4-5 September 2012 Izmir, Turkey

IB Mathematics Higher Level & IB Mathematics Standard Level

Workshop Resource Book

Tim Garry [email protected]

IB Math HL & SL workshop

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** Note ** The 2006 courses had their first exams in May 2006 and will have their last exams in November 2013. The ‘new’ 2014 courses will have their first exams in May 2014 and their last exams in November 2020.

Workshop Schedule (draft) 1

T U E S D A Y W E D N E S D A Y

2

9:00-10:15

Introductions. Workshop schedule & resource book. Overview of changes to ‘new’ 2014 Math HL & SL courses. Calculator technology.

10:15-10:45

coffee break Overview of new internal assessment for HL & SL (Exploration).

10:45-12:00 Management of the new IA – scheduling, ideas, resources, forms. 12:00-13:15

3

lunch Samples of student Explorations. Marking a student Exploration.

13:15-14:30 Teacher Support Material (TSM) for IA. 14:30-15:00

coffee break Technology in teaching and assessment. IB exams and GDC use.

4

15:00-16:15 Resources for Math HL & SL – textbooks, software, websites.

5

9:00-10:15

Theory of Knowledge activity. Math SL & HL mock exams – samples. Writing a mock exam. Use of past exams.

10:15-10:45

coffee break

6

Management of current IA (Portfolio Tasks).

10:45-12:00 Closer look at new syllabus content in SL & HL. 12:00-13:15

7

IB exam structure – Paper 1 & Paper 2. Use of exam markschemes.

13:15-14:30 Marking student samples. 14:30-15:00

8

lunch

coffee break Sharing teaching ideas & materials. Preparing students for exams.

15:00-16:15 Current and future developments in mathematics education.

IB Math HL & SL workshop

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IB Math HL & SL Workshop Booklet

Table of Contents  IBDP & Group 5 IB Learner Profile Aims & Objectives

4 5

 Course Planning ‘New’ SL Course (2014) – Summary of Changes

7

‘New’ HL Course (2014) – Summary of Changes

9

Comparison of SL and HL syllabuses for 2006 course

14

‘New’ SL Syllabus (2014) – Syllabus Content

25

Suggested teaching units – SL & HL (2014 courses)

29

~ Diversion #1 ~ pg.6

 External Assessment – Written Exams External Assessment – format change starting May 2008

30

Points to consider when writing a mock examination

31

Sample SL mock Paper 1 exam & markscheme

32

Sample HL mock Paper 1 exam & markscheme

47

Sample HL mock Paper 2 exam & markscheme

69

Exam Tips / Advice for Students (33)

90

 Internal Assessment (IA) - Exploration & Portfolio IA (2014) – Exploration – Teacher Support

93

Exploration (IA) – FAQs

101

Assessment Criteria for the Exploration (IA)

104

IA (2006) – Portfolio – Important Information

107

Portfolio Task - Student Checklist

112

Portfolio Task Type I Scoring Rubric

113

Portfolio Task Type II Scoring Rubric

114

Portfolio Teacher’s Record – Form A

115

Portfolio Feedback to Student – Form B

116

 Teaching Materials / Ideas Algebra Prep Exercises (SL & HL) + Worked Solutions Set of 13 SL Unit Tests  Theory of Knowledge (TOK) Mathematics – TOK Questions

~ Diversion #3 ~ pg.117

118 123

~ Diversion #4 ~ pg.145 ~ Diversion #5 ~ pg.146

147

TOK Activity – Conjecturing & Proof

149

Is Mathematics Invented or Discovered?

