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IB Mathematical Studies Standard Level Scheme of Work: Year 1 1. Number and Algebra Time : 14 hours 2.1 2.2 2.3 2.4 2.5 2.6 The sets of Natural numbers, N ; Integers, Z ; Rational numbers, Q ; Real numbers, R. Approximation: decimal places and significant figures. Percentage Errors. Estimation. Expressing numbers in Standard Index Form and operations using these numbers. SI units and conversion between basic units of length, area, volume, weight, speed and temperature. Arithmetic sequences and series. Use formulae for nth term and sum of n terms. Geometric sequences and series. Use formulae for nth term and sum of n terms. 2. Graphic Display Calculator (GDC) 1.1 2.7 Time : 3 hours Arithmetic calculations Graph a variety of functions; linear, quadratic, reciprocal etc Explain WINDOW , ZOOM and TRACE facilities to locate points to a given accuracy. Explain INTERSECT , ZERO , MINIMUM and MAXIMUM functions. Enter data in lists. Solve linear equations simultaneously. Solve quadratic equations. Link to factorising. 3. Probability 3.1 3.2 3.3 3.8 3.9 3.10 Time : 3 hours Basic concepts of set theory: subsets, intersection, union and complement. Venn diagrams and simple applications. Include up to 3 subsets of the universal set. Sample Space : Event A and the complementary Event AI. Equally likely events; coin, dice etc Probability of an event A is given by number of successes divided by total number of outcomes. Probability of a complementary event, P(A’) = 1 – P(A) Solve problems using Venn Diagrams, Tree Diagrams and Tables of Outcomes. Include problems with and without replacement. Laws of Probability Combined Events P( A B) P( A) P( B) P( A B) Mutually Exclusive Events P( A B) P( A) P( B) Independent Events P( A B) P( A).P( B) Conditional Probability P( A B) 4. Coordinate Geometry 5.1 5.2 4.2 4.3 4.4 Time : 3 hours Coordinates in 2 dimensions: points ; lines ; mid-points. Distances between points using Pythagoras. Equation of a line in 2 dimensions; of the form y = mx + c and ax + by + d = 0. Find gradient, intercepts, points of intersection, parallel and perpendicular lines. Linear functions and graphs; of the form f : x mx + c and f(x) = mx + c 5. Functions 4.1 P( A B) P( B) Time : 3 hours Concept of a function as a mapping. Domain and range. Mapping diagrams. The Quadratic function. f(x) = ax2 + bx + c Properties of symmetry; Vertex; Intercepts. Completing the square. Quadratic formula. GDC. The exponential expression : ab ; b Q Graphs and properties of exponential functions: 4.5 4.6 4.7 4.8 f(x) = ax ; f(x) = ax ; f(x) = kax + c ; k , a , , c Q Growth and decay; basic concepts of asymptotic behaviour. Graphs and properties of the sine and cosine functions. f(x) = a sin bx + c and f(x) = a cos bx + c ; a , b , c Q . Describe amplitude and period. Accurate graph drawing of all functions above. Use the GDC to sketch and analyse simple and unfamiliar functions. Recognize and identify vertical and horizontal asymptotes. Use the GDC to solve equations involving combinations of simple and unfamiliar functions. 5x = 3x and x – 3 = 1/x for example. IB Mathematical Studies Standard Level Scheme of Work: Year 2 6. Trigonometry Time : 3 hours 5.3 Trigonometry in right-angled triangles using the ratios sine, cosine and tangent. Problems likely to include Pythagoras’ Theorem. 5.4 The Sine Rule The Cosine Rule a c Ambiguous case could be taught but is not examined. sin A sin B sin C a2 = b2 + c2 – 2bc cos A cos A 5.5 b b2 c2 a 2 2bc 1 ab sin C Area of Triangle = 2 Geometry of three dimensional shapes: cuboid; prism; pyramid; cylinder; sphere; hemisphere; cone. Lengths of lines joining vertices, vertices with mid-points, and mid-points with mid-points; sizes of angles between two lines and between lines and planes. 