org/5923">Impossible Triangles?</a> B Finding solutions of sin(kx)=c, cos(kx)=c, tan(kx)=c and equations which can be reduced to these forms within a specified interval. </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Inverse trigonomic relations sin-1, cos -1, tan-1, and their graphs on an appropriate domain. Properties of sec, cosec and cot. Solving equations using: sec2 ? = 1+ tan2 ? cosec2 ? = 1 + cot2 ? expansions of sin(A+B), cos(A+B), tan(A+B) formulae for sin 2A, cos 2A, tan 2A <a href="http://nrich.maths.org/7050">Trig Identity</a> W expression of a sin ? + b cos ? in the form Rsin(?+?) and Rcos(?+?) <a href="http://nrich.maths.org/2693">Loch Ness</a> B </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Definition of sinh, cosh, tanh, sech, cosech and coth in terms of ex. Graphs of simple hyperbolic functions. cosh 2x ? sinh 2x = 1, sinh 2x = 2 sinh x cosh x, etc. Expressing in terms of logarithms the inverse hyperbolic relations sinh-1x, cosh-1x, tanh-1x. </td> </tr> <tr style="height:1.0pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="1061"> Sequences and Series </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Definitions such as un=n2 or un+1=2un, and deducing simple properties from such definitions. ? notation Arithmetic and geometric progressions, finding the sum of an AP or GP, including the formula ½n(n+1) for the sum of the first n natural numbers. <a href="http://nrich.maths.org/6333">Direct Logic</a> A <a href="http://nrich.maths.org/7097">AP Train</a> W <a href="http://nrich.maths.org/7054">Prime APs</a> W <a href="http://nrich.maths.org/7056">Mad Robot</a> W <a href="http://nrich.maths.org/7315">Medicine Half Life</a> W Sum to infinity of a GP with |r|<1. <a href="http://nrich.maths.org/262">Circles Ad Infinitum</a> B Expansion of (a+b)n where n is a positive integer. </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Expansion of (1+x)n where n is a rational number and |x|<1 </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> ?r, ?r2, ?r3 and related sums. Summing finite series using the method of differences. Recognising when a series is convergent, finding the sum to infinity. </td> </tr> <tr style="height:1.0pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="1061"> Algebra and functions </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Factor Theorem and Remainder Theorem. <a href="http://nrich.maths.org/7066">Cubic Roots</a> W Algebraic division of polynomials by a linear polynomial. Sketching y=ax where a>0 Relationship between logarithms and indices. Laws of logarithms. <a href="http://nrich.maths.org/6159">Power Match</a> C <a href="http://nrich.maths.org/6169">Extreme Dissociation</a> B, S Solving ax=b using logarithms. </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Simplifying rational functions. Algebraic division of polynomials by a linear or quadratic polynomial. Expressing rational functions using partial fractions. <a href="http://nrich.maths.org/6960">Rational Request</a> A <a href="http://nrich.maths.org/6959">Inverting Rational Functions</a> A Identifying domain and range. Composition of functions. One-one functions, finding inverses. Graphical illustration of the relation between a one-one function and its inverse. Exponential and logarithmic functions ex and ln x, and their graphs. Exponential growth and decay. </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Partial fractions with (x2+a2) in the denominator, and where the numerator is of higher degree than the denominator. Determining asymptotic behaviour for rational functions. Relationship between graphs of y=f(x) and y2=f(x) </td> </tr> <tr style="height:1.0pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="1061"> Numerical methods </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Locating roots by graphical considerations or sign-change <a href="http://nrich.maths.org/7038">Solve Me!</a> W <a href="http://nrich.maths.org/5876">Root Hunter</a> B Simple iterative methods, xn+1=F(xn), relating such an iterative formula to the equation being solved. <a href="http://nrich.maths.org/7291">Archimedes Numerical Roots</a> W Numerical integration: Simpson's rule. </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Staircase and cobweb diagrams. Properties of successive errors in a converging iteration. Newton-Raphson method for finding roots. </td> </tr> <tr style="height:1.0pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="1061"> Differential Equations </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Forming differential equations from situations involving rate of change First order differential equations with separable variables: general form, and particular solutions from initial conditions. Interpreting solutions to differential equations within the context of a problem being modelled. <a href="http://nrich.maths.org/5874">It's only a minus sign</a> B </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Integrating