150

 Miscellaneous Recommendations / Suggestions

152

IB Math HL & SL workshop

~ Diversion #2 ~ pg.92

~ Diversion #6 ~ pg.154

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IB Math HL & SL workshop

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Aims Group 5 aims The aims of all mathematics courses in group 5 are to enable students to: 1. Enjoy mathematics, and develop an appreciation of the elegance and power of mathematics 2. Develop an understanding of the principles and nature of mathematics 3. Communicate clearly and confidently in a variety of contexts 4. Develop logical, critical and creative thinking, and patience and persistence in problemsolving 5. Employ and refine their powers of abstraction and generalization 6. Apply and transfer skills to alternative situations, to other areas of knowledge and to future developments 7. Appreciate how developments in technology and mathematics have influenced each other 8. Appreciate the moral, social and ethical implications arising from the work of mathematicians and the applications of mathematics 9. Appreciate the international dimension in mathematics through an awareness of the universality of mathematics and its multicultural and historical perspectives

10. Appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge” in the TOK course

Assessment objectives Problem-solving is central to learning mathematics and involves acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics SL course, students will be expected to demonstrate the following. 1. Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts 2. Problem-solving: recall, select and use their knowledge of mathematical skills, results and models in both real and abstract contexts to solve problems 3. Communication and interpretation: transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation 4. Technology: use technology, accurately, appropriately and efficiently both to explore new ideas and to solve problems 5. Reasoning: construct mathematical arguments through use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions 6. Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analysing information, making conjectures, drawing conclusions and testing their validity

IB Math HL & SL workshop

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~ Diversion #1 ~

IB Math HL & SL workshop

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Mathematics Standard Level - 2014 course → 2006 course Revised (‘new’) course – teaching starts in August 2012 with first exams in May 2014 ► Summary of changes from 2006 SL course (last exams 2013) to 2014 SL course ◄ *Note: hours given are approximate number of teaching hours suggested for each component of the course

2006 Maths SL course

(last exams in May/November 2013)

Syllabus content (140 hrs) 1. Algebra (8 hours) 2. Functions and Equations (24 hrs) 3. Circular Functions and Trigonometry (16 hrs) 4. Matrices (10 hrs) 5. Vectors (16 hrs) 6. Statistics and Probability (30 hrs) 7. Calculus (36 hrs)

Internal Assessment – Portfolio (10 hrs)

2014 Maths SL course

(first exams in May/November 2014)

Syllabus content (140 hrs) 1. Algebra (9 hours) 2. Functions and Equations (24 hrs) 3. Circular Functions and Trigonometry (16 hrs) 4. Vectors (16 hrs) 5. Statistics and Probability (35 hrs) 6. Calculus (40 hrs)

Internal Assessment – Mathematical Exploration (10 hrs)

IB Math HL & SL workshop

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Summary of Changes to Maths SL course 2006 SL course

Changes

2014 SL course

Syllabus content (140 hrs) n

1. Algebra (8 hrs)

1. Algebra (9 hrs)

Calculation of binomial coefficient  r  using both GDC and formula.

Encouragement to connect with applications in physics, chemistry economics, etc. Know exact values of trigonometric 3. Circular Functions & 3. Circular Functions &     Trigonometry (16 hrs) Trigonometry (16 hrs) ratios of 0, 6 , 4 , 3 , 2 and their multiples. 2. Functions and Equations (24 hrs)

2. Functions and Equations (24 hrs)



4. Matrices (10 hrs)

5. Vectors (16 hrs)

Content on matrices removed.

4. Vectors (16 hrs)

No changes.

6. Statistics & Probability (30 hrs)

5. Statistics & Probability (35 hrs)

statistical outliers defined; linear correlation of bivariate data including: Pearson’s product-moment correlation coefficient r; scatter diagrams and lines of best fit; equation for regression line of y on x and use of this equation for prediction purposes; no statistical tables in formula booklet

7. Calculus (36 hrs)

6. Calculus (40 hrs)

limit notation; integration by inspection, or substitution of the form  f  g  x  g   x  dx

Internal Assessment (10 hrs) two portfolio tasks

one mathematical exploration

A 6-12 page report written by each student focusing on a topic chosen by them and assessed by the teacher using five criteria.