7. Financial Mathematics Time : 3 hours 8.1 Currency Conversions. Include transactions involving commission. 8.2 Simple Interest I Crn 100 where C = Capital , r = % rate and n = number of time periods , I = Interest n 8.3 8.4 r Compound Interest I C 1 C 100 Depreciation. The value of r can be positive or negative. Include GDC/Calc iterative method to find number of time periods in questions. Construction and use of tables: loan and repayment schemes; investment and saving schemes; inflation. 8. Statistics 6.1 6.2 6.3 6.4 6.5 6.6 Time : 3 hours Classification of data as discrete and continuous. Simple discrete data: frequency tables; frequency polygons. Grouped discrete or continuous data: frequency tables; mid-interval values; upper and lower boundaries. Frequency Histograms. Stem and Leaf diagrams (Stem Plots). Cumulative frequency tables for grouped discrete data and grouped continuous data; cumulative frequency curve. Box and Whisker Plots (Box plots). Percentiles and Quartiles. Measures of central tendency. Simple discrete data: Mean, Median and Mode Grouped discrete and continuous data: approximate Mean, Modal group and 50 th percentile. Measures of dispersion: range, interquartile range and standard deviation. Understand population and sample. Appreciate that sample mean and standard deviation serve as approximations to population mean and standard deviation. 6.7 Scatter diagrams: line of best fit, by eye, passing through mean point. Bivariate data: the concept of correlation. Pearson’s product-moment correlation coefficient: r s xy sx s y Interpretation of positive, zero and negative correlations. sxy will be given in exams if required, use GDC to calculate r from raw data. 6.8 The regression line for y on x : y y s xy sx 2 x x Understand outliers. Use of regression line for prediction. Beware extrapolation beyond data points is unreliable. 6.9 test for independence: 9. Introductory Differential Calculus 7.1 7.2 7.3 7.4 7.5 3.5 3.6 3.7 2 calc 2 Time : 3 hours Explain differentiation from first principles. Find the gradient between two points P and Q on a function, e.g. f(x) = x 2 Examine the behaviour of the gradient of PQ as Q approaches P. Tangents to curves. Explain the derivative is the Gradient Function using dy f ( x h) f ( x) f ( x) lim ; f ( x ) dx h 0 h The principle that f(x) = axn f I (x) = anxn-1 f II (x) = an(n-1)xn-2 n – negative & positive integers The derivative of functions of the form f(x) = axn + bxn-1 + ….. , n Z Find gradients of curves at given values of x. Find values of x where f I(x) is given. Find the equation of a tangent at a given point. (Equation of a Normal is not required.) Where/when are functions ‘increasing’ and ‘decreasing’. Graphical interpretation of f I(x) > 0 , f I(x) = 0 , f I(x) < 0 Find values of x where the gradient is 0 (zero). i.e. Solve f I(x) = 0 Classify as Minimum or Maximum points. (Develop awareness of gradient being ZERO at inflection points – will not be examined.) 10. Sets and Logic 3.4 f0 fe 0 – observed , e – expected. fe Formulation of null and alternative hypotheses; significance levels; contingency tables; expected frequencies. Degrees of freedom, use of tables of critical values and p-values. The 2 Time : 3 hours Basic concepts of symbolic logic. Definition of a proposition and symbolic notation for a proposition. Compound logic statements which involve ‘implication, ’ ; ‘equivalence, ’ ; ‘negation, ’ ; ‘conjunction, ’ ; ‘disjunction, ’ ; ‘exclusive disjunction, ’ Translation between verbal statements, symbolic notation and venn diagrams. Knowledge and use of ‘exclusive disjunction’ and the distinction between it and ‘disjunction’. Use Truth Tables to provide proofs for the properties of connectives and explain the logical contradiction and tautology of compound statements. (MAX 3 propositions) The defintion of Implication and its converse, inverse and contapositive. Logical equivalence.