factors for first order differential equations Reducing a first order differential equation to linear form or variable-separable using substitution. Complementary functions, particular integrals and general solutions of differential equations. Finding particular solutions using initial conditions, interpreting solutions in the context of a problem modelled by a differential equation. <a href="http://nrich.maths.org/4752">Out in Space</a> B <a href="http://nrich.maths.org/5875">Differential Equation Matcher</a> C </td> </tr> <tr style="height:1.0pt"> <td colspan="2" style="width:530.65pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="708"> Vectors </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Vectors and Matrices </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Addition and subtraction of vectors, multiplication of a vector by a scalar, geometrical interpretation of these. Unit vectors, position vectors, displacement vectors <a href="http://nrich.maths.org/6572">Vector Walk</a> B <a href="http://nrich.maths.org/6632">Polygon Walk</a> B Magnitude of a vector Scalar product of two vectors; determining the angle between two vectors <a href="http://nrich.maths.org/439">Flexi Quads</a> A Equation of a straight line in the form r = a + tb Angle between straight lines, point of intersection of straight lines, parallel or skew lines. </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Matrix addition, subtraction and multiplication. Singular and non-singular matrices, finding determinants and inverses. 2x2 matrices as transformations in the x-y plane. Solving linear simultaneous equations using matrices. <a href="http://nrich.maths.org/6875">Square Pair</a> B <a href="http://nrich.maths.org/6876">Matrix Meaning</a> B <a href="http://nrich.maths.org/6877">Nine Eigen</a> B <a href="http://nrich.maths.org/2370">Limiting Probabilities</a> B Equation of a line in the form (x-a)/p = (y-b)/q = (z-c)/r Equation of a plane in the form ax + by + cz = d or (r ? a).n=0 or r = a + ?b + ?c Vector product of two vectors <a href="http://nrich.maths.org/6576">Cross with the Scalar Product</a> B <a href="http://nrich.maths.org/6575">Fix Me or Crush Me</a> B Determining whether a line is in a plane, parallel to a plane or intersects a plane, finding point of intersection. Line of intersection of two planes Perpendicular distance from point to plane or line Angle between two planes or a line and a plane Shortest distance between skew lines </td> </tr> <tr style="height:1.0pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="1061"> Complex Numbers </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Real and imaginary parts, modulus and argument, complex conjugate. Addition, subtraction, multiplication, division, square roots of complex numbers x + iy Conjugate pairs of roots of a polynomial Complex conjugates and addition/subtraction of complex numbers on an Argand diagram, loci of simple equations and inequalities. <a href="http://nrich.maths.org/2374">Thousand Words</a> B Multiplication and division of complex numbers in polar form. de Moivre's theorem sin ? and cos ? in terms of ei? nth roots of unity. </td> </tr> <tr style="height:1.0pt"> <td colspan="3" style="width:796.0pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="1061"> Proof by Induction </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Establishing a given result using induction. Making conjectures based on some trial cases, then proving the conjectures using induction. </td> </tr> <tr style="height:1.0pt"> <td colspan="2" style="width:530.65pt;border:solid windowtext 1.0pt; border-top:none;background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="708"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; background:#FF99CC;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Groups </td> </tr> <tr style="height:1.0pt"> <td style="width:265.3pt;border:solid windowtext 1.0pt; border-top:none;padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> </td> <td style="width:265.35pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:0cm 5.4pt 0cm 5.4pt;height:1.0pt" valign="top" width="354"> Definition of a group Establishing whether a structure is or is not a group <a href="http://nrich.maths.org/2681">Group of Sets</a> A <a href="http://nrich.maths.org/7446">Poison, Antidote, Water</a> C Order of group, order of elements in a group. Subgroups Lagrange's theorem Cyclic groups Isomorphic groups <a href="http://nrich.maths.org/7005">Rose</a> B </td> </tr> </tbody> </table> </mdoxml> 2 3 0 0 0 1 1 Lion hunting A killer lion is causing devastation. From the locations of its reported activity, can you work out where its lair is located? Using, Applying and Reasoning about Mathematics Making and testing hypotheses Advanced Probability and Statistics Random variables Handling, Processing and Representing Data Handling data Using, Applying and Reasoning about Mathematics Trial and improvement