Other changes: ▪ The format of the course syllabus has changed. The current SL course (2006 SL course) had three columns: (1) Content, (2) Amplifications/Inclusions, and (3) Exclusions. The syllabus for the ‘new’ SL course (2014 SL course) has the following three columns: (1) Content, (2) Further Guidance, and (3) Links. The ‘Links’ column in the syllabus provides useful links to the aims of the course containing suggestions for discussion, real-life examples and ideas for further investigation. ▪ The Aims and Objectives for Group 5 (mathematics & computer science) have been revised. ▪ ‘Presumed Knowledge’ is now called ‘Prior Learning Topics’ ▪ There are some minor changes to the external assessment. Although Paper 1 and Paper 2 will continue to be worth 90 marks each, the 90 marks may not necessarily be divided evenly between Section A and Section B. Section A and Section B will each be worth approximately 45 marks. ▪ Linear correlation of bivariate data is not being added to the HL core syllabus (being added to HL Statistics & Probability option topic). Therefore, SL syllabus content is no longer a strict subset of the HL core syllabus content. IB Math HL & SL workshop

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Summary of Changes to Maths HL course

IB Math HL & SL workshop

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IB Math HL & SL workshop

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IB Math HL & SL workshop

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IB Math HL & SL workshop

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Summary of changes to HL options

IB Math HL & SL workshop

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Comparison of SL and HL syllabuses for 2006 course (last exams 2013)

Math SL Syllabus content

Core syllabus content

140 hrs

Topic 1 - Algebra Topic 2 - Functions and equations Topic 3 - Circular functions and trig Topic 4 - Matrices Topic 5 - Vectors Topic 6 - Statistics and probability Topic 7 - Calculus Portfolio

Topic 1 - Algebra Topic 2 - Functions and equations Topic 3 - Circular functions and trig Topic 4 - Matrices Topic 5 - Vectors Topic 6 - Statistics and probability Topic 7 - Calculus Portfolio

8 hrs 24 hrs 16 hrs 10 hrs 16 hrs 30 hrs 36 hrs 10 hrs

Total

Math HL

Option syllabus content

150 hrs

Total

190 hrs 20 hrs 26 hrs 22 hrs 12 hrs 22 hrs 40 hrs 48 hrs 10 hrs 40 hrs 240 hrs

Topic 1 - Algebra

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Math SL

Math HL

Topic 2 - Functions and Equations

IB Math HL & SL workshop

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Math SL

IB Math HL & SL workshop

Math HL

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Math SL

Math HL

Topic 2 - Circular Functions and Trigonometry

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Math SL

Math HL

Topic 4 - Matrices

IB Math HL & SL workshop

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Math SL

Math HL Topic 5 - Vectors

IB Math HL & SL workshop

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Math HL

Topic 6 - Statistics and Probability

IB Math HL & SL workshop

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Math SL

IB Math HL & SL workshop

Math HL

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Math SL

Math HL

Topic 7 - Calculus

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Math SL

IB Math HL & SL workshop

Math HL

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Math HL

credit: Wiley Miller, Universal Press Syndicate

IB Math HL & SL workshop

permission for classroom use only

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Mathematics SL syllabus (incl. Prior Learning Topics) - first exams 2014 0

Prior Learning Topics Number 0.1 Routine use of addition, subtraction, multiplication and division, using integers, decimals and fractions, including order of operations 0.2 Simple positive exponents 0.3 Simplification of expressions involving roots (surds or radicals) 0.4 Prime numbers and factors, including greatest common divisors and least common multiples 0.5 Simple applications of ratio, percentage and proportion, linked to similarity 0.6 Definition and elementary treatment of absolute value (modulus), x 0.7 Rounding, decimal approximations and significant figures, including appreciation of errors 0.8 Expression of numbers in standard form (scientific notation), that is, a 10n , 1  a  10 , n  Sets and Numbers 0.9 Concept and notation of sets, elements, universal (reference) set, empty (null) set, complement, subset, equality of sets, disjoint sets 0.10 Operations on sets: union and intersection 0.11 Commutative, associative and distributive properties 0.12 Venn diagrams 0.13 Number systems: natural numbers, integers, ; rationals, ; and irrationals; real numbers, 0.14 Intervals on the real number line using set notation and using inequalities. Expressing the solution set of a linear inequality on the number line and in set notation 0.15 Mappings of the elements of one set to another. Illustration by means of sets of ordered pairs, tables, diagrams and graphs Algebra 0.16 Manipulation of simple algebraic expressions involving factorization and expansion, including quadratic expressions 0.17 Rearrangement, evaluation and combination of simple formulae. Examples from other subject areas, particularly the sciences, should be included 0.18 The linear function and its graph, gradient and y-intercept 0.19 Addition and subtraction of algebraic fractions 0.20 The properties of order relations: , , ,  0.21 Solution of equations and inequalities in one variable, including cases with rational coefficients 0.22 Solution of simultaneous equations in two variables Trigonometry 0.23 Angle measurement in degrees. Compass directions and three figure bearings 0.24 Right-angle trigonometry. Simple applications for solving triangles 0.25 Pythagoras’ theorem and its converse Geometry 0.26 Simple geometric transformations: translation, reflection, rotation, enlargement. Congruence and similarity, including the concept of scale factor of an enlargement 0.27 The circle, its centre and radius, area and circumference. The terms “arc”, “sector”, “chord”, “tangent” and “segment” 0.28 Perimeter and area of plane figures. Properties of triangles & quadrilaterals, incl. parallelograms, rhombuses, rectangles, squares, kites, trapeziums; compound shapes 0.29 Volumes of prisms, pyramids, spheres, cylinders and cones IB Math HL & SL workshop

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Coordinate Geometry 0.30 Elementary geometry of the plane, including the concepts of dimension for point, line, plane and space. The equation of a line in the form y  mx  c 0.31 Parallel and perpendicular lines, including m1  m2 and m1m2  1 0.32 Geometry of simple plane figures 0.33 The Cartesian plane: ordered pairs  x, y  , origin, axes 0.34 Mid-point of a line segment and distance between two points in the Cartesian plane and in three dimensions Statistics and Probability 0.35 Descriptive statistics: collection of raw data; display of data in pictorial and diagrammatic forms, including pie charts, pictograms, stem and leaf diagrams, bar graphs and line graphs 0.36 Obtaining simple statistics from discrete data and continuous data, including mean, median, mode, quartiles, range, interquartile range 0.37 Calculating probabilities of simple events

► Syllabus Content ◄ 1. Algebra 1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series; sigma notation; applications of arithmetic and exponential sequences (linear and exponential growth/decay) 1.2 Elementary treatment of exponents and logarithms; laws of exponents; laws of logarithms; change of base n 1.3 The binomial theorem: expansion of  a  b  , n  ; calculation of binomial

n coefficients using Pascal’s triangle and the formula for   , also written as n Cr r

2. Functions and Equations 2.1 Concept of function f : x

f  x  ; domain, range; composite functions; identity

function; inverse function f 1 2.2 The graph of a function; its equation y  f  x  ; function graphing skills; investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range; use of technology to graph a variety of functions; the graph of y  f 1  x  as the reflection in the line y  x of the graph of y  f  x  2.3 Transformations of graphs; translations: y  f  x   d ; y  f  x  c  ; reflections (in both axes): y   f  x  ; y  f   x  ; vertical stretch with scale factor p: y  a f  x  ; stretch in

1 : y  f bx  ; composite transformations b ax2  bx  c : its graph, y-intercept  0, c  ; axis of symmetry;

the x-direction with scale factor 2.4 The quadratic function x ‘factored’ form: x

x

a  x  p  x  q  , x-intercepts  p, 0  and  0, p  ; ‘vertex’ form:

a  x  h   k , vertex  h, k  2

2.5 The reciprocal function x function x

1 , x  0 ; its graph and self inverse nature; the rational x

ax  b and its graph; vertical and horizontal asymptotes cx  d

IB Math HL & SL workshop

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2.6 Exponential functions and their graphs: x b x , b  0 ; and x e x ; logarithmic functions and their graphs: x b x , b  0 , x e x ; logarithmic functions and their graphs: x logb x, x  0 , x ln x, x  0 ; relationships between these functions:

b x  e x ln b ; logb b x  x ; blogb x  x, x  0 2.7 Solving equations, both graphically and analytically. Use of technology to solve a variety of equations. Solving ax2  bx  c  0, a  0 ; the quadratic formula; the discrminant   b2  4ac and the nature of the roots, that is, two distinct roots, two equal real roots, no real roots; solving exponential equations 2.8 Applications of graphing skills and solving equations that relate to real-life situations

3. Circular Functions and Trigonometry 3.1

The circle: radian measure of angles; length of an arc; area of a sector

3.2 Definition of sin  and cos  in terms of the unit circle; definition of tan  as exact values of trigonometric ratios of 0,

3.3 3.4

3.5 3.6

   

sin  ; cos 

, , , and their multiples 6 4 3 2 The Pythagorean identity sin 2   cos2   1 ; double angle identities for sine and cosine; relationship between trigonometric ratios The circular functions sin x, cos x and tan x ; their domains and ranges; amplitude, their periodic nature; and their graphs; composite functions of the form f  x   a sin  b  x  c    d ; transformations Solving trigonometric equations in a finite interval, both graphically and analytically; equations leading to quadratic equations in sin x, cos x or tan x Solution of triangles; the cosine rule; the sine rule, including the ambiguous case; area of a triangle 12 ab sin C

4. Vectors 4.1 Vectors as displacements in the plane and in three dimensions; components of a vector;  v1    column representation; v   v2   v1 i  v2 j  v3 k ; algebraic and geometric approaches to v   3 the following: sum and difference of two vectors; zero vector; the vector  v ; multiplication by a scalar kv ; parallel vectors; magnitude of a vector, v ; unit vectors; base vectors; i, j and k ; position vectors OA  a ; AB  OB  OA  b  a 4.2 The scalar product of two vectors; perpendicular vectors; parallel vectors; the angle between two vectors 4.3 Vector equation of a line in two and three dimensions: r  a   b ; the angle between two lines 4.4 Distinguishing between coincident and parallel lines; finding the point of intersection of two lines; determining whether two lines intersect

5. Statistics and Probability 5.1 Concepts of population, sample, random sample, discrete and continuous data; presentation of data: frequency distributions (tables); frequency histograms with equal class intervals; box-and-whisker plots; outliers; grouped data: use of mid-interval values for calculations; interval width; upper and lower interval boundaries; modal class IB Math HL & SL workshop

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5.2 Statistical measures and their interpretations; central tendency: mean, media, mode; quartiles, percentiles; dispersion: range, interquartile range, variance, standard deviation; effect of constant changes to the original data 5.3 Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles 5.4 Linear correlation of bivariate data; Pearson’s product-moment correlation coefficient r; scatter diagrams; lines of best fit; equation of the regression line of y on x; use of the equation for prediction purposes; mathematical and contextual interpretation 5.5 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event; the n  A probability of an event A is P  A  ; the complementary events A and A (not A); n U  use of Venn diagrams, tree diagrams and tables of outcomes 5.6 Combined events, P  A  B  ; mutually exclusive events, P  A  B   0 ; conditional probability; the definition P  A B  

P  A  B ; independent events; the definition P  B

P  A B   P  A  P  A B  ; probabilities with and without replacement

5.7 Concept of discrete random variables and their probability distributions; expected value (mean), E  X  for discrete data 5.8 Binomial distribution; mean and variance of the binomial distribution 5.9 Normal distributions and curves; standardization of normal variables (z-values, z-scores); properties of the normal distribution

6. Calculus 6.1 Informal ideas of limit and convergence; limit notation; definition of derivative from first  f  x  h  f  x  principles as f   x   lim   ; derivative interpreted as gradient function h 0 h   and as rate of change; tangents and normals, and their equations 6.2 Derivative of x n  n   , sin x , cos x , tan x , e x and ln x ; differentiation of a sum and a real multiple of these function; the chain rule for composite functions; the product and quotient rules; the second derivative; extension to higher derivatives 6.3 Local maximum and minimum points; testing for maximum and minimum; points of inflexion with zero and non-zero gradients; graphical behaviour of functions. Including the relationship between the graphs of f , f  and f  ; optimization 6.4 Indefinite integration as anti-differentiation; indefinite integral of x n  n   , sin x ,

1 and e x ; the composites of any of these with the linear function ax  b ; x integration by inspection, or substitution of the form  f  g  x  g   x  dx

cos x ,

6.5 Ant-differentiation with a boundary condition to determine the constant term; definite integrals, both analytically and using technology; areas under curves (between the curve and the x-axis); areas between curves; volumes of revolution about the x-axis 6.6 Kinematic problems involving displacement s, velocity v and acceleration a; total distance travelled

IB Math HL & SL workshop

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Suggested Teaching Units – SL & HL (first exams 2014) * On pages 123-144 of this workshop booklet there is a set of 13 Unit Tests for the SL units

SL units

HL units (core syllabus)

1. Fundamentals (review of prior learning)

1. Fundamentals (review of prior learning)

2. Functions & Equations

2. Functions – Basics

3. Sequences & Series; Binomial Theorem

3. Functions, Equations & Inequalities

4. Exponential & Logarithmic Functions

4. Sequences & Series

5. [ Matrices - optional ]

5. Counting Principles; Binomial Theorem; Induction

6. Trigonometric Functions & Equations

6. Exponential & Logarithmic Functions

7. Triangle Trigonometry

7. [ Matrices - optional ]

8. Vectors

8. Trigonometric Functions & Equations

9. Differential Calculus

9. Triangle Trigonometry

10. Integral Calculus

10. Vectors

11. Statistics

11. Complex Numbers

12. Probability

12. Differential Calculus

13. Probability Distributions

13. Integral Calculus 14. Statistics 15. Probability 16. Probability Distributions

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External Assessment Mathematics HL and SL assessment model changes for May 2008 (first announced 30/08/2006) There are changes to the assessment model for May 2008. These were announced in the March 2006 coordinator notes. Second editions of the subject guides will be published in September 2006. In response to teacher queries, the examining team and IBCA have drafted the following guidance. Paper 1 Students are not permitted access to any calculator. Questions will mainly involve analytic approaches to solutions, rather than requiring the use of a GDC. It is not intended to have complicated calculations, with the potential for careless errors. However, questions will include some arithmetical manipulations when they are essential to the development of the question. Mathematics HL 

Paper 1 and paper 2 will both consist of Section A, short questions answered on the paper, (similar to the current paper 1), and Section B, extended-response questions, answered on answer sheets (similar to the current paper 2).



Calculators will not be allowed on paper 1.



Graphic display calculators (GDCs) will be required on paper 2 and paper 3.

Any references in the subject guide to the use of a GDC will still be valid, for example, finding the inverse of a 3 x 3 matrix using a GDC, this means that this will not appear on Paper 1. Another example of questions that will not appear on paper 1 are statistics questions requiring the use of tables. In trigonometry, candidates are expected to be familiar with the characteristic of the sin, cos and tan curves, their symmetry and periodic properties, and this includes knowledge of the ratios of 0°, 30°, 45°, 60°, 90°, 180° and deriving the ratios of multiples by using the symmetry of the curves eg sin 210°=-sin 30°. Mathematics SL 

Paper 1 and paper 2 will both consist of Section A, short questions answered on the paper, (similar to the current paper 1), and Section B, extended-response questions, answered on answer sheets (similar to the current paper 2).



Calculators will not be allowed on paper 1.



Graphic display calculators (GDCs) will be required on paper 2.

Any references in the subject guide to the use of a GDC will still be valid, for example, finding the inverse of a 3 x 3 matrix using a GDC, obtaining the standard deviation from a GDC, this means that these will not appear on Paper 1. Other examples of questions that will not appear on paper 1 are calculations of binomial coefficients in algebra, and statistic questions requiring the use of tables. In trigonometry, candidates are expected to be familiar with the characteristic of the sin, cos and tan curves, and this includes knowledge of the ratios of 0°, 90°, 180° etc.

IB Math HL & SL workshop

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Mathematics SL – External Assessment Structure (starting May 2008)

Points to consider when writing a mock examination The current external assessment structure (started May 2008) has Paper 1 where no calculator is allowed and Paper 2 where a graphic display calculator (GDC) is required. Both papers consist of two sections – Section A and Section B. Section A will have short answer questions and will have a total of 45 marks. Section B will have longer response questions and will also have a total of 45 marks. There is no set time for each section and students will not be prompted to move from one section to another. A student has 90 minutes to complete each paper. In the previous external assessment structure, both papers started with accessible questions moving on to more discriminating questions nearer the end of the paper. The current exam structure combines the two types of questions - short and long answer. This means there will be some accessible questions in Section A and in Section B. Therefore, questions at the end of Section A will be at a similar level to that of the previous (preMay 2008) questions 14 and 15 on Paper 1, and Section B will start with questions at a similar level to that of the previous questions 1 and 2 on Paper 2. There is not a set number of questions in each section, although the constraints of the mark total and type of question mean that there will not be much variation. Section A must have a mark total of 45 coming from short answer questions, which will have approximately 6 or 7 marks each. Section B will have 45 marks coming from 3 or 4 longer response questions.

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Mathematics Standard Level Paper 1

Mock Exam

1 hour 30 minutes

sample mock Paper 1 exam for Mathematics Standard Level written by William Bradley of Emirates International School – Jumeirah (Dubai, UAE)

IB Math HL & SL workshop

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Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.

Section A Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary.

1.

[Maximum mark: 7] In an arithmetic sequence, u12  31 and S 5  20 . (a)

(b)

Find (i)

the common difference;

(ii)

the first term.

Find S10 .

[4 marks] [3 marks]

……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ………………………………………………………………………………………

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2.

[Maximum mark: 6] The quadratic function f is defined by f ( x)  x 2  2 x  3 . (a)

Write f in the form f ( x)  ( x  h) 2  k .

(b)

On the grid below, sketch the graph of f clearly marking any important points. [4 marks]

[2 marks]

……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ………………………………………………………………………………………

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3.

[Maximum mark: 6] (a)

Given that 2 cos 2   3 sin   3 find the two values for sin  .

(b)

Given that 0    360 and that one solution for  is 30 , find the other two possible values for  . [2 marks]

[4 marks]

……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ………………………………………………………………………………………

IB Math HL & SL workshop

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4.

[Maximum mark: 6] (a)

(b)

(i)

Factorise the equation 2u 2  u  3  0 .

(ii)

Hence, or otherwise, solve the equation 2(2 2 x )  2 x  3  0 . [4 marks]

Solve log x 121  2

[2 marks]

……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ………………………………………………………………………………………

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5.

[Maximum mark: 7] Show that

d  cos x  sin x  2   dx  cos x  sin x  1  sin 2 x

……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ………………………………………………………………………………………

IB Math HL & SL workshop

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6.

[Maximum Mark: 6] (a)

Find

(b)

Find

6

x



3 0

3

dx

x( x  3)dx

[3 marks]

[3 marks]

……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ………………………………………………………………………………………

IB Math HL & SL workshop

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7.

[Maximum mark: 7] A coin is biased so that P(Head) 

2 1 and P(Tail)  3 3

The coin is tossed 5 times.

What is the probability of obtaining:

(a)

exactly 4 heads?

[2 marks]

(b)

zero tails?

[2 mark]

(c)

less than 3 heads?

[3 marks]

……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ………………………………………………………………………………………

IB Math HL & SL workshop

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Section B Answer all the questions on the answer sheets provided. Please start each question on a new page.

8.

[Maximum mark: 12]

(a)

Write out the sixth line of Pascal’s Triangle

(b)

Consider the expansion of (1  x) 5 (i)

[1 mark]

Write down the first four terms of this expansion

(ii) Use your answer to (b)(i) to evaluate (1.003) 5 , giving your answer correct to 7 decimal places. [6 marks] 15

3  (c) Find the term independent of x in the expansion of  2 x 2   , leaving x  n p q your answer in the form C r a b where a, b, p & q are integers. [5 marks]

9.

[Maximum mark: 13] Consider the function f ( x) 

x3 . x2

(a)

Write down the equation of the vertical asymptote.

[1 mark]

(b)

Find the equation(s) of any horizontal asymptotes.

[2 marks]

(c)

Find the y-intercept.

[2 marks]

(d)

Find the x-intercept.

[2 marks]

(e)

Show that the graph has no turning points.

[3 marks]

(f)

Sketch the graph of f (x) showing all relevant detail.

[3 marks]

IB Math HL & SL workshop

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10.

[Maximum mark: 11]

(a) The lifetime of a particularly battery is normally distributed with a mean of 45 hours and a standard deviation of 10 hours. (i) Find the probability that a particular battery lasts less than 40 hours. [4 marks] (ii) A sample of 10000 batteries is chosen. Find the expected number of batteries which last less than 40 hours. [2 marks] (b) A different type of battery is also normally distributed with a mean of 45 hours. In this battery, the standard deviation is unknown. Given that 0.62% of these batteries last longer than 50 hours, find the standard deviation. [5 marks]

11.

[Maximum mark: 9] Consider the function f : x  3x 2  1 (a) Find the area enclosed by the curve, the lines x  1 and x  2 , and the x-axis. [3 marks] (b) Suppose this area is rotated through 360 about the x-axis, find the volume of the solid so generated. [6 marks]

IB Math HL & SL workshop

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sample mock Paper 1 exam for Mathematics Standard Level

MARKSCHEME Section A 1.

(a) (i)

u1  11d  31 {1} 5 (2u1  4d )  20  u1  2d  4 {2} 2 {1}  {2} 9d  27 d 3

2.

[1M 1A]

(ii)

u1  33  31  u1  2

(b)

S10  5(2(2)  9(3))  115

[1M 1A] [2M 1A]

(a)

x 2  2x  3  ( x 2  2 x  1)  1  3  ( x  1) 2  4 (b)

[1M 1A]

Graph is a parabola passing through: (-3,0), (-1,4), (0,-3) and (1,0) Overall shape

3.

[3A] [1A]

(a)

2 cos 2   3 sin   3  2(1  sin 2   3 sin   3  0  2 sin 2   3 sin   3  0  (2 sin   1)(sin   1)  0 1  sin   or 1 [2M 2 (b) 4.

  150 or 90

2 A]

[2 A]

(a) (i)

(2u  3)(u  1)  0

IB Math HL & SL workshop

[1A]

Page 42

(ii)

Let u  2 x u

3 2

 2x  

or 1 3 2

or 1

3 no solution [1A] 2 2x  1  x  0 [1M 1A] 2x  

(b)

x 2  121  x  11

[1M 1A]

5.

du   sin x  cos x dx dv v  cos x  sin x    sin x  cos x dx dy (cos x  sin x)( sin x  cos x)  (cos x  sin x)( sin x  cos x)  dx (cos x  sin x) 2

u  cos x  sin x 

cos 2 x  sin 2 x  2 sin x cos x  cos 2 x  sin 2 x  2 sin x cos x cos 2 x  sin 2 x  2 sin x cos x 2  1  2 sin x cos x 2  [7 M ] 1  sin 2 x 

6. (a)

3  6 x dx 

6 x 2 3 c  2 c 2 x

[2M 1A]

(b) 3

 (x 0

2

 3 x)dx

3 2   x  3x   3 2 

3

0

9 2 However, the negative simply refers to the position of the area in relation to the x-axis, 

Hence required area is

9 2 u 2

IB Math HL & SL workshop

[2M

1A]

Page 43

7. (a) 4

0

80  2 1 P( X  4) C 4      243  3  3 5

32 243

(b)

Zero tails  5 heads =

(c)